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Touch-based object localization in cluttered environments Huy Nguyen, Quang-Cuong Pham School of Mechanical and Aerospace EngineeringNanyang Technological University,SingaporeEmail: [email protected], [email protected] 30, 2023 ======================================================================================================================================================================================================firstpage–lastpage 2017 The Large-Aperture Experiment to Detect the Dark Age (LEDA) was designed to detect the predicted O(100) mK sky-averaged absorption of the Cosmic Microwave Background by Hydrogen in the neutral pre- and intergalactic medium just after the cosmological Dark Age.The spectral signature would be associated with emergence of a diffuse Lyα background from starlight during `Cosmic Dawn'. Recently, <cit.> have reported detection of this predicted absorption feature, with an unexpectedly large amplitude of 530 mK, centered at 78 MHz. Verification of this result by an independent experiment, such as LEDA, is pressing. In this paper, we detail design and characterization of the LEDA radiometer systems, and a first-generation pipeline that instantiates a signal path model. Sited at the Owens Valley Radio Observatory Long Wavelength Array, LEDA systems include the station correlator, five well-separated redundant dual polarization radiometers and backend electronics. The radiometers deliver a 30–85 MHz band (16<z<34) and operate as part of the larger interferometric array, for purposes ultimately of in situ calibration. Here, we report on the LEDA system design, calibration approach, and progress in characterization as of January 2016. The LEDA systems are currently being modified to improve performance near 78 MHz in order to verify the purported absorption feature.telescopes, instrumentation: detectors, cosmology: observation, dark age, reionization, first stars § INTRODUCTION Cosmic Dawn is a cosmological epoch extending between the build up of the very first population of stars ∼ 100 Myr after the Big Bang <cit.>, followed by corresponding generations of black holes <cit.>, to the onset of widespread reionization of the intergalactic medium (IGM) ∼ 500 Myr after the Big Bang <cit.>. This is one of the most interesting and least understood epochs in the history of the Universe <cit.>. Cosmic Dawn is marked by the rise of the earliest populations of sources (stars and black holes), rapid evolution of radiation fields, and the onset of metal enrichment<cit.>.Recently, <cit.> reported detection of the sky-averaged spectral signature of the 21-cm ground-state transition of neutral Hydrogen (HI), placing Cosmic Dawn at redshifts 20>z>15. This signal, predicted by <cit.>, is sensitive to both cosmological and astrophysical processes in the early Universe; as such, it is an excellent probe of the physics between the CMB decoupling and the end of the epoch of reionization. Indeed, if verified, the <cit.> result would constitute the earliest detection of the thermal footprint of the first stars <cit.>. Specifically, <cit.> report detection of an ∼530 mK absorption feature, centered at ∼78.1 MHz, with width ∼18.7 MHz, using a relatively simple—yet exquisitely calibrated—dipole antenna and radiometer system known as the Experiment to Detect the Global EoR Step <cit.>. The amplitude of this absorption feature is, remarkably, 2–3 times higher than that expected with the most optimistic models <cit.>. Also at odds with existing models, the feature is flat-bottomed, as opposed to Gaussian-like. The <cit.> result suggests gas temperatures during Cosmic Dawn were far cooler than previously predicted, and could even point toward interaction between baryons and dark-matter particles <cit.>. An alternative explanation is that there was more radiation than expected, such as a significant contribution from an extragalactic background <cit.>.Nevertheless, some concerns remain that the purported Cosmic Dawn signal could in fact be an artifact, due to an unmodelled periodic instrumental feature, for example <cit.>. If verified, the <cit.> result places virtually the first observational constraints on Cosmic Dawn models. In comparison, the relatively more explored Epoch of Reionization (EoR; z∼ 6–10), is somewhat constrained by (i) the integrated optical depth of Thomson scattering of Cosmic Microwave Background (CMB) radiation <cit.>, (ii) the high-redshift galaxy UV luminosity function probed out to redshift of z∼ 10 <cit.>, (iii) detection of dusty galaxies at redshifts out to z∼ 10 <cit.>,and (iv) supermassive black holes at z∼ 7 <cit.>.Cosmic Dawn is unique in terms of the astrophysical processes and sources that played roles. In contrast to the EoR, which was likely populated by a `mature' population of galaxies residing in ∼ 10^8.5–10^10 M_⊙ halos <cit.> and producing copious ionizing radiation, Cosmic Dawn was populated by pockets of intense star formation hosted in dark matter halos of ∼ 10^6–10^8 M_⊙, which were less efficient in ionizing their surroundings. Sources of X-rays, Lyα and Lyman-Werner (LW, 11.2–13.6 eV) radiation, on the other hand, played major roles during this epoch <cit.>, and direct study of this epoch is anticipated to deliver new knowledge about early stellarpopulations and to constrain formation scenarios for supermassive black holes (complementary to study of the EoR). The preponderance of HI in the diffuse pre- and intergalactic medium (P/IGM) during Cosmic Dawn, and the sensitivity of the transition to radiative backgrounds produced by early stars and black holes makes the 21-cm line a unique tracer of the early Universe.To date, the main focus of radio instruments undertaking `21-cm cosmology' <cit.>, has been detection of the of EoR power spectrum (i.e. large-scale spatial fluctuations). The Giant Meter-wave Radio Telescope <cit.>, the Precision Array for Probing the Epoch of Reionization <cit.>, the Low Frequency Array<cit.> and the Murchison Widefield Array<cit.> have all placed upper limits on the amplitude of the EoR power spectrum. The upcoming Hydrogen Epoch of Reionization Array <cit.>, and the Square Kilometre Array telescope <cit.>, also seek to constrain the EoR power spectrum.Several experiments have been deployed in an attempt to measure the global 21-cm EoR signal. The first constraint on the global 21-cm EoR signal was provided by EDGES <cit.>, which excluded reionization more rapid than Δ z > 0.06 with 95% confidence. The Broadband Instrument for Global HydrOgen ReioNisation Signal <cit.>, and the Shaped Antenna measurement of the background RAdio Spectrum <cit.> also target the global EoR signal, where results from the latter exclude at 68 to 95% confidence some parameter combinations that correspond to late heating by X-rays in tandem with rapid reionization.EDGES is one of several experiments designed to detect the global 21-cm Cosmic Dawn signal. The SARAS 2 experiment <cit.>, SCI-HI <cit.> and the related Probing Radio Intensity at high z from Marion (PRIZM) experiment, follow similar methodology and instrumentation approaches. The Dark Ages Radio Explorer (DARE) concept proposes a satellite-based radiometer in lunar orbit, where earth occultation and absence of ionospheric effects are favorable <cit.>. Here, we detail the Large-Aperture Experiment to Detect the Dark Age instrument <cit.>, which observes at frequencies between theHF (3–30 MHz) and FM (88–108 MHz) radio broadcast bands (30<ν<88 MHz, 16<z<34). LEDA is unique in its embedding of radiometers in a densely interferometric array to enable calibration of radiometric data (in part) with observations of celestial sources (Sec. <ref>) and to create a ready path for exploration of power spectra estimation for Cosmic Dawn. In this paper, we present the design and characterization of the radiometry system for LEDA. The paper is organized as follows. In Section <ref> we provide a broad overview of the physics of Cosmic Dawn, details of the expected 21-cm signal, and outline of experimental requirements needed to observe the Cosmic Dawn signal. We also point out astrophysical scenarios that can be detected or ruled out by LEDA. In Section <ref> we set the stage discussing the site and the architecture of the telescope. LEDA radiometers are discussed in Section <ref>. Calibration is discussed in Section <ref>. In Section <ref> we characterize the instrument, including gain linearity, reflection and transmission coefficients, receiver temperature, noise diode thermal stability, and temporal stability. Results, including absolute calibration, RFI occupancy, spectral index measurements, and comparison to extant sky models appear in Section <ref>. Discussion folows in Section <ref>. § SCIENCE DRIVER The 21-cm line is a tracer of HI at all stages of cosmic evolution. Before the end of EoR, the signal is mainly produced by the intergalactic neutral medium; at lower redshifts, galactic HIdominates. The signal is sensitive to both cosmological and astrophysical processes, and as such, is arguably the best probe of the Universe at the intermediate redshift range between the CMB decoupling and the end of reionization.The observed differential brightness temperature relative to the CMB is T_21≈ 27 x_ HI√(1+z/10)(T_S-T_CMB/T_S) [mK],where we have ignored terms of 𝒪(δ), with δbeing spatial density fluctuations.In Eq. (<ref>)x_ HI is the HI fraction, T_CMB is the CMB temperature, and T_S is the spin temperature defined as T_S^-1 = T_CMB^-1+x_α T_c^-1+x_cT_K^-1/1+x_α +x_cwhere T_K and T_c arekinetic and collisional temperatures, and x_α and x_c are Lyα and collisional coupling factors, respectively. Evolution of T_S with redshift is complex anddepends on the intensity of the Lyα radiative background throughthe Wouthuysen-Field (WF) effect <cit.> which sets x_α, and on thethermal history of the Universe via T_K, T_c (withT_c≈ T_K) and x_c. The temperature T_21 may be positive or negativedepending on the sign of (T_S - T_CMB). For instance, when the transition is coupled to the gas temperature and the P/IGM is colder than the CMB, T_S<T_CMB and the signal is seen in absorption (T_21<0).When the gas is hotter than the CMB, T_S>T_CMB and the signal is seen in emission (T_21>0). For any particular scenario of structure and star formation,the evolution of T_21 can be used as a `cosmic clock' that tracks the evolution of the Universe. In what follows, we focus on the zero-mode of the signalfromEq. <ref>,a.k.a.,the global signal[Higher order modes are linked tospatial fluctuations and are outside of the scope of this paper <cit.>.]. The predominant feature of the signal (e.g., the black curve in Fig. <ref> which is a model with a strong signature in the LEDA observing band) is a Gaussian-like absorption trough signifyingsufficient Lyα coupling and a cold diffuse medium. The centroid and amplitude of the trough depend directly on the balance between the processes of star formation and P/IGM heating. In particular, the low-frequency maximum(located at ν = 46 MHz for the black curve in Fig. <ref>) denotes the onset of star formation giving rise to the Lyα background. These photons coupled T_S to T_K via the WF effect, creating an absorption signature because at that epochgas was colder than the CMB.The strongest absorption(at ν = 68 MHz for the black curve in Fig. <ref>) marks approximately the moment at which the IGM has reached its minimum temperature and a growing X-ray background due to compact sources becomes significant.As cosmic heating progresses, contrast between T_S and T_CMB decreases untilthe moment when the gas reaches the temperature of the background radiation, and the signal vanishes (at ν = 87 MHz for the black curve in Fig. <ref>). If this happens prior to reionization by ultraviolet sources, the signal will appear in emission for lower z. The 21-cm spectra for an ensemble ofastrophysical model parameters combinations (M_min, f_*, L_X, X-ray SED and the total CMB optical depth τ)permitted by extant data and theoretical studies exhibit a large scatteras is shown in Fig. <ref>;the recent <cit.> result is overlaid in red. The amplitude of the CD trough varies between 25 and 240 mK, with the absorption trough located between 40<ν<120 MHz <cit.>. The <cit.> result is inconsistent with these models, exhibiting a much larger amplitude of 530 mK. Given this discrepancy, coupled with some outstanding concerns that the signal is an artifact <cit.>, and that the result is yet to be verified, we do not as of yet rule these scenarios out.For a large fraction of physically motivated models, and also for the purported <cit.> signal, the absorption minimum falls within the LEDA observing band, and, thus, could be detected by the instrument. This is discussed further in the next section. §.§ Observational ProspectsRadiometric detection requires separation of foreground signals and the (background) 21-cmsignal. The diffuse and continuum foreground sources are know to be spectrally smooth; that is, they exhibit power-law spectra over the 30–88 MHz band. As such, they are separable from the background signal, which is expected to manifest as an absorption trough. <cit.> showed a basic demonstration of concept, by convolving a Global Sky Model <cit.> with an analytical model for a simple dipole antenna to form simulated measurements; this approach is followed by several Cosmic Dawn experiments <cit.>, including LEDA. The spectral smoothness of foregrounds allows retrieval of 21-cm features by modeling the brightness temperature of the foreground, T_fg, with a low-order log-polynomial. <cit.> expounded on this by including instrumental effects; <cit.> showed that the angular structure and frequency dependence of a more realistic broadband dipole (modeled on the design of the Long Wavelength Array) increases the required polynomial order but not necessarily so much so as to confound detection.For this approach to work, any spectral structure introduced by the measurement apparatus must be accounted for and calibrated out. For this reason, zero-mode radiometer experiments have preferred simple, low-gain dipole antennas over high-gain single dishes that exhibit more complex gain patterns <cit.>. We discuss the approach of LEDA to calibrationin Section <ref>.Significance of the detection by LEDA is determined by tiny deviations in the shape of the actual sky temperature from the smooth foreground curve in the LEDA band. To estimate which part of the astrophysical parameter space is actually targeted by LEDA, we use the signal to noise ratio (SNR) defined as SN^2 = ∑_i (T_sky-T̃_fg)^2/σ_i^2 where the sum is over frequency channels in the LEDA band, the mock data is defined as T_sky = T_fg + T_cosm and theforeground signal is modeled as a seventh order polynomial in logν <cit.>. We fit out the foreground component by calculating T̃_fg as a best fitto the mock data of the shape provided by <cit.>. The residual signal is then compared to the rms noise given by the radiometer equation σ≈ 2.6×T_sys/√(Δν t)whereΔν is the bandwidth over which the signal is measured, and t is the integration time, and we assumed that the system temperature, T_sys, is dominated by the temperature of the sky. The factor 2.6 is an approximation that takes into account thermal uncertainties after calibration. (Here we assumed hot and cold reference diode temperatures of 6500 K and 1000 K, as is explained in detail in Sec. <ref>, Eq. <ref>). Thus for a sky temperature of 2000 K and using Δν=1 MHz, a 5σ detection of the <cit.> feature could be made in under 45 minutes of observation with LEDA.Alternatively, one maycalculate SNR for each of the ∼ 200 models shown inFig. <ref>, assuming Δν = 1 MHz and integration time of 1000 hours. Out of ∼ 200 different astrophysical scenarios (Fig. <ref>) the models with the highest SNR are those with strongest variation within the LEDA band.These models typically share high star formation efficiency (often in low-mass halos) and high X-ray efficiency, which suggests that LEDA should have considerable leveragein constraining (i) star formation during cosmic dawn, and in particular the roles of small halos, and (ii) the timing of X-ray heating and properties of high-redshift X-ray sources (e.g., XRB, mini-quasars). The model with the highest signal to noise of SNR = 9.2 (heavy solid curve in Fig. <ref>) shows both a strong absorption and an early emission signal within the LEDA band. These features are hard to mimic with smooth foregrounds. The underlying astrophysical model assumes high star formation efficiency of f_* = 50% in heavy halos above circular velocity of 35.5 km s^-1 and a very luminous XRB populationshining at the luminosity ofL_X = 15 × 10^41 erg s^-1 per unit star formation rate in M_⊙ yr^-1 (i.e., 50 times brighter than the low-redshift counterparts).In another detectable scenario (heavy dashed curve) only the absorption trough is located within the LEDA band which makes the detection a bit more challenging. The underlying astrophysical model has moderate star formation efficiency, f_* = 5%,stars form via cooling of atomic hydrogen, X-ray heating is due to XRB with L_X = 2.4 × 10^41 erg s^-1 per unit star formation rate in M_⊙ yr^-1, and the total CMB optical depth of τ = 0.066. The moderate star formation and heating result in a moderate SNR = 4 in the LEDA band. Finally, in Fig. <ref> we also show an astrophysical scenario that cannot be separated from the foregrounds, and, thus, isundetectable by LEDA (SNR = 0.1, dotted line in the figure). This case has low star formation efficiency, f_* = 0.5%, star formation via molecular hydrogen cooling subjected to strong LW feedback, and weak X-ray heating with L_X = 0.03 × 10^41 erg s^-1. § INSTRUMENT OVERVIEWMotivated by limitations of single-antenna experiments, LEDA is a multi-antenna experiment, co-installed on the Long Wavelength Array stations at Owens Valley Radio Observatory (OVRO-LWA, 37.24^∘N, 118.28^∘W), and the National Radio Astronomy Observatory in Socorro, New Mexico (LWA1, 34.07^∘N, 107.63^∘W). Initial work, based at LWA1, is presented in (Schinzel et al., LWA Memo #208); here we focus on work carried out at OVRO-LWA. Further details about the LWA systems may be found in <cit.>.At both the LWA1 and OVRO-LWA stations, an additional 5 outrigger stands (Fig. <ref>) were installed and outfitted with the LEDA frontend receiver card (see Sec. <ref>). The LEDA outrigger stands, detailed further in Sec. <ref>, are placed at a distance from the core to minimize mutual coupling effects. <cit.> argue that detection will require precise knowledge of the antenna radiation pattern, which may not be deliverable by EM simulation. <cit.> are pessimistic that a detection may be made without an accurate model of the ionosphere. There is also concern that terrestrial radio frequency interference (RFI) could confound detection. The OVRO-LWA core region consists of 251 stands within a 100-m radius(Fig <ref>), roughly double the radius of LWA1.The OVRO-LWA station (Fig. <ref>) was built in 2013 and utilizes the same antenna design and analog systems as LWA1, with a different digital system designed for wide-bandwidth cross-correlation <cit.>. An additional 5 outrigger stands (Fig <ref>) were installed and outfitted with the LEDA frontend receiver card—the characterization and design of which is the focus of this article. A more complete overview of OVRO-LWA may be found in Hallinan et al., in prep.§.§ Observational strategy OVRO-LWA allows for LEDA to monitor the ionosphere, characterize the foreground sky, and measure antenna gain patterns in-situ, all while radiometric measurements are being taken. Gain patterns are measured via cross-correlation of LEDA radiometer antennas against the dense core of OVRO-LWA. The OVRO-LWA array is capable of imaging the radio sky from horizon to horizon; by doing so, the position and apparent brightness of celestial sources can be monitored. By monitoring the passage of celestial sources, a beam model can be inferred; similarly, a model of ionospheric-induced refractive offsets may be formed by monitoring offsets in the apparent positions of sources.The LEDA observational strategy is implemented to avoid complications that arise due to synchronization requirements between the noise diode switching and correlator integration. In order to derive gain patterns, once a week a 24-hour interferometric observation of the sky is performed at a low duty cycle (a 10 s integration every 100 s). These data are also used to form a model of the sky. During these observations, the radiometer antennas do not switch into diode states. In contrast, during radiometric observations the ionosphere is monitored using the core antennas and cross-correlations with the outriggers are discarded.The primary LEDA observational window is during night-time hours over December-March. The Sun is a potential source of interference, and RFI is known to be more prevalent during daylight hours. Furthermore, at low frequencies the galactic plane has a significant contribution to the overall system temperature of a radiometer, so it is desirable to observe when the galactic plane is low.§.§ LEDA correlator and spectrometer OVRO-LWA is operated as a radio interferometer, which requires the cross correlation of all antenna pairs. Cross correlation is an O(N^2) operation, which is computationally challenging for the N=512 inputs of OVRO-LWA. The cross-correlation of all 256 dual-polarization antenna pairs is performed by the LEDA correlator, detailed in <cit.>. Briefly, the LEDA correlator is a FX-style system where the data are channelized (by `F-engines') before cross-correlation by the `X-engine'. The F-engines run on ROACH-2 field-programmable gate array(FPGA) boards from the CASPER collaboration <cit.>. The F-engines are connected via 10 Gb Ethernet to compute servers running the xGPU cross-correlation X-engine code <cit.>. The firmware for the LEDA F-engine has been modified from that detailed in <cit.> to generate autocorrelation spectra at higher bit depth. The LEDA correlator requantizes the output of the polyphase filterbank down to 4 bits, for data transport to the compute servers. The modified firmware generates autocorrelation spectra from the (18-bit) F-engine output before requantization, yielding higher dynamic range and lowering quantization-induced non-linearity compared to the 4-bit data stream. We refer to this system as the LEDA spectrometer.All spectral data products presented here are from the LEDA spectrometer. Corresponding cross-correlation data from the LWA core were also recorded for ionospheric monitoring, but are not used further within this article. The spectrometer is implemented using a 4096-channel, 4-tap, Hamming-windowed polyphase filterbank (PFB). The PFB provides ∼50 dB of isolation between neighboring channels, which prevents leakage of narrowband RFI signals between channels. The digitizer is clocked at 196.608 MHz, resulting in a 24 kHz channel bandwidth. The output of each PFB channel is squared and accumulated for 1 s. An external pulse-per-second (PPS) signal is used to trigger each new accumulation. Accumulated data are read from the ROACH-2 board's Ethernet control interface. After every accumulation, data are timestamped and written to a hierarchical data format (HDF5) file.§.§ Outrigger antennas The outrigger antennas (Fig. <ref>) are of the same design as the LWA cross-dipole antennas, detailed in <cit.>. Each dual-polarization antenna consists of four triangular 'blades' of length 1.4 m, which form two pairs of orthogonal antennas (single-polarization). The blades are attached to a central pole of height 1.5 m, at the top of which is a weatherproof box for the FE. The two antenna pairs are oriented North-South and East-West. The blades are angled down to improve response at the horizon and beam symmetry. A wire-grid 3×3-m ground screen isolates the antennas from the earth ground, whose characteristics may change with moisture content.The outrigger antennas are physically isolated from the core antennas and other metallic objects. Each antenna has a 3×3 m ground screen, and is protected from grazing cattle by a wooden fence. The antennas are connected to the shelter via buried lengths of LMR400 coaxial cable. These cables are fed up through the central pole of the antenna, through to the receiver.§ LEDA RADIOMETERSIn this section we provide further detail about a single radiometer system; in total, there are 5×2 complete radiometers within LEDA. A block diagram of a LEDA radiometer is given in Fig. <ref> for a single polarization. As seen in the diagram, there are four main components: the antenna, front-end receiver, back-end analog systems, and digital systems. Each dual-polarization antenna is connected directly at its terminals to the front-end electronics (FE). The FE converts the balanced antenna terminal pair to unbalanced 50 Ω via a 4:1 balun, then amplifies and filters the signal, outputting the conditioned signal over a buried coaxial cable to the OVRO-LWA electronics shelter. The entry bulkhead of the shelter connects the buried coaxial to an FM bandstop and lightning arrestor installed at the bulkhead; a length of coaxial cable connects the bulkhead to the back-end analog systems (CRX).The CRX applies further amplification and signal filtering, in preparation for digitization. The CRX systems also provide power to the FE, via the coaxial cable lengths that connect the antenna to the shelter. The signal from the CRX is converted from 50 Ω unbalanced to 100 Ω balanced, and Category-7A Ethernet cables are used to transport the signals to the digitizer. Further details of these systems are given below. §.§ Deployment historyThe LEDA radiometer systems are under active development; iterative upgrades and improvements are made after each field deployment (Tab. <ref>). A brief historical summary of deployments is as follows. The LEDA correlator system achieved first light in August 2013, using an early revision of the FE card (ver 2.0) along with the standard LWA analog receiver system (ARX), with spectra formed from 4-bit data sent to the correlator. In December 2013, the correlator F-engine firmware was modified to add an independent autocorrelator spectrometer with higher bit-depth. An updated version of the FE with added MS147 test ports (ver 2.5) was installed in April 2014. Due to concerns of potential crosstalk, a fully shielded switching controller (Sec. <ref>) was installed in November 2014, and the LWA analog receiver was replaced by a fully connectorized system in December 2014. Major improvements were made to the FE over the course of 2015; the FE version 2.9 was deployed in January 2016. In this paper, we detail the system as installed at OVRO-LWA in January 2016; details of the FE version 2.5 and LWA1 results are presented in Schinzel et. al. (in prep).§.§ Front-end receiver board The LEDA FE (Fig. <ref>) connects to the antenna terminals (Fig. <ref>) and applies first-stage signal amplification and conditioning. The FE is a two-sided, four-layer circuit board of dimensions of 11.5×11.5 cm, installed in the weatherproof box at the antenna's apex. The FE provides signal paths for both stand polarizations, one per side. A differential 200 Ω line connects each blade pair to a Mini-Circuits ADT4-6T transformer (balun), that converts the balanced signal to 50 Ω unbalanced. A four-throw switch (Mini-Circuits GSWA-4-30DR) allows for selection between the antenna path and two calibration reference paths (Sec. <ref>). The switch output then connects to the first-stage low noise amplifier (LNA, Mini-Circuits Gali-74+, Tab. <ref>). A bandpass filter (Mini-Circuits BPF-C45+) suppresses signals outside of 25–90 MHz, attenuating HF and FM RFI sources outside the band of interest. Mini-Circuits LAT attenuators are used between components to improve impedance matching.Second-stage signal amplification is done after filtering, using two Mini-Circuits Gali-6+ amplifiers connected in cascade, each with 12.2 dB gain. While the Gali-74+ has a lower noise figure than the Gali-6+, the Gali-6+ was chosen for its flatter gain response over the LEDA band. Along the signal path, several buffering attenuators are installed to improve impedance match between components. Tab. <ref> shows the cascaded gain (ßGrx) and receiver temperature (ßTrx) after each component in the FE analog path. The overall receiver temperate ßTrx=432 K, and the receiver gain ßGrx=31.2 dB. Note that these values are calculated from specifications provided in component data sheets; actual measurements are provided in Sec. <ref>. Losses before the first-stage LNA and the LNA's noise temperature dominate ßTrx. The overall ßTsys of a LEDA radiometer is nonetheless dominated by the antenna temperature, which is >1000 K across the LEDA band. §.§.§ DC power and state controlThe FE receives DC power via its SMA output jack. The RF signal and DC power are separated by an on-board bias tee. The switch state is controlled by changing the DC voltage supplied to the board, between 17 V (sky), 20 V (cold reference) and 23 V (hot reference). This allows FE state to be controlled remotely from the electronics shelter.Many of the components on the board require 12 V or lower. Regulation to 12 V is conducted on an external board, which connects to the FE via a ribbon cable; the thermal load of the regulators would produce undesired thermal gradients if placed directly on the FE. §.§.§ Test ports The ability to measure reflection coefficients of components is essential for absolute calibration of the LEDA radiometers. To facilitate this, Hirose MS147 test ports have been added to the circuit. When a MS147 cable is connected to the test port, an internal mechanical switch within the test port reroutes the circuit to the MS147 cable. The MS147 ports are used to measure the reflection coefficients, ßΓant and ßΓrx, as introduced in Sec. <ref>.§.§.§ Calibration subcircuitAt LEDA frequencies, the intrinsic sky noise(>1000 K) is much greater than ambient temperature (∼300 K). As such, comparison to an ambient 50 Ω load results in a large swing in LNA input power; this is undesirable for two reasons. Firstly, digitizer dynamic range requirements are reduced, allowing more overhead to deal with radio interference. Secondly, large swings in power change which bits in the digitizer are being exercised, and are consequently more likely to be affected by non-linear quantization gain. Additionally, the signal to noise ratio after applying three-state switching (Eq. <ref>) is improved by using stronger references. As such, the reference calibration states on the FE are provided by a noise diode based subcircuit, with equivalent noise temperatures better matched to the sky temperature. The calibration subcircuit is shown in Fig. <ref>. This circuit is based on a NoiseWave NC501-12/SM noise diode package, which we have modified to improve its stability. The top plastic cover of the NC501 was removed and replaced by a DN505 heater seated uponan aluminium block. The DN505 is set to maintain a constant temperature of 60 ^∘C, to mitigate ambient temperature variations. The circuit includes a constant voltage regulator (LM3480) and constant current regulator (LT3092) to ensure the noise diode receives a stable DC supply.A resistive splitter is used to provide two calibration paths, upon which we place different attenuators to yield `hot' and `cold' references. Although a resistive splitter gives 3 dB of loss as compared to an ideal power divider, we found that thermal stability—the change in noise power output as a function of ambient temperature—was notably worse when a power divider was used; see Sec. <ref> for measurement details. §.§ Analog backendSecond-stage signal conditioning for the LEDA radiometers is performed using a connectorized analog receiver chain (CRX, Fig. <ref>). While the station is outfitted with an LWA 512-input second-stage analog receiver system (ARX) <cit.>, used to amplify and filter antenna signals prior to correlation, this system was not suitable forprecision radiometry: the ARX demonstrated crosstalk between neighboring channels, pickup of radiation emitted by the densely packed signal paths inside the shielded analog rack, and reflection along the signal path that affected the noise floor.§.§.§ Connectorized receiver system The first two components of the CRX are situated at the RF-shielded shelter's entry bulkhead: a Lightning arrestor (PolyPhaser GT-NFF-AL) and a FM bandstop filter (Mini-Circuits NBSP-108+). A RG316-DS coaxial cable connects the bulkhead components to the rest of the CRX, which is located in an RF-shielded rack. A bias tee (Mini-Circuits ZFBT-282-1.5A+) supplies DC voltage back to the antenna, while blocking DC on the output port. A highpass filter (Mini-Circuits SHP-50+) filters out VHF interference, this is connected to a Mini-Circuits ZFL-500LN+ amplifier (G=24 dB). A further 19 dB of amplification is provided by a ZFL-500HLN+ amplifier, which is optimized for higher power than the preceding 500LN+ model. Final-stage filtering of the FM band and above is then provided by two Mini-Circuits SLP-90+ filters in cascade. For improved impedance matching, Mini-Circuits VAT-6+ 6 dB attenuators are placed before, between, and after the two amplifiers.§.§.§ Switching Assembly In order to select between sky and reference diode states on the FE, different DC power levels are supplied via the CRX bias tees. The DC power supplied to the CRX bias tees is controlled by the switching assembly (SAX), shown in Fig <ref> as the receiver board voltage controller. The SAX system consists of a custom voltage regulation circuit controlled by a Rabbit 3000 Microprocessor via a wired Ethernet connection. The SAX accepts DC input power from a 28 V, 11 A Acopian Gold linear supply (A28H1100-230), and outputs power at 17, 20 and 23 V, as required by the FE. Tunable potentiometers may be used to adjust the output power, to account for power drop over the coaxial cables (1–1.5 V).The SAX accepts a PPS signal, which may be used to trigger state changes on the FE. Alternatively, FE state may be controlled manually, by issuing commands to the Rabbit microprocessor over Ethernet. As the SAX is located in the analog rack, it is encased in an RF-tight box, to shield any microprocessor-generated RF power from the analog systems.§.§ Inter-rack signal transport The digital and analog systems are housed in separate RF-shielded racks that share a common bulkhead wall.Consistent with the LWA engineering model, CRX signals in the analog rack are converted from unbalanced 50 Ωto balanced 100 Ω for transmission to the digital rack on Category-7a (CAT7A) Ethernet cable.A custom-made balun module converts every 4×SMA inputs into a single Belfuse ARJ45-ended output (oneconductor pair per RF signal path).Custom RF-tight, CAT-7A CONEC feedthroughs route signals through bulkhead plates.ARJ45 was substituted for LWA-standard RJ45 CAT5e during an upgrade cycle, motivated by superior near-end crosstalk (NEXT).Although the digitizer cards were not upgraded, VNA S21 bench measurements demonstrated cross-talk of < -60 dBfor a CAT7A signal path including a passthrough and single CAT5e plug and jack combination at one end.With this, residual cross talk in the signal path is dominated by the ADC card. Bench measurements have demonstrated reduction to between -30 and -40 dB across the science band when only two of four conductor pairs are used on each cable, corresponding to CAT5e 8P8C connector pins 1, 2 and 7, 8 which are physically the most widely separated.§ CALIBRATION EQUATIONSThe output power of a radiometer system is given byP_out = G k_B Δν (T_ant + T_rx)where G is the total gain of the radiometer's analog systems (amplifiers, filters, etc), k_B is Boltzmann's constant, and T_rx and T_ant are the noise-equivalent temperatures of the receiver and antenna, respectively.At frequencies corresponding to the CD trough, radiometer systems are generally sky-noise dominated; that is, T_ant≫ T_rx.Eq. <ref> is the fundamental measurement made by a radiometer.Antenna temperature is given by the average of the actual sky brightness T_sky(θ,ϕ) as seen from the antenna's location, weighted by the antenna's gain pattern B(θ,ϕ):T_ant(ν)=∫ dΩ B(θ,ϕ,ν)T_sky(θ,ϕ,ν)/∫ dΩ B(θ,ϕ,ν).If one separates the sky temperature T_sky into a "foreground" component, T_fg, and the cosmological, T_cosm, term consisting of the sky-averaged 21-cm emission from Eq. <ref> and the background CMB radiation, thenT_sky(θ, ϕ, ν) = T_fg(θ, ϕ, ν) + T_cosm(ν),such that Eq. <ref> becomesT_ant(ν)=∫ dΩ B(θ,ϕ,ν)T_fg(θ,ϕ,ν)/∫ dΩ B(θ,ϕ,ν) + T_cosm(ν).In practice, impedance mismatch between the antenna and receiver must also be taken into consideration, as must noise waves generated by the receiver's first-stage amplifier. These factors are detailed further in Sec. <ref>. The LEDA radiometer employs a three-state switching calibration technique <cit.>, where the receiver cycles between the sky and two calibration references (a "hot" and "cold" state). Three-state switching allows the removal of variations in system gain G = G(ν, t) and receiver temperature ßTrx = ßTrx(ν,t), and allows for a temperature scale to be imposed on the data. The LEDA outrigger antennas switch between the sky and two calibration reference paths. A noise diode in series with attenuators is used to provide a reference with an equivalent noise temperature of ßThot and ßTcold. The power measured by the radiometer in each state (Eq. <ref>) is given byP_ant= Gk_BΔν(ßTant+ßTrx)P_hot= Gk_BΔν(ßThot+ßTrx)P_cold= Gk_BΔν(ßTcold+ßTrx),where P_ant,P_hot and P_cold are powers measured in antenna, hot reference, and cold reference states;T_ant,T_hot and T_cold are antenna, hot and cold reference noise-equivalent temperatures. The three-state switch calibrated temperature T_ant may then be recovered viaßTant = (ßThot - ßTcold)P_ant-P_cold/P_hot-P_cold+T_cold.Example spectra for the three states, is shown in Fig. <ref>.As presented in <cit.>, the true antenna temperatureT_cantis related to the three-state calibratedT_ant by T_cant=T_ant(1-|Γ|^2),whereΓ is the reflection coefficient: a measure of impedance mismatch between the receiver and the antenna. However, Eq. <ref> is not strictly accurate for two reasons. Firstly, one must take care to use an appropriate definition for power gain G, for which there are several (see e.g., <cit.>). Here, we are interested in the power delivered to the load from a given source, for which the transducer power gain should be used. As shown in <cit.>, for a given amplifier with S_12 (reverse isolation, see Section <ref> for more details) negligibly small, when connected to a source with reflection coefficient Γ_S and a load with reflection coefficient Γ_L, the transducer power gain is given by:G_T=|S_21|^2(1-|Γ_S|^2)(1-|Γ_L|^2)/|1-S_11Γ_S|^2|1-S_22Γ_L|^2,whereS_21is a parameter equivalent to forward gain, and S_11 and S_22 are reflection coefficients. Note that in the ideal case, Γ_L=Γ_S=0 and Eq. <ref> yields G_T=|S_21|^2.Secondly, the noise temperature of the receiver, T_ rx, also depends upon the source. For an amplifier with optimal noise figure F_ opt, the noise figure F for a given Γ_S is given byF=F_ opt+4R_N/Z_0|Γ_S-Γ_ opt|^2/(1-|Γ_S|^2)|1+Γ_opt|^2where R_N is the equivalent noise resistance of the amplifier, Z_0 is the characteristic impedance, and Γ_opt is the amplifier's reflection coefficient at which its noise figure is the lowest. The receiver temperature—defined as T_rx=T_0(F-1) with T_0=290 K—is thus dependent upon the source. An alternative approach to modeling noise within an analog system is to use noise correlation matrices <cit.>, or `noise wave' analysis <cit.>; the EDGES formalism uses the latter approach.The magnitude of the inaccuracy of Eq. <ref> due to neglect of Eq. <ref> and Eq. <ref> is primarily dependent upon Γ_S. The formalism of <cit.> (and <cit.>) is therefore only accurate in the case that the diode and load states are well matched to the receiver; in both the LEDA and EDGES instruments, Γ_S is lower than -30 dB for reference states, so this requirement is satisfied. For the calibration detailed here, we follow the formalism of <cit.> and <cit.>, but nonetheless highlight that improvement of the formalism is an area deserving future examination. Following this formalism, Γ in Eq. <ref>, is calculated from reflection coefficients of the antenna and receiver, Γ_antand Γ_rx respectively (Fig. <ref>), measured with a vector network analyzer (VNA), andßTcant= ßTskyßHant|F|^2ßHrx^-1 + ßTu|ßΓant|^2|F|^2ßHrx^-1 +(ßTc cos(ψ) + ßTs sin(ψ))|ßΓant||F|ßHrx^-1.The terms of Eq. <ref> are: * ßΓant is the reflection coefficient of the antenna, as measured at the output of the balun.* ßΓrx is the reflection coefficient of the receiver. The first component in the receiver is a low noise amplifier (LNA), which will be a main cause of reflections between the antenna and the receiver.* ßHant=1-|ßΓant|^2 and ßHrx=1-|ßΓrx|^2 are gain terms arising due to antenna / receiver mismatch.* F=(1-|ßΓrx|^2)^1/2(1-ßΓantßΓrx)^-1 is another complex gain factor encompassing receiver and antenna mismatch.* T_u is the uncorrelated `noise wave' power, emitted from the LNA and reflected back by the antenna <cit.>. * ßTccos(ψ) and ßTssin(ψ) are correlated noise waves that depend on the amplitude and phase of the antenna reflection, whereψ is the phase of the noise wave reflected from the antenna.From Eq. <ref>, one may solve for the true sky temperature ßTsky. If the antenna is not lossless, a further correction must be applied:ßTcsky =(ßTsky - ßTamb(1-L))/L,where L=10^-l/10 for a loss l in dB, and T_amb is the ambient temperature of the antenna.§.§ Thermal uncertainties The LEDA receiver uses two reference diode states, in contrast to the load and diode approach used in EDGES. Here we show that the dual diode approach optimizes measurement signal to noise. An estimate of thermal noise present in the three-state switched spectrum may be found by propagating the uncertainties of Eq. <ref>:dT_ant^2=(∂ T_ant/∂ P_ant)^2dP_ant^2+(∂ T_ant/∂ P_cold)^2dP_cold^2+(∂ T_ant/∂ P_hot)^2dP_hot^2.where the uncertainty of measurement in each state is given by the radiometer equation. This yieldsdT_ ant= A √(dP_ ant^2+B(dP_ hot)^2+C(dP_ cold)^2)A =T_ hot-T_ cold/P_ hot-P_ coldB =(P_ ant-P_ cold/P_ hot-P_ cold)^2C =(P_ ant-P_ hot/P_ hot-P_ cold)^2It follows that to optimize measurement signal to noise, the two references should be as high power as possible, while maintaining a large difference in power between them. In tension with this, the finite dynamic range of the ADC motivates diode temperatures comparable to the sky brightness. Fig. <ref> compares the measurement uncertainty as a function of observation time for a dual diode system with fiducial values T_hot=6500 K and T_cold=1000 K, against a system with T_hot=450 K and T_cold=300 K. § INSTRUMENT CHARACTERIZATION Of particular importance to calibration is a sound understanding of the characteristics of the FE. In this section, we present detailed measurements of the characteristics of the FE and antenna for a single polarization (antenna 252A). Nonetheless, all FE boards undergo the same characterization process; comparison between antennas is presented in following sections.§.§ Gain linearity To ensure the amplifiers on the FE are operating in a nominal regime, we tested the gain linearity of the FE using an Agilent E4424B signal generator and Agilent N9000A spectrum analyzer. The signal generator was used to produce a 50 MHz tone with amplitudes covering -100 to -15 dBm. We found the 1 dB compression point occurs at an input power of -17.2 dBm (Fig. <ref>). Note that the expected power from the antenna is well under this 1 dB compression point: for a 10,000 K sky over 100 MHz (an overestimation), one would expect -78.6 dBm input power to the FE. The first harmonic of the 50 MHz tone was not apparent on the spectrum analyzer until an input power of -40 dBm (far above expected input power), when it appeared above the spectrometer's noise floor with an output power of -72.8 dBm. From our data, we extrapolate the IP2 intercept to be ∼37 dBm. §.§ Scattering parameters The reflection and transmission characteristics of the LEDA FE and LWA antenna were measured using an Anritsu MS2034B Vector Network Analyzer (VNA). Between the MS147 test ports and the SMA output, the LEDA FE can be treated as a 2-port network. This allows us to measure its scattering parameters (S-parameters) relating incident and reflected voltage waves.§.§.§ Antenna + balun A VNA measurement of the antenna cannot be made without the use of a balun; the characteristics of the balun must therefore be known for one to de-embed its effect. The MS147 connector directly after the FE balun allows the reflection coefficient ßΓant to be measured, see Fig. <ref>. Given the far-field distance of the antennas is many hundreds of meters, and that the surrounding environment affects the beam characteristics, data were necessarily taken in-situ at OVRO. Care was taken to ensure that during measurement, all equipment was placed low to the ground as far away as possible. The VNA was placed on the ground, at a distance of 20 m away from the antenna, orthogonal to the antenna blade pair under test. Low-loss coaxial cable was laid from the VNA, across the ground, and up the antenna's central mast to the FE MS147 connector that connects to the balun.The magnitude and phase of ßΓant for antenna 252A is shown in Fig. <ref>. The magnitude and phase of ßΓant are seen to vary smoothly as a function of frequency, varying between -4 to -6 dB over the 40–85 MHz band. §.§.§ Front-end receiver S-parameter measurements of the FE were taken by connecting the VNA to the MS147 connector directly preceding the switch. This port is labelledßΓsw0 in Fig. <ref>; we will refer to this as ßΓrx, as it is the main VNA measurement presented for the receiver board. The board was characterized over 10–100 MHz, using a low VNA port power (-25 dBm) such that the FE was operating in a linear gain regime. So that the FE could be measured as a single device-under-test (DUT), the board was powered via the regulator daughter board in lieu of using an external bias tee. The S-parameters for the FE are shown in Fig. <ref>. The overall gain (S_ 21) is 32.1±0.2 dB over 30–80 MHz. The 3 dB and 10 dB rolloff points occur at (27.7, 85.5) MHz and (26.0, 88.7) MHz, respectively; primarily due to the bandpass filter. The S_11(i.e. ßΓant), is better than -30 dB across the LEDA science band of 40–85 MHz, with S_11 increasing outside of the passband. Similarly, the S_22 is also better than -30 dB across the LEDA science band.§.§.§ Calibration parameters From the VNA measurements of the FE and antenna, we are able to form the calibration parameters ßHant, ßHrx and |F|^2 (Fig. <ref>). In the top panels of Fig. <ref>, the VNA measurements are shown in red, and smoothly-varying fitted models are shown in black; the bottom panel shows the residual between the VNA data and the fit. We have fit ßHrx with an 11-term polynomial, |F|^2 with a 21-term Fourier series, and ßHant with a combined 5-term polynomial and 21-term Fourier series. As such, ßHant has the largest effect upon calibration; the LNA's low reflection coefficient ßΓrx means that ßHrx has only a small (<0.1%) effect upon the overall calibration. §.§ Receiver temperatureWe determined the receiver temperature ßTrx using the Y-factor method <cit.>.For an accurate measurement, noise contribution and any loss from the cables and connectors between the reference source and DUT must be included in ßThot and ßTcold. Further, precise measurement requires Y ≫ 1, meaning the hot and cold references should be as different as possible.To measure the receiver temperature of the LEDA receiver boards (DUT), we applied the Y-factor method using a calibrated HP 346C noise source as a reference. Hot and cold references states were created by inserting a 10 dB and 6 dB pad between the 346C and the DUT; the resulting noise temperature was computed usingT_ cal=(1-L_att)T_346C + 290 L_attwhere L_att is the combined loss of the attenuator and coaxial cable, as measured using a VNA, and T_346C is the manufacturer specified noise temperature of the HP 346C source.We measured the ßTrx of all FE boards in the laboratory immediately prior to installation, using an Agilent 9000A spectrum analyzer; the HP346C reference source was connected at the MS147 test port between the balun and the switch. We find the receiver temperatures to be in good agreement with Tab. <ref> within 30–80 MHz, increasing rapidly outside the receiver passband (<ref>). For data analysis, we fit a line to the measured ßTrx between 40 and 80 MHz. §.§.§ Noise diode temperatures Once the receiver temperature is known, the temperature of the hot and cold references can be calculated with reference to the external HP346C. Fig. <ref> shows the equivalent noise temperature of the hot (red) and cold (blue) noise diode reference states. A linear model is fitted to both states (black), which is used in subsequent calibration. §.§ LNA noise wave analysis As discussed in Sec. <ref>,calibration requires that the noise waves emitted by the receiver are accounted for (Eq. <ref>). Characterization of the receiver's emitted noise wave <cit.> requires multiple measurements of output power with varying impedances at the receiver's input. A convenient method for characterization of the noise wave is by applying Eq. <ref> to a system where the antenna is replaced by an open (or shorted) coaxial cable; the wrapping of phase ϕ as a function of frequency over a cable of suitable length allows deduction of the phase of the emitted noise wave. The process we employed was as follows: * The reflection coefficient ßΓcoax of an open coaxial cable at room temperature was measured using a VNA, along with ßΓrx. * The coaxial cable was connected to the FE at the MS147 test port, and power spectra for ßPcoax, ßPhot and ßPcold were measured using a spectrum analyzer. * A three-state calibrated spectrum was computed via application of Eq. <ref> (top panel of Fig. <ref>). * Replacing the terms ßΓant and ßHant with ßΓcoax and ßHcoax, we applied least-squares fitting to estimate scalar values T_c, T_s, T_u and ψ.For the FE corresponding to antenna 252A, we measure T_U=194.67, T_c=-174.39, and T_s =-1.14. The overall magnitude of the noise wave from the receiver, when connected to antenna 252A, is shown in the bottom panel of Fig. <ref> and is at a level of a few percent of the sky temperature.§.§ Noise diode thermal stability The characteristics of many active RF devices, including noise diodes, are dependent upon ambient temperature. We used a thermal control chamber (Test Equity 1000) to characterize the effect of ambient temperature upon the output power of the FE noise diode. The chamber allows control of the ambient temperature from -15^∘ to 45^∘ C, with ±0.1^∘ C precision.The FE was placed in the chamber inside an RF-shielded box; a coaxial cable connected the FE output to an HP436A power meter via a bias tee that supplied DC voltage to the FE. The power meter also outputs a 0 dBm 50 MHz reference tone; this was connected to the FE input MS147 port, with 60 dB of attenuation was added at the reference output. We waited 20 minutes between temperature changes, to allow the FE and RF-tight box to equilibrate. We found a temperature coefficient of -0.00815 dB/K for the hot reference path, and -0.00585 dB/K for the cold path (Fig. <ref>). The fractional stability of the two paths is 0.00874%; as such, temperature dependence of the noise diode is not expected to be a significant source of error. §.§ Allan deviationThe Allan variation, σ_y^2(τ), and the Allan deviation, σ_y(τ)=√(σ_y^2(τ)), are common measures of stability over time. Allan deviation may be used to differentiate between different types of noise within a system. Notably, for random Gaussian noise, such as that in a radiometer, the Allan deviation will decrease as τ^-1/2. To characterize the stability of the LEDA radiometer system in the field, we took spectrometer data overnight with a 50 Ω load connected to the FE MS147 input port. Data were calibrated using the three-state switching method outlined above. We used the allantools package <cit.> to compute the Allan deviation for the calibrated overnight data (Fig. <ref>), finding a maximum integration time of τ=2000 s, before other system instabilities become significant.By Eqn. 4, for Δν=1 MHz and sky temperatures of 5000 K, 3000 K and 1000 K, the corresponding rms noise levels are 290 mK, 174 mK, and 58 mK, respectively. As such, the radiometer is stable enough to reach the level required for validation of the <cit.> result. Nevertheless, as will be discussed later, other systematics currently dominate the noise budget.We believe the main source of instability in this field test is the change in the load's ambient temperature overnight, and that the intrinsic stability is higher than that presented here; thermal isolation of the load will be required for future tests. § RESULTS Three LEDA FE boards, as described in Sec. <ref>, were deployed at OVRO-LWA in January 2016, to antennas 252, 254 and 255 (Fig. <ref>). On 2016-01-27, on-sky data were recorded for 24-hours using the LEDA digital spectrometer systems (Sec. <ref>). For these observations, we switched between the sky and the reference diode states every 5 seconds. For reproducibility, these data, along with analysis scripts used to generate plots in this paper are available online[<http://github.com/telegraphic/leda_analysis_2016>]. During the January 2016 deployment, the `B' polarization of board 252 was found to have poor characteristics, so have been excluded from analysis here. In this section, we first present detailed results from a single antenna, before comparing results across antennas in Sec. <ref>.§.§ Absolute calibration Data were calibrated using Eq. <ref>, following the measurement procedures outlined in Sec. <ref>. Fig. <ref> shows the dynamic spectra for antenna 252A over the 24-hour period on 2016-01-26, after RFI flagging (see Sec. <ref>). The corresponding antenna temperature spectrum at LST 11:00, is shown in the top panel of Fig. <ref>; the bottom panel shows the change in system temperature over the 24-hour period at 60 MHz.§.§ RFI environment To identify and flag RFI, we apply the sumthreshold algorithm <cit.>, which we have ported to a Python package called dpflgr. Dynamic spectra from antenna 252A post-flagging are shown in Fig. <ref>; the flagged data fractions for day and night are shown in Fig. <ref>. The RFI environment is seen to be quieter at night, but nevertheless several bright narrowband sources are omnipresent. We choose to completely flag channels or timesteps with high occupancy (>40%). The presence of increased RFI during the day, along with increased air traffic, on-site human activity, and potential solar flare events, motivate primary LEDA observations to be conducted at night. §.§ Comparison to sky models To compare against our measurements, we simulated the expected antenna temperature spectra using a model of the LWA antennawith several sky models: the aforementioned GSM <cit.>, the `updated' GSM released in 2016 <cit.>, and the Low-frequency Sky Model <cit.>. For the antenna gain pattern, we used an empirical model (valid between 40–80 MHz) based on LWA1 data <cit.>; the response at 50 MHz is shown in Fig. <ref>.Simulated antenna temperature spectra for the three models are shown in Fig. <ref>, for an observer at OVRO, LST 12:00. While the models are in agreement to the 10% level, the LFSM exhibits an unexpected dip at ∼45 MHz. After subtraction of a 5th order polynomial in logν (bottom panel), a discontinuity can be seen in the GSM2016 residual data. The residuals for both the GSM2016 and LFSM are of order ∼ 100 K, notably larger than the GSM. The behavior can be traced to the inclusion of data from the <cit.> 45-MHz survey, suggesting a systematic offset in the underlying data from which the sky model is generated. Due to the unexpected discrepancies in the GSM2016 and LFSM data, we use the GSM2008 as our reference model. A comparison between calibrated spectrum and that expected from the GSM2008 is shown in Fig. <ref>. The ratio between data and model lies between 0.85–0.92 across the 40–80 MHz band.§.§ Spectral index To compute the spectral index α, we perform a least-squares minimization onχ^2=∑_i^N[T_i^ meas-T_70(ν_i/70 MHz)^α ]^2/σ_i^2where T^meas_i are our measured sky temperature data per frequency channel ν_i, and σ_i^2 are per-channel estimates of the thermal noise. We used the lmfit Python package perform the minimization of χ^2 over fit parameters T_70 and α. We find the spectral index varies between -2.28 to -2.38 over LST (Fig <ref>). These values are consistent with other Northern hemisphere experiments (Tab. <ref>). The effect of beam chromaticity <cit.> is not considered here, and is left for future work.§.§ Comparisons across antennasThe fractional difference between spectra integrated for 20 minutes around LST 12:00 are shown in Fig. <ref>; measurements are consistent to ±5% between 40–83 MHz. Above ∼83 MHz, the attenuation due to bandpass filters gives rise to non-linear ADC gain effects, which act to artificially attenuate the sky temperature <cit.>. As shown in Fig. <ref>, the receiver temperature also increases out-of-band. Improving the response above 83 MHz is an ongoing effort toward verification of the purported <cit.> absorption feature. §.§ Residuals across antennas As detailed in <cit.> and <cit.>, beam chromaticity must be accounted for to mitigate frequency-dependent structure introduced to the global signal. The frequency-dependent response of the antenna must therefore be either simulated using EM software packages such as HFSS and FEKO, or measured directly; LEDA employs the latter approach. In-situ measurement of the gain pattern of LEDA antennas via cross-correlation with the OVRO-LWA core antennas is beyond the scope of this paper and will be detailed in a future publication. Nevertheless, it is illustrative to subtract a log-polynomial sky model from the calibrated data to produce residuals. Fig. <ref> shows the residuals after subtraction of log-polynomial fits for LEDA data between 50–80 MHz, averaged over a one-hour observation period centered an LST of 11:00, 2016-01-26. The calibration and reduction procedure was as follows. Data were calibrated following the absolute calibration approach of Sec. <ref>, after which RFI events were flagged (Sec. <ref>). After flagging, data were averaged in time (1 hr total) and frequency (1.008 MHz bins) to form mean observed spectra, T^ meas(ν) for each. From top to bottom panel, Fig. <ref> shows the residuals of calibrated data after subtraction of 1, 3, 5 and 7-term polynomial fits. We attribute differences between antenna stands primarily to beam chromaticity due to differences in surrounding terrain and differences in as-built antenna geometries. Antenna 252A exhibits the best performance (between -5 to 5 K after 7-term fit), with antennas 255A and 255B exhibiting notably higher residual values.§ DISCUSSIONIn this paper, we have presented the design and preliminary characterization results for the LEDA radiometer systems. The path toward detection of the 21-cm CD trough will require iterative improvements of the analog systems and analysis methods, as knowledge of the instrumental systematics improve. By comparison to the GSM2008, we find our antenna temperature is within 10-15% of that predicted for an empirical model of the LWA antenna. Unaccounted for losses in the antenna, or inaccuracy of the maunfacturer-supplied specifications of the HP346C noise source could account for this;if a multiplicative scale factor of 1.12 is applied, measured data agree with the model to within ±3%.We measure the spectral index of the sky to vary between -2.28 (LST 11:00) to -2.38 (LST 17:00, when the galaxy is high). While in agreement with other observations, we note that beam chromaticity has not been accounted for, which would improve the measurement. Pickup from the ground due to an imperfect ground screen, and from the Sierra Nevada mountain range on the horizon, potentially flatten the true spectral index of the radio sky. Improved measurement of the spectral index is the subject of future work. §.§ In-situ beam measurements An important outstanding step is empirical measurement of the antenna gain pattern for each outrigger antenna. As discussed in <cit.> and <cit.>, gain-pattern-induced chromaticity limits foreground subtraction: this motivated a complete redesign of the EDGES antenna to improve chromatic performance. While an empirical beam model for (closely-packed) LWA antennas has been derived by <cit.>, calibration requirements motivate a per-antenna model. Measurements between three antennas (Sec. <ref>) show agreement to within ±5% between 40-83 MHz; variation in antenna gain pattern may account for much of this. Indeed, these data highlight the advantage that the redundancy offered by multiple measurements of the sky with different radiometer systems provides.§.§ Future improvements We identify several areas in which our instrument characterization can be improved. Firstly, periodic measurement of scattering parameters measured in the field would allow longitudinal monitoring of the receiver and antenna's reflection coefficients. Additionally, the emitted noise waves could also be measured in the field. Measurement of the noise waves using a commercial impedance tuner in lieu of an open cable would offer an alternative characterization of the LNA noise waves.Here, we applied the calibration formalism of <cit.>. Other approaches, such as the matrix-based calibration approach of <cit.> offer an alternative approach based on more modern formalisms of noise characteristics. The approach of <cit.>, in which extra calibration parameters are included to better fit the data, also offers an alternative avenue toward improved instrument modeling.Our absolute temperature calibration relies upon an HP346C noise source with manufacturer-supplied characterization. Cross-calibration with other calibration standards, and/or experimental verification of the manufacturer-supplied parameters, may provide improved accuracy of the absolute temperature scaling. §.§ Validation of EDGES absorption feature Validation of the absorption feature reported by <cit.> is pressing. As reported here, the LEDA systems exhibit the required radiometric stability, but other systematics, namely the direction-dependent gain of the antennas, confound measurement. Further characterization work is ongoing. Upgrades to the LEDA systems are made on a rolling basis. Since the January 2016 campaign—as detailed here—several upgrades have been made to the LEDA systems. These improvements will be discussed further in a future paper. Briefly, radiometric receivers have been installed on all 5 outrigger antennas, modifications to further improve the stability of the noise diode have been made, and a logging system for measurement of the ambient temperature at the antenna has been added. Of importance to validation of the <cit.> signal, bandstop filters with sharper roll-off have been sourced to allow access to frequencies of up to 87.5 MHz, while still strongly attenuating the 88–108 MHz FM band.An observation campaign with the upgraded LEDA system was undertaken over November 2016–March 2017; analysis of these data, along with data from 2018, is ongoing.§ CONCLUSIONSMeasurement of the 21-cm emission from the early Universe via radiometric methods requires exquisite calibration and comprehensive knowledge of the radiometer systems. The purported detection of a 21-cm absorption feature during Cosmic Dawn by <cit.> suggests that the radiometric approach does indeed offer a window into Cosmic Dawn. Validation of the <cit.> signal is pressing, particularly given the concerns raised by <cit.>, and is an exciting opportunity for radiometric Cosmic Dawn experiments such as LEDA, SARAS 2 and PRIZM.In this paper, we have presented the design and characteristics of the LEDA radiometer systems. Comparison of the system performance with predictions based on the GSM2008 sky model and LWA antenna gain pattern are in agreement to the 15% level over 40–83 MHz. Between antennas, data agree to ±5%. Above 83 MHz, the rolloff of the filter for FM-band rejection (88–108 MHz) becomes significant.Upgrades to increase the LEDA observation window cutoff from 83 MHz to 87.5 MHz are underway. Further characterization work is also ongoing, in order to place limits on the 21-cm emission during Cosmic Dawn. In particular, individual characterization of the antenna's direction-dependent gain may be needed to account for the frequency dependence of the beam. Work on this characterization is underway, using interferometric measurements with the combined LEDA radiometer antennas and OVRO-LWA core antennas.§ ACKNOWLEDGMENTS This work has benefited from open-source technology shared by the Collaboration for Astronomy Signal Processing and Electronics Research (CASPER). We thank the Xilinx University Program for donations; NVIDIA for proprietary tools, discounts, and donations; Digicom for collaboration on manufacture and testing of samplers and ROACH2 processors; and Y. Belopolsky (Bel-Stewart R&D) for collaboration in development of CAT-7A ARJ45 pass-through hardware and cable assemblies.We thank R. Blundell, Ed Tong, and P. Riddle of the Smithsonian Astrophysical Observatory Submillimeter Receiver Lab for collaboration on development and fabrication of receivers and other LEDA signal path and control elements.The great dedication, innovation, and exemplary skill of the Caltech Owens Valley Radio Observatory staff, demonstrated in constructing the LWA array, having created a purpose-built facility in no time deserves special mention.LEDA research has been supported in part by NSF grants AST/1106059, PHY/0835713, and OIA/1125087. The OVRO-LWA project was enabled by the kind donation of Deborah Castleman and Harold Rosen. GB acknowledges support from the Royal Society and the Newton Fund under grant NA150184. This work is based on the research supported in part by the National Research Foundation of South Africa under grant 103424. GH acknowledges the support of NSF CAREER award AST-1654815.mnras | http://arxiv.org/abs/1709.09313v3 | {
"authors": [
"D. C. Price",
"L. J. Greenhill",
"A. Fialkov",
"G. Bernardi",
"H. Garsden",
"B. R. Barsdell",
"J. Kocz",
"M. M. Anderson",
"S. A. Bourke",
"J. Craig",
"M. R. Dexter",
"J. Dowell",
"M. W. Eastwood",
"T. Eftekhari",
"S. W. Ellingson",
"G. Hallinan",
"J. M. Hartman",
"R. Kimberk",
"T. J. W. Lazio",
"S. Leiker",
"D. MacMahon",
"R. Monroe",
"F. Schinzel",
"G. B. Taylor",
"E. Tong",
"D. Werthimer",
"D. P. Woody"
],
"categories": [
"astro-ph.IM"
],
"primary_category": "astro-ph.IM",
"published": "20170927024926",
"title": "Design and characterization of the Large-Aperture Experiment to Detect the Dark Age (LEDA) radiometer systems"
} |
Azimuthal angle correlations at large rapidities: revisiting density variation mechanism [ December 30, 2023 =========================================================================================== We describe the algorithm behind our PACE 2017 submission to the heuristic tree decomposition computation track. It was the only competitor to solve all instances and won a tight second place. The algorithm was originally developed in the context of accelerating shortest path computation on road graphs using multilevel partitions. We illustrate how this seemingly unrelated field fits into tree decomposition and parameterized complexity theory.§ INTRODUCTION Tree decompositions are an established graph decomposition methodology. They can be used to measure how “close” a graph is to a tree. The width of a decomposition indicates how “tree-like” a graph is. We refer to <cit.>, <cit.>, and <cit.> for an overview of the field.Tree decompositions are often used in combination with parameterized complexity. Many NP-hard problems can be solved in linear time on tree graphs. The corresponding linear time algorithms can often be generalized to tree decompositions. The generalized algorithms' running times are usually linear in the graph size but super-polynomial in the width of the decomposition.A lot of theoretical research with in-depth results into this field exists. However, it is unclear whether these results translate into algorithms that are efficient in practice. A core issue consists of finding decompositions of small size. Without an algorithm that is fast in practice and computes tree decompositions with a reasonably small width, no algorithm parameterized in the tree width is usable in practice.To bridge this gap between theory and practice, the PACE implementation challenge was created in 2016 <cit.>. Track A2 focuses on computing tree decompositions of small width within a given amount of time. Because of its practical relevance, the challenge was repeated in 2017. In this paper, we describe the FlowCutter submission to the PACE 2017 Track A2 contest. It was the only submission so solve all test instances and won a close second place.FlowCutter was originally developed to accelerate shortest path computations on road graphs. Fortunately, it is applicable in a significantly broader context. The corresponding PACE 2016 and 2017 submissions demonstrate this. For a survey of shortest path computation algorithms, we refer to <cit.>. Tree decompositions are rarely used explicitly in this community. However, very similar concepts are used.Many shortest path acceleration techniques exist. They work in two phases:In the preprocessing phase, the graph is transformed. In the query phase, shortest paths are computed using the transformed graph. The preprocessing phase may be slow. The query phase should be fast. The motivation is that the preprocessing only needs to run when the map is updated. This is assumed to be rare. For example, in the preprocessing phase one can construct a small tree decomposition. In the query phase, the decomposition can be used to compute shortest paths.These shortest path computation algorithms are an example of parameterized complexity. The algorithms assume that a small tree decomposition is given in the input. The complexity of the shortest path problem is bounded in terms of the tree width. The difference to regular fixed parameter tractability theory is that the considered problem is not NP-hard.Two common shortest path techniques are Contraction Hierarchies (CH) <cit.> and Multilevel Dijkstra (MLD) <cit.>. None of the original publications mention tree decompositions. However, very similar concepts are used. The similarity is a comparatively recent discovery. We illustrate the relationship in this paper.A question often asked is what algorithms big companies with mapping services use. Unfortunately, most companies do not publicly share this information. A welcome exception is Microsoft. They stated that the Bing routing service uses MLD <cit.>. This illustrates that research in this domain has direct practical implications.The relation between multilevel graph partitions and tree decompositions is not well known as none of the cited survey articles mention it <cit.>. The relationship was hinted in <cit.>. In this paper, we make the connection clearer and more explicit. There is a one-to-one correspondence between multilevel graph partitions and rooted tree decompositions. Using this one-to-one relationship, we can also establish a link between CH and MLD via tree decomposition theory. We believe that this connection is useful also in other contexts than shortest path computations. It allows to interpret tree decompositions as multilevel graph partitions.Our PACE 2017 submission uses the multilevel graph view. It is based upon recursive bisection respectively nested dissection. It uses FlowCutter <cit.> to bisect graphs. §.§ Outline We first define the terminology used in this paper. Afterwards, we present a high level overview of related shortest path algorithms. We then present in the next step, how multilevel partitions and tree decompositions relate. Using the multilevel partition view, we describe our PACE 2017 algorithm. Finally, we present the PACE 2017 results.§ DEFINITIONS In this section, we start by defining standard tree decomposition and chordal graph concepts.Afterwards, we formally define the notion of multilevel partition.An edge cut C of a graph G=(V,E) is a non-empty edge set. C decomposes G into the connected components of G∖ C. Similarly, a node separator S of a graph G=(V,E) is a non-empty node set.S decomposes G into the connected components of G∖ S. Usually, one requires there to be at least two connected components in G∖ S. However, we allow for the degenerate case of there only being one connected component. We say that a cut or separator separates two nodes, if they are in different components. An undirected graph is chordal, if for every cycle Z with at least four nodes, there exists an edge between two nodes of Z that are not adjacent within Z. A chordal supergraph G' of a graph G is a supergraph of G that is chordal. Triangulated graph is a synonym for chordal graph, also used in the literature. An elimination order of an undirected graph G is an order O of the nodes of G.A supergraph G' is obtained by iteratively contracting the nodes of O, i.e., removing a node and adding a clique among its neighbors. In the shortest path literature, the term contraction order is used. A perfect elimination order of a graph G is an order where no node contraction inserts edges. A graph has a such an order, if and only if, it is chordal <cit.>. We therefore refer to G' as chordal supergraph of G.A tree decomposition of a graph G=(V,E) is a pair (B,T), where B is the set of bags and T is the tree backbone. Every bag b∈ B is a set of nodes, i.e., b⊆ V.T is a tree where the bags are the nodes, i.e., B is the set of nodes of T. A tree decomposition must fulfill three criteria to be valid: * Every node is in a bag, i.e., ⋃_b∈ B b = V.* For every edge {x,y} of G, there must be a bag b∈ B such that both end points are in b, i.e., x∈ b and y ∈ b.* For every node x, the subgraph of the tree backbone T induced by all bags that contain x is a tree.A rooted tree decomposition is a tree decomposition, where the tree backbone is directed towards a root bag r. A rooted tree decompositions is illustrated in Figure <ref>. The width of a tree decomposition is the maximum size of a bag plus one. The tree width tw of a graph is the minimum width over all tree decompositions.A cell c of a graph G=(V,E) is a node subset c⊆ V. Two cells a and b touch, if there is an edge with endpoints in a and b or when a and b share a node. The touching relationship is illustrated in Figure <ref>. The boundary b of c is the set of nodes adjacent to a node in c but not in c.A multilevel partition P is a set of cells. Two criteria must be fulfilled for P to be valid: * V is a cell. We refer to V as the toplevel cell.* Touching implies nesting. If two cells a and b touch then a⊆ b or b⊆ a.p is a parent cell of c if c⊂ p and no other cell q exists with c⊂ q ⊂ p. Analogous, c is a child of p. In a bilevel partition, only the toplevel cell has children. A bilevel partition is illustrated in Figure <ref> and a multilevel partition in Figure <ref>.The term “multilevel partition” is often used in experimental algorithms papers <cit.> but rarely formally defined. It usually describes an informal algorithm design pattern.The exact details therefore significantly vary between application-focused papers. However, there is a significant difference between our formalization and many other papers. We separate cells using node separators. Many application-focused papers use edge cuts. We use separators as it enables a tighter coupling with tree decompositions.§ APPLICATION: SHORTEST PATHS IN ROAD GRAPHS Multilevel Dijkstra (MLD) and Contraction Hierarchies (CH) are two very successful shortest path acceleration techniques. In this section, we briefly outline the algorithms. §.§ Multilevel Dijkstra The core idea of Multilevel Dijkstra (MLD) was introduced in <cit.> using a bilevel partition. It was extended to multiple levels in <cit.>. The algorithm was refined and popularized by <cit.>. A parameterized analysis of the algorithm using tree width is provided by <cit.>. We describe the initial bilevel algorithm as it is the simplest. This technique is also often called multilevel overlay graph (MLO) or CRP which references the title of <cit.>.The first step of the preprocessing consists of computing a set of non-touching cells. This is illustrated in Figure <ref>. In the second step, an overlay is computed for every cell c.Denote by G_c the subgraph induced by c and its boundary. Further, denote by H_c the overlay graph. H_c is substitution for G_c that maintains the shortest path distances among the boundary nodes. H_c must contain at least the boundary nodes. In theory, H_c can be very complex graphs. However, usually a clique among the boundary nodes is used. The weights of the clique edges are computed by running Dijkstra's algorithm from every boundary node restricted to G_c. Figure <ref> illustrates the overlays.The query consists of running a bidirectional variant of Dijkstra's algorithm. It explores the cells of s and t and replaces all other cells with their overlays. Figure <ref> illustrates the explored subgraph.The idea of this algorithm can easily be applied recursively. The extension uses a multilevel partition as input.In <cit.>, the MLD distance query running time with clique overlays was bounded by O(tw^2 (log n) (log(twlog n))). §.§ Contraction Hierarchies The algorithm was originally described in <cit.>. A newer variant of the algorithm is called Customizable Contraction Hierarchies (CCH) <cit.>. We outline the later variant, because it is tighter coupled with tree decompositions.The preprocessing consists of several steps. In the first step, a contraction order O is computed using nested dissection. In the second step, the nodes are contracted in this order. The inserted edges are called shortcuts. The graph plus the shortcuts is the CCH. In different contexts, the contraction order is called elimination order, the shortcuts are the fill-in, and the CCH is the chordal supergraph.The order O is usually interpreted as ordering the nodes from bottom to top. The top node is the last node of O and contracted last. A node x is lower than y, if x comes before y in O.Denote by w(x,y) the weight of edge {x,y}. The objective is to enforce the lower triangle inequality. For every triangle {x,y,z} in the CCH, it requires that w(x,z)+w(z,y) ≥ w(x,y), where z is lower than x and y. The lower triangle inequality is enforced using a triangle listing algorithm. All edges already present in the input graph are assigned their input weight. Shortcuts are assigned ∞. Triangles {x,y,z} are enumerated ordered by the position of the lowest node z in O. If w(x,z)+w(z,y)< w(x,y), then w(x,y) is set to w(x,z)+w(z,y).A query for a st-path consists of a bidirectional graph search from s and t. From a node x both searches only follow edges to nodes higher than x. Once both searches are terminated, a shortest up-down st-path P is found. P is a path v_1⇝ v_i ⇝ v_k such that the nodes between v_1 and v_i are ascending according to O. The nodes from v_i to v_k are descending according to O.The query is correct, if there always exists an up-down path that is as long as a shortest path. To show this consider a shortest path P.If P is an up-down path, we are done. Otherwise, there exists a subpath x→ z → y such that z is lower than x and y. As z is contracted before x and y, there exists a shortcut from x to y. Further, because of the lower triangle inequality, the shortcut's weight is as long as the path x→ z → y. We can thus remove z from P. Either P is now an up-down path or we apply the argument iteratively. As we eventually always end up with a shortest up-down path, the query is correct.The maximum cliques in a CCH/chordal supergraph correspond to the tree decomposition. It was shown in <cit.> that the number of nodes visited by one side is bounded by O(twlog n). As the explored subgraphs can be dense, the distance query runs in O(tw^2 log^2 n) time.§ CONNECTION BETWEEN TREE DECOMPOSITIONS AND MULTILEVEL PARTITIONS In this section, we prove a one-to-one correspondence between multilevel partitions and rooted tree decompositions. A consequence is that both concepts are just two different views onto the same object.We first describe the algorithm to translate a multilevel partition into a rooted tree decomposition. In the next step, we describe the algorithm to perform the inverse operation. Finally, we show that the outputs of both algorithms are valid and that chaining both algorithms is the identity function. §.§ From Multilevel Partition to Rooted Tree Decomposition. For every cell c in the multilevel partition, we construct a bag b in a tree decomposition. b is the union of the boundary and interior nodes of c minus the interior nodes of all children. The parent-child relation between cells induces a tree on the bags.This is the tree backbone. The top level cell is the root of the rooted tree backbone. The so obtained tree decomposition can be degenerate, i.e., it is possible that bags exist that are subsets of other bags.§.§ From Tree Decomposition to Multilevel Partition A tree decomposition T does not uniquely define a multilevel partition P because the tree backbone does not have a root. The transformation from T must therefore start by picking a root bag r. With respect to r, we can construct for every bag b except the root a cell as follows: Denote by p the parent of b, i.e., the first node on the unique path from b to r.We set the boundary of cell c to b∩ p. The interior of c is set to the union of all direct or indirect children bags of c minus c's boundary. We additionally construct a cell for the root bag. This cell's boundary is empty and its interior is the whole graph. §.§ ExampleFigure <ref> depicts a multilevel partition with 16 cells. Every cell (except the top level cell) is depicted as closed curve.The color of a curve indicates the recursion depth. Orange indicates depth 1, blue depth 2, green has depth 3, and red indicates depth 4. Grey nodes are not part of any separator. The color of the remaining nodes indicates the depth of the separator that they are part of. Table <ref> enumerates all cells in the multilevel partition and the derived bags. Together with the parent-child relation of the cells, we obtain the rooted tree decomposition depicted in Figure <ref>.There are cells, such as the cell with interior {a,b}, which is “divided” along the separator {b} into one part, namely the cell with interior {a}. Sufficiently large cells are usually divided into more than one part.However, for tiny cells, it often occurs that such awkward separators are used.[15]o7cm < g r a p h i c s > Largest Bag in Road Graph Tree Decompositon.Figure <ref> depicts the nodes in a mostly German road graph at their geographical positions. Every dot is a node. The nodes in the largest bag of a tree decomposition are highlighted. This bag has 369 nodes. The Figure depicts an area surrounded by orange dots. This area is divided into two parts along separator consisting of further orange dots.The nodes in the area are the nodes in the cell. The orange separator nodes are also in the cell. However, the orange nodes that surrounded the area are not inside the cell. They are in its boundary.§.§ CorrectnessThe constructed tree decomposition is valid. We need to show that the three conditions laid out in the tree decomposition definition are fulfilled. We need to show that every node is in a bag. To prove this, we observe that every node is interior to the top level cell. A node v interior to a cell c is either in v's bag or interior to a child of c. As every cell has a finite number of descendants, we cannot build infinite chains of nested cells. We have thus proven that every node is in a bag.Further, we need to show that for every edge {x,y} there exists a bag b such that x and y are part of b. As touching cells are ordered by inclusion and as there are only finitely many cells, we know that there exists a smallest cell c_x that has x in its interior. x is in the bag of c_x because x is in c_x but not in the interior of a child of c_x. Let c_y be the analogous smallest cell for y. If c_x = c_y, then c_x is the required b. Otherwise, we observe that the existence of {x,y} implies that c_x and c_y touch each other. They are therefore ordered by inclusion. Assume without loos of generality that c_x ⊆ c_y. The existence of {x,y} implies that y is on the boundary of c_x and therefore in the bag of c_x. c_x is therefore the required bag b.Finally, we need to show that for every node x the set of bags that include x forms a subtree of the backbone. Consider again the smallest cell c_x that contains x. All cells that contain x are ancestors of c_x. As x is in c_x, x cannot be in the bag of any ancestor of c_x. We therefore know that c_x is the only cell that has x in its interior and bag. Pick another cell d whose bag contains x. As d ≠ c_x, we know that x is on d's boundary. Denote by p the parent cell of d. x is in p or on the boundary of p. We can thus conclude that x is in the bag of p. From d we can iteratively follow the parent relation. As the parent-child relation is acyclic, we eventually arrive at c_x. As for every d the corresponding path ends at c_x, we have proven that the set of bags that include x forms a subtree of the backbone.As we have proven all three properties, we have proven that the constructed tree decomposition is valid.The constructed multilevel partition is valid. We need to show that touching cells are ordered by inclusion.Cells touch if one of two conditions is fulfilled: * They share a node.* There exists an edge with endpoints in both cells.We show the required ordering property independently for both cases.Consider some arbitrary node x and denote by T_x the subtree of the backbone induced by x. T_x contains a unique bag b_x that is closest to the root. The bags in the tree decomposition fall into three categories: * They are in T_x but are different from b_x.* They lie on the unique path from b_x to the root.* They are neither in T_x nor on the path.We show that only the cells corresponding to the bags of category 2 contain x. The cells along a path are trivially ordered by inclusion. Let q be a bag from category 1. As it is different from b_x, it cannot be the root. Therefore, there exists a parent bag p of q. By construction x is also contained in p. We have that x∈ q∩ p. x is in the boundary and not in the interior of the cell corresponding to q.Now let q be a bag from category 3. The corresponding cell is constructed by forming the union of bags that do not contain x. The union thus also does not contain x.Finally, let q denote a bag on the path. The constructed union contains all bags of T_x and thus also contains x. It remains to show that x is not on the boundary of the corresponding cell. This follows from the fact that b_x is the only bag that contains x. As b_x is the bag the farthest away from the root, no parent bag of a bag on the path is b_x. x is thus not part of any boundary.This completes the first part of the proof. Next consider the case where the cells corresponding to two bags b_x and b_y touch because there exists an edge {x,y} between them. By convention, we set x∈ b_x and y∈ b_y. We know from the second property of the tree decomposition definition that there exists a bag b_xy that contains x and y.We know that the tree backbone must contain the following four paths: * There is a path D_x from b_xy to b_x along T_x because trees are connected. * Using an analogous argument, we know that there exists a path D_y from b_xy to b_y along T_y.* We can follow the parent relation from b_x to the root and obtain a path U_x.* Analogously, there exists a path U_y from b_y to the root.By concatenating all four paths U_x, U_y, D_y, and D_x, we obtain a cycle in the tree backbone. As the tree backbone is a tree, we conclude that the cycle must be degenerate. The set of the root r, b_x, b_y, and b_xy can therefore only contain at most two elements. We conclude that one of the following conditions must hold as otherwise the set would contain three or more elements: b_x = b_y, b_x = r, or b_y = r. In the first case, the cells are equal and thus ordered. In the second and third cases, one of the cells is the root cell and thus by definition a superset of the other cell.This completes the second and last part of the proof. We have proven that the constructed multilevel partition is valid.Applying both algorithms after another is the identity. As both algorithms essentially just copy the tree backbone, it is clear that it is not modified. It remains to show that the contents of the bags and cells remains unchanged.Pick a cell c from the multilevel partition and denote by b the corresponding bag. b is by construction a subset of the union c's boundary and its interior. The missing nodes must be in the interior of a child cell of c and therefore in the bag of a descendant of b. The union over the bags of the subtree rooted at b is thus equal to the union of c's boundary and interior. Removing the boundary from this union yields the interior, as the interior and the boundary are disjoint. The interior of c is thus left unchanged after applying both algorithms. Applying both algorithms is therefore the identity. § PACE 2017 FLOWCUTTER SUBMISSION Our algorithm is based on the nested dissection paradigm <cit.>. In the graph partitioning literature, this approach is often called recursive bisection. Compared to the original algorithms, ours does not recurse until all graph parts are smaller than a given threshold. Instead, it aborts early, when it detects that the decomposition width can no longer be improved.Internally, our algorithm uses the multilevel partition representation described in the previous section. It translates the data to a tree decomposition when formatting the result. For every cell c, it stores two sets B_c and I_c. Both sets contain nodes of the input graph. B_c contains the nodes on the boundary of c. I_c contains the interior nodes of c that are not in a child cell. The bag corresponding to c is B_c ∪ I_c. We refer to |B_c| + |I_c| as the bag size of c. The maximum bag size over all cells in a multilevel partition is the corresponding tree decomposition width minus one.Our algorithm maintains a multilevel partition. The cells are organizes in two sets ℱ and 𝒪. ℱ is the set of final cells. 𝒪 is the set of open cells. ℱ contains the cells that can no longer be modified. The cells in 𝒪 can be modified using a cell split operation. Splitting c yields one final cell and several possibly zero open cells. No new cell has a bag size larger than the one of c.The set 𝒪 is organized as maximum priority queue using the bag size as key. Our algorithm further maintains the maximum bag size over all final cells. Initially, the set ℱ is empty. We define the maximum bag size over an empty ℱ as 0. Initially,𝒪 contains a single toplevel cell c that contains all nodes. This means that B_c = ∅ and I_c = V.Our algorithm iteratively splits cells. It picks the cell c from 𝒪 that maximizes the bag size. If c's bag size is smaller or equal to the maximum bag size over all final cells, our algorithm terminates. Otherwise, it removes c from 𝒪 and splits c. This yields a final cell c_f and several possible zero open cells c_1… c_k. Our algorithm adds c_f into ℱ and updates the stored maximum bag size. The cells c_1… c_k are added into 𝒪. §.§ Splitting a Cell In this subsection, we describe how a cell c is split. Consider the subgraph G_c induced by I_c. Our algorithm computes a balanced separator S_c using FlowCutter. The resulting cells are derived from this separator. The final cell c_f is similar to c except that nodes are removed from I_c. Formally, B_c_f = B_c and I_c_f = S_c. The resulting open cells correspond to the connected components of G_c∖ S_c. For every component there is a cell c_c. Our algorithm sets I_c_c to the nodes in the corresponding component. B_c_c is a subset of B_c ∪ S_c. It only contains the nodes adjacent to a node in I_c_c.To test the adjacency, we maintain an array that maps every node onto a bit. Initially, we set each bit to false. For all nodes in I_c_c, we set the bit to true. Next we iterate over all nodes x in B_c ∪ S_c. We iterate over the neighbors of x in the input graph. If the bit of one neighbor is set, we add x to B_c_c. Finally, we reset the bit of all nodes in I_c_c to false.§.§ FlowCutter FlowCutter is a graph bisection algorithm described in <cit.>. In this subsection, we present the high level ideas. For all details, we refer to <cit.>. The base algorithm computes balanced edge cuts. It can be extended to compute balanced node separators. Our algorithm interprets the graphs as symmetric unit flow network. It repeatedly solves multi-source multi-target maximum flow problems.Initially, there is only one source and one target node. They are picked uniformly at random. Our algorithm starts by computing a maximum flow. From this flow, it derives the least balanced minimum cut C. If C is sufficiently balanced, it terminates. Otherwise, it iteratively improves the cut in rounds. New source or target nodes are added in each round.Suppose that C is unbalanced. The algorithm thus continues with an additional round. Further, assume without loose of generality that the source's side is smaller than the target's side. Our algorithm marks all nodes on the source's side as additional sources. Additionally, it marks one node on the target's side as source node. This node is called piercing node. It is an endpoint of a cut edge. Choosing the piercing node is difficult. If there is a choice that does not increase the cut size, our algorithm picks it. Otherwise, it employs a heuristic based on the distances from the original source and target nodes. After adding the additional sources, C no longer separates all source nodes. In the next round a different cut is found.Usually, we run several FlowCutter instances in parallel with different initial source and target pairs. We refer to the number of instances as the number of cutters.In each round FlowCutter finds a new cut.During its execution our algorithm thus does not only compute a single balanced cut. Instead, it computes a set of cuts that heuristically optimize balance and cut size in the Pareto-sense. This allows us to optimize complex criteria heuristically. For example, we can pick a cut that minimizes the ratio between the cut size and the smaller side size, i.e., with minimum expansion. From our experience, minimizing the expansion subject to a bounded balance, heuristically yields good results.In flow problems, node capacities can usually be reduced to edge capacities <cit.>. Using such a reduction, FlowCutter can be used to compute node separators. Alternatively, node separators can be derived from edge cuts by picking for every cut edge an endpoint. The former can achieve smaller separators but is slower as the graph needs to be expanded. §.§ Submission Details Our submission runs FlowCutter in an endless loop until the maximum execution time of 30 min is reached. It uses node capacities and minimizes the expansion subject to a bounded balance. In each iteration, our submission varies some parameters such as the number of cutters, the minimum required balance, or the random seed.For small instances, we compute heuristic elimination orders before running FlowCutter. From these orders, we derive tree decompositions. We compute an order that greedily picks a minimum degree node. Another order greedily picks a minimum fill-in node. We compute these orders because the derived tree decompositions are sometimes slightly smaller than those computed by FlowCutter. However, on large instances computing them using our implementation is not possible within the maximum execution time of 30 min.On the large instances, we configure FlowCutter to compute edge cuts and minimize the cut size subject to a bounded balance. From the edge cuts, we derive node separators by picking an endpoint of every edge. This is necessary to assure that at least one iteration finishes within the allowed execution time.Our implementation finds a new tree decomposition each time a cell is split. Formatting the textual output corresponding to the new tree decomposition can be slower than performing the split operation. To avoid these costs, we produce a new textual output only every 30 s.§ EVALUATION To evaluate our algorithm, we submitted it the PACE 2017 challenge. We entered track A2. The objective is to compute a tree decomposition of small size within 30 min. All competitors were evaluated on the same set of test graphs. The algorithms that find a smallest decomposition win the instance. Entries were ranked using the Schulze method. This usually implies that the competitor that wins the most instances wins the competition. Our submission won the second place. The submission by Tamaki et al. won the competition. Abseher et al. is ranked third. The difference between the first two places is small. In the following, we look at the results in depth.Denote by t^* the best solution found for an instance and by t the solution found by a competitor. A weakness of the PACE evaluation method is that is does not consider how large t-t^* is. t-t^* = 1 is weighted in the same way as t-t^* = 1000. Only the relative ranking among competitors matters. To investigate the impact of this effect, we present in Table <ref> alternative rankings.We report relative approximation ratios. Denote by t^* the best solution found for an instance and by t the solution found by a competitor. We report the average and maximum t/t^* over all non-trivially[A non-trivial solution must have width at most |V|-5. Criterion was defined by PACE organizers.] solved instances. With respect to both criteria, our submission clearly wins. Interestingly, the competition winner is only ranked fifth with respect to the average t/t^*.Additionally, we report how often t-t^* is below a threshold. For instances that were not or only trivially solved, we set t=∞. “=0” is the number of instances where a competitor produced a best solution. “≤ X” indicates how often it was off by X. “< ∞” is the number of non-trivially solved instances.The algorithm by Tamaki et al. clearly wins with respect to “=0”. The gap with our algorithm is only 2% for “≤ 1”. For “≤ 2” our algorithm wins. Further, our algorithm is the only competitor that finds non-tririval solutions on all instances.The algorithm by Tamaki et al. finds on many small instances solutions that are one or two nodes smaller than our algorithm. For large instances, their algorithm often does not find a solution.§ CONCLUSION We conclude that the winning algorithm by Tamaki et al. is good, if the instances are small and achieving the smallest width is of utmost importance. If the instances are large or finding nearly the best solutions is good enough, our algorithm is better. Further, we illustrated that viewing tree decompositions as multilevel partitions is a useful algorithm design tool.Acknowledgements. I thank the PACE organizers for providing a good testing infrastructure. It made the experimental evaluation easier and less error-prone. I thank Michael Hamann for fruitful discussions.10acdgw-o-16 Ittai Abraham, Shiri Chechik, Daniel Delling, Andrew V. Goldberg, and Renato F. Werneck. On dynamic approximate shortest paths for planar graphs with worst-case costs. In SODA'16, pages 740–753. SIAM, 2016.amo-nf-93 Ravindra K. Ahuja, Thomas L. Magnanti, and James B. Orlin. Network Flows: Theory, Algorithms, and Applications. Prentice Hall, 1993.bdgmpsww-rptn-16 Hannah Bast, Daniel Delling, Andrew V. Goldberg, Matthias Müller–Hannemann, Thomas Pajor, Peter Sanders, Dorothea Wagner, and Renato F. Werneck. Route planning in transportation networks. In Algorithm Engineering - Selected Results and Surveys, volume 9220 of LNCS, pages 19–80. Springer, 2016.bcrw-s-16 Reinhard Bauer, Tobias Columbus, Ignaz Rutter, and Dorothea Wagner. Search-space size in contraction hierarchies. Theoretical Computer Science, 645:112–127, 2016.bp-aicgc-93 Jean Blair and Barry Peyton. An introduction to chordal graphs and clique trees. In Graph Theory and Sparse Matrix Computation, volume 56 of The IMA Volumes in Mathematics and its Applications, pages 1–29. Springer, 1993.b-atgt-93 Hans L. Bodlaender. A tourist guide through treewidth. Acta Cybernetica, 11:1–21, 1993.b-tsa-07 Hans L. Bodlaender. Treewidth: Structure and algorithms. In Proceedings of the 14th International Colloquium on Structural Information and Communication Complexity, volume 4474 of LNCS, pages 11–25. Springer, 2007.bmsss-ragp-13 Aidın Buluç, Henning Meyerhenke, Ilya Safro, Peter Sanders, and Christian Schulz. Recent advances in graph partitioning, 2013. arXiv:1311.3144 [cs.DS].dhjkkr-tfpac-16 Holger Dell, Thore Husfeldt, Bart M. Jansen, Petteri Kaski, Christian Komusiewicz, and Frances Rosamond. The first parameterized algorithms and computational experiments challenge. In IPEC'16, pages 30:1–30:9, 2016.dgpw-crprn-13 Daniel Delling, Andrew V. Goldberg, Thomas Pajor, and Renato F. Werneck. Customizable route planning in road networks. Transportation Science, 51(2):566–591, 2017.bingcrp Bing Developers. Bing maps new routing engine. Website, 2012. Online at <https://blogs.bing.com/maps/2012/01/05/bing-maps-new-routing-engine>.dsw-cch-15 Julian Dibbelt, Ben Strasser, and Dorothea Wagner. Customizable contraction hierarchies. ACM Journal of Experimental Algorithmics, 21(1):1.5:1–1.5:49, April 2016.fg-imig-65 Delbert R. Fulkerson and O. A. Gross. Incidence matrices and interval graphs. Pacific Journal of Mathematics, 15(3):835–855, 1965.gssv-erlrn-12 Robert Geisberger, Peter Sanders, Dominik Schultes, and Christian Vetter. Exact routing in large road networks using contraction hierarchies. Transportation Science, 46(3), 2012.g-ndrfe-73 Alan George. Nested dissection of a regular finite element mesh. SIAM Journal on Numerical Analysis, 10(2):345–363, 1973.hs-gbpo-16 Michael Hamann and Ben Strasser. Graph bisection with pareto-optimization. In ALENEX'16, pages 90–102. SIAM, 2016.hsw-emlog-08 Martin Holzer, Frank Schulz, and Dorothea Wagner. Engineering multilevel overlay graphs for shortest-path queries. ACM JEA, 13(2.5):1–26, December 2008.sww-daola-00 Frank Schulz, Dorothea Wagner, and Karsten Weihe. Dijkstra's algorithm on-line: An empirical case study from public railroad transport. ACM JEA, 5(12):1–23, 2000. | http://arxiv.org/abs/1709.08949v1 | {
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A new approach for short-spacing correction of radio interferometric data sets S. Faridani1Corresponding author: [email protected] ,F. Bigiel2,L. Flöer1,J. Kerp1 S. Stanimirović3Received — ; accepted — =================================================================================================================================empty emptyIn many industrial robotics applications, such as spot-welding, spray-painting or drilling, the robot is required to visit successively multiple targets. The robot travel time among the targets is a significant component of the overall execution time. This travel time is in turn greatly affected by the order of visit of the targets, and by the robot configurations used to reach each target. Therefore, it is crucial to optimize these two elements, a problem known in the literature as the rtsp. Our contribution in this paper is two-fold. First, we propose a fast, near-optimal, algorithm to solve rtsp. The key to our approach is to exploit the classical distinction between task space and configuration space, which, surprisingly, has been so far overlooked in the rtsp literature. Second, we provide an open-source implementation of the above algorithm, which has been carefully benchmarked to yield an efficient, ready-to-use, software solution. We discuss the relationship between rtsp and other tsp variants, such as the gtsp, and show experimentally that our method finds motion sequences of the same quality but using several orders of magnitude less computation time than existing approaches.§ INTRODUCTIONIn many industrial robotics applications, such as spot-welding, spray-painting or drilling, the robot is required to visit successively multiple targets. Consider for instance the drilling task depicted in fig:setup, which was proposed at the Airbus Shopfloor Challenge held during ICRA 2016 in Stockholm, Sweden. The task, mimicking the actual drilling process in aircraft manufacturing, consisted in drilling as many holes as possible in one hour, from a given pattern of 245 hole positions. The robot travel time between the holes is a significant component of the overall execution time. This travel time is in turn greatly affected by the order of visit of the holes, and by the robot configurations used to reach each hole. Therefore, it is crucial to optimize these two elements, a problem known in the literature as the rtsp, see <cit.> for a recent review. Finally, note that, since the position of the panel is unknown at the beginning of the round, the planning time is included within the one hour limit. Thus, in this Challenge as in many practical applications, there is a need for an algorithm that can find near-optimal plans within minutes, not hours.rtsp is closely related to tsp, a classical problem in Computer Science. In tsp, a set of locations is given on a map, and one is interested in finding the order to visit all the locations while minimizing the total travel distance, see fig:layoutssubfig:tsp. The key difference between rtsp and tsp is that, in rtsp, each of the targets (e.g. a hole position) may be reached by multiple robot configurations, also known as ik solutions.One simple work-around may consist in assigning a fixed robot configuration for each target: rtsp then becomes a classical tsp among the assigned robot configurations. Such a work-around is however sub-optimal. Another approach consists in formulating rtsp as an instantiation of gtsp: in gtsp, the locations are split into bins, and one is required to visit exactly one location per bin, see fig:layoutssubfig:gtsp. Here, each bin will contain the different robot configurations corresponding to the same target. While there have been many works devoted to gtsp and some efficient solutions exist, the sheer size of real-world rtsp instances [For instance, in the set-up of the Airbus Shopfloor Challenge, using a discretization of π/2 radians for the free-dof, one obtains 3,779 different configurations, grouped into 245 bins, which cannot be solved in practical times by any existing gtsp solver. See also sub:other_methods for a detailed comparison.] make this approach inapplicable in practice. sec:related provides a more detailed discussion on existing approaches to solve rtsp.Our contribution in this paper is two-fold. First, we propose a fast, near-optimal, algorithm to solve rtsp. The key to our approach is to exploit the classical distinction between task space and configuration space, which, surprisingly, has been so far overlooked in the rtsp literature. Specifically, we propose a three-step algorithm [see fig:layoutssubfig:rtsp for illustration]:* Find a (near-)optimal visit order of the targets in a task-space metric (e.g. Euclidean distance between the hole positions), using classical tsp algorithms; * Given the order found in Step 1, find for each target the optimal robot configuration, so that the total path length through the configurations is minimized in a configuration-space metric (e.g. Euclidean distance between the robot configurations – collisions are ignored at this stage), using a graph shortest path search algorithm; * Compute the final collision-free configuration-space trajectories by running classical motion planning algorithms (e.g. rrt) through the robot configurations found in Step 2 and in the order given by Step 1.Our second contribution is to provide an open-source implementation [The open-source implementation of the proposed method is accessible at <www.github.com/crigroup/robotsp>] of the above algorithm. In particular, we carefully benchmark different key components of the algorithm (underlying task-space tsp solver, configuration-space metric, discretization step-size for the free-dof), so as to come up with an efficient, ready-to-use, software solution.The remainder of the paper is organized as follows. In sec:related, we discuss existing approaches to rtsp. In sec:method, we introduce and describe in detail the proposed method. In sec:experiments, we present the experimental results, showing in particular that our method finds motion sequences of the same quality but using several orders of magnitude less computation time than existing approaches. Finally, in sec:conclusions, we conclude with a few remarks. § RELATED WORKSA recent survey on strategies to solve rtsp can be found in <cit.>. We recall some of the main results below.In <cit.>, the authors plan the motions for a 3-dof robot to visit 6 targets, each of which can be reached with two different robot configurations. For this, they formulate a gtsp with 6 bins and 2 configurations per bin. They then convert the gtsp into a tsp, which can be solved efficiently.In <cit.>, the authors consider a fruit picking task, which has up to 250 targets, but with only one robot configuration per target. The problem can then be directly formulated as a tsp, which the authors solved using the Nearest-Neighbor heuristic.In <cit.>, the authors consider a rtsp with one configuration per target, which can then be formulated as a regular tsp. The emphasis here is on collision-free trajectories, which are difficult to find when the environment includes obstacles. First, the authors approximate the travel cost between configurations as the Euclidean distance in the configuration space. Then, a minimum spanning tree is computed using Prim's algorithm to find a near-optimal tour under the approximated cost. Next, collision-free paths are calculated given this near-optimal tour. The collision-free paths yield an updated travel cost, which is then used to iteratively refine the tour. This idea of computing first a good tour with a simple metric (without considering collisions) before computing collision-free trajectories is reused in our present work (but without the iterative refinement step).An extension of this work is proposed in <cit.>. Here the authors consider multiple robot configurations per target. Instead of a spanning tree, they compute a near-optimal group spanning tree <cit.>. On a task involving 50 targets with 5 configurations per target, a near-optimal solution could be found in 9,600 s.Following a different approach, the authors of <cit.> propose to use ga to solve gtsp. The optimization criteria is the task cycle time. On a task with 3-dof and 6-dof robots involving 50 targets, a near-optimal solution is found in 1,800 s. The quality of the solution depends on several control parameters (related to the ga) and the number of iterations. This approach has been further extended to include collision-free path planning for 2D and 3D environments <cit.>.Yet another approach consists in formulating rtsp as a multi-objective constraint optimization problem. In <cit.>, a robotic spray painting task is considered. The optimization criteria is set to minimize the task planning and execution time while maximizing the painting quality. Three constraints were defined: process, resources and quality constraints. The multi-objective problem then is solved using the dfs algorithm. On a task involving 8 targets, with 4 configurations per target, an optimal solution is found in 10,000 s.The main limitation of all the works discussed above is the large computation time they require. In particular, none of these works could have been applied to the setting of the Airbus Shopfloor Challenge presented in sec:introduction, which involves 245 targets, with tens of configurations per target, and which has to be solved within a few minutes – since the planning time is counted in the one hour limit of the challenge. § ROBOTSP ALGORITHM §.§ Setting Consider n targets in the task-space. A tour in the task-space that visits each target exactly once is called a task-space tour [Strictly speaking, a tour requires to return to the first target, so we are making a slight abuse of vocabulary here.]. We first compute ik solutions for each target – using a suitable discretization for the free-dof if necessary. A tour in the configuration-space that starts from the robot home configuration, visits, for each target, exactly one ik solution associated with that target, and returns to the home configuration is called a configuration-space tour. Our objective is to find the fastest collision-free configuration-space tour subject to the robot constraints (e.g. velocity and acceleration bounds).Let m_i be the number of ik solutions found for target i.If we do not take into account obstacles, there are (n-1)!(∏_i=1^n m_i) possible configuration-space tours (with straight paths) for this task. One cannot therefore expect to find the optimal sequence by brute force in practical times. §.§ Proposed algorithm As presented in sec:introduction, our method consists in: * Finding a (near-)optimal task-space tour in a task-space metric; * Given the order found in Step 1, finding, for each target, the optimal robot configuration, so that the corresponding configuration-space tour has minimal length in a configuration-space metric – collisions are ignored at this stage; * Computing fast collision-free configuration-space trajectories by running classical motion planning algorithms (e.g. rrt-Connect with post-processing <cit.>) through the configurations found in Step 2 and in the order given by Step 1.Implementation details and benchmarking results for Steps 1 and 3 are given in sec:experiments.Regarding Step 2, we first construct an undirected graph as depicted in fig:cspace_graph. Specifically, the graph has n layers, each layer i contains m_i vertices representing the m_i ik solutions of target i (the targets are ordered according to Step 1), resulting in a total of ∑_i=1^nm_i vertices. Next, for i∈[1,…,n-1], we add an edge between each vertex of layer i and each vertex of layer i+1, resulting in a total of ∑_i=1^n-1m_im_i+1 edges. Finally, we add two special vertices: Start and Goal, which are associated with the robot home configuration, and connected respectively to the m_1 vertices of the first layer and the m_n vertices of the last layer.The edge costs are computed according to a configuration-space metric: for instance, the cost for the edge joining vertices *q and *q' can be given by the Euclidean distance in the configuration space √(∑_k=1^(q_k-q'_k)^2). sub:metrics_benchmark examines in detail how the choice of such metrics influences the quality of the final path. One can note here that the metric should be fast to compute – in particular, collisions are ignored at this stage – since the costs must be computed for all m_1 + ∑_i=1^nm_im_i+1 + m_n graph edges.Finally, we run a graph search algorithm to find the shortest path between the Start and Goal vertices. By construction, any path between the Start and Goal vertices will visit exactly one vertex in each layer, in the order specified by Step 1. Conversely, for any choice of ik solutions for the n targets, there will be a path in the graph between the Start and Goal vertices and going through the vertices representing these ik solutions. Therefore, Step 2 will find the true optimal selection of ik solutions that minimize the total cost, according to the specified configuration-space metric, given the order of the targets. §.§ Complexity analysis For Step 1, it is well-known that tsp is NP-complete, which means that finding the true optimal tour for n targets has in practice an exponential complexity. Many heuristics have been developed over the years to find near-optimal tours. For instance, 2-Opt <cit.> and lkh <cit.> can find tours in practical times with an optimality gap bellow 5% and 1% respectively <cit.>.For Step 2, let M be an upper-bound of the number of ik solutions m_i per target. The number of graph vertices is then nM and the number of the graph edges is nM^2. Since Dijkstra's algorithm (with binary heap) has a complexity in ElogV where E and V are respectively the number of edges and vertices, Step 2 has a complexity in nM^2log(nM).For Step 3, one has to make n-1 queries to the motion planner, yielding a complexity in n. However, as the constant in the(average computation time per motion planning query) is large, the overall computation time is dominated by that of Step 3 in our setting. In general, the computation time of motion planning queries depends largely on the environment (obstacles), see <cit.> for recent benchmarking results showing the CPU time required when planning practical robot motions. § EXPERIMENTSThis section evaluates the proposed method when applied to the drilling task shown in fig:setup. Our system is formed by a Denso VS060 6-dof industrial manipulator equipped with a commercial off-the-shelf hand drill. All benchmarks were executed in a system with Intel® Core™ i7 processor and 24 GB RAM, GeForce GTX 960M video card, running Ubuntu 16.04 (Xenial), 64 bits. §.§ Benchmarking task-space tsp solversTo solve the task-space tsp (Step 1 of our algorithm), one may use exact or near-optimal solvers. The choice depends on the trade-off between the available CPU time and the solution quality. Here we evaluate three tsp solvers. * Exact: cip can be used to find true optimal tsp tour <cit.>. Here, we used the scip Optimization Suite <cit.> to implement an exact solver; * 2-Opt: We re-implemented this simple, yet efficient, algorithm to find a near-optimal solution to tsp. The algorithm iteratively improves an initial guess by repeatedly replacing pairs of edges that cross over <cit.>. * RNN: We re-implemented this algorithm, which consists in iteratively selecting the nearest neighbor as the next target to visit. This process is repeated until all the targets are visited <cit.>. One can do several restarts from different initial targets, and choose the tour with the lowest cost from all the restarts. The drawback of this method is that it tends to corner itself, which requires long edges to get back to unvisited targets.fig:tsp_solvers shows a benchmark of the three methods. We run the tsp solvers on task-space subsets of 25, 50, 100, 150, 200 random targets as well as on the total 245 targets.One can observe that the 2-Opt solver yields high-quality tours (less than 5% of sub-optimality) with low CPU time usage (less than 1 s). As for the Exact solver, it is not practical for more than 150 targets. Therefore, for all the subsequent experiments, we shall use 2-Opt as our near-optimal task-space tsp solver. §.§ Benchmarking configuration-space metricsThe configuration-space metric that defines the edge cost in Step 2 of our algorithm is the key component for the overall performance of the method. Given two robot configurations *q and *q', the ideal cost of the edge c^*(*q, *q') is the duration of a time-optimal collision-free trajectory between them. However, since there are thousands of such edges, running full-fledged motion planning algorithms (with collision checks) for every edge would not be tractable. Therefore, one must consider approximate metrics, which should be fast to compute, yet give a good prediction of the corresponding time-optimal collision-free trajectory duration. Here we evaluate three such metrics. * Weighted Euclidean joint distance: The cost c(*q, *q') is estimated as the weighted ^2 norm:c(*q, *q') := √(∑_k=1^w_k(q'_k-q_k)^2),where w_k is a positive weight for joint k. The weights are chosen in proportion to the maximum possible distance (Euclidean distance in the task-space) traveled by any point on the robot, when moving along the corresponding joint. Similar to <cit.>, in our experiments this metric outperforms consistently the Euclidean joint distance. * Maximum joint difference: The cost c(*q, *q') is estimated as follows:c(*q, *q') := max_k(q'_k-q_k)/q̇^max_k.The intuition of this metric is to determine the maximum joint displacement when moving from *q to *q' by simply computing the joint difference, (q'_k-q_k), over the joint k velocity limit, for k ∈[1,…,]. Then the maximum value is used; * Linear trajectory interpolation: the cost c(*q, *q') is given by the duration of a trajectory obtained by linear interpolation. It only requires to specify the positions, *q and *q', and guarantees continuity at the position level subject to the robot velocity and acceleration bounds. Moreover, this metric does not consider obstacles which greatly reduces its computing time.fig:metrics_benchmark shows the benchmarking results for the three proposed configuration-space metrics. One can observe that the Maximum joint difference metric takes the lowest CPU time and yields task execution times comparable to, in some cases even better than, the Euclidean and Linear Interpolation metrics. Therefore, for all the subsequent experiments, we shall use Maximum joint difference as our metric. §.§ Benchmarking discretization step size for the free dofMany industrial tasks such as spot-welding, spay-painting or drilling involve less than 6 degrees of freedom. Therefore, a classical 6-dof industrial robot has more joints than strictly required to execute such tasks. Specifically, the drilling task at hand involves 5 dof, since the rotation θ about the drilling direction is irrelevant. One approach to tackle this redundancy can consist in setting a specific value for the irrelevant dof: for instance θ∈{0, π/2, π, 3π/2} for a π/2 discretization step size. For each of the discretized value of θ, we then have a full 6-dof ik problem. To solve the full 6-dof ik, we next use OpenRAVE's IKFast <cit.>, which outputs all the ik solutions (here we have a discrete redundancy situation – think of the elbow up and elbow down configurations). We finally group all the IK solutions corresponding to all the discretized values of θ into a single list, which is the list of all IK solutions that will be considered for a given target. tab:discret_effect gives the total number of IK solutions considered as a function of the discretization step size and of the number of targets. One can see that the choice of the discretization step size is governed by a trade-off between speed and optimality. fig:discrete shows the computation time and task execution time as a function of the discretization step size. As expected, the computation time increases as the discretization step size decreases, but interestingly, the task execution time does not change significantly for step sizes below π/4, which thus yields a good trade-off between CPU time and task execution time. Therefore, for all the subsequent experiments, we shall use π/4 as our discretization step size. §.§ Comparison to other methodsAs none of the methods described in sec:related provide public implementations, we were unable to reproduce their results and perform a fair comparison with the method herein presented. However, we can note that the computation times reported in previous works are several orders of magnitude higher than ours, yet for problem instances that are smaller [The problem instance size is considered in terms of targets and configurations per target] than what we consider in this work.In the following, we compare our method to two of the existing methods, which we could re-implement.* tsp in C-space <cit.>: when only one configuration is considered per target, rtsp is reduced to a regular tsp in the configuration space. Here, for each target, we consider the ik solution with the best manipulability index <cit.>;* glkh: here the rtsp is formulated as a gtsp <cit.>. To solve that gtsp, we use the state-of-the-art glkh solver <cit.> which makes use of the lkh heuristic <cit.>. fig:other_methods shows the comparison of our method (RoboTSP) to the two methods just described. While the tsp in C-space method has a similar running time as RoboTSP (indeed both run a tsp on the same number of targets), the time durations of the trajectories it produces are higher than those of RoboTSP, since it does not optimize the ik choice per target.glkh produces trajectories with similar total duration as RoboTSP but the computing time is higher several orders of magnitude.A visualization with the trajectories produced by these three methods to visit all the 245 targets is available at <https://youtu.be/w33QfRjKFs8>. § CONCLUSIONSWe have proposed a method to determine a near-optimal sequence to visit n targets with multiple configurations per target, also known as the rtsp. For a complex drilling task, which requires visiting 245 targets with an average of 28.5 configurations per target, our method could compute a high-quality solution in less than a minute. To our knowledge, no existing approach could have solved the same problem in practical times. We have also provided a carefully benchmarked open-source software solution, which can be readily used in complex, real-world, applications such as drilling, spot-welding or spray-painting. § ACKNOWLEDGMENTThis work was supported in part by NTUitive Gap Fund NGF-2016-01-028 and SMART Innovation Grant NG000074-ENG.6IEEEtran | http://arxiv.org/abs/1709.09343v2 | {
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On The Statistical Properties of Cospectra]On The Statistical Properties of Cospectra1Center for Data Science, New York University, 65 5h Avenue, 7th Floor, New York, NY 10003 2Center for Cosmology and Particle Physics, Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA 3DIRAC Institute, Department of Astronomy, University of Washington, 3910 15th Ave NE, Seattle, WA 98195 4INAF-Osservatorio Astronomico di Cagliari, via della Scienza 5, I-09047 Selargius (CA), ItalyIn recent years, the cross spectrum has received considerable attention as a means of characterising the variability of astronomical sources as a function of wavelength. While much has been written about the statistics of time and phase lags, the cospectrum has only recently been understood as means of mitigating instrumental effects dependent on temporal frequency in astronomical detectors, as well as a method of characterizing the coherent variability in twowavelength ranges on different time scales. In this paper, we lay out the statistical foundations of the cospectrum, starting with the simplest case of detecting a periodic signal in the presence of white noise. This case is especially relevant for detecting faint X-ray pulsars in detectors heavily affected by instrumental effects, including ,and . We show that the statistical distributions of both single and averaged cospectra differ considerably from those for standard periodograms. While a single cospectrum follows a Laplace distribution exactly, averaged cospectra are approximated by a Gaussian distribution only for more than ∼ 30 averaged segments, dependent on the number of trials. We provide an instructive example of a quasi-periodic oscillation inand show that applying standard periodogram statistics leads to underestimated tail probabilities for period detection. We also demonstrate the application of these distributions to aobservation of the X-ray pulsar Hercules X-1.§ INTRODUCTION Time series analysis is one of the primary ways to understand the physical properties of astronomical object in our universe, from exoplanets and stars to black holes and Active Galactic Nuclei (AGN).Fourier analysis, especially through the periodogram[We distinguish in this paper between the power spectrum, which describes the process at the source generating variable time series, and the periodogram, which denotes a realization of said power spectrum, i.e. the time series we actually observe. In line with the signal processing literature, we also use the term periodogram or power spectral density for the square of the absolute value of Fourier amplitudes derived from an observed light curve. We use the terms cospectrum or cospectral density to denote the real component of the cross-spectrum, i.e. the result of a multiplication of the Fourier amplitudes of one light curve with the complex conjugate of the Fourier amplitudes of a second light curve (see also Section <ref> for an exact definition)], has long been used to find periodic and quasi-periodic signals as well as characterize the stationary stochastic processes often present in accreting systems.While in principle, the statistics of the periodogram is well understood and characterized in the literature <cit.>, the periodogram is often subject to instrumental effects like dead time that change its statistical properties and thus make statistical inference difficult in practice <cit.>.In the past years, the field of spectral timing has enjoyed significant success by making it possible to combine both temporal and spectral information in a single model. Within this framework, the complex cross spectrum—defined as the Fourier transform of one time series with the complex conjugate of the Fourier transform of a second time series–holds a central position. The cross spectrum is commonly used to compute phase lags, which are easily converted to time lags and are central for understanding e.g. reverberation mapping in accreting black holes <cit.>. The real part of the cross spectrum, also named cospectrum, has gained less attention, but can be just as useful. It has long been used to study gravity waves in the Earth's atmosphere <cit.>, models of the Martian atmosphere <cit.>, the solar heliosphere <cit.>, surface elevation of arctic sea ice <cit.> and drifting snow <cit.> as well as surface gravity waves in beaches <cit.> and eddy heat flux in the Earth's troposphere <cit.>.Within astronomy, and in particular the field of X-ray timing, it has recently been recognized as a powerful solution to two different problems. One problem is the reliable estimation of the variability contained in certain features of the power spectrum like quasi-periodic oscillations (QPOs) via the fractional rms amplitude. The rms amplitude crucially depends on reliable estimation of the Poisson noise level, which is sometimes difficult to calculate in practice. For instruments with two independent, identical detectors, the cospectrum of the light curves measured in each detector, respectively, will not contain any white noise contribution, since Poisson counting statistics are effects local to the instrument, and thus the observed data sets are independently sampled in each detector. This makes estimation of source-intrinsic variability much more reliable than comparable measurements using the periodogram.The cospectrum is also effectively used as as one approach to mitigate instrumental effects, in particular an effect called dead time. In many X-ray detectors, after detection of a photon there is a time interval during which the detector (or in imaging detectors the individual pixel) is blocked from detecting a second photon. This time interval is generally called dead time (the detector is effectively “dead”, and incoming photons will not produce a signal) and its characteristic time scale is set by the details of the detector and can range from very short (2μs in theGamma-Ray Burst Monitor) to very long (2.5ms in ). It leads to frequency-dependent changes in the mean and variance of the statistical distributions governing the periodogram, which cannot be mitigated by averaging periodograms of multiple segments.Where in standard analysis, light curves of multiple detectors are summed before Fourier transforming the summed light curve, it is possible to instead Fourier-transform the signal of two independent detectors within the same instrument observing the same source strictly simultaneously in the same energy band.The resulting co-spectral densities will be less affected by dead time (see details in ), because the latter is introduced in each detector independently and tends to cancel out in the cospectrum. This approach relevant for current X-ray missions carrying at least two identical detectors such asand , but will also be relevant to future missions with multiple detectors like . It has recently been used instudies of millisecond pulsars <cit.>, Ultraluminous X-ray Sources <cit.> and X-ray binaries <cit.>. It is particularly important for instruments likewhich have a fairly long dead time and thus exhibit dead time effects in the periodogram at frequencies comparable to those where signals are expected, and at fairly low count rates. Similarly, the cospectrum of two time series taken with the same instrument, but in different wavelength ranges, can be used to characterize the coherent, phase-aligned variability in both time series as a function of frequency.While much has been written on the subject of the statistics of cross spectra and time lags <cit.>, the derivation of cospectral statistics is notably absent from the astronomy literature, and most publications utilizing the approach of forming co-spectra from identical, but independent detectors observing the same source assume that either the χ^2_2 distribution used for standard periodograms or a Gaussian distribution for averaged spectra is appropriate. That these assumptions are valid has not been shown until now. In this paper, we lay out the basic statistical distributions for detecting periodic and narrow quasi-periodic signals in the presence of detector noise (e.g. photon counting statistics) for both single and averaged cospectra. We show below that unlike for the periodogram, the statistical distribution for a single cospectrum reduces to a Laplace distribution, while the distribution for averaged cospectra is considerably more complex. We also find that for averaged cospectra consisting of more than ∼30 averaged individual segments, the assumption of a Gaussian distribution is indeed appropriate for single-trial tail probabilities and reduces computation overhead. However, for averaged cospectra of fewer segments, more stringent significance thresholds or large numbers of trials, the statistical distribution–and hence the derived p-values used for period detection–deviate significantly from a Gaussian distribution. We note that the results below hold only for cases where the signal to be detected is phase-aligned in both light curves. This must be true by definition for analyses utilizing data from independent, identical detectors observing the same source simultaneously (unless detectors are so far apart that light travel time effects might be an issue and photon arrival times are not barycentre-corrected to the gravitational centre of the solar system), but might not strictly be true for observations of the same source e.g. in different energy bands. We caution the reader that for signals that are not phase-aligned, the resulting power may decrease significantly and result in a non-detectable signal.The paper is laid out as follows. In Section <ref>, we derive the PDF of a single cospectrum, and show associated simulations and detection thresholds for period detection in Section <ref>. Section <ref> extends the derivation to the common case where the cospectra of several time series are averaged. We show that the statistical distribution indeed changes from a Laplace distribution once multiple cospectra are averaged and becomes consecutively more Gaussian as a larger number of cospectra are included in the average in Section <ref>. Finally, Section <ref> presents two real-world examples: the first uses simulateddata of a quasi-periodic oscillation (QPO) as commonly found in accreting neutron star X-ray binaries. The second example comprises a realobservation of the bright X-ray pulsar Hercules X-1. We end in Section <ref> with a short discussion and conclusion. All figures and results are reproducible, and the associated code can be found online[<https://github.com/dhuppenkothen/cospectra-paper>]. In a second, forthcoming paper, we will treat the considerably more complex case of cospectra where the time series consist of stochastic variability and show how to model the cospectrum in both a Maximum Likelihood and Bayesian framework.For the reader looking for the statistical distributions of relevance, who may be only casually interested in the mathematical background, we point to Equations <ref> and <ref> for the probability density functions (PDFs) for a single cospectrum and averaged cospectrum, respectively, and Equations <ref> and <ref> for the cumulative distribution functions (CDFs) in both cases. § THE STATISTICAL DISTRIBUTIONS OF COSPECTRAL DENSITIESIn the following, we will consider the example of detecting a strictly periodic or very narrow quasi-periodic signal in the presence of simple white noise, as is commonly the case for example in pulsar searches in X-ray data. We assume that both the white noise and the periodic signal are wide-sense stationary (i.e. the mean and the autocovariance of the time series do not change with time) and the light curve is evenly sampled simultaneously in two identical detectors, or that photon arrival times are binned in intervals of equal length. While the white noise is also assumed to be strictly stochastic, the signal to be detected may be either deterministic (in the case of a strictly periodic signal) or stochastic (for a quasi-periodic oscillation). We then aim to reject a null hypothesis where the observed cospectral density at a given frequency can be explained by white noise alone. §.§ Statistical Distribution for a Single CospectrumConsider two independently distributed, evenly-sampled, constant, stationary time series, 𝐱 = {x_k}_k=1^N𝐲 = {y_k}_k=1^N with N data points taken at simultaneous time intervals {t_k}_k=1^N with a constant time resolution Δ t and a total duration T = NΔ t. Assume for simplicity that the measurements x_k and y_k are normally distributed, such thatx_k∼ 𝒩(x, w_x^2)y_k∼ 𝒩(y, w_y^2)with means x, y and variances w_x^2, w_y^2. The data points in the time series 𝐱 and 𝐲 can be expressed in terms of a Fourier series, x_k =1/N∑_jℱ_x(j)x_y =1/N∑_jℱ_y(j) where ℱ_x(j)=1/2 (A_xj - i B_xj) e^-i( 2 π j t/T) ℱ_y(j)=1/2 (A_yj - i B_yj) e^-i( 2 π j t/T). Here, i = √(-1), and A_xj, A_yj and B_xj, B_yj describe the real and imaginary parts of the Fourier amplitudes, respectively (for a pedagogical introduction into Fourier analysis, see ). We restrict ℱ_x(j) and ℱ_y(j) to frequencies between ν_j=0 = 1/T and the Nyquist frequency ν_j=N/2 = 1/(2Δ t). The complex cross spectrum is then calculated by multiplying the Fourier transform of light curve 𝐱 with the complex conjugate of the Fourier transform of light curve 𝐲 (, see alsofor a recent review of spectral timing techniques): ℱ_x(j) ℱ_y^*(j) =1/2 (A_xj - i B_xj) e^i 2 π j t/T1/2 (A_yj + i B_yj) e^i -2 π j t/T =1/4 [ (A_xjA_yj + B_xjB_yj) +i (A_xjB_yj - A_yjB_xj) ] Note that for strictly real-valued time series, as light curves in astronomy always are, A_j = A_-j and B_j = - B_-j, such thatℱ_x(j) ℱ_y^*(j) = 1/2{ (A_xjA_yj + B_xjB_yj) + i (A_xjB_yj - A_yjB_xj) }. The real part of this equation is the cross-spectral equivalent of the power spectral density, also called the cospectrum: C_j = 1/2(A_xjA_yj + B_xjB_yj) . For evenly sampled, normally distributed light curves (and indeed for uncertainties coming from a wide range of statistical distributions, including the Poisson distribution, if N is large), the real and imaginary amplitude components are distributed as A_xj, B_xj∼𝒩(0, σ_x^2) with σ_x =√(∑_k=1^Nx_k/2) andA_yj, B_yj∼𝒩(0, σ_y^2) with σ_y = √(∑_k=1^Ny_k/2). Note that these distributions for the Fourier amplitudes only hold if the underlying process producing the observations is stochastic and wide-sense stationary. This includes many processes commonly observed in astrophysical sources such as white noise observed from a constant background, as well as red noise processes such as shot noise and other (broken) power-law power spectra often seen in AGN, and quasi-periodic oscillations with stochastic variations in amplitude and period of the observed signal, commonly observed in black hole X-ray binaries. For strictly deterministic processes (e.g. strictly periodic variations), the distributions of the Fourier amplitudes will not be centered on μ = 0, and the distributions below will not be correct.For standard periodograms, A_xj = A_yj and B_xj = B_yj, and the power spectral density reduces to P_j = 1/2 (A_j^2 + B_j^2), which is well-known to follow a χ^2 distribution with 2 degrees of freedom. Because this condition is not fulfilled for cospectra, we need to derive the probability distribution of the sum of the products over independent Gaussian random variables. The probability distribution of the product of two random variables[We continue the following derivation using the amplitudes A_j, but the same arguments apply exactly to the imaginary amplitudes B_j.] Z = A_xjA_yj is called the product distribution, defined asp_Z(z) =∫_-∞^+∞p_X(x) p_Y(z/x) 1/|x| dx =∫_-∞^+∞1/2 πσ_x σ_yexp-x^2/2σ_x^2exp-(z/x)^2/2σ_y^21/|x| dx. It can be shown <cit.> that the integral in Equation <ref> above can be reduced to P_Z(z) = K_0( |z|/σ_x σ_y)/πσ_x σ_y, where K_0(x) = ∫_0^+∞cos(xt)/√(t^2 + 1) dt is the Bessel function of the second kind of order 0. We can now use this result to derive the probability density function of C_j. In particular, we find that both random variables Z_j = A_xj A_yj and Q_j = B_xj B_yj follow the Bessel distribution defined in Equation <ref>. Our task is therefore to find the PDF of the sum of two Bessel distributions. The PDF of this sum requires the convolution of the PDFs of each individual random variable being summed. This convolution is difficult to calculate directly for the Bessel distribution defined in Equation <ref> above. We instead consider the moment-generating function of the PDF, generally defined asM_Z(t) := 𝔼[e^tZ] for a random variable Z. Consider the sum of any two independent random variables, S = Z + Q. While the PDF of S can be found via the convolution of the individual PDFs, it is often simpler to consider the moment-generating function, where the convolution reduces to a simple multiplication operation: M_S(t) = M_Z(t) M_Q(t) .The moment-generating function of the Bessel distribution in Equation <ref> above is, in the general case <cit.> where the means μ_x and μ_y are non-zero and the random variables may have unequal variances σ_x ≠σ_y:M_Z(t) = exp( tμ_x μ_y + 0.5 (μ_y^2 σ_x^2 + μ_x^2 σ_y^2) t/1 - t^2 σ_x^2 σ_y^2)/√(1 - t^2 σ_x^2 σ_y^2),but since μ_x = μ_y = 0 for the Fourier amplitudes of a stationary stochastic process, this reduces toM_Z(t) =1/√(1 - t^2 σ_x^2 σ_y^2), Thus, the moment-generating function for the sum of Z_j and Q_j becomes M_C(t) = M_Z(t) M_Q(t) = 1/1 - t^2 σ_x^2 σ_y^2. We note that the Laplace distribution is defined asp_Laplace(x | μ, b) = 1/2bexp(-|x - μ|/b) and its moment-generating function as M_Laplace(t) = e^tμ/1 - b^2 t^2. Comparing this last equation with Equation <ref>, we find that that Equation <ref> is equal to the moment-generating function of the Laplace distribution with μ = 0 and b = σ_x σ_y, and hence the cospectral densities follow a Laplace distribution: p(C_j | 0, σ_xσ_y) = 1/σ_x σ_yexp(- |C_j|/σ_xσ_y) with σ_x =√(∑_k=1^Nx_k/2) and σ_y =√(∑_k=1^Ny_k/2).§.§.§ Detection ThresholdsDetection thresholds for cospectra will generally be different from those of classical periodograms, because the Laplace distribution tends to be narrower than the equivalent χ^2_2 distribution for single periodograms. To show how the distributions and the corresponding detection thresholds differ, we simulated simple Poisson-distributed light curves. First, we simulated two light curves with a duration of 10 s and 10^6 data points each, corresponding to a time resolution of 10^-5 s. The light curves have an identical mean count rate of 10^6 counts/s, corresponding to 10 counts per bin. In order to simulate typical measurement uncertainties in X-ray detectors, we sampled from a Poisson distribution for each time bin with a rate parameter λ = 10, corresponding to the average counts per bin.We then calculated both the cospectrum of the two light curves and the periodogram of only the first light curve for comparison. For simplicity, both spectra were computed in Leahy normalization <cit.>, which is typically used when searching for (quasi-)periodic signals in time series. In order to normalize the cospectrum correctly, we used 2/√(N_ph, xN_ph, y), where N_ph, x and N_ph, y are the number of photons of light curves 𝐱 and 𝐲, respectively, as prescribed by <cit.>. In this normalization the densities are distributed as χ^2_2 exactly for the periodogram, and following a Laplace distribution with μ=0 and σ = 1 for the cospectrum. In Figure <ref>, we plot the resulting distribution of densities. While the periodogram is only defined for positive values, the Laplace distribution is symmetric around zero, and in general the cospectrum will comprise both positive and negative densities. It is also immediately visible from Figure <ref> that the probability of obtaining a certain (positive) noise power is always lower for the Laplace distribution than for the χ^2 distribution. In practice, this implies that using the latter where the former is appropriate, we may miss significant periodic signals, because we assume them to be weaker than they are in reality. To demonstrate this, we plot the survival function in Figure <ref>. The survival function, defined in terms of the CDF as SF(x) = 1 - CDF(x), encodes the tail probability of seeing at least a value x ≥ X. This tail probability is often considered to be the p-value of rejecting the null hypothesis that a certain candidate for a periodic signal could be reasonably produced by the noise powers. The CDF for the Laplace distribution with μ=0 is defined asF_C_j(x)) =1/2exp(C_j/σ_x σ_y) ifC_j < 0 1 - 1/2exp-( C_j/σ_x σ_y) ifC_j ≥ 0 Much like the PDF, the tail probability is always smaller for the Laplace distribution, indicating that for a given candidate signal, the p-value for rejecting the null hypothesis will be stronger than for χ^2-distributed variables. To reinforce this statement, we again simulated two light curves, each again with a duration of 10 s, but this time with only 1000 data points for simplicity and speed, and a time resolution of 0.01 s. For this simulation, we assumed a mean count rate of 1000 counts/s or 10 counts per bin, and additionally introduced a sinusoidal signal with a period of 0.1 s and a fractional rms amplitude of a_frac = 0.055. Again, this template was used to produce two Poisson-distributed light curves with a rate parameter equal to the number of counts in each bin as defined by the flat continuum and the periodic signal. In Figure <ref>, we show the cospectral densities along with trial-corrected 0.99 detection thresholds for both the Laplace and χ^2 distribution. If the densities are assumed to follow a χ^2 distribution, as for the periodogram, the candidate at 10Hz would be discounted at the 99% detection threshold, whereas correctly applying the Laplace distribution yields a correct rejection of the null hypothesis at the same detection threshold.Note that for light curves affected by dead time, the resulting cospectrum will still follow the Laplace distribution above, but with a variable variance that changes as a function of frequency <cit.>. In practice, the cospectrum can be straightforwardly corrected for this effect using the differences in Fourier amplitudes derived from the light curves of two detectors (the Fourier Amplitude Difference (FAD) technique), and <cit.> show that the corrected cospectrum will closely follow the Laplace distribution derived here, allowing for unbiased significance tests for periodicity detection.§.§ Averaged CospectraThe χ^2 distribution used for periodograms has the simple property that sums of χ^2-distributed variables again follow the same distribution, with a different number of degrees of freedom. The same is not true for the Laplace distribution. For n independent and identically distributed (i.i.d.) random variables distributed following a standard Laplace distribution with a mean of μ = 0 and a width of b = 1, the distribution of the sums of these random variables can be derived using the fact that a single Laplace random variable X can be rewritten as the difference of two exponential random variables,X = Z - Z' ,and thus for n summed random Laplace random variables, T = ∑_i=1^n X_i = ∑_i=1^nZ_i - ∑_i=1^n Z'_i = G_1 - G_2 , where G1 and G2 are i.i.d. standard gamma random variables with a distribution g(x) = 1/Γ(ν) x^ν-1 e^-x and a shape parameter ν = n. For the full derivation of the density, we refer the reader to <cit.> and simply state the end result for the PDF for n averaged standard Laplace random variables, X_n (see , Equations 2.3.25 and 2.3.26): f_X_n(x) = n e^-|nx|/(n-1)! 2^n∑_j=0^n-1(n-1+j)!/(n-1-j)! j!|nx|^n-1-j/2^j , x ∈ R . For practical purposes, evaluating this PDF for averaged spectra above n∼ 85 is difficult numerically, because the factorials and exponents in the sum become very large and small, respectively. However, as we will show in Section <ref> below, we expect that for large n, the Central Limit Theorem implies that the PDF of averaged cospectral densities tends towards a normal distribution. We find that in practice, when n ≳ 30, detection thresholds derived from Equation <ref> provide only a negligible difference over that derived from a normal distribution N(0, √(2/(n+1)), depending on the significance threshold required and the number of trials. In order to derive tail probabilities useful for hypothesis testing, we require the CDF rather than the PDF. In order to correctly account for the absolute values in the PDF, we split the CDF into two parts: a case where x < 0 and a case where x ≥ 0. The integral F_X_n(x) = P ( X ≤ x) = ∫_∞^x f_X_n(t) dt then becomes: F_X_n(x)) =∑_j=0^n-1 D 1/n(2Γ(-j+n) - γ(-j+n, nx)),x ≥ 0∑_j=0^n-1 D1/nγ(-j+n, -nx),x < 0 where Γ(l)= (l-1)! is the gamma function, γ(l+1, m) = lγ(l,m) - l^m e^-m is the incomplete upper gamma function, and the pre-factor constant D is defined asD = n(n-1+j)!/(n-1-j)! j! (n-1)! 2^n+j. As laid out in Section <ref>, the tail probability can easily be calculated via the survival function, SF(x) = 1 - CDF(x).§.§.§ Detection ThresholdsIn order to show the way the probability distribution changes as a function of averaged cospectra, we simulate light curves of 10^5 data points and a mean count rate of 100 counts/s consisting of pure white noise. We compute n such light curves and average their cospectral densities in order to show the distribution of those densities compared to the expected probability distributions. In Figure <ref>, we show the simulated distribution of Leahy-normalized power spectral densities, along with the distributions that describe them. For a single, non-averaged spectrum, we use the Laplace PDF described in Equation <ref>. When averaging 10 cospectra, we use Equation <ref> and show that the theoretical predictions agree with the simulated densities . Finally, for an averaged cospectra consisting of 100 individual light curves, Equation <ref> becomes difficult to compute numerically, and we use a Gaussian PDF instead, which describes the distribution of simulated densities well. In order to assess the effect on the p-values derived from averaged cospectra, we calculate the tail probabilities for the simulated data sets and compare them with the theoretically expected survival function as defined in Equation <ref> as well as a simple Gaussian distribution (Figure <ref>). Similar to the results derived by <cit.>, we find that for cospectra of more than 30 averaged light curves, a Gaussian distribution yields a reasonably good approximation to the true distribution up to p ≈ 10^-4 with lower overhead. Note, however, that this holds for single-trial probabilities. In general, one will wish to correct for calculating the significance of multiple trial frequencies, requiring the use of more stringent significance threshold. As shown in Figure <ref>, the tail probabilities diverge as a function of power, and thus the Gaussian approximation will increasingly overestimate the significance of the signal the higher the threshold is set. Depending on the number of trials used, it is hence advisable to use Equation <ref> as long as it remains numerically stable (depending on implementation, up to ∼85 averaged cospectra). § CAVEATS All caveats applying more generally to Fourier analysis and to periodograms specifically also apply to cospectra. In particular, as for the χ^2 distribution used for period searches in periodograms, the distributions derived here assume that the underlying process is wide-sense stationary and the light curves are simultaneously and evenly sampled. These conditions are typically met in most observations taken with X-ray timing instruments likeandfor problems such as timing of X-ray binaries, AGN and X-ray pulsar searches, but this may not be the case for e.g. optical observations of main-sequence or binary stars. Note that we also implicitly assume that the time scales of interest are much shorter than outburst timescales in X-ray binaries or changes in the X-ray background in pulsar searchers, both of which may not be stationary.Non-stationarity (as is e.g. observed in flares) will impose a window function onto the cospectrum and shift the mean of the Fourier amplitudes, thus invalidating the assumptions above (for an illustrative example of this effect and its implications on period searches in periodograms of magnetar bursts, see ). Uneven sampling, on the other hand, introduces covariances between adjacent frequencies and renders many of the statistical assumptions underpinning period searches using both in the periodogram and the cospectrum invalid. For unevenly sampled data, other methods that do not depend on a regular sampling pattern must be employed (for example the Lomb-Scargle periodogram <cit.> or the Bayesian period search in <cit.>) and dead-time effects must be forward-simulated.An additional assumption is the phase-alignment of the periodic signal to be detected in both light curves. This is always true when the same source is observed simultaneously in the same energy range in independent, but identical detectors on the same spacecraft, as is common practice for instruments like ,and . Because the phase of the periodic signal can more generally be energy-dependent, we caution the reader to be careful when producing cospectra of light curves taken for example in different energy ranges. Similarly, the phase of a quasi-periodic signal can shift with times, thus observations at different points in time, or of different sources must be analyzed similarly carefully. In all these cases, period searches will only be sensitive to signals that are phase-aligned, and will in the worst case be undetectable if the phase-shift is 90 or 270 degrees.In realistic applications, in particularly where dead time is a major concern, the cospectrum will show a frequency-dependent variation in the local variance, whose strength depends positively on the overall count rate of the object observed. It is imperative that this should be corrected before using the distributions used here by applying the Fourier Amplitude Differencing (FAD) technique <cit.>. This method, while powerful, is not without caveats. In particularly, FAD-corrected cospectra tend to overestimate the integrated rms when both count rate and variability amplitude are both very high. This should be accounted for when deriving estimates of the fractional rms amplitude.§ EXAMPLE: In the following, we will consider two more realistic examples in turn. First, we will consider a more realistic simulation of a QPO as expected to be observed in a typicalobservation. Subsequently, we present an example of realdata containing a coherent pulsed signal from the X-ray neutron star source Hercules X-1. §.§ A Simulated QPO inIn order to show the difference of the detection limits with the cospectrum and the power spectrum, we show how a QPO at 200 Hz from a very bright source would appear in .We simulate a light curve of T=3000s duration with a time resolution of δ t = 0.5ms and an average count rate of 200counts s^-1. To this constant background we add a quasi-periodic oscillation with a period of 5 ms, a fractional rms amplitude of f_rms = 0.15 and phases randomized using a normal distribution with a mean of 0 and a width of σ_qpo = 0.01. After producing this light curve, the procedure is similar to that followed by <cit.>. We simulated photon events using rejection sampling from this light curve using the software package stingray[<https://github.com/StingraySoftware/stingray>], running the functiontwice in order to produce two light curves that are statistically independent, but have the same signal and properties, as we would expect from an instrument with two independent detectors observing the same object. Subsequently we simulated variable deadtime for both light curves with an average time scale of 2.5ms as commonly seen indata <cit.> using HENDRICS[<https://github.com/StingraySoftware/HENDRICS>] <cit.>. We then produced the periodogram of the summed light curves, the averaged periodogram of the two individual light curves, and the cospectrum of the two light curves. Note that in all three cases we produced averaged periodograms and co-spectra by splitting the light curves into 600 segments of 5 s length each in order to suppress the variance in the powers and show the effects of dead time more clearly. As mentioned above, cospectra with dead time are subject to frequency-dependent changes in variance, and we thus corrected the simulated cospectra using the FAD technique.The results are shown in Figure <ref>. While in all three cases, the QPO is clearly visible, the two periodograms show strong deviations from the expected flat power spectrum. The shape is distorted and requires a precise model of the non-linearly increasing baseline with a non-linearly increasing rms. While in principle, the periodogram of the combined light curves would have a higher significance by a factor of √(2), the modeling requirements complicate the calculation of the significance of the QPO.The baseline of the cospectrum, conversely, is not distorted by dead time, and requires only an estimate of the local rms in order to calculate the significance of the QPO using Equation <ref>: it is sufficient to multiply the cospectrum around the feature by an estimate of the local standard deviation of the white noise (which is 1 in the ideal case) to use the equations above with no modifications. §.§Observations of Her X-1Hercules X-1 (Her X-1) is a well-studied persistent X-ray binary pulsars <cit.> in an X-ray binary comprising the neutron star itself and a stellar companion HZ Herculis <cit.> with a mass of ∼2.2M_⊙ <cit.> whose type varies between late-type A and early-type B with orbital phase <cit.> . The neutron star spins with a period of P = 1.23 s <cit.> and the system overall exhibits an orbital period of P_orb = 1.7 days <cit.>, along with super-orbital variations on a ∼35-day timescale <cit.>.Her X-1 has been observed by the Nuclear Spectroscopic Telescope Array () multiple times.For this work, we considered the observation taken from UT 2012-09-19 to UT 2012-09-20, one of those used by <cit.> to characterize the cyclotron resonance scattering features in the spectrum of the source. We downloaded the observation directory for observation ID 30002006002 from the HEASARC and used the FTOOLon the L2 cleaned science event files (file name ending with ) to correct the photon arrival times to the solar system barycenter. For our analysis, we considered photons from 3 to 79 keV at most 50 from the nominal position of the source, extracted from the two identical Focal Plane Modules A and B (FPMA and FPMB, respectively) onboard the spacecraft. For this work, we used a total of 32.67 ks of good time intervals (GTIs), only selecting intervals longer than 10 s. In Figure <ref> (upper panel), we present the light curve of the Her X-1 observations. The source varied substantially in brightness during the observations on fairly long time scales (10s to 1000s of seconds), indicating a significant source of red noise at low frequencies.In the lower panel of Figure <ref>, we present an averaged periodogram created from a total of 3260 light curve segments, each 10 s in duration. The periodogram shows very strong peaks at the 1.24 s rotational period of the pulsar, along with its first five harmonics. While at low frequencies, the deviation from the expected noise distribution (a χ^2 distribution centered on 2) can be explained with the longer-term variability within the source, at frequencies above 5Hz or so, the periodogram displays the typical oscillatory pattern associated with dead time (see also ), suggesting that the standard distributions usually applied to periodograms will not produce unbiased results for these observations.In Figure <ref> (upper panel), we present a cospectrum of the Her X-1observation. Because the periodic signal generated by the pulsar dominates the cospectrum, we plot the cospectrum of a short segment of 10 s duration starting at MET = 85740109.54169 for illustrative purpose to highlight the noise properties. Unlike the periodogram, the cospectrum is not modulated by dead time, and the statistical distributions defined in Section <ref> apply once the cospectra are corrected using the FAD technique (see also Figure <ref>, lower left panel and below). We also show the trial-corrected detection threshold under the null hypothesis that the cospectrum consists solely of white noise. The highest cospectral density of P = 55.16 occurs at the frequency of the pulsar, ν_rot = 0.806 Hz; the probability of observing this power under the null hypothesis is effectively p ∼ 0 within the limits of numerical accuracy. In order to test our theoretical predictions for the distributions governing (averaged) cospectra, we first produced 3260 individual FAD-corrected cospectra out of each of the 10 s segments, and for each cospectrum extracted the cospectral densities in the range 50 Hz to 400 Hz, where we do not expect strong contamination by the pulsar signal and its harmonics, or by the low-frequency red noise component.We plot a histogram of cospectral densities between 50 Hz and 400 Hz for all 3260 segments (a total of 11406500 cospectral densities) in Figure <ref> (lower left panel), along with the theoretically expected Laplace distribution. We find a generally high agreement between the Laplace distribution and the distribution of observed cospectral densities. In order to repeat the same process for averaged cospectra, we take the 7 GTIs that are longer than 3000 sand extract a light curve of exactly 3000 s from each. From each of these light curves, we produce an averaged cospectrum by averaging 15 consecutive light curve segments of 200 s duration. This yields 7 averaged cospectra. As above, we extract cospectral densities between 50 Hz and 400 Hz and plot the histogram of cospectral densities extracted from all averaged cospectra (910000 cospectral densitites) in Figure <ref> (lower right panel). As with the single cospectrum, the histogram for the averaged cospectral densities matches the theoretical prediction very closely. This indicates that there is no evidence in the data that would argue against the observed cospectral densities being drawn from the distribution in Equation <ref>.§ DISCUSSION AND CONCLUSIONS We have derived the statistical distribution for the cospectrum, defined as the real part of the cross spectrum. We show that because the Fourier amplitudes being multiplied to derive the cospectrum are now no longer identical (as is the case in the periodogram), the statistical distributions no longer reduce to a simple χ^2 distribution with two degrees of freedom. Instead, we find that the densities in a single cospectrum follow a Laplace distribution with a mean of μ=0 and a width of σ=1. This has important consequences for period detection. Most importantly, the Laplace distribution is considerably narrower than the χ^2_2 distribution expected for periodograms, and thus the significance of a candidate periodic signal will generally be underestimated when using the latter. Using the correct distribution therefore helps correctly identifying weak signals, which the χ^2_2 distribution would ignore as false negatives.Similarly, we find that the sums of Laplace distributions do not follow a similarly simple expression as in the periodogram case, but is considerably more computationally expensive, and may be difficult to estimate numerically when the number n of averaged light curves in the final cospectrum is large. However, we find that for n ≳ 30, the statistical distribution can be well approximated by a Gaussian distribution, and the resulting tail probabilities used for period detection are very nearly the same as those derived from the exact distribution, up to a tail probability of p ≈ 10^-4. This conclusion, however, depends sensitively on the detection threshold as well as the number of trials: for very small tail probabilities, the two distributions may still deviate significantly. For practical purposes, we suggest using Equation <ref> for at least up to ∼30 averaged cospectra, but also for averages of more spectra if significance thresholds for tail probabilities are smaller than 10^-4, or the number of trials is large. As an example, we have simulated how a QPO would appear inin the presence of dead time, and have shown that the shape of the periodogram is strongly distorted, whereas that of the cospectrum is not (for a longer introduction into the cospectrum and how it can be used in the presence of dead time, see also ). The significance of the QPO is difficult to assess in the periodogram, because of the non-linearities introduced by the variable dead time. The standard χ^2_2 distributions may either overestimate or underestimate the significance, depending on the shape of the underlying power spectrum and its modification due to dead time at a given frequency, adding complexity to the detection process in the form of finding a non-linear model for the dead time. The cospectrum, on the other hand, only requires an estimate of the local variance in order to use the equations derived above, making it a far more convenient choice for periodicity detection. By applying a straightforward correction using the differences between Fourier amplitudes derived from two independent detectors, cospectra can be corrected such that they follow the distributions derived here very closely.We show that the same is true for observations of the bright neutron star X-ray binary Her X-1, where the intrinsic brightness of the source leads to strong modulations in the periodogram due to dead time. For sources observed withand other instruments with a similar set-up of at least two redundant, identical detectors, one can take advantage of the independent, simultaneous observations in two detectors to form a cospectrum. While any (quasi-)periodic signal will appear weaker in the cospectrum than in the periodogram (even in the case, as presented here, where the signals in the two detectors are exactly in phase), it has the substantial advantage that it is less affected by dead time and other similar detector effects. We show that the cospectrum of the observations observed withfollow the expected theoretical distributions very closely once the FAD correction is applied.The distributions laid out above allow for detecting periodic and narrow quasi-periodic signals in the presence of detector white noise, and especially important in the context of pulsar detection in X-rays, where faint sources may yield marginal detections even in the best of cases. At the same time, as instruments likeandallow for observations with higher sensitivity, incorporating an accurate treatment of instrumental biases becomes increasingly important, and the cospectral statistics laid out here provide powerful tools to do so.Notably absent from this discussion, however, is the much more common case where a source exhibits stochastic variability in the form of red noise or notably broadened quasi-periodic oscillations. In this case, the goal is either estimation of the precise properties of the underlying stochastic process, or detection of periods against a background varying stochastically. As has been shown above, the fact that the cross spectrum consists of two different time series complicates the statistical distributions considerably, and this is similarly true cospectra with variability. The exact treatment of this case is beyond the scope of this paper, and will be considered in depth in a forthcoming publication. Acknowledgements The authors thank the anonymous referee for their helpful comments. The authors also thank Thomas Laetsch for helpful suggestions regarding the mathematical derivations, and Peter Bult for spotting a mathematical error. DH is supported by the James Arthur Postdoctoral Fellowship and the Moore-Sloan Data Science Environment at New York University.DH acknowledges support from the DIRAC Institute in the Department of Astronomy at the University of Washington. The DIRAC Institute is supported through generous gifts from the Charles and Lisa Simonyi Fund for Arts and Sciences, and the Washington Research Foundation. MB is supported in part by the Italian Space Agency through agreement ASI-INAF n.2017-12-H.0 and ASI-INFN agreement n.2017-13-H.0. apj | http://arxiv.org/abs/1709.09666v2 | {
"authors": [
"D. Huppenkothen",
"M. Bachetti"
],
"categories": [
"astro-ph.IM",
"astro-ph.HE"
],
"primary_category": "astro-ph.IM",
"published": "20170927180000",
"title": "On the Statistical Properties of Cospectra"
} |
address1]Carlos E. Alvarado-Rodríguez [email protected],address3]Jaime Klappmycorrespondingauthor [mycorrespondingauthor]Corresponding author [email protected]]Leonardo Di G. Sigalotti [email protected]]José M. Domínguez [email protected]]Eduardo de la Cruz Sánchez [email protected][address1]Departamento de Ingeniería Química, DCNyE, Universidad de Guanajuato, Noria Alta S/N, 36000 Guanajuato, Guanajuato, Mexico[address2]Departamento de Física, Instituto Nacional de Investigaciones Nucleares (ININ), Carretera México-Toluca S/N, La Marquesa, 52750 Ocoyoacac, Estado deMéxico, Mexico[address3]ABACUS-Centro de Matemáticas Aplicadas y Cómputo de Alto Rendimiento, Departamento de Matemáticas, Centro de Investigación y de Estudios Avanzados (Cinvestav-IPN), Carretera México-Toluca km. 38.5, La Marquesa, 52740 Ocoyoacac, Estado de México, Mexico[address4]Área de Física de Procesos Irreversibles, Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana-Azcapotzalco (UAM-A), Av. San Pablo 180, 02200 México D.F., Mexico[address5]EPHYSLAB (Environmental Physics Laboratory), Facultad de Ciencias, Campus de Ourense, Universidad de Vigo, 32004 Ourense, Spain In this paper we implement a simple strategy, based on Jin and Braza's method, to dealwith nonreflecting outlet boundary conditions for incompressible Navier-Stokes flows using the method of smoothed particle hydrodynamics (SPH). The outflow boundaryconditions are implemented using an outflow zone downstream of the outlet, where particles are moved using an outgoing wave equation for the velocity field so that feedback noise from the outlet boundary is greatly reduced. For unidirectional flow across the outlet, this condition reduces to Orlanski's wave equation. The performance of the method is demonstrated through several two-dimensional test problems, including unsteady,plane Poiseuille flow, flow between two inclined plates, the Kelvin-Helmholtz instability in a channel, and flow in a constricted conduit, and in three-dimensions for turbulent flow in a 90^∘ section of a curved square pipe. The results show that spurious waves incident from the outlet are effectively absorbed and that steady-state laminar flows can be maintained for much longer times compared to periodic boundary conditions. In addition, time-dependent anisotropies in the flow, like fluid recirculations, are convected across the outlet in a very stable and accurate manner.Smoothed particle hydrodynamics (SPH); Boundary conditions; Open-boundary flows; Outflow; Flows in ducts, channels, nozzles, and conduits § INTRODUCTION A standard procedure in engineering fluid dynamics for simulating internal flows is to artificially truncate their actual physical domain into a short region to reduce the computational cost. However, this demands the use of one or more open boundary conditions that must be specified at the extremes of the computational domain. Typical examples of this type of flows include pipe and channel flows, constricted flows in a pipe section, flows around bluff bodies, and external flows, in which case the flow develops in a practically infinite free stream, as in rising fire and hydrothermal seawater plumes, among others. Such open boundary conditions must allow the fluid to enter (inlet) and leave (outlet) the computational domainwhile reasonably minimizing non-physical feedbacks <cit.>, such as the artificial build-up of fluid near the outlet <cit.> and the reflection of outgoing waves in recirculating flows at high and moderate Reynolds numbers<cit.>. In addition, numerical instabilities may well develop when strong vortices are convected through the outlet <cit.>.In this paper, we restrict ourselves to weakly compressible flows and introducea practical methodology for handling nonreflecting outlet boundary conditions with smoothed particle hydrodynamics (SPH), when flow anisotropies are present near the outlet. Because of its Lagrangian character, SPH presents inherent disadvantages in the treatment of open boundary conditions compared to traditional Eulerian methods, where inlet and outlet boundary conditions for a stationary flow are more naturally described <cit.>. Earlier attempts of handling inlet/outletboundaries with SPH for simple flows, such as Poiseuille and Couette flows, have been performed using periodic boundary conditions <cit.>. In this case, the particle distribution is continually recycled so that any time a particlecrosses the outlet, it is forced to re-enter the domain through the inlet. Although this interpolation has been widely employed in SPH simulations of internal, incompressible flows <cit.>, it is known to degradethe numerical solution over time because disturbances in the particle distribution are re-introduced into the computational domain. For steady-state flows this problem is mitigated by just placing the outlet plane sufficiently far from the inlet to allow the numerical oscillations to be dissipated <cit.>.However, this may incur in an excessive increase of the computational burden.A number of approaches has been presented in the literature to prescribe thecorrect inlet and outlet boundary conditions for weakly compressible and incompressible flows in SPH, with all of them aimed at reducing the level of disturbance at the outflow and therefore the reflections that may significantly alter the flow upstream. Truly incompressible SPH schemes (ISPH) employ a pressure-correction projection scheme to compute the pressure from a Poisson equation, which is then used to make the velocity field divergence-free<cit.>. With these methods, solution of the Poisson equation typically requires implementing a homogeneous Neumann condition for the pressure <cit.>. An improved algorithm, based on a non-homogeneous pressure boundary condition, the so-called rotational pressure-correction scheme <cit.>, was implemented in ISPH by Hosseini and Feng <cit.>. Other strategies based on the use of a time-dependent driving force<cit.> and the influx of kinetic energy into the domain through the outflow boundaries <cit.> have also been designed.On the other hand, weakly compressible SPH (WCSPH) solves the Navier-Stokesequations by defining the pressure through an algebraic relation so that the sound speed is artificially set to achieve accurate results in fluid propagation <cit.>. It is common practice in SPH to deal with spatially fixed (Eulerian) inlet and outlet boundaries by defining inflow and outflow regions that are external to the computational domain <cit.>. These regions are filled with particles and their widths are comparable to or greater than the smoothing length of the fluid particles to avoid truncation of the kernel function. Thus, as a particle pertaining to the inflow region enters the fluid domain, a new one is automatically created to compensate it. According to the flow rates across the inlet and outlet, inflow and outflow particles are being added and removed, respectively. Here we describe a methodology, based on inflow and outflow zone particles, that conserves the global mass of the system and minimizes the reflection of disturbances from the outlet back into the fluid domain. To this end, the velocity vector of particles in the outflow zone is evolved by means of an anisotropic propagation wave equation following the procedure described by Jin and Braza <cit.>. In addition to adapting their procedure for use in SPH simulations, the method is extended to three-space dimensions. We validate the method against several benchmark test cases for the simulation of two-dimensional (2D) internal flows, including unsteady, plane Poiseuille flow, flow along a divergent duct, the Kelvin-Helmholtz instability of flow with discontinuous shear in a channel, and plane choked flow through a constrictedconduit. Validation of the method in three dimensions (3D) is shown against physical experiments for the turbulent flow in a 90^∘ section of a curved square pipe.§ WCSPH FORMULATION For viscous incompressible flows, the governing equations are given by the Navier-Stokes equationsd v/dt=-1/ρ∇ p+ν∇ ^2 v,where ρ is the density, p the pressure, v the velocity field, and ν the kinematic viscosity. In a WCSPH formulation, where the pressure is given as a function of the density, local variations of the pressure gradient may induce local density fluctuations in the flow. Therefore, the flow is modelled by an artificial fluid that is approximately incompressible. This is done by defining the total pressure gradient in Eq. (1) as <cit.>-1/ρ∇ p=-1/ρ∇ p_d-1/ρ∇ p_h= -1/ρ∇ p_d+ F,where p_d is the dynamical pressure as calculated from the equation of state, p_h is the hydrostatic pressure, and the term -∇ p_h/ρ is treated as a body force F to be determined. In a WCSPH scheme, the mass of a fluid element remains constant and only its associated density fluctuates. Such density fluctuations are calculated by solving the continuity equationdρ/dt=-ρ∇· v.The dynamical pressure p_d, which for simplicity we shall denote by p, is calculated using the relation <cit.>p=p_0[(ρ/ρ _0)^γ-1],where γ =7, p_0=c_0^2ρ _0/γ, ρ _0 is areference density, and c_0 is the sound speed at the reference density. This equation enforces very low density fluctuations since the speed of sound can be artificially slowed with accurate results in fluid propagation. By restricting the sound speed to be at least 10 times higher than the maximum expected fluid velocity, the density fluctuations will be within 1%.In order to capture coherent turbulent structures within turbulent flows, the standard SPH viscous formulation is replaced by a sub-particle scaling (SPS) technique <cit.>. This is achieved by Favre-averaging Eqs. (1) and (3) over a length scale comparable to the particle sizes, where the velocity field v can be decomposed into a mean part ṽ and a fluctuating part v^' such that v=ṽ+ v^', where the mean part is defined by a density weightedavarage, ṽ=ρ v/ρ, and the overbars denote an arbitrary spatial filtering. Applying a flat-top spatial-filter to Eqs. (1) and (3), they become <cit.>dṽ/dt = -1/ρ∇p+ ν/ρ[∇·(ρ∇)]ṽ+ ν/ρ∇· T, dρ/dt = -ρ∇·ṽ,respectively, where T is the SPS stress tensor defined in component form asT_ij=ρ(2ν _tS̃_ij-2/3S̃_kkδ _ij)-2/3ρC_IΔ ^2δ _ij,whereS̃_ij=-1/2(∂ṽ_i/∂ x_j+ ∂ṽ_j/∂ x_i),is the Favre-averaged strain rate tensor, C_I=0.00066,ν _t=(C_sΔ)^2|S̃|, with C_s=0.12, is the Smagorinsky eddy viscosity, |S̃|=(2S̃_ijS̃_ij)^1/2 is the local strain rate, δ _ij is the Kronecker delta, and Δ is a measure of the initial particle spacing.§ SPH SOLVER The computer code used in this work is based on standard SPH methods <cit.>, where the density of particle a is given by the usual kernel summationρ _a=∑ _b=1^nm_bW_ab.In this expression, m_b is the mass of particle b, W_ab=W(| x_a- x_b|,h) is the kernel function, wherex_a- x_b is the distance between particles a and b and h is the width of the kernel or smoothing length, and the summation is taken over all n neighbour particles within the kernel support. Note that the density in Eq. (9) may be either the local density ρ or the particle-scale density ρdepending on whether we are dealing with laminar or turbulent (rotational) flows. The same is true for the velocity field, where v may represent a local value (for laminar flows) or the Favre-averaged velocity ṽ (for turbulent flows).In Eqs. (1) and (5) the pressure gradient is written in SPH form using the symmetric representation proposed by Colagrossi and Landrini <cit.>, which ensures numerical stability at the interface between two media with large density differences, while the laminar viscous term and the SPS stresses are discretized according to the formulations given by Lo and Shao <cit.>. Therefore, in SPH form Eq. (5) readsd v_a/dt= - 1/ρ _a∑ _b=1^nm_b/ρ _b(p_a+p_b)∇ _aW_ab+4ν∑ _b=1^nm_b v_a- v_b/ρ _a+ρ _b x_ab·∇ _aW_ab/| x_ab|^2+ϵ ^2+ ∑ _b=1^nm_b( T_a/ρ _a^2+T_b/ρ _b^2)·∇ _aW_ab,where x_ab= x_a- x_b, ϵ ^2=0.01h^2, and W_ab is evaluated according to the symmetrized kernel function <cit.>W_ab=1/2[W(| x_a- x_b|,h_a)+ W(| x_a- x_b|,h_b)],which has the correct limiting behaviour when h_a=h_b. For laminar flows the SPH representation of Eq. (1) can be recovered from the discrete Eq. (10) by dropping the SPS stress term and keeping in mind that the fluid variables will now correspond to local quantities. Coupled to Eqs. (9) and (10), the equationd x_a/dt= v_a+β/M∑ _b=1^Nm_b x_ab/( x_ab· x_ab)^3/2 x_0v_ max,is solved for the particle positions, where the second term on the right-hand side is the shifting vector of particle a <cit.>, which modifies the position of particles in order to prevent magnification of the SPH discretization errors due to anisotropies in their distribution. Here, β is a dimensionless parameter which is chosen to be β =0.04, v_ max is the maximum velocity in the system, M is the total massM=∑ _b=1^Nm_b,and x_0=1/N∑ _b=1^N( x_ab· x_ab)^1/2.Note that the summations in the above two expressions are over all particles filling the fluid domain. The addition of the shifting vector on the right-hand side of Eq. (12) does not affect momentum preservation.Since direct evaluation of second-order derivatives of the kernel is not required, we adopt a low-order, Wendland C^2 function <cit.> as the interpolation kernelW(q,h)=7/4π h^2(1-q/2)^4(2q+1),for 0≤ q<2 and zero otherwise, where q=| x- x^'|/h. A Verlet algorithm is used for the time integration of Eqs. (10) and (12), where the velocities and positions of particles are advanced from time t^n to time t^n+1=t^n+Δ t according to the difference formulaev_a^n+1 =v_a^n-1+2Δ t(d v_a/dt)^n,x_a^n+1 =x_a^n+Δ t v_a^n+0.5Δ t^2(d v_a/dt)^n.In order to improve the coupling of Eqs. (10) and (12) during the entire evolution, the above time integration is replaced every 50 time steps by the alternative difference formsv_a^n+1 =v_a^n+Δ t(d v_a/dt)^n,x_a^n+1 =x_a^n+Δ t v_a^n+0.5Δ t^2(d v_a/dt)^n.This prevents the time integration to produce results that diverge from the actual solution. The time step, Δ t, is calculated using the Courant-Friedrichs-Lewy (CFL) and the viscous diffusion conditions such thatΔ t_f,a = min _a(h|dv_a/dt|^-1)^1/2, Δ t_v,a = max _b|h x_ab· v_ab/( x_ab· x_ab+ϵ ^2)|, Δ t_cv,a = min _a[h(c_a+Δ t_v,a)^-1], Δ t = 0.3min _a(Δ t_f,a,Δ t_cv,a),where the maximum and minima are taken over all particles in the system, v_a=( v_ab· v_ab)^1/2, and c_a is the sound speed for particle a. §.§ Solid boundary conditions No-slip boundary conditions are implemented at contact with solid surfaces. A stable and accurate approach is achieved here using the method of image particles <cit.>, where imaginary particles are initially created by simply reflecting actual fluid particles across the solid surface. Such particles are external to the fluid domain and serve to remove the kernel truncation in the proximity of the surface. Unlike actual fluid particles, imaginary particles are not allowed to move relative to the solid surface and are forced to maintain their initial distribution during the timeevolution. However, a velocity needs be assigned to each imaginary particle in order to evaluate the compressional and viscous forces in Eq. (10).§ INLET AND OUTLET BOUNDARY CONDITIONS We consider flow through a truncated section of a pipe, or channel, and assume that the open boundaries at the entrance and exit of the pipe section are perfectly planar. Since SPH particles cannot be spatially fixed at the planar boundaries, we must allow them to flow in and out consistently with the flow rate across the inlet and outlet planes, respectively.The method as presented here distinguishes among three zones, which are external to the computational domain, i.e., an inflow zone, which is placed in front of the pipe inlet, an outflow zone, which is placed downstream of the pipe exit plane, and a reservoir zone, where inert particles are temporarily stored. The inflow region consists of five columns of uniformly spaced particles and has a length equal to 5Δ x_0, where Δ x_0 is the initial uniform spacing of the fluid particles in the direction of the flow. As in Refs. <cit.>, inflow particles are allowed to cross the inlet plane with a prescribed velocity that may vary in time and/or space. Scalar variables, such as density and pressure, are also prescribed for inflow particles. Since particles close to the inlet, but inside the flow domain, are updated according to Eqs. (9), (10), and (12), it always happen that some inflow particles that are close to the inlet fall within the kernel support of the near-boundary fluid particles. This will allow boundary information to be propagated into the flow domain <cit.>.The main differences between this and previously reported methods for WCSPH flows in SPH lie on the treatment of the outflow zone particles and the use of a reservoir buffer to ensure conservation of both the total mass and the total number of particles. The length of the outflow zone is chosen to be the same of the inflow zone. It is common practice to put this boundary sufficiently far from any sources of flow anisotropy, as may be the case of flow past a backward facing step, where the flow becomes essentially unidirectional and approaches a steady-state regime. In this case, classical “do-nothing” conditions <cit.>, where d v/dt=0 andT· n=0,at the outlet, have become the most widely used outflow conditions for the Navier-Stokes equations. Since these conditions are strictly valid for steady-state,fully-developed flows, they may present the problem of upstream wave propagation from the outlet if anisotropies are being convected into the outflow zone. An extension of the “do-nothing” conditions that enhances the stability properties against non-physical feedbacks has recently been proposed by Braack and Mucha <cit.>.Here we implement a type of outflow boundary condition that simulates the propagation of waves out of the computational domain by allowing the flow to cross the outlet without being significantly reflected back. To do so we adopt the procedure described in Ref. <cit.>, which is based on a wave equation and allows for anisotropic wave propagation across the outlet. The velocity vector of particles crossing the outlet and entering the outflow zone is considered as a transported wave quantity incident on the boundary. To this end, we consider the wave equation∂ ^2 v/∂ t^2-c_x^2∂ ^2 v/∂ x^2-c_y^2∂ ^2 v/∂ y^2-c_z^2∂ ^2 v/∂ z^2= 0,where c_x, c_y, and c_z are the characteristic velocities of wave propagation in the x-, y-, and z-directions, respectively. Introducing the differentialoperator L asL=c_x^2∂ ^2/∂ x^2+c_y^2∂ ^2/∂ y^2+c_z^2∂ ^2/∂ z^2-∂ ^2/∂ t^2,Eq. (20) can be rewritten asL v= L^+ L^- v= 0,where L^+ and L^- are factorization operators providing information on the outgoing and ingoing (reflected) waves, respectively. The decomposition of L into the product L^+ L^- yields the formsL^+ = c_x∂/∂ x+∂/∂ t(1-s^2)^1/2,L^- = c_x∂/∂ x-∂/∂ t(1-s^2)^1/2,where s^2=c^2_y(∂/∂ y)^2(∂/∂ t)^-2+c^2_z(∂/∂ z)^2(∂/∂ t)^-2.Application of the equation L^- v= 0 to the outflow particles results in a total non-reflecting condition. Now, using the approximation (1-s^2)^1/2≈ 1-s^2/2 for s small and makingc_y≈ c_z=c, gives for the outgoing wave∂ v/∂ t+c_x∂ v/∂ x -1/2c^2(∂/∂ t)^-1( ∂ ^2 v/∂ y^2+∂ ^2 v/∂ z^2)= 0,where the anisotropic term containing the coefficient c is a diffusion-like term. Noting that c^2(∂ /∂ t)^-1 has the same dimensions of the kinematic viscosity ν, a matching of Eq. (26) with the Navier-Stokesequations can be made by applyingthe following equivalences 2ν→ c^2(∂ /∂ t)^-1 and c_x→ v_x, where v_x is the x-component of the velocity field. In this way, the outgoing wave equation becomes∂ v/∂ t+v_x∂ v/∂ x -ν(∂ ^2 v/∂ y^2+ ∂ ^2 v/∂ z^2)= 0,where v=(v_x,v_y,v_z). If the diffusion term is dropped, Eq. (27) reduces to∂ v/∂ t+v_x∂ v/∂ x= 0,which is the Orlanski equation for unidirectional monochromatic travelling waves <cit.>. Note that setting v=(v_x,v_y) with v_x and v_y depending only on x and y, Eq. (27) reduces to the 2D form derived by Jin and Braza <cit.>.Particles in the outflow zone are evolved using either Eq. (27) or (28) until they flow past its downstream limit. When a particle leaves the outflow zone its velocity is automatically zeroed and it is temporarily stored in a reservoir buffer. Figure 1 shows a schematic diagram of the inflow and outflow boundary zones. Every time an inflow particle enters the fluid domain, a particle is removed from the reservoir buffer and inserted in the upstream side of the inflow zone with the desired prescribed velocity and density. This is a necessary step because in most problems of interest the inlet and outlet mass rates and cross-sections may differ. At the beginning of a calculation the number of reservoir particles depends on the flow model and can be as large as needed.In SPH form Eq. (27) for an outflow zone particle “o” is written as follows∂ v_o/∂ t = -v_x,o∑ _b=1^nm_b/ρ̅_ob( v_b- v_o)∂ W_ob/∂ x_o+ 2ν∑ _b=1^nm_b/ρ _b( v_b- v_o)/| x_ob|^2+ϵ ^2(y_ob∂ W_ob/∂ y_o+z_ob∂ W_ob/∂ z_o), where x_ob= x_o- x_b, y_ob=y_o-y_b, and z_ob=z_o-z_b. Flow particles next to the outlet plane are updated according to the usual SPH procedures so that some outflow particles may fall inside the compact support of the near-boundary fluid particles. The same is true for outflow particles close to the outlet in Eq. (29) where some neighbours b may actually be fluid particles, allowing fluid information to be propagated into the outflow zone. In order to ensure stability of Eq. (29), the velocity v_x,o is smoothed according tov_x,o=∑ _b=1^nm_b/ρ _bv_x,bW_ob,where the summation is taken over all neighbours of outflow zone particle “o”. This is, in fact, equivalent to averaging the convective velocity at the outlet. The position of outflow particles is updated according to d x_o/dt= v_o, which together with Eq. (29), is integrated in time using the Verlet algorithm described by Eqs. (16) and (17).§ NUMERICAL TESTS§.§ Unsteady plane Poiseuille flow As a first test we consider the Poiseuille flow between stationary infinite plates placed at distances y=± 5× 10^-4 m from the centre y=0. For this test, the Navier-Stokes equations admit the exact solutionv_x(y,t) = F/2ν(y^2-d^2)+ ∑ _n=0^∞16(-1)^nd^2F/νπ ^3(2n+1)^3cos[ (2n+1)π y/2d]exp[-(2n+1)^2π ^2ν t/4d^2],where d is half the distance between the parallel plates, ν =η /ρ is the kinematic viscosity, and F=-2ν v_0/d^2 is a driving force proportional to the pressure difference (Δ p) between the inlet and outlet, and v_0=-d^2Δ p/(2ρν L) is a constant asymptotic velocity, where L is the length of the pipe section. In the limit when t→∞ the above solution tends to the well-known steady-state, parabolic profile v_x(y)=v_0(1-y^2/d^2).For this test problem we choose ρ _0=1000 kg m^-3, v_0=1.25× 10^-5 m s^-1, and ν =1.0× 10^-6 m^2 s^-1, corresponding to a Reynolds number Re=2dv_0/ν =0.0125. The fluid domain is filled with 1942 particles initially at rest and regularly distributed in the spanwise direction between x=0 and x=L=2.33× 10^-4 m. The particles are given a smoothing length h≈ 2.4× 10^-5 m and Eq. (4) is used as the pressure-density relation with c_0=2 m s^-1. The transient behaviour as calculated with the outlet boundary condition treatment is shown in Fig. 2, where the SPH solution (dots) is compared with the theoretically predicted one (solid lines) at selected times. The solution is depicted up to t=1.0 s when a steady-state regime has been already established. Figure 3 shows the relative errors between the numerical and analytical peak velocities for the run of Fig. 2 with the outflow boundary condition method (dots) as compared with an identical run using periodic boundary conditions (circles). The error in the periodic simulation grows rapidly during the first 0.4 s and reaches values that are an order of magnitudegreater than the error carried by the nonreflecting outlet simulation. This difference is the result of cumulative errors due tothe recycling of numerical disturbances in the periodic simulation. Figure 4 also shows the numerical y-component of the velocity for both calculations. With the present method (top frame), the numerically induced y-component of the velocity reaches maximum absolute values above and below y=0 of ≈ 4.0× 10^-14m s^-1, while in the periodic case (bottom frame) v_y exhibits erratic oscillations after ∼ 0.4 s and reaches values that are about 7 orders of magnitude higher. The asymmetry of the oscillations with respect to the central plane y=0 is indicative of the presence of noise due to the periodic recycling of particles. Evidently, the nonreflecting outlet boundary conditions are doing a superior job for this test as the flow remains laminar with a very good accuracy. §.§ Flow between inclined plates As a second test, we consider the case of laminar flow between two inclined plates, where the inlet and outlet cross-sections differ as shown in Fig. 5. If the inclination angle α is small and the flow is driven by a pressure difference between the inlet and outlet planes, an analytical solution can be derived for the streamwise velocity after a steady-state flow is reached <cit.>, namelyv_x=-Δ p/η L[y^2-(l_1+xtanα)^2] l_1^2(l_1+Ltanα)^2/(2l_1+Ltanα) (l_1+xtanα)^3,where η =ρν is the shear viscosity, l_1 is half the separation of the inclined plates at the inlet, and L is the distance between the inlet and outlet planes. Following the procedure described by Liang et al. <cit.>, the body force at the inlet x=0 is given byF(x=0)=2Δ ptanα/ρ l_1^3[1/(l_1+Ltanα)^2-1/l_1^2]^-1,while at any streamwise position x it obeys the relationF(x)=[l_1/l(x)]^3F(x=0),where l(x)=l_1+xtanα. This gives F(x=L)=F(x=0)/(1+Ltanα /l_1)^3 at the outlet. Equations (33)–(35) are valid if the Reynolds number Re<1 and L≫ l_1. For this test case we choose the same parameters as in Liang et al. <cit.>, that is, Δ p≈ -1.217× 10^-3 N m^-2, L=4 mm, 2l_1=0.5 mm, ν =1.0× 10^-6 m^2 s^-1, ρ =1000 kg m^-3, andα =3.503^∘, except for the sound speed which is taken to be c=5.0 m s^-1 in order to keep the density fluctuations below 1% with the use of Eq. (4). In contrast, Liang et al. <cit.> used an equation of state of the form p=c^2p with c=2.5× 10^-4 m s^-1. Initially the particles are at rest and distributed on a regular Cartesian mesh and the initial smoothing length is set to h=1.1Δ, where Δ is the initial interparticle distance along the x- and y-directions. With the above parameters, the Reynolds number of the flow is Re=2l_1v_0/ν =0.0125, where v_0=2.5× 10^-5 m s^-1 is the velocity at the inlet plane. According to Eq. (34), the body force at the inlet is F(x=0)=8.0× 10^-4 m s^-2, while the body force entering in Eq. (2) is F= xF(x), which is always parallel to the x-axis and zero otherwise.Figure 6 depicts the x-component of the fluid velocity after about 0.15 s when the flow reaches a steady state. The numerically obtained profiles are compared with the analytical solution (solid line) as given by Eq. (33). Three different runs are shown with identical initial parameters but varied initial spatial resolution: N=6592 (crosses), 15984 (circles), and 103168 (dots), corresponding to initial interparticle separations of 0.25, 0.125, and 0.0625 mm, respectively. Figure 6 shows that the results obtained with the present method converge to the theoretical solution as the resolution is increased. In terms of the root-mean-square error (RMSE)RMSE(v_x)=√(1/N∑ _a=1^N(v_x,a^ anal- v_x,a^ SPH)^2),where v_x,a^ anal represents the analytical solution (33) at the position of particle a and v_x,a^ SPH the corresponding SPH calculated value, the numerical errors decrease with decreasing initial particle size withRMSE(v_x)≈ 1.49× 10^-6 m s^-1 (for Δ =0.25 mm),≈ 1.06× 10^-6 m s^-1 (for Δ =0.125 mm), and ≈ 1.54× 10^-7 m s^-1 (for Δ =0.0625 mm).The pressure constant p_0 in Eq. (4) governs the relative density fluctuations |Δρ|/ρ _0, with Δρ =ρ -ρ _0. Since |Δρ|/ρ _0∼ M^2, where M is the Mach number, density fluctuations in the flow can be kept of the order of 1%, or less, by choosing M≤ 0.1. To enforce this condition p_0 must be equal to c_0^2ρ _0/γ, where c_0 is the sound speed at the reference density ρ _0 which is chosen large enough to guarantee that M≤ 0.1. Figure 7 shows the maximum value of |∇· v| in the flow as a function of time for the three resolutions tried. During the first 0.05 s, peaks of the velocity divergence as high as ∼ 2.7 s^-1 and ∼ 0.6 s^-1 arise in the low resolution run. At later times, the maximum velocity divergence decreases and oscillates about 0.2 s^-1. As the resolution is increased to Δ =0.1250 mm, the peak intensity at the beginning is reduced to less than ∼ 0.8 s^-1 and the maximum value of the divergence oscillates about ∼ 0.12 s^-1. This mean value improves to ∼ 0.08 s^-1 for the high resolution run. In this case, the divergence achieves a peak of ∼ 0.4 s^-1 at the very beginning. The actual maximum density fluctuations associated with these deviations from exact incompressibility (calculated as the product max(|∇· v|_a)Δ t) correspond to mean values of 3.2× 10^-7 (for Δ =0.25 mm), 2.6× 10^-7 (for Δ =0.125 mm), and 9.1× 10^-8 (for Δ =0.0625 mm). §.§ Kelvin-Helmholtz instability in a channel We now assess the ability of our method to inhibit feedback noises when convectingflow anisotropies across the outlet. The test case concerns the onset of theKelvin-Helmholtz (KH) instability at the interface between two shearing fluids of different velocities when velocity perturbations perpendicular to the interface grow to eventually mix the layers <cit.>. We consider a two-dimensional setup similar to that reported by Price <cit.>, using N=10880 equal mass particles filling the domain 0≤ x≤ 0.4 m and 0≤ y≤ 0.1442 m. The particles are initially placed on a uniform Cartesian array and the density is set to ρ =1000 kg m^-3 everywhere. A shear flow is setup in the x-direction with velocity v_x=1 m s^-1 for 0≤ y<0.103 m and v_x=2 m s^-1 for 0.103≤ y≤ 0.1442 m, so that the tangential fluid velocity has a discontinuous jump across the interface between the streams. This flow corresponds to Re=10000. For this test we use Eq. (4) with c_0=40 m s^-1. This configuration is known to be susceptible to a KH instability at all wavelengths. The instability is seeded byintroducing a small velocity in the y-direction given byv_y=Asin[-2π/λ(x+1/2)],for 0.09<y<0.116 m and zero elsewhere, with A=0.5 m s^-1 and λ =0.1 m. For this setup the linear KH growth time-scale for the sinusoidal mode defined byτ _ KH=2λ/|v_x,1-v_x,2|,is τ _ KH=0.2 s. No-slip boundary conditions are applied at contact with the walls of the channel. For this test the inlet consists of an upstream section of 0.8 m long, while the actual channel has a length of 0.4 m and the outflow zone is 0.1 m long. Initially, the inlet and the channel sections are filled with particles, which are then evolved from the above initial conditions using SPH.As the flow proceeds, the inlet section becomes progressively depleted of particles, resembling a moving piston boundary condition. The calculation is halted immediately before the inlet becomes completely depleted of particles. At the exit of the channel, the outlet boundary conditions are employed. For this test calculation, Eq. (5) is used with the viscous force term replaced by an artificial viscosity using the scheme proposed by Monaghan <cit.> with a coefficient α _ν=0.01. In order to test the performance of the nonreflectingoutlet boundary conditions, a second run using periodic boundary conditions at the inlet and outlet in the x-direction was performed for direct comparison. Figure 8 shows the results at different times up to 5.0 s. At each time, the top and bottom frames correspond to nonreflecting outlet and periodic simulations, respectively. In the former case 5508 inflow particles were needed to follow the evolution up to 5.0 s by which time the inflow zone was almost depleted. With the nonreflecting boundary conditions the linear growth phase is similar to the periodic simulation. The instability grows at the shear layer and the peaks of each fluid phase penetrate into each other (t=0.25 s). After further penetration of the fluid phases, non-linear shear leads them to roll up into the well-known KH whorls (t=1.0 s). We may see that the whorl height is nearly identical in both caculations. However, at t=1.0 s the rolling appears to be slightly more pronounced in the nonreflecting outlet simulation. At later times, the interface rolls up into a sequence of spiral vortices (t=2.5 s). As time progresses, the turns are elliptically deformed (t=5.0 s). While a vortex field is formed with the nonreflecting boundary conditions, which then amplifies and eventually leads to mixing, the solution with periodic boundary conditionslooks highly degraded by t=2.5 s because of the continued re-entry of numerical perturbations.Figure 9 depicts the time evolution of the absolute value of the maximum velocity divergence for the nonreflecting outlet simulation. At the very beginning the velocity divergence drops sharply, decaying from ∼ 67 s^-1 to less than ∼ 0.3 s^-1 during the first second of the evolution. After this time, it decreases slowly to less than ∼ 0.2 s^-1 by t=6.0 s. In addition, Fig. 10 shows the time evolution of the maximum kinetic energy. During the first two seconds, the maximum kinetic energy is seen to decreaserapidly by an order of magnitude and then at a much slower rate during the spiraling and elliptical deformation of the vortex sheet, reaching a value of ∼ 4.0× 10^-6 kg m^2 s^-2 by 6.0 s, when the calculation is terminated because of particle depletion in the inlet section. §.§ Flow through a constricted channel As a further test we consider the flow between two parallel walls with a sharp-edged, narrow passage (or throat) of length l at the centre of the channel, as shown in the top view of Fig. 11. The main flow direction is taken along the x-axis and the depth of the channel is assumed to be infinite so that the flow is in the (x,y)-plane. Three separate simulations are considered. Two of them use identical parameters except that in one run (R1) particles in the outflow zone are evolved solving Eq. (29) with the anisotropic term dropped to mimic an Orlanski type outlet boundary condition, while the second run (R2) solves Eq. (29) including the anisotropic term. A third run (R3) is identical to R2 but with a longerdownstream pipe section. This test problem is more stringent than the previous examples because downstream the throat anisotropic flow develops as in the case of flow past a backward facing step. In addition, if the throat is modelled as a very narrow passage, its cross-section can be made to strongly differ from that of the outlet as desired. For these simulations we take ρ =1000 kg m^-3, ν =1.0× 10^-6 m^2 s^-1, and a time-varying plane Poiseuille velocity profile in the inflow zone given byv_ in=v_0(t/t_0)(1-4y^2/D_1^2),where t_0=0.2 s and v_0=4.45× 10^-2 m s^-1. The inlet flow is exactly zero at t=0 and increases linearly with time in the course of the evolution. This is equivalent to applying a pressure difference by suction on the outlet. When t≥ t_0, we set t/t_0=1 and the inlet flow becomes time-independent. The sound speed is taken to be c_0=1.0 m s^-1 and Eq. (4) is used as the pressure-density relation. The throat has a length of l=7.78 cm and an opening width of d≈ 1.33 cm. The upstream section has a width of D_1≈ 8.89 cm and a length l_1=20 cm, while the downstream section has a width of D_2≈ 7.56 cm and a length l_2=20 cm for models R1 and R2 and l_2=36 cm for model R3. For these simulations we use a total of 37467 (for models R1 and R2) and 50346 regularly distributed particles (for model R3) filling the entire channel.The influence of the computational domain size and type of nonreflecting boundary conditions as given by Eqs. (27) and (28) are now examined. Figures 12 and 13 display the velocity field in the throat and downstream sections at t=4.65 and 6.0 s, respectively. The top frame shows the flow structure for model R1 with the Orlanski type outlet boundary, while the other two frames correspond to models R2 (middle frame) and R3 (bottom frame) using nonreflecting conditions with the anisotropic term in Eq. (29) included and a different size of the downstream section. The Reynolds number in the throat conduit can be defined as Re=v_md/ν, where v_m is the mean velocity there. This gives Re≈ 4531 (at t=4.65 s) and ≈ 4552 (at t=6.0 s) for cases R2 and R3, while Re≈ 4516 (at t=4.65 s) and Re≈ 4546 (at t=6.0 s) for model R1. At t=6.0 s, the maximum velocity at the exit of the throat is v_max≈ 0.43 m s^-1 for model R1 against v_max≈ 0.40 m s^-1 for models R2 and R3, while the mean pressure drop through the throat is Δ p≈ 6.51× 10^-3 Pa for model R1 compared to Δ p≈ 1.22× 10^-2 Pa for the other two cases.In the downstream section, a jet forms just behind the throat exit surrounded by recirculatory flow, which extends along the full length of the section. Winding of the jet downstream is due to its interaction with the moving smallest vortices. Details of this recirculatory flow are displayed in Fig. 14, which show blowup views of the flow just above the vena contracta at t=4.65 s for the models of Fig. 12. Inspection of these figures shows that the vortices appearing downstream on both sides of the vena contracta are damped in model R1 (top frame) compared to models R2 (middle frame) and R3 (bottom frame), implying that neglecting the anisotropic term in Eq. (29) affects the structure of the flow. Also the reattachment length at t=4.65 and 6.0 sis much shorter in model R1 (≈ 0.10 m) compared to model R3 (≈ 0.24 m). As shown in Figs. 12 and 13, the flow structure for model R2 closely follows that shown for model R3 with a longer downstream channel, demonstrating that the feedback noise from the outlet boundary is also greatly reduced for this test.Figures 15 and 16 show longitudinal velocity profiles for models R1, R2, and R3 at successive streamwise stations along the downstream section for the same times of Figs. 12 and 13, respectively. In both figures, the squares depict the profiles for model R1, while the dots and triangles correspond to the profiles for models R2 and R3, respectively. It can be clearly seen that there is a very good correspondence between the profiles for models R2 and R3 for all stations at both times. In contrast, the profiles of model R1 substantially deviate from those of models R2 and R3 on both sides of the centreline and towards the channel walls,with the magnitude of the deviations increasing close to the outlet. The good correspondence between the results of models R2 and R3 proves the efficiency of the nonreflecting outlet boundary conditions when the anisotropic diffusion term is accounted for in Eq. (29), which allows us to work with smaller sizes of the computational domain.§ FLOW IN A SQUARE-SECTIONED 90^∘ PIPE BEND We now assess the performance of the nonreflecting outlet boundary conditions on a full 3D test problem. We simulate the steady, turbulent flow in a 90^∘ section of a curved square pipe at Re=40000. The numerical results are compared with experimental measurements <cit.> and numerical simulations carried out with the software package FLUENT 6.2 <cit.> for the same parameters.The geometrical model and parameters are the same employed by Sudo et al. <cit.> in their experimental investigation. The pipe geometry is shown in Fig. 17. The pipe has a square cross-sectionmeasuring l× l=80 mm × 80 mm and a 90^∘ bend of curvature radius R=160 mm connected at its both ends with a horizontal straight duct upstream of L_ h=2 m long and a vertical straight duct downstream of length L_ v=1.6 m. At the inlet, a flat velocity profile with v_ c=7.4 m s^-1 is assumed in correspondence with the experimental bulk mean velocity. With these parameters, the curvature radius ratio is 2R/l=4 and the Dean number is D= Re√(l/2R)=2× 10^4, with Re=v_ inl/ν =4× 10^4, where ν is the kinematic viscosity. To achieve a comparable spatial resolution to the simulations by Rup et al. <cit.>, we fill the pipe volume with 1.03 million particles initially at rest and uniformly spaced in all three coordinate directions (Δ x=Δ y=Δ z=3 mm). The particles are given an initial smoothing length h≈ 6.06 mm and Eq. (4) is used as the pressure-density relation with c_0=5 m s^-1.In order to provide direct comparison with the experimental data of Sudo et al. <cit.> and the numerical calculations of Rup et al. <cit.>, Fig. 18 depicts profiles of the longitudinal mean velocity in the horizontal plane including the duct axis at three different streamwise stations: (a) in the horizontal duct at 0.08 m from the entrance of the bend (corresponding to x=1.92 m from the inlet, i.e., z^'/d=-1 in Sudo et al.<cit.> notation), (b) within the bend at θ =60^∘, and (c) down the vertical duct at y=0.8 m from the bend exit (i.e., z^'/d=10 in Sudo et al. notation). We may see that the SPH profiles (solid lines) are in reasonably good agreement with the experimental data (dots) and the FLUENT simulations (dashed lines). Because of the assumption of a flat velocity profile at the inlet, the flow in the SPH and FLUENT simulations is not fully developed at x=1.92 m from the inlet (top frame) and therefore the velocities around the pipe centreline are underestimated compared to the experimental data. As the flow enters the bend, the longitudinal velocity profile distorts as a secondary flow grows. At θ =60^∘ within the bend (middle frame), the fluid flow is faster towards the inner wall due to the larger pressure gradients there. The SPH calculation reproduces reasonably well the asymmetric profile and closely matches the experimental and FLUENT profiles at this station. Away from the bend exit at y=0.8 m (bottom frame), the secondary flow attenuates and the vortex breaks down. At this station, the SPH simulation reproduces very well the experimental flow velocity front, meaning that the nonreflecting outlet boundary conditions are not influencing the flow in the elbow and along the verticalduct. In contrast, the FLUENT calculation underestimates the front velocity by about 10%. Given the good matching of the FLUENT results with the experimental measurements atθ =60^∘, the 10% deviation in the front velocity at y=0.8 m from the bend exit may be caused by some influence from the outlet boundary condition.§ CONCLUSIONS In this paper, we have described a procedure, based on Jin and Braza's <cit.> method, for modelling nonreflecting outlet boundary conditions for incompressible Navier-Stokes flows using the method of smoothed particle hydrodynamics (SPH). The method, which was originally developed for two-dimensional (2D) flows, was also generalized to three-space dimensions (3D). As it is common practice in SPH, the method involves inflow and outflow zones of particles, which are external to the fluid domain. A reservoir zone is designed to temporarily store particles, which is useful in most applications where the rate of outflowing and inflowing particles is not the same. Nonreflecting outlet boundary conditions are implemented here by allowing the particles that leave the computational domain and enter theoutflow zone to move according to an outgoing wave equation for the velocity field so that feedback noises from the boundary are effectively reduced. For unsteady, unidirectional flows, the method reduces to the well-known Orlanski wave equation, while for steady-state flows it takes the form of a zero diffusive boundary condition.The performance and accuracy of the method was assessed against several 2D tests, including the unsteady, plane Poiseuille flow, flow between inclined plates, the Kelvin-Helmholtz instability in a channel, and flow in a constricted channel. The performance of the method was also assessed for a 3D test problem involving the turbulent flow in a square-sectioned 90^∘ pipe bend. For this test, the numerical SPH results were compared with experimental measurements and previous numerical analysis obtained using the software package FLUENT 6.2. In general, the results show that spurious waves incident on the outlet are effectively absorbed, inhibiting feedback noises and allowing us to reduce the length of the computational domain. In addition, steady-state laminar flows can be maintained stably for much longer times compared to periodic boundary conditions. The method is stable and has the advantage of being easily implemented for other types of incompressible flows at low and moderate Reynolds numbers, as may be the case of flows aroundobstacles and free shear layer flows with transition towards turbulence, among others.§ ACKNOWLEDGEMENTWe thank the reviewers who have provided a number of comments and suggestions that have greatly improved the style and content of the paper. The calculations of this paper were performed using the computing facilities of ABACUS-Cinvestav. This work was partially supported by ABACUS under CONACyT grant EDOMEX-2011-C01-165873 and by the Departamento de CienciasBásicas of the Universidad Autónoma Metropolitana–Azcapotzalco (UAM-A) through internal funds.§ REFERENCES | http://arxiv.org/abs/1709.09141v1 | {
"authors": [
"Carlos E. Alvarado-Rodríguez",
"Jaime Klapp",
"Leonardo Di G. Sigalotti",
"José M. Domínguez",
"Eduardo de la Cruz Sánchez"
],
"categories": [
"physics.flu-dyn",
"physics.comp-ph"
],
"primary_category": "physics.flu-dyn",
"published": "20170926172444",
"title": "Nonreflecting outlet boundary conditions for incompressible flows using SPH"
} |
*theorem*Theorem theoremTheorem[section] lemma[theorem]Lemma example[theorem]Example proposition[theorem]Proposition corollary[theorem]Corollary problem[theorem]Problem definition[theorem]Definition assumption[theorem]Assumption remark[theorem]Remark claim[theorem]Claim criterion[theorem]Criterion observation[theorem]Observationequationsection | http://arxiv.org/abs/1709.09481v1 | {
"authors": [
"Erwin Adriaans",
"Julia Komjathy"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170927130534",
"title": "Weighted distances in scale-free configuration models"
} |
A Simple Reinforcement Learning Mechanism for Resource Allocation in LTE-A Networks with Markov Decision Process and Q-LearningEinar C. SantosFederal University of GoiasAv. Dr. Lamartine Pinto de Avelar, 1120Catalao - GO - [email protected] 30, 2023 ================================================================================================================================================================================================ Resource allocation is still a difficult issue to deal with in wireless networks. The unstable channel condition and traffic demand for Quality of Service (QoS) raise some barriers that interfere with the process. It is significant that an optimal policy takes into account some resources available to each traffic class while considering the spectral efficiency and other related channel issues. Reinforcement learning is a dynamic and effective method to support the accomplishment of resource allocation properly maintaining QoS levels for applications. The technique can track the system state as feedback to enhance the performance of a given task. Herein, it is proposed a simple reinforcement learning mechanism introduced in LTE-A networks and aimed to choose and limit the number of resources allocated for each traffic class, regarding the QoS Class Identifier (QCI), at each Transmission Time Interval (TTI) along the scheduling procedure. The proposed mechanism implements a Markov Decision Process (MDP) solved by the Q-Learning algorithm to find an optimal action-state decision policy. The results obtained from simulation exhibit good performance, especially for the real-time Video application. § INTRODUCTION Wireless networks are remarkably known due to its unpredictable physical conditions. Its channel suffers intense variation caused by numerous aspects: signal pathloss; fading; et cetera. Additionally, is critical nowadays the offering of Quality of Service (QoS) support for applications since the resource demand is actually becoming far more stringent. The machine learning is emerging as an attractive choice among the variety of techniques suited to optimize resource allocation for recent wireless networks. Reinforcement learning is particularly unique to help the achievement of an optimal performance by the system orienting it from a resultant output after a performed action. In this context, it is proposed a reinforcement learning mechanism for Long Term Evolution Advanced (LTE-A) networks designed to determine and restrict the number of Resource Blocks (RBs) available for each traffic class. The proposal models the problem of choosing the number of RBs as a Markov Decision Process (MDP) and is solved running the Q-Learning algorithm. The mechanism is attractive because of its simplicity. The Q-Learning algorithm is simple, straightforward and efficient to solve finite state MDPs. Concerning complexity, it is computationally cheap and easy to implement <cit.>. In fact, the MDP is a popular tool for modeling agent-environment interaction <cit.>. Several works in the literature implement MDP in wireless networks for various applications, raising the interest in the concept and its comprehensiveness. In a MDP every decision corresponds to an action taken towards a represented state of the process, helping the system to evaluate its condition. The proposed mechanism directs its effort in analyzing the system state subject to the QoS parameters of applications as throughput, delay, and packet loss rate, instead of evaluating physical information of the wireless system. The mechanism is evaluated at the system level through simulation and compared it with some scheduling algorithms utilized in LTE-A networks. This paper is arranged as follows: in Section <ref> there is some discuss some of the related works and their resemblances with the proposal; the fundamentals of MDPs are presented in Section <ref>; the description of Q-Learning algorithm is shown in Section <ref>; in Section <ref> the proposed mechanism is exposed; simulation parameters are shown in Section <ref> while results are presented in Section <ref> with some discussion; finally, in Section <ref> it brings some conclusion about the whole work. § RELATED WORK In <cit.> the authors propose a reinforcement learning method aimed to improve QoS provisioning for adaptive multimedia applications in cellular wireless networks defining policies for Call Admission Control (CAC) and Bandwidth Adaptation (BA). They adopted Semi-Markov Decision Process (SMDP) – which treats continuous-time problems as discrete-time <cit.> – to model and solve the problem. It is important to take into account the CAC procedure in order to control the system resources, but this also can be accomplished at the resource allocation level. The authors in <cit.> devised a distributive reinforcement learning mechanism for joint resource allocation and power control on femtocell networks. Each femtocell seeks to maximize its capacity while maintaining QoS. The Q-Learning algorithm is adopted, and the information about each independent learning procedure is shared among the femtocells to speed up the overall learning process. However, they not consider traffic differentiation and some action to avoid the services may harm each other. The Scheduling-Admission Control (SAC) for a generic wireless system is appropriately investigated in <cit.>. Authors also propose two online learning algorithms in order to optimize the SAC procedure with low complexity and convergence faster than the Q-Learning algorithm. They approached the problem with a model-based solution, however, ignoring QoS issues. In contrast, it is important to implement the model-free approach because it can cover any technology in the field, independently. § MARKOV DECISION PROCESS The MDP is a control process model with stochastic, memoryless and discrete time properties. It is formally described as a tuple <S, A, P, R> where S is the state set, A is the action set, P is a set comprising the transition probabilities among states, and R is a reward set containing a r value for each action a taken <cit.>. Each p value, with p ∈ P, measures the probability of an action a ∈ A be performed at a decision epoch. An action a changes the process state from a s to a new s' value and represents a decision-making. Figure <ref> depicts a MDP according to the given definition. For simplicity, transition probabilities P and the reward values R are not displayed. The decision epoch is a discrete time unit adopted for decision-making. If the number of decision epochs is finite, the MDP formulation is referred as finite-horizon. The horizon also can be infinite or undefined, when a MDP stops if a final state is reached. A decision taken at an epoch k is a function d_k that maps the rule d_k(s): S ↦ A which also corresponds to an action (d_k(s) = a_k) <cit.>. A policy π is defined as the collection of decision rules, given as: π = {d_0, d_1, …, d_Z-1}. The variable Z is the total number of decision epochs. An optimal policy is the one that maximizes the measure of long-run expected rewards. It can be obtained from the optimal value of total reward function u^*_k(s), given as follows <cit.>: u^*_k(s_k)= max_d_k(s) ∈ A{r_k(s_k,d_k(s))+ γ∑_s' ∈ S^p(s_k, d_k(s), s')u_k+1^*(s')} The γ value is the discounting factor used to weight immediate rewards. In order to store values for every state-action pair the MDP should consider the Q(s,a) function, usually called action-value function or simply Q-function <cit.>: Q(s,a) = r(s,a) + γmax_a'Q(s',a')§ Q-LEARNING ALGORITHM Q-Learning <cit.> is a straightforward and model-free reinforcement learning algorithm adopted to define the values of transition probabilities and to find an optimal policy for a MDP. It also converges to an optimal policy given a finite action-state MDP <cit.>. The Q-Learning algorithm iteratively updates the Q-table for each (s,a) pair visited. The formula for updating the values at each step t, with the learning rate α, is: Q(s_t,a_t) = Q(s_t,a_t) + α[r_t+1 + γmax_aQ(s_t+1,a) - Q(s_t,a_t)] The algorithm is given as follows <cit.>: In a deterministic model, to ensure that all state-action pair is going to be visited, the algorithm must randomly select the (s_t,a_t) pair and run the for loop during a sufficient total number of steps T previously chosen. However, considering the case of a stochastic model (with unpredictable reward values), a nice option is to implement online the ϵ-greedy method, which selects the (s,a) pair randomly with ϵ probability, balancing the system and finding a way to circumvent the exploitation versus exploration dilemma <cit.>. The second option is also suitable when the system will run during an infinite (or unknown) number of steps. § MECHANISM DESCRIPTION The proposed mechanism is just referred as MDP with Q-Learning (MDP-QL). For QoS guarantee the system should look at the traffic QoS Class Identifier (QCI) priority values <cit.> and also to classification from these values. The number of resources is selected from the current state of MDP at each Transmission Time Interval (TTI). Meaning that every MDP state denotes a portion of available RBs and one TTI is equivalent to one decision epoch. If the limit of available resources is reached for a traffic class, the system must advance to the next class observing QCI priority values until all resources or traffic have been exhausted in allocation procedure. RB metrics for every User Equipment (UE) are calculated as usual but restricting the usable load regularly for each class according to the selected proportion. Firstly, the state-action table Q(s,a) should be created and initialized for the MDP so that the system can perform the selection from state-action values. At the early steps the chosen values will not serve nicely for the needed quantity, but as long as the algorithm runs the state-action table, it will converge to more appropriate values. §.§ Model Definition The MDP adopted in this proposal has a finite number of states Σ and is infinite-horizon once the mechanism should operate indefinitely. However, for simulation purposes the horizon can be assumed as finite, so the number of decision epochs is constrained by the simulation time. Of course, the Σ value must be carefully chosen. A large number of states consequently makes the running impracticable. This is also referred as the curse of dimensionality <cit.>. Each state s_k,i contains the value that delimitates the maximum number of RBs for a treated traffic class. The total number of RBs N in the system, the state index i and the total number of states Σ settled should be considered: s_k,i=⌈N · i/Σ⌉ i ={1,2,…,Σ},N > Σ. In order to obtain reward at some decision epoch, the mechanism has to monitor some indicators. In this proposal the goal is to improve the system as follows: r_k = log(R̅_̅k̅/δ̅_̅k̅ρ̅_̅k̅) The reward value r_k takes into account the average throughput R̅_̅k̅, the average delay δ̅_̅k̅ and the average packet loss rate ρ̅_̅k̅ of all running applications in the system at the decision epoch k. Thus, it is established as overall goal the system throughput maximization while reducing its delay and packet loss rate. The log() function is used to compensate the scale. The mechanism scheme is depicted in Figure <ref>. § SIMULATION PARAMETERS This Section is dedicated to present the values of selected parameters contemplated in simulation definition to perform mechanism evaluation at the system level. In this particular case solely the downlink channel was analyzed. However, the mechanism can be studied from the uplink channel perspective as well. The parameters values are presented in Table <ref>. Every UE is randomly positioned in the cell before simulation starting and has one running instance of each application: real-time Video, VoIP, and Web. It was chosen the following scheduling algorithms for comparison: Proportional Fair (PF); Round Robin (RR) and Frame Level Scheduling (FLS) <cit.>. § RESULTS The following results present the behavior of evaluated algorithms including the proposed mechanism. It was analyzed throughput, delay, jitter, fairness index and packet loss. Figure <ref> presents the throughput performance obtained for the Video application. The average throughput for Video application running MDP-QL maintains the needed rate for application up to 50 UEs. The MDP-QL also keeps the average throughput higher than the other algorithms when the system has more than 40 UEs. To ensure that this performance is not occurring to the detriment of the VoIP traffic the Figure <ref> shows the average throughput for VoIP application is still preserved with MDP-QL as well as in the other algorithms up to 70 UEs. When analyzing the throughput sharing among UEs is possible to see in Figure <ref> that the fairness index for MDP-QL is above 0.9 up to 60 UEs in the system and slightly above the other algorithms from 60 UEs, too. It means that MDP-QL, in addition to maintaining a good throughput, still balances it in terms of sharing. Such statement is corroborated by results presented in Figure <ref>, showing the curves of Cumulative Distribution Function (CDF) for average throughput with 40 UEs in the system. The MDP-QL allocates just over 60% of UEs with throughput higher than 400 Kbps. Figure <ref> exhibit CDF for average throughput with 100 UEs present. The graphic in Figure <ref> is useful to demonstrate the behavior of MDP-QL and its evolution from 40 UEs, which still conserves good distributed throughput in such condition. Figure <ref> presents the values for average delay. The average delay for MDP-QL is not so efficient from 50 UEs in the system. Although, all algorithms maintain delay under the limit for the considered Video application. A significant result is displayed in Figure <ref>. Keeping low packet loss rate is essential to guarantee QoS for real-time traffic. So, Figure <ref> shows that MDP-QL can still preserve some QoS level for Video up to 50 UEs. From 50 UEs, the mechanism still maintains the packet loss rate below the achieved by other algorithms. The average jitter is presented in Figure <ref>. The MDP-QL is quite stable regarding jitter, exhibiting values below all the other algorithms for almost all evaluated scenarios. Finally, the Figure <ref> presents the results of average throughput obtained for the Web application. It is noticeable the FLS does not meet the application demand even with low traffic load, which indicates that all the effort of the algorithm is directed to support multimedia application in detriment of the best effort traffic. On the other hand, the MDP-QL can support a proper service level, given the adaptive behavior of the reinforcement learning technique. Indeed, the proposed mechanism is capable of improving the operation of real-time Video application. It achieves good QoS levels for Video but preserving performance for VoIP and maintaining basic service level for Web application. As the mechanism is based on a technique that aims to optimize some given task, it is expected at least that the proposal can ameliorate some aspects of resource allocation in LTE-A networks. § CONCLUSION Hitherto was presented a simple reinforcement learning mechanism employed into resource allocation in wireless networks, more specifically the LTE-A technology. The proposed mechanism applies MDP to model the problem of selecting and restricting the number of resources available for each traffic class. In addition, it implements Q-Learning in order to solve and enhance the proposed model. Simulation results show good performance measured for Video application that was achieved by the mechanism, which evidences its importance in offering QoS. The mechanism also reaches favorable levels of packet loss rate, average throughput, and average jitter, standing out the proposal in comparison with the analyzed algorithms. It would be interesting take advantage of the proposed mechanism integrating it with some technique employed for traffic classification. It is expected therefore some performance improvement with a better traffic classification procedure. ieeetr | http://arxiv.org/abs/1709.09312v1 | {
"authors": [
"Einar Cesar Santos"
],
"categories": [
"cs.AI",
"cs.NI"
],
"primary_category": "cs.AI",
"published": "20170927024229",
"title": "A Simple Reinforcement Learning Mechanism for Resource Allocation in LTE-A Networks with Markov Decision Process and Q-Learning"
} |
Atmospheric tides and their consequences on the rotation of planetsLAB, Université de Bordeaux, CNRS UMR 5804, Université de Bordeaux - Bât. B18N, Allée Geoffroy Saint-Hilaire, CS50023, 33615 Pessac Cedex, France IMCCE, Observatoire de Paris, CNRS UMR 8028, PSL, 77 Avenue Denfert-Rochereau, 75014 Paris, France Laboratoire AIM Paris-Saclay, CEA/DRF - CNRS - Université Paris Diderot, IRFU/SAp Centre de Saclay, F-91191 Gif-sur-Yvette Cedex, France LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Université, UPMC Univ. Paris 6, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, F-92195 Meudon, France Atmospheric tides can have a strong impact on the rotational dynamics of planets. They are of most importance for terrestrial planets located in the habitable zone of their host star, where their competition with solid tides is likely to drive the body towards non-synchronized rotation states of equilibrium, as observed in the case of Venus. Contrary to other planetary layers, the atmosphere is sensitive to both gravitational and thermal forcings, through a complex dynamical coupling involving the effects of Coriolis acceleration and characteristics of the atmospheric structure. These key physics are usually not taken into account in modelings used to compute the evolution of planetary systems, where tides are described with parametrised prescriptions. In this work, we present a new ab initio modeling of atmospheric tides adapting the theory of the Earth's atmospheric tides <cit.> to other terrestrial planets. We derive analytic expressions of the tidal torque, as a function of the tidal frequency and parameters characterizing the internal structure (e.g. the Brunt-Väisälä frequency, the radiative frequency, the pressure heigh scale). We show that stratification plays a key role, the tidal torque being strong in the case of convective atmospheres (i.e. with a neutral stratification) and weak in case of atmosphere convectively stable. In a second step, the model is used to determine the non-synchronized rotation states of equilibrium of Venus-like planets as functions of the physical parameters of the system. These results are detailed in <cit.> and <cit.>. Atmospheric tides and their consequences on the rotational dynamics of terrestrial planets S. Mathis^3, December 30, 2023 ========================================================================================== § INTRODUCTION Like solid and oceanic tides, atmospheric tides are responsible for internal variations of mass distribution and dissipation of energy, and thus contribute to the dynamical and rotational evolution of planetary systems <cit.>. This contribution is often negligible compared to the two formers because of the weak energy of tides in the atmosphere relatively to the one in solid and oceanic layers. For example, the major part of the current total tidal dissipation of the Earth is due to the ocean. However, the atmospheric tidal dissipation can be strong in the case of Venus-like planets, i.e. planets basically composed of a solid core and a massive atmosphere, and rotating slowly. Indeed, as showed by early studies of the rotational dynamics <cit.>, the observed retrograde rotation rate of Venus is a state of equilibrium resulting from the competition between atmospheric and solid tides. Without the action of atmospheric tides, gravitational tides would torque Venus to spin-orbit synchronization.The large number of new planetary systems discovered during the two past decades thanks to the CoRoT and Kepler space missions <cit.>, and particularly those hosting terrestrial planets in the habitable zone, has enhanced the importance of this effect of atmospheric tides on the rotational dynamics. Among the discovered bodies, configurations such as Venus' one are probable. As a consequence, atmospheric tides are likely to have a strong impact on the rotational dynamics of these planets. They can also affect indirectly their climate and atmospheric general circulation (e.g. differential rotation, tides induced flows, heating), which are of great importance regarding their surface habitability. For all these reasons, computing the long term orbital and rotational evolution of planetary systems and characterizing the atmosphere dynamics of the detected planets now requires to take into account the effects of atmospheric tides in a realistic way.Yet, this is not the case of simplified modelings commonly used in celestial dynamics. Some of them, like the so-called Kaula's model <cit.>, set the dissipation to a constant, and thus do not describe the dependence of tidal dissipation on the tidal frequency, although this makes the dissipated energy vary over several orders of magnitude. Others include considerations on fluid dynamics based on a geophysical modeling <cit.> but have to introduce a parametrised regularization to prevent the dissipated energy to diverge at spin-orbit synchronization. By using a General Circulation Model (GCM) to compute thermal tides numerically, <cit.> recover this needed regular behaviour described by these modeling in the case of Venus with a physical approach. However, this method can be heavy to implement to explore the full domain of parameters, because of its possibly high computational cost. Moreover, it is important to understand in details the complex physical mechanisms involved in the atmospheric tidal dissipation. To address this question in the general case, we developed a new global ab initio modeling based on the state of the art work of <cit.>, who treat the case of the Earth's atmospheric tides. In this model, the internal dissipative processes which are responsible for the behaviour of the tidal response at the vicinity of synchronization are taken into account using a Newtonian cooling <cit.>. We present here the outlines of the modeling and show how it can be applied to provide a prediction about the atmospheric structure and its impact on the rotational evolution of planets. For technical details, the reader may refer to <cit.> and <cit.>. § TIDAL WAVES DYNAMICSWe give here an overview of the dynamics of the modeling of the atmospheric tidal response. The results presented in this section are detailed in <cit.>. We consider a terrestrial planet of radius R rotating at the angular velocity Ω (the corresponding spin vector being Ω), and tidally excited by the thermal and gravitational forcings of the host star (Fig. <ref>). To study the tidal response of the planet's atmosphere, we introduce the equatorial frame co-rotating with the planet ℛ_ E: {X_ E , Y_ E, Z_ E}, where X_ E and Y_ E define the equatorial plane and Z_ E = Ω / | Ω| is aligned with the spin axis of the planet. In this frame, the position of a point M is described by the usual system of spherical coordinates ( r , θ , φ) and the associated basis ( e_r , e_θ , e_φ), r being the radius, θ the colatitude and φ the longitude (r = re_r stands for the corresponding position vector); the time is denoted t. The physics of the atmosphere are represented by the pressure (p), density (ρ), temperature (T), velocity (V) and gravity (g) distributions, a given quantity f being written as f ( r , t ) = f_0 ( r) + δ f ( r , t ), where the superscript _0 refers to background distributions and δ to the tidal perturbation. The analytic treatment of the dynamics is simplified by assuming the approximations listed below: * Solid body rotation: the atmosphere is supposed to rotate uniformly with the planet, so that mean flows are ignored (V_0 = 0).* Moderate rotation: the centrifugal acceleration due to the rotation of the body is negligible compared to its self-gravity. This implies that Ω≪Ω_ c, the parameter Ω_ c = √(g / r) being the so-called critical Keplerian angular velocity. * Spherical symmetry of the background: the background distributions of gravity, pressure, density and temperature are assumed to vary along the radial direction only. Thus, the horizontal variations of the atmospheric structure are neglected.* Perfect gas approximation: the fluid is supposed to follow the law of perfect gas and to be characterized by the constant adiabatic exponent Γ_1 = 1.4. We also introduce the parameter κ = ( Γ_1 - 1 ) / Γ_1.* Uniform composition: the atmosphere is uniform in composition, this later being characterized by the specific gas constant ℛ_ s = ℛ_ GP / M (where ℛ_ GP designates the perfect gas constant and M the molecular weight of the atmosphere). The thermal capacity per unit mass of the atmosphere is defined by C_ p = ℛ_ s / κ.* Cowling approximation <cit.>: the self-gravitational perturbation resulting from the tidal variations of mass distribution is not taken into account. * Traditional approximation: the latitudinal components of Coriolis accelerations (2 Ωsinθ V_r and 2 Ωsinθ V_φ) are ignored. The conditions of validity of this approximation are discussed in details in <cit.> and <cit.>.* Linear approximation: the perturbation is assumed to be small enough to neglect non-linear couplings, i.e. δ f / f_0 ≪ 1. * Newtonian cooling: the zero–order effect of radiative cooling and thermal diffusion is described with the sink power per unit mass J_ NC = σ_0 C_ pδ T, where the radiative frequency σ_0 corresponds to the inverse of the effective radiative time of the atmosphere.The compressibility of the atmosphere is characterized by the sound frequency c_s and its stratification by the Brunt-Väisälä frequency N, defined byN^2 = g [ 1/Γ_1d ln p_0/dr - d lnρ_0/dr]. The tidal perturbation being periodic in time and longitude, the gravitational potential U and the thermal power per unit mass J of the forcing, as well as the perturbed quantities are expanded in Fourier series. Thanks to the traditional approximation, Fourier coefficients can be themselves expanded in series of functions with separated coordinates. Hence, a fluctuation δ f of any quantity writesδ f = ∑_n,m,σδ f_n^m,σ( r ) Θ_n^m,ν( θ) e^i ( σ t + m φ),where we have introduced the tidal frequency σ,the longitudinal and latitudinal wavenumbers, m and n respectively, the spin parameter ν = 2 Ω / σ, the Hough functions Θ_n^m,ν <cit.> and the corresponding vertical profiles δ f_n^m,σ. We will not develop here technical aspects concerning solutions δ f_n^m,σ, which are described in <cit.>. However, the physical setup allows us to identify the possible regimes for the atmospheric tidal response (Fig. <ref>):* Mixed-acoustic regime for tidal frequencies greater than the Lamb frequencies σ_s ; n = √(Λ_n^m,ν) c_s / R of the gravest acoustic modes (Λ_n^m,ν designates the eigenvalue associated to the Hough function Θ_n^m,ν).* Dynamic regime for σ_0 ≪| σ| ≪| σ_ s ; n|. In this regime, which corresponds to the Earth's semi-diurnal tide typically <cit.>, the effect of dissipation on the structure of tidal waves can be ignored.* Radiative regime for low tidal frequencies (| σ| ≲σ_0). This regime corresponds to the vicinity of synchronization. The atmospheric tidal response is regular thanks to dissipative processes, which damp tidal waves.As showed by Fig. <ref>, we recover with this modeling the observed lag and amplitude of the Earth's semidiurnal surface pressure oscillations (i.e. oscillations corresponding to the tidal frequency σ = 2 ( Ω - n_ orb), where n_ orb designates the orbital frequency of the Earth). Such oscillations are mainly due to the first quadrupolar gravity mode (m=2, n = 0).§ TIDAL TORQUE: THE KEY ROLE PLAYED BY STRATIFICATIONThe rotational evolution of the planet is affected by the tidal torque exerted on the atmosphere with respect to the spin axis of the planet, denoted 𝒯^2,σ in the case of a quadrupolar perturbation. This torque is deduced straightforwardly from the variations of mass distribution <cit.>. Hence, for a slowly rotating planet with a convective atmosphere, one obtains <cit.> 𝒯_ neutral^2,σ = 2 π R^2 κρ_ s/gU_2 J_2 σ/σ^2 + σ_0^2, where ρ_ s designates the surface density of the atmospheric layer, and U_2 and J_2 the quadrupolar components of the gravitational tidal potential and thermal forcing respectively. This torque, plotted on Fig. <ref> in the case of a Venus-like planet (blue dashed line), is similar to those obtained by <cit.> with numerical simulations using GCMs and early parametrised modelings <cit.>. It corresponds to the horizontal (H) hydrostatic adjustment of the atmosphere with a lag depending on its thermal inertia and dissipative properties. The expression of the torque in the case of a stably stratified isothermal atmosphere (N^2 ≫| σ|) is given by <cit.> 𝒯_ strat^2,σ = - 2 π R^2 H U_2 ∑_n C_2,n,2^2,ν{𝒜_n J_2 + ℬ_n U_2}, wheredesignates the imaginary part of a complex number, H = p_0 / ( g ρ_0 ) the pressure height scale of the atmosphere, the C_2,n,2^2,ν the projection coefficients quantifying the effect of the Coriolis acceleration on the structure of the perturbation, and η_s ; n, 𝒜_n and ℬ_n parameters depending on the tidal frequency and vertical wavenumber of the ( n , 2 )-mode. In the vicinity of synchronization, the Archimedean force inhibits tidal motions in the vertical direction. The integral of the resulting variations over the air column vanishes whenσ→ 0. As a consequence, the tidal torque (Fig. <ref>, red continuous line), which receives the contribution of vertical (V) displacements (green dashed line), is weak compared to 𝒯_ neutral^2,σ (H, blue dashed line). Considering planetary rotational evolution, the planet is thus likely to be driven towards non-synchronized states of equilibrium in the convective case and towards spin-orbit synchronization in the stably-stratified one, as illustrated by Fig. <ref>. § CONSEQUENCE ON THE EQUILIBRIUM ROTATION OF VENUS-LIKE PLANETSBy using the torque computed in the case of a convective atmosphere and the so-called Maxwell rheology to describe the tidal response of the solid part, we derive the final rotation states of equilibrium of a Venus-like planet as functions of the physical parameters of the system <cit.>. In this framework, the stability of the states of equilibrium appears to depend on the hierarchy of frequencies associated to dissipative processes, i.e. σ_0 for the atmosphere, and the Maxwell relaxation frequency of the material σ_ M for the solid part. These results are illustrated by Fig. <ref>, where the total torque exerted on the planet and its sign are plotted as functions of the difference to synchronization and distance to the host star. Note that the Maxwell model underestimates the tidal torque of the solid part for | σ| ≫σ_ M, which can be detrimental to a quantitative prediction. In order to address this point, the Maxwell rheology could be replaced by the Andrade rheology <cit.>, this later giving a better description of the solid tidal response. Four non-synchronized states instead of two would thus be obtained, as demonstrated by <cit.>.§ CONCLUSIONSIn this work, we developed an ab initio modeling for the atmospheric tides of terrestrial planets and applied it to the case of Venus-like planets. We established the dependence of the tidal response on the parameters of the internal structure, and particularly the stability of the vertical stratification. If this later is convective, we recover the tidal torque given by GCMs and early modelings. Otherwise, stable stratification decreases the amplitude of the torque of several orders of magnitude, which leaves the planet evolve towards spin-orbit synchronization. The chosen analytic treatment provides both a diagnostic of the physics involved in atmospheric tides and a predictive tool to explore the full domain of possible parameters. It will be improved in the future with the introduction of general circulation, means flows being able to modify significantly the structure of tidal waves <cit.>. P. Auclair-Desrotour and S. Mathis acknowledge funding by the European Research Council through ERC grants WHIPLASH 679030 and SPIRE 647383. This work was also supported by the Programme National de Planétologie (CNRS/INSU) and CoRoT/Kepler and PLATO CNES grant at CEA-Saclay.aa | http://arxiv.org/abs/1709.09478v1 | {
"authors": [
"Pierre Auclair-Desrotour",
"Jacques Laskar",
"Stéphane Mathis"
],
"categories": [
"astro-ph.EP",
"85-06"
],
"primary_category": "astro-ph.EP",
"published": "20170927125254",
"title": "Atmospheric tides and their consequences on the rotational dynamics of terrestrial planets"
} |
V-cycle algorithms for DG methods on non-nested polytopic meshes P. F. AntoniettiMOX-Laboratory for Modelling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Tel.: (+39) 02 2399 [email protected] G. Pennesi MOX-Laboratory for Modelling and Scientific Computing, Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy. Tel.: (+39) 02 2399 4604 [email protected] V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes This work has been supported by the research grant PolyNuM founded by Fondazione Cariplo and Regione Lombardia, and by the SIR Project n. RBSI14VT0S funded by MIUR.P. F. Antonietti G. Pennesi.December 30, 2023 ==================================================================================================================================================================================================================================================================== In this paper we analyse the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations arising from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopal meshes. Here, the sequence of spaces that stands at the basis of the multigrid scheme is possibly non nested and is obtained based on employing agglomeration with possible edge/face coarsening. We prove that the method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large.65F10 65M55 65N22§ INTRODUCTION The Discontinuous Galerkin (DG) method was introduced in 1973 by Reed and Hill for the discretization of hyperbolic equations <cit.>. Extensions of the method were quickly proposed to deal with elliptic and parabolic problems: some of the most relevant works include Arnold <cit.>, Baker <cit.>, Nitsche <cit.> and Wheeler <cit.>, whose contributions put the basis for the development of the interior penalty DG methods.In the last 40 years the scientific and industrial community has shown an exponentially growing interest in DG methods - see for example <cit.> for an overview. On one side, the features of DG methods have been naturally enhanced by the recent development of High Performance Computing technologies as well as the growing request for high-order accuracy. In particular, as the discrete polynomial space can be defined locally on each element of the mesh, DG methods feature a high-level of intrinsic parallelism. Moreover, the local conservation properties and the possibility to use meshes with hanging nodes make DG methods interesting also from a practical point of view.Recently, it has been shown that DG methods can be extended to computational grids characterized by polytopic elements, cf. Ref. <cit.>. In particular, the efficient approach presented in <cit.> is based on defining a local polynomial discrete space by making use of the bounding box of each element <cit.>: this technique together with a careful choice of the discontinuity penalization parameter permits the use of polytopal elements which can be characterized by faces of arbitrarily small measure and as shown in <cit.>, see also <cit.>, possibly by an unbounded number of faces. On the other hand, the development of fast solvers and preconditioners for the linear system of equations arising from high-order DG discretization is been developed. A recent strand of the literature has focused on multilevel techniques, including Schwarz domain decomposition methods, cf. Ref. <cit.>, and two-level and multigrid techniques, cf. Ref. <cit.>. The efficiency of those methods is more evident in the case of polygonal grids, because the flexibility of the element shape couples very well with the possibility to easily define agglomerated meshes, which is the key ingredient for the developing of multigrid algorithms. In <cit.> a two-level scheme and W-cycle multigrid method is developed to solve the linear system of equations arising from high-order discretization introduced in <cit.>. One iteration of the proposed methods consists of an iterative application of the smoothing Richardson operator and the subspace correction step. In particular, the latter is based on a nested sequence of discrete polynomial spaces where the underlying polytopal grid of each subspace is defined by agglomeration. While being faster than other classical iterative methods, the agglomeration approach presents itself some limitations. When the finest grid is unstructured and characterized by polytopic elements, there is the possibility that its very small edges could be inherited by the coarser levels until the one where the linear system is solved with a direct method. In this case the presence of small faces negatively affects the condition number of the associated matrix: indeed, according to <cit.>, the discontinuity penalization parameter is defined locally in each face as the inverse of its measure.In this paper we aim to overcome this issue by solving the same linear system through a multilevel method characterized by a sequence of non-nested agglomerated meshes in order to make sure that the number of faces of the agglomerates does not blows up as the number of levels of our multigrid method increases. This can be achieved for example based on employing edge-coarsening techniques in the agglomeration procedures. The flexibility in the choice of the computational sub-grids leads to the definition of a non-nested multigrid method characterized by a sequence of non-nested multilevel discrete spaces, cf. Ref. <cit.>, and where the discrete bilinear forms are chosen differently on each level, cf. Ref. <cit.>. The first non-nested multilevel method was introduced by Bank and Dupont in <cit.>; a generalized framework was developed by Bramble, Pasciak and Xu in <cit.>, and then widely used in the analysis of non-nested multigrid iterations, cf. Ref. <cit.>. The method of <cit.>, to whom we will refer as the BPX multigrid framework, is able to generalize also the multigrid framework that we will develop in this paper, but the convergence analysis relies on the assumption that [j][I_j-1^j u][I_j-1^j u] ≤[j-1][u][u], which might not be guaranteed in the DG setting, as we will see in Sect. <ref>. Here [j] and [j-1] are two bilinear forms suitably defined on two consecutive levels, and I_j-1^j is the prolongation operator whose definition is not trivial, differently from the nested case. For this reason the convergence analysis will be presented based on employing the abstract setting proposed by Duan, Gao, Tan and Zhang in <cit.>, which permits to develop a full analysis of V-cycle multigrid methods in a non-nested framework relaxing the hypothesis [j][I_j-1^j u][I_j-1^j u] ≤[j-1][u][u]. We will prove that our V-cycle scheme with non-nested spaces converges uniformly with respect to the discretization parameters provided that the number of smoothing steps, which depends on the polynomial approximation degree p, is chosen sufficiently large. This result extends the theory of <cit.> where W-cycle multigrid methods for high-order DG methods with nested spaces where proposed and analyzed.The paper is organized as follows. In Sect. <ref> we introduce the interior penalty DG scheme for the discretization of second-order elliptic problems on general meshes consisting of polygonal/polyhedral elements. In Sect. <ref>, we recall some preliminary analytical results concerning this class of schemes. In Sect. <ref> we define the multilevel BPX framework for the V-cycle multigrid solver based on non-nested grids, and present the convergence analysis of the algorithm. The main theoretical results are validated through a series of numerical experiments in Sect. <ref>. In Sect. <ref> we propose an improved version of the algorithm, obtained by choosing a smoothing operator based on a domain decomposition preconditioner.§ MODEL PROBLEM AND ITS DG DISCRETIZATION We consider the weak formulation of the Poisson problem, subject to a homogeneous Dirichlet boundary condition: find u∈ V = H^2(Ω)∩ H_0^1(Ω) such that𝒜(u,v) =∫_Ω∇ u ·∇ v dx =∫_Ω f v dx∀ v∈ V,with Ω⊂ℝ^d, d = 2,3, a convex polygonal/polyhedral domain with Lipschitz boundary and f ∈ L^2(Ω). The unique solution u ∈ V of problem (<ref>) satisfies [u][2][] ≤ C [f][2][].In view of the forthcoming multigrid analysis, let {[T][j]}_j=1^J be a sequence of tessellation of the domain , each of which is characterized by disjoint open polytopal elements of diameter , such that = ⋃_∈[T][j], j=1,…,J. The mesh size of [T][j] is denoted by j =max_∈[T][j]. To each [T][j] we associate the corresponding discontinuous finite element space V_j, defined asV_j ={v∈ L^2():v|_∈[P][p_j](),∈[T][j]},where [P][p_j]() denotes the local space of polynomials of total degree at most p_j≥1 on ∈[T][j].For the sake of brevity we use the notation x ≲ y to mean x ≤ Cy, where C>0 is a constant independent from the discretization parameters. Similarly we write x ≳ y in lieu of x ≥ Cy, while x ≈ y is used if both x ≲ y and x ≳ y hold. A suitable choice of {[T][j]}_j=1^J and { V_j }_j=1^J leads to the hp-multigrid non-nested schemes. This method is based on employing, from one side, a set of non-nested partitions {[T][j]}_j=1^J, such that the coarse level [T][j-1] is independent from [T][j], with the only constrainj-1≲j≤j-1∀ j = 2,…, J, from the other side we assume that the polynomial degree vary from one level to another such thatp_j-1≤ p_j ≲ p_j-1∀ j = 2,…, J.Additional assumptions on the grids {[T][j]}_j=1^J are outlined in the following paragraph. §.§ Grid assumptionsFor any [T][j], we define the faces of the mesh [T][j], j=1,…,J, as the intersection of the (d-1)-dimensional facets of neighbouring elements. This implies that, for d=2, a face always consists of a line segment, however for d=3, the faces of [T][j] are general shaped polygons. Thereby, we assume that each facets of an element ∈[T][j] may be subdivided into a set of co-planar (d-1)-dimensional simplices and we refer to them as faces.In order to introduce the DG formulation, it is helpful to distinguish between boundary and interior element faces, denoted as ℱ_j^B and ℱ_j^I, respectively. In particular, we observe that F ⊂∂Ω for F ∈ℱ_j^B, while for any F ∈ℱ_j^I we assume that F ⊂∂^±, where ^± are two adjacent elements in [T][j]. Furthermore, we denoted as ℱ_j = ℱ_j^I ∪ℱ_j^B the set of all mesh faces of [T][j]. With this notation, we assume that the sub-tessellation of element interfaces into (d-1)-dimensional simplices is given. Moreover, assume that the following assumptions hold, cf. <cit.>. For any j=1,…,J, given ∈[T][j] there exists a set of non-overlapping d-dimensional simplices T_l ⊂, l=1,…,n_, such that for any face F ⊂∂ it holds that F = ∂∩∂T_l for some l, it holds ∪_l=1^n_T_l⊂, and the diameter h_ ofcan be bounded by h_≲d |T_l |/|F|∀ l=1,…,n_. For any ∈[T][j], j=1,…,J, we assume that h_^d ≥ || ≳ h_^d, where d=2,3 is the dimension of . Every polytopic element ∈[T][j], j=1,…,J, admits a sub-triangulation into at most m_ shape-regular simplices {𝔰_i }_ i=1^m_, for some m_∈ℕ, such that = ∪_i=1^m_𝔰_i and |𝔰_𝔦| ≳ || ∀ i=1,…,m_,Let [T][j]^#= {𝒦}, denote a covering of Ω consisting of shape-regular d dimensional simplices 𝒦. We assume that, for any ∈[T][j], there exists 𝒦∈[T][j]^# such that ⊂𝒦 and max_∈[T][j] card {' ∈[T][j]: ' ∩𝒦∅ , 𝒦∈[T][j]^# such that ⊂𝒦}≲ 1. Assumption <ref> is needed in order to obtain the trace inequalities of Lemma <ref> and Lemma <ref>. Assumption <ref> and <ref> are required for the inverse estimates of Lemma <ref> and Theorem <ref>. Assumption <ref> guarantees the validity of the approximation result and error estimetes of Lemma <ref> and Theorem <ref>, respectively.Assumptions <ref> allows to employ polygonal and polyhedral elements possibly characterized by face of degenerating Hausdorff measure as well as unbounded number of faces, cf. <cit.>, see also <cit.>.§.§ DG formulationIn order to introduce the DG discretization of (<ref>), we firstly need to define suitable jump and average operators across the faces F ∈[F][j], j=1,…,J.Let τ and v be sufficiently smooth functions. For each internal face F∈[F][j]^I, such that F ⊂∂^±, let 𝐧^± be the outward unit normal vector to ∂^±, and let τ^± and v^± be the traces of the functions τ and v on F from ^±, respectively. The jump and average operators across F are then defined as follows:τ = τ^+ ·𝐧^+ + τ^- ·𝐧^-, τ = τ^++τ^-/2,F∈[F][j]^I, v = v^+ 𝐧^+ + v^-𝐧^-, v = v^++v^-/2,F∈[F][j]^I, τ =τ, v = v 𝐧, F∈[F][j]^B,cf. <cit.>. With this notation, the bilinear form [j]: V_j× V_j→ℝ corresponding to the symmetric interior penalty DG method on the j-th level is defined by[j][u][v] =∑_∈[T][j]∫_ (∇ u + ℛ_j(u ) ) · ( ∇ v + ℛ_j(v))dx +∑_F∈[F][j]∫_Fσ_ju·v ds,where σ_j∈ L^∞([F][j]) denotes the interior penalty stabilization function, which is defined by σ_j(x)= C_σ^j max_∈{^+,^-}{p_j^2/h_},x∈ F, F∈[F][j]^I, F ⊂∂^+∩∂^-,C_σ^j p^2/h_, x∈ F, F∈[F][j]^B, F ⊂∂^+∩∂Ω,with C_σ^j>0 independent of p, |F| and ||, and ℛ_j:[L^1([F][j])]^d → [V_j]^d is the lifting operator on the space V_j, defined as∫_ℛ_j(𝐪) ·η = - ∫_[F][j]𝐪·η ds ∀ η∈ [V_j]^d.We refer to <cit.> for more details.Here, the formulation with the lifting operators ℛ_j allows to introduce the discrete gradient operator 𝒢_j: V_j → [V_j]^d, defined as 𝒢_j(v) = ∇_j v + ℛ_j(v)∀ j=1,…,J,where ∇_j is the piecewise gradient operator on the space V_j. The role of 𝒢_j will be clarified in Sect. <ref>. The goal of this paper is to develop non-nested V-cycle multigrid schemes to solve the following problem posed on the finest level V_J: find u_J∈ V_J such that[J][u_J][v_J] = ∫_ fv_J dx∀ v_J∈ V_J.By fixing a basis for V_J, i.e. V_J = span{ϕ_J^k}_k, formulation (<ref>) results in the following linear system of equations 𝐀_J 𝐮_J = 𝐟_J, where 𝐮_J is the vector of unknowns. § PRELIMINARY RESULTS In this section we recall some preliminary results which form the basis of the convergence analysis presented in the next section.Assume that the sequence of meshes {[T][j]}_j=1^J, satisfies Assumption <ref> and let ∈[T][j], then the following bound holds[v][2][∂][2]≲ϵ/h_[v][2][][2] + h_/ϵ | v |_H^1()^2∀ v ∈ H^1(),where h_ is the diameter ofand ϵ>0 is a positive number.The proof of Lemma <ref> is given in Appendix <ref>.Assume that the sequence of meshes {[T][j] }_j=1^J satisfies Assumption <ref> and let ∈[T][j]. Then, the following bound holds[v][2][∂][2]≲p_j^2/h_[v][2][][2] ∀ v ∈[P][p_j](). We refer to <cit.> for the proof. On each discrete space {V_j}, j=1,…,J, we consider the following DG norm:[w][j][2]=∑_∈[T][j]∫_ |∇ w|^2 dx + ∑_F∈[F][j]∫_F σ_j|w|^2 ds.The well-posed of the DG formulation is established in the following lemma. The following continuity and coercivity bounds, respectively, hold[j][u][v] ≲[u][j][v][j]∀u,v∈V_j, [j][u][u] ≳[u][j][2]∀u∈V_j,Next, we recall the following approximation result, which is an analogous bound presented in <cit.>.Let Assumption <ref> be satisfied, and let v ∈ L^2(Ω) such that, for some k ≥ 0, v|_∈ H^k() for each ∈[T][j]. Then there exists a projection operator Π_j: L^2(Ω)→ V_j such that[v-Π̃_jv][q][Ω] ≲h_j^s-q/p_j^k-q[v][k][Ω],for0 ≤ q ≤ k,where s=min{p_j+1,k} and p_j ≥ 1. The result presented in Lemma <ref> leads to the following error bounds for the underlying interior penalty DG scheme. The error in the energy norm has been proved in <cit.>, see also <cit.>. L^2-estimates can be found in <cit.>.Assume that Assumptions <ref> and <ref> hold. We denote byu_j∈ V_j, j=1,…,J, the DG solution of problem (<ref>) posed on level j, i.e., [j][u_j][v_j] = ∫_ fv_j dx∀ v_j∈ V_j.If the solution u of (<ref>) satisfies u|_∈ H^k(), k ≥ 2, then [u-u_j][j] ≲h_j^(s-1)/p_j^(k-3/2)[u][k][],[u-u_j][2][] ≲j^s/p_j^k-1[u][k][],where s=min{p_j+1,k} and p_j ≥ 1. We point out that the bounds in Theorem <ref> are optimal in h and suboptimal in p of a factor p^1/2 and p for the DG-norm and the L^2-norm, respectively. Optimal error estimates with respect to p can be shown, for example, by using the projector of <cit.> for quadrilateral meshes providing the solution belongs to a suitable augmented Sobolev space. The issue of proving optimal estimates as the ones in <cit.> on polytopic meshes is an open problem and it is under investigation. In the following, we will write:[u-u_j][j] ≲h_j^(s-1)/p_j^(k-1-μ/2)[u][k][],[u-u_j][2][] ≲j^s/p_j^k-μ[u][k][],where s=min{p_j+1,k}, p_j≥ 1, and μ∈{ 0,1 } for optimal and suboptimal estimates, respectively. We also need to introduce an appropriate inverse inequality, cf. <cit.>. Assume that Assumptions <ref> and <ref> hold. Then, for any v∈ V_j, j=1,…,J, the following inverse estimate holds[∇ u][2][][2]≲p_j^4 h_^-2[u][2][][2] ∀∈[T][j]. Thanks to the inverse estimate of Lemma <ref>, it is possible to obtain the following upper bound on the maximum eigenvalue of . We refer to <cit.> for a similar result on standard grids, and to <cit.> for its extension to polygonal grids. Let Assumptions <ref>, <ref> and <ref> be satisfied. Moreover, we assume that h_j = max_∈[T][j] h_≈ h_ ∀∈[T][j], for j=1,…,J. Then [j][u][u]≲p_j^4/j^2[u][2][][2]∀ u∈ V_j, j=1,…,J.§ THE BPX-FRAMEWORK FOR THE V-CYCLE ALGORITHMS The analysis presented in this section is based on the general multigrid theoretical framework already employed and developed in <cit.> for non-nested spaces and non-inherited bilinear forms. In order to develop a geometric multigrid, the discretization at each level V_j follows the one already presented in <cit.>, where a W-cycle multigrid method based on nested subspaces is considered. The key ingredient in the construction of our proposed multigrid schemes is the inter-grid transfer operators.Firstly, we introduce the operators A_j:V_j→ V_j, defined as(A_ju,v)=[j][u][v]∀ u,v∈ V_j,j=1,…,J,and we denote as Λ_j∈ℝ the maximum eigenvalue of A_j ∀ j=2,…,J. Moreover, let Id_j be the identity operator on level V_j.The smoothing scheme, which is chosen to be the Richardson iteration, is then characterized by the following operators:B_j = Λ_jId_jj=2,…,J. The prolongation operator connecting the coarser space V_j-1 to the finer space V_j is denoted by I_j-1^j. Since the two spaces are non-nested, i.e. V_j-1⊄V_j, it cannot be chosen as the ”natural injection operator”. The most natural way to define the prolongation operator is the L^2-projection, i.e. I_j-1^j : V_j-1→ V_j(I_j-1^j v_H, w_h)_L^2() = ( v_H, w_h)_L^2() ∀ w_h ∈ V_j, The restriction operator I_j^j-1 : V_j→ V_j-1 is defined as the adjoint of I_j-1^j with respect to the L^2(Ω)-inner product, i.e.,(I_j^j-1 w_h,v_H)_L^2(Ω) = (w_h,I_j-1^j v_H)_L^2(Ω)∀ v_H ∈ V_j-1. For our analysis, we also need to introduce the operator P_j^j-1:V_j→ V_j-1 such that:[j-1][P_j^j-1 w_h][v_H] = [j][w_h][I_j-1^j v_H]∀ v_H ∈ V_j-1, w_h ∈ V_j. According with (<ref>), problem (<ref>) can be written in the following equivalent form: find u_J ∈ V_J such thatA_J u_J = f_J,where f_J ∈ V_J is defined as (f_J,v)_L^2() = ∫_ f v dx ∀ v∈ V_J. Given an initial guess u_0 ∈ V_J, and choosing parameters m_1,m_2 ∈ℕ, the multigrid V-cycle iteration algorithm for the approximation of u_J is outlined in Algorithm <ref>. In particular, 𝖬𝖦_𝒱 (J,f_J,u_k,m_1,m_2) represents the approximate solution obtained after one iteration of our non-nested V-cycle scheme, which is defined by induction: if we consider the general problem of finding z ∈ V_j such thatA_j z = g,with j∈{2,…,J} and g ∈ L^2(), then 𝖬𝖦_𝒱 (j,g,z_0,m_1,m_2) represents the approximate solution of (<ref>) obtained after one iteration of the non-nested V-cycle scheme with initial guess z_0 ∈ V_j and m_1, m_2 number of pre-smoothing and post-smoothing steps, respectively. The recursive procedure is outlined in Algorithm <ref>, where we also observe that on the level j=1 the problem is solved by using a direct method.§.§ Convergence analysisWe first define the following norms on each discrete space V_jv_s,j=√((A_j^sv,v)_L^2())∀ s∈ℝ, v∈ V_j, j=1, …,J.To analyze the convergence of the algorithm, for any j=2,…,J we set G_j = Id_j - B_j^-1A_j and let G_j^* be its adjoint respect to [j][][]. Following <cit.>, we make three standard assumptions in order to prove the convergence of Algorithm <ref>:0.3cm*Stability estimate: ∃ C_Q>0 such that(Id_j - I_j-1^j P_j^j-1)u_h _1,j≤ C_Q u_h_1,j∀ u_h ∈ V_j,j=2,…,J;*Regularity-approximation property: ∃ C_1>0 such that| [j][(Id_j - I_j-1^j P_j^j-1)u_h][u_h] | ≤ C_1u_h_2,j^2/Λ_j∀ u_h ∈ V_j,j=2,…,J,where Λ_j = maxλ_i(A_j) ; *Smoothing property: ∃ C_R>0 such that[u_h][2][]/Λ_j≤ C_R ( ℛ u_h, u_h )∀ u_h ∈ V_j,j=2,…,J,where ℛ = ( Id_j - G_j^* G_j ) A_j^-1. The convergence analysis of the V-cycle method is described by the following theorem that gives an estimate for the error propagation operator, which is defined as𝔼_1,m_1,m_2^𝖵v= 0,j=1, 𝔼_j,m_1,m_2^𝖵v = (G_J^*)^m_2(Id_j - I_j-1^j P_j^j-1 + I_j-1^j 𝔼_j-1,m_1,m_2^𝖵 P_j^j-1 )G_j^m_1 v, j >1. If Assumptions <ref>, <ref> and <ref> hold, then| [j][ 𝔼_j,m,m^𝖵u ][u] | ≤δ_j [j][u][u] ∀ u ∈ V_j,j=2,…,Jwhere δ_j = C_1 C_R/m - C_1 C_R<1, provided that m > 2C_1C_R.We refer to <cit.> for the proof of Theorem <ref> in an abstract setting. In the following, we prove the validity of Assumptions <ref>, <ref> and <ref> for the algorithm presented in this section. We start with a two-level approach, i.e. J=2, so we will consider the two-level method for the solution of (<ref>), based on two spaces V_J-1⊄V_J. The generalization to the V-cycle method will be given at the end of this section.§.§ Verification of Assumption <ref> In order to verify Assumption <ref> for the two-level method we first show a stability result of the prolongation operator I_J-1^J. In the following, we also consider the L^2-projection operator on the space V_J defined asQ_J : L^2() → V_J , such that(Q_Ju,v_J)_L^2(Ω) = (u,v_J)_L^2(Ω)∀ v_J ∈ V_J.From the definition of I_J-1^J given in (<ref>), it holds I_J-1^J = Q_J |_V_J-1. Moreover, we need the following approximation result which shows that any v_j ∈ V_j, j=J-1,J, can be approximated by an H^1-function, see <cit.>. Let 𝒢_j be the discrete gradient operator (<ref>) introduced in Remark <ref>, and consider the following problem: ∀ v_j ∈ V_j, find ℋ(v_j) ∈ H^1_0() such that∫_∇ℋ(v_j) ·∇ w dx = ∫_𝒢_j(v_j) ·∇ w dx ∀ w ∈ H^1_0(). It is shown in <cit.> that ℋ(v_j) possesses good approximation properties in terms of providing an H^1-conforming approximant of the discontinuous function v_j:Letbe a bounded convex polygonal/polyhedral domain in ℝ^d, d=2,3. Given v_j ∈ V_j, we write ℋ(v_j) ∈ H^1_0() to be the approximation defined in (<ref>). Then, the following approximation and stability results hold:[v_j - ℋ(v_j)][2][] ≲h_j/p_j[σ_j^1/2v_j][2][ℱ_h],| ℋ(v_j) |_H^1()≲[v_j][j]. We make use of the previous result in order to show the following stability result of the prolongation operator: There exists a positive constant 𝖢_𝗌𝗍𝖺𝖻, independent of the mesh size such that[I_J-1^Jv_H][J]≤𝖢_𝗌𝗍𝖺𝖻(p_J) [v_H][J-1] ∀ v_H ∈ V_J-1,here 𝖢_𝗌𝗍𝖺𝖻(p_J) ≈ p_J.Let v_H ∈ V_J-1, by the definition of the DG-norm (<ref>), we need to estimate:[I_J-1^J v_H][J]^2 = [∇_J (I_J-1^Jv_H)][2][𝒯_J]^2 + [ σ_J^1/2 |I_J-1^J v_H|][2][ℱ_J]^2 .We next bound each of the two terms on the right hand side. For the first one let be ℋ_H = ℋ(v_H) defined as in (<ref>). Then:∇_J(I_J-1^J v_H) _L^2(𝒯_J)^2≤[∇_J(I_J-1^J v_H - Π_J(ℋ_H) )][2][𝒯_J]^2 +[∇_J(ℋ_H - Π_J(ℋ_H) )][2][𝒯_J]^2 + |ℋ_H|_H^1()^2,where we have added and subtracted the terms ∇_J Π_J(ℋ_H) ) and ∇ℋ_H. The second term of the right hand above side can be estimated using the interpolation bounds of Lemma <ref>, the Poincaré inequality for ℋ_H ∈ H^1_0(Ω) and the second bound of (<ref>):[∇_J(ℋ_H - Π_J(ℋ_H) )][2][𝒯_J]^2≲ | ℋ_H |_H^1(Ω)^2 ≲[v_H][J-1]^2.In order to estimate the first term on the right hand side in (<ref>) we observe that, since I_J-1^J v_H - Π_J(ℋ_H) ∈ V_J, it is possible to make use of the inverse inequality of Lemma <ref>, that leads to the following bound:∇_J(I_J-1^J v_H -Π_J(ℋ_H) )_L^2(𝒯_J)^2 ≲ p_J^4 h_J^-2[I_J-1^J v_H -Π_J(ℋ_H)][2][𝒯_J][2].By adding and subtracting ℋ_H to [I_J-1^J v_H -Π_J(ℋ_H)][2][𝒯_J][2] we obtain[I_J-1^J v_H -Π_J(ℋ_H)][2][𝒯_J][2] ≲[I_J-1^J v_H -ℋ_H][2][𝒯_J][2]+ [ℋ_H -Π_J(ℋ_H)][2][𝒯_J][2].Using Lemma <ref> and the Poincaré inequality we have[ℋ_H -Π_J(ℋ_H)][2][𝒯_J][2] ≲h_J^2/p_J^2ℋ_H _H^1(Ω)^2≲h_J^2/p_J^2[v_H][J-1]^2,whereas the term [I_J-1^J v_H -ℋ_H][2][𝒯_J][2] can be estimate as follow:I_J-1^J v_H - ℋ_H _L^2(𝒯_J)^2≲[I_J-1^J v_H -Q_J(ℋ_H)][2][𝒯_J][2] +[ℋ_H -Q_J(ℋ_H)][2][𝒯_J][2]Using Remark <ref>, the continuity of Q_J with respect to the L^2-norm, Lemma <ref> and (<ref>) we haveI_J-1^J v_H -ℋ_H _L^2(𝒯_J)^2≲[Q_J(v_H - ℋ_H)][2][𝒯_J][2]+ [ℋ_H -Q_J(ℋ_H)][2][𝒯_J][2]≲[v_H - ℋ_H][2][𝒯_J][2] + [ℋ_H - Π_J(ℋ_H) ][2][𝒯_J][2] ≲h_J^2/p_J^2[σ_J^1/2 |v_H][2][ℱ_J]^2 +h_J^2/p_J^2[ℋ_H][1][]^2≲h_J^2/p_J^2[v_H][J-1]^2.Thanks to the previous estimates and inequalities (<ref>), it holdsI_J-1^J v_H - Π_J(ℋ_H)_L^2(𝒯_J)^2 ≲h_J^2/p_J^2[v_H][J-1]^2, the previous estimate, together with (<ref>), (<ref>) and the bound |ℋ_H|_^2 ≲[v_H][J-1]^2 leads to ∇_J (I_J-1^J v_H) _L^2(𝒯_J)^2 ≲ p_J^2 [v_H][J-1]^2. Next we bound the second term on the right hand side in (<ref>). By the definition of the jump term and remembering that ℋ_H=0 ∀ F ∈[F][J] since ℋ_H ∈ H_0^1(Ω), it holdsσ_J^1/2 I_J-1^J v_H_L^2(ℱ_J)^2 ≲∑_∈𝒯_Jp_J^2/h_( [ I_J-1^J v_H - Π_J( ℋ_H)][2][∂]^2 + [ Π_J( ℋ_H) - ℋ_H ][2][∂]^2),where we also used the definition of σ_J. Now, we first observe that we could use the trace inequality of Lemma <ref> in order to obtain[ I_J-1^J v_H - Π_J( ℋ_H)][2][∂]^2 ≲p_J^2/h_J[ I_J-1^J v_H - Π_J( ℋ_H)][2][]^2.To bound the second term on the right hand side in (<ref>) we make use of the continuous trace inequality on polygons of Lemma <ref> with ϵ = p_J, the approximation property of Lemma <ref> and the Poincaré inequality:Π_J(ℋ_H) - ℋ_H _L^2(∂)^2≲p_J/h_J[Π_J( ℋ_H) - ℋ_H][2][][2] + h_J/p_J | Π_J( ℋ_H) - ℋ_H |_H^1()^2≲p_J/h_Jh_J^2/p_J^2ℋ_H _L^2()^2 + h_J/p_Jℋ_H _H^1()^2≲h_J/p_J | ℋ_H |_H^1()^2.From the previous inequality and the bound (<ref>), (<ref>) becomes: σ_J^1/2I_J-1^J v_H_L^2(ℱ_J)^2 ≲p_J^4/h_J^2[ I_J-1^J v_H - Π_J( ℋ_H)][2][𝒯_J]^2 +p_J | ℋ_H |_H^1(Ω)^2 ≲ p_J^2[v_H][J-1]^2,where we also used inequality (<ref>). This estimate together with (<ref>) lead to[I_J-1^Jv_H][J] ≤𝖢_𝗌𝗍𝖺𝖻 (p_J) [v_H][J-1] ∀ v_H ∈ V_J-1.where 𝖢_𝗌𝗍𝖺𝖻(p_J) ≈ p_J. We can use the previous result in order to prove that Assumption <ref> holds. We first observe that also the operator P_J^J-1 satisfies a similar stability estimate as the one of I_J-1^J, that isP_J^J-1v_h _DG,J-1^2≲[J-1][P_J^J-1 v_h][P_J^J-1 v_h] = [J][v_h][I_J-1^J P_J^J-1 v_h] ≲[v_h][J] [I_J-1^J P_J^J-1 v_h][J] ≲𝖢_𝗌𝗍𝖺𝖻 (p_J) [v_h][J] [P_J^J-1 v_h][J],from which it follows [P_J^J-1 v_h][J-1] ≲𝖢_𝗌𝗍𝖺𝖻 (p_J) [v_h][J].Assumption <ref> holds with C_Q ≈ p_J^2.Let v_H ∈ V_J-1, making use of Lemma <ref> we have [J][I_J-1^J v_H][I_J-1^J v_H]≲[I_J-1^J v_H][J]^2 ≲ p_J^2 [v_H][J-1]^2 ≲ p_J^2 [J-1][v_H][v_H],and similarly it holds [J-1][P_J^J-1 u_h][P_J^J-1 u_h] ≲ p_J^2 [J][u_h][u_h]∀ u_h ∈ V_J. Let u_h ∈ V_J and fix v_H = P_J^J-1 u_h, then the following inequality holds:[J][I_J-1^J P_J^J-1 u_h][I_J-1^J P_J^J-1 u_h] ≲ p_J^2 [J-1][P_J^J-1 u_h][P_J^J-1 u_h].By adding and subtracting u_h to both arguments of [J] on the left hand side of (<ref>), and using (<ref>) we obtain 𝒜_J((Id_J - I_J-1^J P_J^J-1) u_h, (Id_J - I_J-1^J P_J^J-1) u_h) _=(Id_J - I_J-1^J P_J^J-1)u_h_1,J^2≲( p_J^2 ( p_J^2 - 2 ) + 1 )_≤ p_J^4[J][u_h][u_h],that concludes the proof. §.§ Verification of Assumption <ref> In order to show the validity of Assumption <ref> we need the following standard approximation result, which is proved in Appendix <ref>. Let Assumptions <ref> - <ref> hold. Then[(Id_J - I_J-1^J P_J^J-1)v_J][2][] ≲h_J^2/p_J^2-μ v_J _2,J∀ v_J ∈ V_J. Thanks to Lemma <ref>, it is possible to show the following theorem:The regularity-approximation property <ref> holds with C_1 ≈ p_J^2+μ. Theorem <ref> gives the following bound of the maximum eigenvalue of A_J: Λ_J ≲p_J^4/J^2. Using Lemma <ref>, the above bound on Λ_J, and the symmetry of [J][][] we have, for all v ∈ V_J:[J][(Id_J - I_J-1^J P_J^J-1)v][v] ≤ v _2,J (Id_J - I_J-1^J P_J^J-1)v _0,J≲h_J^2/p_J^2-μ v _2,J^2≲ p_J^2+μ v _2,J^2/Λ_J. that concludes the proof. §.§ Verification of Assumption <ref> Assumption <ref> holds with C_R=1. We have:ℛ = ( Id_J - G_J^* G_J ) A_J^-1 = ( 2/Λ_J A_J - 1/Λ_J^2 A_J A_J ) A_J^-1 = 1/Λ_J( Id_J + ( Id_J - 1/Λ_J A_J ) ),and so( ℛ u,u )_L^2(Ω) = [u_h][2][]/Λ_J + ( ( Id_J - 1/Λ_J A_J )u, u )_L^2(Ω).We now prove that ( Id_J - 1/Λ_J A_J ) is a positive definite operator. By contradiction, let us suppose that there exists a function u∈ V_J, u 0, such that ( ( Id_J - 1/Λ_J A_J )u, u)_L^2(Ω) < 0, then Λ_J (u, u)_L^2(Ω) < [J][u][u],by Lemma <ref> and the symmetry of the bilinear form [J][][], the eigenfunctions {ϕ^J_k}_k=1^N_J satisfy [J][ϕ^J_k][v] = λ^J_k (ϕ^J_k, v)_L^2(Ω)∀ v ∈ V_J, where 0 < λ^J_1≤λ^J_2≤…≤λ^J_N_J = Λ_J. The set of eigenfunctions is an orthonormal basis for the space V_J, i.e. (ϕ^J_i,ϕ^J_j)_L^2(Ω) = δ_ij, and they satisfy [J][ϕ^J_i][ϕ^J_j] = λ^J_iδ_ij, where δ_ij is the Kronecker symbol. Since {ϕ^J_k}_k=1^N_J is a basis of the space V_J, we can write u = ∑_k=1^N_J c_k ϕ^J_k, so that (<ref>) becomesΛ_J ∑_i,j=1^N_Jc_j(ϕ^J_j, ϕ^J_i)_L^2(Ω) c_i < ∑_i,j=1^N_J c_j [J][ϕ^J_j][ϕ^J_i] c_i = ∑_i,j=1^N_J c_j λ^J_i (ϕ^J_i,ϕ^J_j)_L^2(Ω) c_i,⇒Λ_J ∑_i=1^N_Jc_i^2 < ∑_i,j=1^N_J c_i^2 λ^J_i,which is a contradiction. We then deduce that ( Id_J - 1/Λ_J A_J ) is a positive definite operator.We observe that, as we need to satisfy the condition m > 2C_1C_R of Theorem <ref>, we can guarantee the convergence of the method choosing the number m of smoothing steps such that m ≳ p_J^2+μ, which is in agreement to what proved for W-cycle algorithms in <cit.> and <cit.> on nested grids.The analysis of this section can be generalized to the full V-cycle algorithm with J>2 as follows: Assumption <ref> is verified with C_R=1 also on the arbitrary levels j,j-1, because each level j satisfies Assumption <ref> with constant C_R^j=1. Assumptions <ref> and <ref> are satisfied with C_1 = max_j{ C_1^j } and C_Q = max_j{ C_Q^j }, respectively, where C_1^j and C_Q^j are the same as the ones defined in the previous analysis but on the level j.§ NUMERICAL RESULTSIn this section we present several numerical results to test the theoretical convergence estimates provided in Theorem <ref>. We focus on a two dimensional problem on the unit square Ω = (0,1)^2. For the simulations, we consider the sets of polygonal grids shown in Figure <ref>. Each polygonal element mesh is generated through the Voronoi Diagram algorithm by using the software package<cit.>. In particular the finest grids (Level 4) of Figure <ref> consist of 512 (Set 1), 1024 (Set 2), 2048 (Set 3) and 4096 (Set 4) elements. Starting from the number of elements of each initial mesh, a sequence ofcoarse, non-nested partitions is generated: each coarse mesh is built independently from the finer one, with the only constrain that the number of element is approximately 1/4 of the finer one. An example of sequence of non-nested partitions is shown in Figure <ref>.First of all, we verify the estimate of Lemma <ref>, numerically evaluating 𝖢_𝗌𝗍𝖺𝖻(p) ≈ p, where p is the polynomial approximation degree. To this aim we consider three pairs of non-nested grids, where the number of elements of the coarser grid is the number of the finer divided by 4: for each pair, we compute the value of 𝖢_𝗌𝗍𝖺𝖻(p) as a function of p. Figure <ref> show that, as expected, 𝖢_𝗌𝗍𝖺𝖻(p) depends linearly on p and is independent of the mesh-size h. We now consider the grids shown in Set 1 and in Set 2 of Figure <ref>, and numerically evaluate the constant δ_j in Theorem <ref> based on selecting the Richardson smoother with m_1=m_2=m=3p^2, cf. Figure <ref>. Here, we observe thatδ_2 and δ_3 are asymptotically constant, as the polynomial degree p increases showing that our two-level and V-cycle algorithms are uniformly convergent also with respect to p provided that m ≈ p^2.Next, we investigate the performance of the iterative Multigrid non-nested V-cycle algorithm presented in Sect. <ref>. In order to do that, we solve the Poisson problem with homogeneous Dirichlet boundary conditions on the unit square = (0,1)^2, and we compute the number of iterations needed by our V-cycle algorithm to reduce the relative residual error below a given tolerance of 10^-8, by varying the polynomial degree of approximation and the granularity of the finest grid. In Table <ref> we report the convergence factors ρ_J = exp(1/N_it,Jln𝐫_N_it,J/𝐫_0),where N_it,J is the iteration counts needed to attain convergence of the h-version of the V-cycle scheme with J levels, where J=2,3,4, while 𝐫_N_it,J and 𝐫_0 are the final and initial residual vectors, respectively. Here the polynomial approximation degree on each level is chosen as p_j =1, j=1,…,J, while we vary the number of elements of the finest grid and the number of smoothing steps (m_1=m_2=m). According to Theorem <ref>, the convergence factor is independent from the spatial discretization step h, indeed, for a fixed J∈{2,3,4} ad a fixed number of pre-smoothing steps m, the convergence factor is roughly constant between the 4 sets of grids. In particular, this means that the number of iterations needed from the proposed V-cycle method to attain the convergence is not influenced by the mesh refinement, contrarily of what we observe for the Conjugate Gradient (CG) method. As expected, the convergence factor is reduced by increasing the number of smoothing step.We have repeated the same set of experiments employing p_j=3, ∀ j=1,…,J; the results are reported in Table <ref>, where we also have reported the iterations count (between parenthesis). Firstly, a comparison between Table <ref> and Table <ref> confirms that the convergence factor increases as p grows up if the number of smoothing steps is kept fixed. Secondly, we observe that if the number of smoothing step is kept too small then the convergence of the method could not be guaranteed: indeed, according to Theorem <ref>, a uniformly convergent (also with respect to p) solver require a number of smoothing steps m > 2C_1 C_Q ≳ p^2+μ as shown in Figure <ref>. If m is big enough, we observe that also in this case the number of iterations does not depend from the granularity of the underlying mesh, while the iterations count of the Conjugate Gradient method is growing if h decrease. § ADDITIVE SCHWARZ SMOOTHERIn order to improve the convergence properties of the V-cycle algorithm studied above, we define in this section a domain decomposition preconditioner that we will use as a smoothing operator instead of the Richardson iteration. To this end, let [T][j] and [T][j-1] berespectively the finer and the coarser non-nested meshes, satisfying the grid assumptions given in Sect. <ref>. We then introduce the local and coarse solvers, that are the key ingredients in the definition of the smoother on the space V_j, j=2,…,J. Local Solvers. Let us consider the finest mesh [T][j] with cardinality N_j, then for each element _i ∈[T][j], we define a local space V_j^i as the restriction of the DG finite element space V_j to the element _i ∈[T][j]:V_j^i = V_j|__i≡[P][p_j](_i) ∀ i = 1,...,N_j,and for each local space, the associated local bilinear form is defined by𝒜_j^i: V_j^i × V_j^i →ℝ, 𝒜_j^i(u_i,v_i) = [j][R_i^T u_i][R_i^T v_i] ∀ u_i,v_i ∈ V^i,where R_i^T:V_j^i → V_j denotes the classical extension by-zero operator from the local space V_j^i to the global V_j. Coarse Solver. The natural choice in our contest is to define the coarse space V^0_j to be exactly the same used for the Coarse grid correction step of the V-cycle algorithm introduced in Sect. <ref>, that is V^0_j = V_j-1≡{v∈ L^2():v|_∈[P][p_j-1](),∈[T][j-1]},the bilinear form on V^0_j is then given by 𝒜_j^0: V_j^0 × V_j^0 →ℝ, 𝒜_j^0(u_0,v_0) = [j-1][u_0][v_0] ∀ u_0,v_0 ∈ V_j^0.We also define the injection operator from V_j^0 to V_j: conversely with respect to the case where the coarser mesh is obtained by agglomeration, here the injection operator is not trivial, and it is defined as the prolongation operator introduced in Sect. <ref>, that is R_0^T:V_j^0 → V_j, R_0^T = I_j-1^j. By introducing the projection operators P_i = R_i^T P̃_̃ĩ: V_j → V_j, i=0,1,…,N_j, where P̃_̃ĩ:V_j → V^i_j, 𝒜_j^i(P̃_̃ĩv_h, w_i) = [j][v_h][R_i^T w_i] ∀ w_i ∈ V^i_j,i=1,…,N_j,P̃_̃0̃:V_j → V^0_j, 𝒜_j^0(P̃_̃0̃v_h, w_0) = [j][v_h][R_0^T w_0] ∀ w_0 ∈ V^0_j,the additive Schwarz operator is defined by P_ad = ∑_i=0^N_j (R_i^T (A^i_j)^-1 R_i) A_j ≡ B^-1_adA_j, where B^-1_ad = ∑_i=0^N_j (R_i^T (A^i_j)^-1 R_i) is the preconditioner. Then, the Additive Schwarz smoothing operator with m steps consists in performing m iterations of the Preconditioned Conjugate Gradient method using B_ad as preconditioner. In Algorithm <ref> we outline the V-cycle multigrid method using P_ad as a smoother. Here, 𝖬𝖦_𝒜𝒮 (j, g, z_0,m_1,m_2) denotes the approximate solution of A_jz=g obtained after one iteration, with initial guess z_0 and m_1, m_2 pre- and post-smoothing steps, respectively. Here, the smoothing step is performed by the algorithm ASPCG, i.e., z = ASPCG(A, z_0, g, m) represents the output of m steps of Preconditioned Conjugate Gradient method applied to the linear system of equations Ax = g, by using B_as as preconditioner and starting with the initial guess z_0.The numerical performance of Algorithm <ref> are reported in Tables <ref>, <ref> and <ref>, for the corresponding V-cycle algorithm with J=2,3,4 levels. The simulations are similar to the ones described in the previous section: here we used the grids of Set 2, 3 and 4 of Figure <ref>, and we varied the polynomial degree p ∈{1,3,5}. Firstly, we observe that, also in this case, the number of iteration does not increase with the number of elements in the underlying mesh for a fixed number of smoothing steps m; moreover, we does not observe the constrain from below required to the number of smoothing steps with respect to the degree of approximation: the method converges also with high degree of approximation and a small number of smoothing steps. Finally, Table <ref> shows the numerical results relatives to an example of hp-multigrid, characterized by a choice of different polynomial degrees of approximation between non-nested space: also in this case we observe that the number of iterations is independent of the granularity of the finest mesh, and we have convergence for any choice of smoothing steps m. § PROOF OF LEMMA <REF> We follow the idea of <cit.>. First of all, we observe that [v][2][∂][2] = ∑_F ⊂∂[v][2][F][2]. For each face F ⊂∂ let T_F ⊂ be a d-dimensional simplex sharing the face F withand satisfying the Assumption <ref>: in T_F we define a function σ_F as follow: σ_F: x∈T_F↦σ_F(x) = | F |/d |T_F| (x - v_F), where v_F is the vertex of the simplex T_F opposite to the face F. We observe that: * σ_F(x) ·n_F = | F |/d |T_F|h̃ ∀x∈ F, where h̃ is the height of the simplex respect to the face F, that is also h̃ = d |T_F|/| F|, then σ_F|_F ·n_F = 1;* σ_F|_F'·n_F' = 0 ∀ facesF' ⊂∂ T_F, F'F; then we have:[v][2][F][2]= ∫_F | v |^2 dσ = ∫_∂ T_F | v |^2 σ_F ·n_F dσ = ∫_T∇· (|v|^2 σ_F) dx= ∫_T 2v∇ v ·σ_F dx + ∫_T |v|^2 ∇·σ_F dx;now the following properties hold for σ_F: * ∇·σ_F = ∇·| F |/d |T_F| (x - v_F) = | F |/d |T_F|∇·x = | F |/|T_F|;* σ_F _[L^∞(T_F)]^d = | F |/d |T_F| h_T≤| F |/d |T_F| h_, which implies:[v][2][F][2]≤ 2 σ_F _[L^∞(T_F)]^d v ∇ v _[L^1(T_F)]^d + | F |/|T_F| v _L^2(T_F)^2, ≤ 2 | F |/d |T_F| h_ v _L^2(T_F) | v |_H^1(T_F) + | F |/|T_F| v _L^2(T_F)^2,using the Assumption <ref> we have[v][2][F][2] ≤ 2Cv _L^2(T_F) | v |_H^1(T_F) + Cd/h_ v _L^2(T_F)^2.By using Young Inequality we could bound v _L^2(T_F) | v |_H^1(T_F)≤1/2( ϵ/h_ v _L^2(T_F)^2 + h_/ϵ | v |_H^1(T_F)^2 ),where we have chosen ϵ≥ 1. Using the previous inequality we have[v][2][F][2] ≤ 2Cd( ϵ/h_ v _L^2(T_F)^2 + h_/ϵ | v |_H^1(T_F)^2 ).we observe that (<ref>) holds ∀ F ⊂∂. Then, thanks to (<ref>), we have[v][2][∂][2] = ∑_F ⊂∂[v][2][F][2] ≤∑_F ⊂∂ 2Cd( ϵ/h_ v _L^2(T_F)^2 + h_/ϵ | v |_H^1(T_F)^2)= 2Cd ( ϵ/h_∑_F ⊂∂ v _L^2(T_F)^2 + h_/ϵ∑_F ⊂∂| v |_H^1(T_F)^2 )≤ 2Cd ( ϵ/h_ v _L^2()^2 + h_/ϵ | v |_H^1()^2 ), where in the last inequality we have used the fact that the simplices of the set {T_F: F ⊂∂} satisfy Assumption <ref>, in the sense that they are disjoints and∪_F ⊂∂T_F⊂. § PROOF OF LEMMA <REF>In order to show Lemma <ref> we follow the analysis presented in <cit.>, by firstly showing two preliminary results making use of the properties presented in Sect. <ref>. Let Assumptions <ref> - <ref> hold, and let Π_j be the projection operator on V_j as defined in Lemma <ref>, for j=J,J-1. Then[Π_J w - I_J-1^J Π_J-1 w][2][] ≲h_J^2/p_J^2[w][2][] ∀ w ∈ H^2(). Using the triangular inequality, Remark <ref> and the approximation estimates of Lemma <ref> we have:Π_Jw -I_J-1^JΠ_J-1 w_ L^2()≤≤Π_J w -w_L^2()+w - Q_Jw_L^2() +Q_J w - I_J-1^J Π_J-1 w_L^2() = Π_J w -w_L^2()+ min_z_h ∈ V_Jw - z_h_L^2() + Q_J (w -Π_J-1 w)_L^2()≤[Π_J w -w][2][]+ [ w - Π_J w][2][] + [w -Π_J-1 w][2][] ≲h_J^2/p_J^2[w][2][] +h_J-1^2/p_J-1^2[w][2][] ≲h_J^2/p_J^2[w][2][],where in the last inequality we also used hypothesis (<ref>) and (<ref>).Let Assumptions <ref> - <ref> hold. Let be g ∈ L^2() and denote by w_j ∈ V_j the solution of [j][w_j][v] = (g,v)_L^2(Ω) ∀ v ∈ V_j with j=J-1,J. Then the following inequality holds:[w_J - I_J-1^Jw_J-1][2][] + [w_J-1 - P_J^J-1w_J][2][] ≲h_J^2/p_J^2-μ[g][2][]. Consider the unique solution w ∈ V of the problem 𝒜(w,v) = (g,v)_L^2(Ω)∀ v ∈ V.Using Theorem <ref> we have [w - w_j][2][] ≲h_j^2/p_j^2-μ[w][2][],j = J-1,J.Using the triangular inequality and Remark <ref> we have:w_J - I_J-1^Jw_J-1 _L^2()≤[w_J - w][2][] + [w - Π_J w][2][] + [Π_Jw - I_J-1^J Π_J-1 w][2][] + [I_J-1^J Π_J-1w - Q_J w][2][]+ [Q_J w - I_J-1^Jw_J-1 ][2][]= [w_J - w][2][] + [w - Π_J w][2][] + [Π_Jw - I_J-1^J Π_J-1 w][2][]+ [Q_J (Π_J-1w - w)][2][] + [Q_J(w - w_J-1) ][2][]≤[w_J - w][2][] + [w - Π_J w][2][] + [Π_Jw - I_J-1^J Π_J-1 w][2][]+ [ Π_J-1w - w][2][] + [ w - w_J-1 ][2][].Using (<ref>), Lemma <ref> and Lemma <ref>, we have [w_J - I_J-1^Jw_J-1][2][]≲h_J^2/p_J^2-μ[w][2][] +h_J^2/p_J^2[w][2][] +h_J^2/p_J^2[w][2][]+h_J-1^2/p_J-1^2[w][2][] +h_J-1^2/p_J-1^2-μ[w][2][].From the elliptic regularity assumption (<ref>) and hypothesis (<ref>) and (<ref>), we can write[w_J - I_J-1^Jw_J-1][2][]≲h_J^2/p_J^2-μ[g][2][].Now, let z_j ∈ V_j be the solution of:[j][z_j][q] = (w_J-1 - P_J^J-1 w_J, q_j)_L^2(Ω)∀ q_j ∈ V_j,j = J-1,J;Using (<ref>) we get the following estimate:[z_J-1 - I_J-1^J z_J-1][2][] ≲h_J^2/p_J^2-μ[w_J-1 - P_J^J-1w_J][2][].Then, we have:[w_J-1 - P_J^J-1 w_J][2][]^2= [J-1][z_J-1][w_J-1 - P_J^J-1 w_J] = [J-1][z_J-1][w_J-1] - [J][I_J-1^J z_J-1][w_J]= (z_J-1,g) - (I_J-1^J z_J-1,g) = (g,z_J-1 - I_J-1^J z_J-1) ≲[g][2][] h_J^2/p_J^2-μ[w_J-1 - P_J^J-1 w_J][2][],from which, together with (<ref>), inequality (<ref>) follows.For any v_J ∈ V_J we consider the following equality:[(Id_J - I_J-1^J P_J^J-1)v_J][2][] = sup_0 ϕ∈ L^2()( ϕ, (Id_J - I_J-1^J P_J^J-1)v_J )_L^2()/[ϕ][2].Next, consider the solution z_j of the following problems[j][z_j][v_j] = ( ϕ, v_j ) ∀ v_j ∈ V_j,for j=J,J-1.By using the definition of P_J^J-1 and Lemma <ref>, we have:(ϕ, (Id_J - I_J-1^J P_J^J-1) v_J))_L^2() = [J][z_J][v_J] - [J-1][P_J^J-1z_J][P_J^J-1v_J]= [J][z_J - I_J-1^J z_J-1][v_J] + [J][I_J-1^J (z_J-1 - P_J^J-1z_J)][v_J]≤ v_J _2,J( [z_J - I_J-1^J z_J-1][2] + [z_J-1 - P_J^J-1 z_J][2] ) ≲ v_J _2,Jh_J^2/p_J^2-μ[ϕ][2].Using the last inequality together with (<ref>) we get (<ref>). 10 urlstyleAnBrMa2008 Antonietti, P.F., Brezzi, F., Marini, L.D.: Stabilizations of the Baumann-Oden DG formulation: the 3D case. 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"authors": [
"P. F. Antonietti",
"G. Pennesi"
],
"categories": [
"cs.NA",
"65F10, 65M55, 65N22"
],
"primary_category": "cs.NA",
"published": "20170926173243",
"title": "V-cycle multigrid algorithms for discontinuous Galerkin methods on non-nested polytopic meshes"
} |
theoremTheorem[section] prop[theorem]Proposition lemma[theorem]Lemma cor[theorem]Corollary definition[theorem]Definition conj[theorem]Conjecture rmk[theorem]Remark claim[theorem]Claim defth[theorem]Definition-TheoremIndian Statistical Institute, Kolkata, India. Email: [email protected] a Moebius homeomorphism f : ∂ X →∂ Y between boundaries of proper, geodesically complete CAT(-1) spaces X,Y, we describe an extension f̂ : X → Y of f, called the circumcenter map of f, which is constructed using circumcenters of expanding sets. The extension f̂ is shown to coincide with the (1, log 2)-quasi-isometric extension constructed in <cit.>, and is locally 1/2-Holder continuous. When X,Y are complete, simply connected manifolds with sectional curvatures K satisfying -b^2 ≤ K ≤ -1 for some b ≥ 1 then the extension f̂ : X → Y is a (1, (1 - 1/b)log 2)-quasi-isometry. Circumcenter extension of Moebius maps is natural with respect to composition with isometries. Circumcenter extension of Moebius maps to CAT(-1) spaces Kingshook Biswas========================================================§ INTRODUCTION Let X be a CAT(-1) space. There is a positive function called the cross-ratio on the space of quadruples of distinct points of the boundary at infinity ∂ X of X, defined for ξ, ξ', η, η' ∈∂ X by[ξ, ξ', η, η'] = lim_a→ξ, b→ξ', c→η,d→η'exp( 1/2( d(a,c)+d(b,d)-d(a,d)-d(b,c)))(where a,b,c,d ∈ X converge radially towards ξ,ξ',η,η'). A map between boundaries of CAT(-1) spaces is called Moebius if it preserves cross-ratios. Any isometry between CAT(-1) spaces extends to a Moebius homeomorphism between their boundaries. A classical fact which turns out to be crucial in many rigidity results including the Mostow Rigidity theorem is that a Moebius map from the boundary of real hyperbolic space to itself extends to an isometry. More generally Bourdon showed (<cit.>) that if X is a rank one symmetric space of noncompact type with maximum of sectional curvatures equal to -1 and Y a CAT(-1) space then any Moebius embedding f : ∂ X →∂ Y extends to an isometric embedding F : X → Y. In <cit.> the problem of extending Moebius maps was considered for general CAT(-1) spaces, where it was shown that any Moebius homeomorphism f : ∂ X →∂ Y between boundaries of proper, geodesically complete CAT(-1) spaces X, Y extends to a (1, log 2)-quasi-isometry F : X → Y. The proof of this theorem uses an isometric embedding of a proper, geodesically complete CAT(-1) space into a certain space of Moebius metrics on the boundary of the space. A nearest point projection to the subspace of visual metrics is used to construct the extension. We show that this nearest point is unique, and can be constructed as a limit of circumcenters of certain expanding sets. The extension constructed in <cit.> is thus uniquely determined. We call the extension the circumcenter map of f. It is readily seen to satisfy naturality properties with respect to composition with isometries. We have:Let X, Y be proper, geodesically complete CAT(-1) spaces, and f : ∂ X →∂ Y a Moebius homeomorphism. Then the circumcenter extension f̂ : X → Y of f is a (1, log 2)-quasi-isometry which is locally 1/2-Holder continuous: d(f̂(x), f̂(y)) ≤ 2 d(x,y)^1/2for all x,y ∈ X such that d(x,y) ≤ 1. When the spaces X, Y are also assumed to be manifolds with curvature bounded belowwe have the following improvement on the main result of <cit.>:Let X, Y be complete, simply connected Riemannian manifolds with sectional curvatures satisfying -b^2 ≤ K ≤ -1 for some constant b ≥ 1. For any Moebius homeomorphism f : ∂ X →∂ Y, the circumcenter extension f̂ of f is a (1, (1 - 1/b)log 2)-quasi-isometry f̂ : X → Y with image 1/2(1 - 1/b)log 2-dense in Y.We mention that one of the motivations for considering the problem of extending Moebius maps is the marked length spectrum rigidity problem. This asks whether an isomorphism ϕ : π_1(X) →π_1(Y) between fundamental groups of closed negatively curved manifolds which preserves lengths of closed geodesics (recall that in negative curvature each homotopy class of closed curves contains a unique closed geodesic) is necessarily induced by an isometry F : X → Y. Otal (<cit.>) proved that this is indeed the case in dimension two. The problem remains open in higher dimensions. It is known however to be equivalent to the geodesic conjugacy problem, which asks whether the existence of a homeomorphism between the unit tangent bundles ϕ : T^1 X → T^1 Y conjugating the geodesic flows implies isometry of the manifolds. Hamenstadt (<cit.>) proved that equality of marked length spectra is equivalent to existence of a geodesic conjugacy.Bourdon showed in <cit.>, that for a Gromov-hyperbolic group Γ with two quasi-convex actions on CAT(-1) spaces X, Y, the natural Γ-equivariant homeomorphism f between the limit sets Λ X, Λ Y is Moebius if and only if there is a Γ-equivariant conjugacy of the abstract geodesic flows 𝒢Λ X and 𝒢Λ Y compatible with f. In particular for X̃, Ỹ the universal covers of two closed negatively curved manifolds X, Y (with sectional curvatures bounded above by -1), the geodesic flows of X, Y are topologically conjugate if and only if the induced equivariant boundary map f : ∂ X →∂ Y is Moebius. Thus an affirmative answer to the problem of extending Moebius maps to isometries would also yield a solution to the equivalent problems of marked length spectrum rigidity and geodesic conjugacy.Finally we remark that in <cit.> it is proved that in certain cases Moebius maps between boundaries of simply connected negatively curved manifolds do extend to isometries (more precisely, local and infinitesimal rigidity results are proved for deformations of the metric on a compact set).§ SPACES OF MOEBIUS METRICS We recall in this section the definitions and facts from <cit.> which we will be needing.Let (Z,ρ_0) be a compact metric space with at least four points. For a metric ρ on Z we define the metric cross-ratio with respect to ρ of a quadruple of distinct points (ξ, ξ', η, η') of Z by[ξξ' ηη']_ρ := ρ(ξ, η) ρ(ξ', η')/ρ(ξ, η')ρ(ξ', η)We say that a diameter one metric ρ on Z is antipodal if for any ξ∈ Z there exists η∈ Z such that ρ(ξ, η) = 1. We assume that ρ_0 is diameter one and antipodal. We say two metrics ρ_1, ρ_2 on Z are Moebius equivalent if their metric cross-ratios agree:[ξξ' ηη']_ρ_1 = [ξξ' ηη']_ρ_2for all (ξ, ξ', η, η'). The space of Moebius metrics on Z is defined to beℳ(Z, ρ_0) := {ρ : ρ is an antipodal, diameter one metric onZMoebius equivalent to ρ_0 }We will write ℳ(Z, ρ_0) = ℳ. We have the following from <cit.>:For any ρ_1, ρ_2 ∈ℳ, there is a positive continuous function dρ_2/dρ_1 on Z, called the derivative of ρ_2 with respect to ρ_1, such that the following holds (the "Geometric Mean Value Theorem"):ρ_2(ξ, η)^2 = dρ_2/dρ_1(ξ) dρ_2/dρ_1(η) ρ_1(ξ, η)^2for all ξ, η∈ Z.Moreover for ρ_1, ρ_2, ρ_3 ∈ℳ we havedρ_3/dρ_1 = dρ_3/dρ_2dρ_2/dρ_1anddρ_2/dρ_1 = 1/(dρ_1/dρ_2) max_ξ∈ Zdρ_2/dρ_1(ξ) ·min_ξ∈ Zdρ_2/dρ_1(ξ) = 1Moreover if dρ_2/dρ_1 attains its maximumat ξ and ρ_1(ξ, η) = 1 then dρ_2/dρ_1 attains its minimumat η, and ρ_2(ξ, η) = 1. Proof: Let λ, μ denote the maximum and minimum values of dρ_2/dρ_1 respectively, and let ξ, ξ' ∈ Z denote points where the maximum and minimum values are attained respectively. Given η∈ Z such that ρ_1(ξ, η) = 1, we have, using the Geometric Mean-Value Theorem,1 ≥ρ_2(ξ, η)^2 = dρ_2/dρ_1(ξ) dρ_2/dρ_1(η) ≥λ·μwhile choosing η' ∈ Z such that ρ_2(ξ', η') = 1, we have1 ≥ρ_1(ξ', η')^2 = 1/(dρ_2/dρ_1(ξ') dρ_2/dρ_1(η')) ≥ 1/(λμ)hence λ·μ = 1.By the above we havedρ_2/dρ_1(η) ≤ 1/dρ_2/dρ_1(ξ) = 1/λ = μhence dρ_2/dρ_1(η) = μ. By the Geometric Mean Value Theorem this givesρ_2(ξ ,η)^2 = ρ_1(ξ, η)^2 dρ_2/dρ_1(ξ) dρ_2/dρ_1(η) = 1 ·λ·μ = 1 ♢For ρ_1, ρ_2 ∈ℳ, we defined_ℳ(ρ_1, ρ_2) := max_ξ∈ Zlogdρ_2/dρ_1(ξ) From <cit.> we have:The function d_ℳ defines a metric on ℳ. The metric space (ℳ, d_ℳ) is proper. § VISUAL METRICS ON THE BOUNDARY OF A CAT(-1) SPACE Let X be a proper CAT(-1) space such that ∂ X has at least four points.We recall below the definitions and some elementary properties of visual metrics and Busemann functions; for proofs we refer to <cit.>:Let x ∈ X be a basepoint. The Gromov product of two points ξ, ξ' ∈∂ X with respect to x is defined by(ξ | ξ')_x = lim_(a,a') → (ξ, ξ')1/2(d(x,a) + d(x,a') - d(a,a'))where a,a' are points of X which converge radially towards ξ and ξ' respectively. The visual metric on ∂ X based at the point x is defined byρ_x(ξ, ξ') := e^-(ξ|ξ')_xThe distance ρ_x(ξ,ξ') is less than or equal to one, with equality iff x belongs to the geodesic (ξξ').If X is geodesically complete then ρ_x is a diameter one antipodal metric. The Busemann function B : ∂ X × X × X →ℝ is defined byB(x, y, ξ) := lim_a →ξ d(x,a) - d(y,a)where a ∈ X converges radially towards ξ.We have |B(x,y,ξ)| ≤ d(x,y) for all ξ∈∂ X, x,y ∈ X. Moreover B(x,y,ξ) = d(x,y) iff y lies on the geodesic ray [x, ξ) while B(x,y,ξ) = -d(x,y) iff x lies on the geodesic ray [y, ξ). We recall the following Lemma from <cit.>:For x, y ∈ X, ξ, η∈∂ X we haveρ_y(ξ, η)^2 = ρ_x(ξ, η)^2 e^B(x,y,ξ) e^B(x,y,η)An immediate corollary of the above Lemma is the following:The visual metrics ρ_x, x ∈ X are Moebius equivalent to each other anddρ_y/dρ_x(ξ) = e^B(x,y,ξ)It follows that the metric cross-ratio [ξξ'ηη']_ρ_x of a quadruple (ξ, ξ',η,η') is independent of the choice of x ∈ X. Denoting this common value by [ξξ'ηη'], it is shown in <cit.> that the cross-ratio is given by[ξξ'ηη'] = lim_(a,a',b,b') → (ξ, ξ',η,η')exp(1/2(d(a,b)+d(a',b') - d(a,b') - d(a',b)))where the points a,a',b,b' ∈ X converge radially towards ξ,ξ',η,η' ∈∂ X.We assume henceforth that X is a proper, geodesically complete CAT(-1) space. We let ℳ = ℳ(∂ X, ρ_x) (this space is independent of the choice of x ∈ X). From <cit.> we have: The mapi_X : X→ℳ x↦ρ_xis an isometric embedding and the image is closed in ℳ. For k > 0 and y,z ∈ X distinct from x ∈ X let ∠^(-k^2) y x z ∈ [0, π] denote the angle at the vertex x in a comparison triangle xyz in the model space ℍ_-k^2 of constant curvature -k^2.For ξ, η∈∂ X, the limit of the comparison angles ∠^(-k^2) y x z exists as y,z converge to ξ, η along the geodesic rays [x,ξ), [x, η) respectively. Denoting this limit by ∠^(-k^2)ξ x η, it satisfiessin( ∠^(-k^2)ξ x η/2) = ρ_x(ξ, η)^kProof: A comparison triangle in ℍ_-k^2 with side lengths a = d(x,y), b = d(x, z), c = d(y,z) and angle θ = ∠^(-k^2) y x z at the vertex corresponding to x corresponds to a triangle in ℍ_-1 with side lengths ka, kb, kc and angle θ at the vertex opposite the side with length kc. By the hyperbolic law of cosine we havecosh kc = cosh ka cosh kb - sinh ka sinh kb cosθAs y →ξ, z →η, we have a, b, c →∞, and a + b - c → 2(ξ|η)_x, thuscosθ= cosh ka cosh kb/sinh ka sinh kb - cosh kc/sinh ka sinh kb→ 1 - 2 e^-2k(ξ|η)_xhence the angle θ converges to a limit. Denoting this limit by ∠^(-k^2)ξ x η, by the above it satisfiescos(∠^(-k^2)ξ x η) = 1 - 2ρ_x(ξ, η)^2kand hencesin( ∠^(-k^2)ξ x η/2) = ρ_x(ξ, η)^k♢For x,y ∈ X, ξ∈∂ X and k > 0, the limit of the comparison angles ∠^(-k^2) y x z exists as z converges to ξ along the geodesic ray [x,ξ). Denoting this limit by ∠^(-k^2) y x ξ, it satisfiese^kB(y,x,ξ) = cosh(kd(x,y)) - sinh(kd(x,y)) cos(∠^(-k^2)y x ξ)Proof: A comparison triangle in ℍ_-k^2 with side lengths a = d(x,y), b = d(x, z), c = d(y,z) and angle θ = ∠^(-k^2) y x z at the vertex corresponding to x corresponds to a triangle in ℍ_-1 with side lengths ka, kb, kc and angle θ at the vertex opposite the side with length kc. By the hyperbolic law of cosine we havecosh kc = cosh ka cosh kb - sinh ka sinh kb cosθAs z →ξ, we have b, c →∞, and c - b → B(y, x, ξ), thuscosθ= cosh ka cosh kb/sinh ka sinh kb - cosh kc/sinh ka sinh kb→cosh ka/sinh ka - e^kB(y,x,ξ)/sinh kahence the angle θ converges to a limit. Denoting this limit by ∠^(-k^2)ξ x η, by the above it satisfiese^kB(y,x,ξ) = cosh(kd(x,y)) - sinh(kd(x,y)) cos(∠^(-k^2)y x ξ)♢We now consider the behaviour of the derivatives dρ_y/dρ_x as t = d(x,y) → 0 and the point y converges radially towards x along a geodesic. For functions F_t on ∂ X we write F_t = o(t) if ||F_t||_∞ = o(t). We have the following formula from <cit.>, which may be thought of as a formula for the derivative of the map i_X along a geodesic:As t → 0 we havelogdρ_y/dρ_x(ξ) = t cos(∠^(-1)y x ξ) + o(t)§ CONFORMAL MAPS, MOEBIUS MAPS AND GEODESIC CONJUGACIES We start by recalling the definitions of conformal maps, Moebius maps, and the abstract geodesic flow of a CAT(-1) space. A homeomorphism between metric spaces f : (Z_1, ρ_1) → (Z_2, ρ_2) with no isolated points is said to be conformal if for all ξ∈ Z_1, the limitdf_ρ_1, ρ_2(ξ) := lim_η→ξρ_2(f(ξ), f(η))/ρ_1(ξ, η)exists and is positive. The positive function df_ρ_1, ρ_2 is called the derivative of f with respect to ρ_1, ρ_2. We say f is C^1 conformal if its derivative is continuous.Two metrics ρ_1, ρ_2 inducing the same topology on a set Z, such that Z has no isolated points, are said to be conformal (respectively C^1 conformal) if the map id_Z : (Z, ρ_1) → (Z, ρ_2) is conformal (respectively C^1 conformal). In this case we denote the derivative of the identity map by dρ_2/dρ_1.A homeomorphism between metric spaces f : (Z_1, ρ_1) → (Z_2, ρ_2) (where Z_1 has at least four points) is said to be Moebius if it preserves metric cross-ratios with respect to ρ_1, ρ_2. The derivative of f is defined to be the derivative df_*ρ_2/ρ_1 of the Moebius equivalent metrics f_* ρ_2, ρ_1 as defined in section 2 (where f_* ρ_2 is the pull-back of ρ_2 under f). From the results of section 2 it follows that any Moebius map between compact metric spaces with no isolated points is C^1 conformal, and the two definitions of the derivative of f given above coincide. Moreover any Moebius map f satisfies the geometric mean-value theorem,ρ_2(f(ξ), f(η))^2 = ρ_1(ξ,η)^2 df_ρ_1,ρ_2(ξ) df_ρ_1,ρ_2(ξ)Let (X, d) be a CAT(-1) space. The abstract geodesic flow space of X is defined to be the space of bi-infinite geodesics in X,𝒢X := {γ : (-∞,+∞) → X | γ is an isometric embedding}endowed with the topology of uniform convergence on compact subsets. This topology is metrizable with a distance defined byd_𝒢X(γ_1, γ_2):= ∫_-∞^∞ d(γ_1(t), γ_2(t)) e^-|t|/2 dtWe define also two projectionsπ : 𝒢X→ Xγ ↦γ(0)andp : 𝒢X→∂ Xγ ↦γ(+∞) It is shown in Bourdon <cit.> that π is 1-Lipschitz.For x ∈ X, the unit tangent sphere T^1_x X ⊂𝒢X is defined to beT^1_x X := π^-1(x) The abstract geodesic flow of X is defined to be the one-parameter group of homeomorphismsϕ_t : 𝒢X→𝒢Xγ ↦γ_tfor t ∈ℝ, where γ_t is the geodesic s ↦γ(s+t).The flip is defined to be the mapℱ : 𝒢X→𝒢Xγ ↦γwhere γ is the geodesic s ↦γ(-s). We observe that for a simply connected complete Riemannian manifold X with sectional curvatures bounded above by -1, the map𝒢X→ T^1 Xγ ↦γ'(0)is a homeomorphism conjugating the abstract geodesic flow of X to the usual geodesic flow of X and the flip ℱ to the usual flip on T^1 X.Let f : ∂ X →∂ Y be a conformal map between the boundaries of CAT(-1) spaces X, Y equipped with visual metrics. Then f induces a bijection ϕ_f : 𝒢X →𝒢Y conjugating the geodesic flows, which is defined as follows:Given γ∈𝒢X, let γ(-∞) = ξ, γ(+∞) = η, x = γ(0), then there is a unique point y on the bi-infinite geodesic (f(ξ),f(η)) such that df_ρ_x, ρ_y(η) = 1. Define ϕ_f(γ) = γ^* where γ^* is the unique geodesic in Y satisfying γ^*(-∞) = f(ξ), γ^*(+∞) = f(η), γ^*(0) = y. Then ϕ_f : 𝒢X →𝒢Y is a bijection conjugating the geodesic flows. From <cit.> we have:The map ϕ_f is a homeomorphism if f is C^1 conformal. If f is Moebius then ϕ_f is flip-equivariant. § CIRCUMCENTERS OF EXPANDING SETS AND ℱK-CONVEX FUNCTIONS Let X be a proper, geodesically complete CAT(-1) space. Recall that for any bounded subset B ofX, there is a unique point x which minimizes the functionz ↦sup_y ∈ B d(z, y)The point x is called the circumcenter of B, and the number sup_y ∈ B d(x,y) is called the circumradius of B. We will denote these by c(B) and r(B) respectively.Given K ≤ 0, a function f : X → is said to be ℱK-convex if it is continuous and its restriction to any geodesic satisfies f” + Kf ≥ 0 in the barrier sense. This means that f ≤ g if g coincides with f at the endpoints of a subsegment and satisfies g” + Kg = 0. We have the following from <cit.>: Let y ∈ X, ξ∈∂ X. Then:(1) The function x ↦cosh(d(x,y)) is ℱ(-1)-convex.(2) The function x ↦exp(B(x,y,ξ)) is ℱ(-1)-convex.Let f be a positive, proper, ℱ(-1)-convex function on X. Then f attains its minimum at a unique point x ∈ X. Proof: Since f is continuous, bounded below, and proper, f attains its minimum at some x ∈ X. If x' ≠ x is another point where f attains its minimum, let γ : [-d, d] → X be the geodesic joining x to x' where d = d(x,x')/2 > 0. Then g(t) = f(x) cosh t / cosh d satisfies g” - g = 0, and agrees with f at the endpoints of γ, hence f(γ(0)) ≤ g(0) = f(x) / cosh d < f(x), a contradiction. ♢Let f_n, f be positive, proper, ℱ(-1)-convex functions on X such that f_n → f uniformly on compacts. If x_n, x denote the points where f_n, f attain their minima, then x_n → x. Proof: We first show that {x_n} is bounded. If not, passing to a subsequence we may assume d(x, x_n) → +∞. For n sufficiently large we have f_n(x) ≤ 2f(x). Thus f_n(x_n) ≤ f_n(x) ≤ 2f(x) as well. Let γ_n : [-d_n, d_n] → X be the unique geodesic joining x to x_n, where d_n = d(x, x_n)/2. Then the functiong(t) = 1/sinh(2d_n)[ (sinh d_n)(f_n(x)+f_n(x_n)) cosh t + (cosh d_n)(f_n(x_n) - f_n(x)) sinh t ]satisfies g” - g = 0, and agrees with f_n the endpoints of γ_n. Thus for any s > 0, for n large such that s < d_n, letting y_n = γ_n(-d_n + s), we havef_n(y_n)≤ g(-d_n + s) ≤1/2 sinh d_n cosh d_n[ (sinh d_n cosh (d_n - s))(4f(x)) + cosh d_n sinh (d_n - s) (4f(x)) ] Since d_n → +∞ this implies that for n sufficiently large we havef_n(y_n) ≤1/2 e^-s(1 + o(1)) (8f(x)) < f(x)/2for s > 0 large enough. Fixing such an s, the points y_n = γ_n(-d_n+s) lie in the closed ball B of radius s around x, so passing to a subsequence we may assume that y_n → y ∈ B. Since f_n → f uniformly on B, f_n(y_n) → f(y), hence f(y) ≤ f(x)/2 < f(x), a contradiction.Thus the sequence { x_n } is bounded. To show x_n → x, it suffices to show that the only limit point of {x_n} is x. Let K be a compact containing {x_n}. Suppose x_n_k→ y. Then f_n_k(x_n_k) ≤ f_n_k(x) for all k. Since f_n_k→ f uniformly on K, letting k tend to infinity gives f(y) ≤ f(x). By the previous proposition this implies y = x. ♢Let K be a compact subset of 𝒢X. Define the functionu_K(z) = sup_γ∈ Kexp(B(z, π(γ), γ(+∞))) The function u_K is a positive, ℱ(-1)-convex function. It is proper if p(K) ⊂∂ X is not a singleton. Proof: For each γ∈ K, the function z ↦exp(B(z, π(γ), γ(+∞))) is ℱ(-1)-convex. Thus u_K, being the supremum of a family of ℱ(-1)-convex functions, satisfies the ℱ(-1)-convexity inequality. It remains to show that u_K is continuous.Let z_n → z in X. Define functions h_n, h : K → byh_n(γ) := B(z_n, π(γ), γ(+∞)) , h(γ) := B(z, π(γ), γ(+∞))Then |h_n(γ) - h(γ)| = |B(z_n, z, γ(∞)| ≤ d(z_n, z), so h_n → h uniformly on K. It follows thatu_K(z_n) = ||e^h_n||_∞→ ||e^h||_∞ = u_K(z)Thus u_K is continuous.Now suppose p(K) is not a singleton, sothere exist γ_1, γ_2 ∈ K such that the endpoints ξ_i = γ_i(+∞), i = 1,2 are distinct. Let x_n be a sequence in X tending to infinity. Suppose u_K(x_n) does not tend to +∞. Passing to a subsequence we may assume u_K(x_n) ≤ M for all n for some M > 0. Passing to a further subsequence we may assume x_n →ξ∈∂ X. We can choose a ξ_i ≠ξ.Let x = π(γ_i), then by Lemma <ref> we haveexp(B(x_n, x, ξ_i)) = cosh(d(x_n, x)) - sinh(d(x_n, x)) cos(∠^(-1)x_n x ξ_i)= e^-d(x_n, x) + 2 sinh(d(x_n, x)) sin^2(∠^(-1)x_n x ξ_i/2) → +∞since ∠^(-1)x_n x ξ_i →∠^(-1)ξ x ξ_i > 0.Hence u_K(x_n) ≥exp(B(x_n, x, ξ_i)) → +∞, a contradiction. This shows that u_K is proper. ♢ Let K be a compact subset of 𝒢X such that p(K) ⊂∂ X is not asingleton. The asymptotic circumcenter of K is defined to be the unique x in X where the function u_K attains its minimum. We denote the asymptotic circumcenter by x = c_∞(K). The reason for the name 'asymptotic circumcenter' is explained by the following proposition: Let K be a compact subset of 𝒢X such that p(K) is not a singleton.Define for t > 0 bounded subsets A_t of X by A_t = π(ϕ_t(K)), where ϕ_t denotes the geodesic flow on 𝒢X. Thenc(A_t) → c_∞(K)as t → +∞, i.e. the circumcenters of the sets A_t converge to the asymptotic circumcenter of K. Proof: Let u = u_K, and for t > 0 define u_t : X → byu_t(z) = (sup_y ∈ A_tcosh(d(z,y))) · 2e^-tIt is easy to see that u_t is a positive, proper, ℱ(-1)-convex function, and that the circumcenter of A_t is the unique minimizer of the function u_t. Since c_∞(K) is the unique minimizer of u, by the previous proposition it suffices to show that u_t → u uniformly on compacts as t →∞.Noteu_t(z) = (sup_γ∈ Kcosh(d(z, γ(t)))) · 2 e^-tNow for z in a compact ball B and γ in the compact K,d(z, γ(t)) - t → B(z, π(γ), γ(+∞))as t → +∞ uniformly in z ∈ B, γ∈ K. It follows thatcosh(d(z, γ(t))) · 2 e^-t→exp(B(z, π(γ), γ(+∞)))as t → +∞ uniformly in z ∈ B, γ∈ K. Since the convergence in z, γ is uniform, the supremums over γ∈ K converge, uniformly for z ∈ B:u_t(z) = (sup_γ∈ Kcosh(d(z, γ(t)))) · 2 e^-t→sup_γ∈ Kexp(B(z, π(γ), γ(+∞))) = u(z)uniformly in z ∈ B.♢§ CIRCUMCENTER EXTENSION OF MOEBIUS MAPS AND NEAREST POINT PROJECTIONS Let f : ∂ X →∂ Y be a Moebius homeomorphism between boundaries of proper, geodesically complete CAT(-1) spaces X, Y, and let ϕ_f : 𝒢X →𝒢Y denote the associated geodesic conjugacy. The circumcenter extension of the Moebius map f is the map f̂ : X → Y defined byf̂(x) := c_∞(ϕ_f(T^1_x X)) ∈ Y(note that p(ϕ_f(T^1_x X)) = ∂ Y is not a singleton, so the asymptotic circumcenter ofϕ_f(T^1_x X) ⊂𝒢Y exists). In <cit.>, a (1, log 2)-quasi-isometric extension F : X → Y of the Moebius map f is constructed as follows. Since f is Moebius, push-forward by f of metrics on ∂ X to metrics on ∂ Y gives a map between the spaces of Moebius metrics f_* : ℳ(∂ X) →ℳ(∂ Y), which is easily seen to be an isometry. For each ρ∈ℳ(∂ Y), we can choose a nearest point to ρ in the subspace of visual metrics i_Y(Y) ⊂ℳ(∂ Y). This defines a nearest-point projection r_Y : ℳ(∂ Y) → Y. The extension F is then defined byF = r_Y ∘ f_* ∘ i_X We show below that if ρ∈ℳ(∂ Y) is the push-forward of a visual metric on ∂ X, ρ = f_* ρ_x for some x ∈ X, then in fact there is a unique visual metric ρ_y ∈ℳ(∂ Y) nearest to ρ, given by y = f̂(x), the asymptotic circumcenter of ϕ_f(T^1_x X). It follows that the extension F defined above is uniquely determined and equals the circumcenter extension f̂.Let x ∈ X and let ρ = f_* ρ_x ∈ℳ(∂ Y). Then y = f̂(x) is the unique minimizer of the function z ∈ Y ↦ d_ℳ(ρ, ρ_z). In particular, f̂ = F, so f̂ is a (1, log 2)-quasi-isometry. Proof: Fix a z ∈ Y. Given ξ∈∂ X, let γ∈ T^1_x X be such that γ(+∞) = ξ. Let p = π(ϕ_f(γ)) ∈ Y. Then by definition of ϕ_f, we haved f_* ρ_x/dρ_p(f(ξ)) = 1It follows from the Chain Rule for Moebius metrics thatd f_* ρ_x/dρ_z(f(ξ)) = d f_* ρ_x/dρ_p(f(ξ)) ·dρ_p/dρ_z(f(ξ))= exp(B(z, p, f(ξ)))= exp(B(z, π(ϕ_f(γ)), ϕ_f(γ)(+∞))) Moreover, for any γ∈ T^1_x X, the same argument shows that if ξ = γ(+∞), thenexp(B(z, π(ϕ_f(γ)), ϕ_f(γ)(+∞))) = d f_* ρ_x/dρ_z(f(ξ))Thussup_ξ∈∂ Xd f_* ρ_x/dρ_z(f(ξ)) = sup_γ∈ϕ_f(T^1_x X)exp(B(z, π(γ), γ(+∞)))which gives, using the definition of the metric d_ℳ,exp(d_ℳ(ρ, ρ_z)) = u_K(z)where K = ϕ_f(T^1_x X). Since the unique minimizer of u_K is given by y = f̂(x), it follows that the function z ↦ d_ℳ(ρ, ρ_z) also has a unique minimizer given by f̂(x). ♢The circumcenter extension has the following naturality properties with respect to composition with isometries:Let f : ∂ X →∂ Y be a Moebius homeomorphism.(1) If f is the boundary map of an isometry F : X → Y then f̂ = F.(2) If G : X → X, H : Y → Y are isometries with boundary maps g, h, thenh ∘ f ∘ g = H ∘f̂∘ GProof: Let x ∈ X.(1) If f is the boundary map of an isometry F, then f_* ρ_x = ρ_F(x), so the nearest point to f_* ρ_x is ρ_F(x), so by the previous proposition f̂(x) = F(x).(2) Note f_* g_* ρ_x = f_* ρ_G(x). Let z = f̂(G(x)), so ρ_z is the nearest point to f_* ρ_G(x). Since h_* : ℳ(∂ Y) →ℳ(∂ Y) is an isometry which preserves the subspace of visual metrics, h_* ρ_z = ρ_H(z) is the nearest point to h_* f_* ρ_G(x) = (h ∘ f ∘ g)_* ρ_x, hence by the previous proposition H(z) = h ∘ f ∘ g(x), and H(z) = H(f̂(G(x))) so we are done. ♢The key to Theorem <ref> is the following proposition:Let X be a proper, geodesically complete CAT(-1) space. Given ρ∈ℳ(∂ X), if x ∈ X minimizes z ∈ X ↦ d_ℳ(ρ, ρ_z), then for any y ∈ X ∪∂ X distinct from x, there exists η∈∂ X maximizing ζ∈∂ X ↦dρ/dρ_x(ζ) such that ∠^(-1)y x η≥π/2. Proof: Let K ⊂∂ X be the set where dρ/dρ_x attains its maximum value e^M, where M = d_ℳ(ρ, ρ_x), and suppose there is a y ∈ X ∪∂ X such that ∠^(-1)y x η < π/2 for all η∈ K. Then we can choose ϵ, δ > 0 and a neighbourhood N of K such that ∠^(-1)y x η≤π/2 - ϵ for all η∈ N, and such that logdρ/dρ_x≤ M - δ on ∂ X - N.Let z be the point on the geodesic ray [x,y) at a distance t > 0 from x. As t → 0, for η∈ N we have, noting that ∠^(-1)z x η≤∠^(-1)y x η, by Lemma <ref>,logdρ/dρ_z(η) = logdρ/dρ_x(η) - logdρ_z/dρ_x(η) ≤ M - t cos(∠^(-1)z x η) + o(t) ≤ M - t cos(∠^(-1)y x η) + o(t) ≤ M - t sinϵ + o(t)< Mfor t small enough depending only on ϵ, while for η∈∂ X - N we havelogdρ/dρ_z(η) = logdρ/dρ_x(η) - logdρ_z/dρ_x(η) ≤ (M - δ) + t< M for t < δ, thus for t > 0 small enough we have d_ℳ(ρ, ρ_z) < M = d_ℳ(ρ, ρ_x),a contradiction. ♢Theorem <ref> now follows from the following proposition:Let f : ∂ X →∂ Y be a Moebius homeomorphism between boundaries of proper, geodesically complete CAT(-1) spaces X, Y. Then the circumcenter extension f̂ : X → Y satisfiescosh(d(f̂(x), f̂(y))) ≤ e^d(x,y)for all x,y ∈ X. In particular f̂ is locally 1/2-Holder continuous: d(f̂(x), f̂(y)) ≤ 2 d(x,y)^1/2for all x, y ∈ X such that d(x,y) ≤ 1. Proof: Given x,y ∈ X, let x' = f̂(x), y' = f̂(y). We may assume x' ≠ y' (otherwise the above inequality holds trivially), and also (interchanging x, y if necessary) thatd_ℳ(f_* ρ_x, ρ_x') ≥ d_ℳ(f_* ρ_y, ρ_y').Let ρ = f_* ρ_x ∈ℳ(∂ Y). By Proposition <ref>, x' minimizes z ∈ Y ↦ d_ℳ(ρ, ρ_z). Hence by the previous Proposition <ref>, there exists η∈∂ Y maximizing ζ∈∂ Y ↦dρ/dρ_x'(ζ) such that ∠^(-1)y' x' η≥π/2. By Lemma <ref>, we havee^B(y',x', η) = cosh(d(x', y')) - sinh(d(x',y')) cos(∠^(-1)y' x' η) ≥cosh(d(x', y'))Also,e^B(y',x', η)= dρ_x'/dρ_y'(η)= dρ_x'/d f_* ρ_x(η) d f_* ρ_x/d f_* ρ_y(η) d f_* ρ_y/dρ_y'(η) ≤exp(-d_ℳ(f_* ρ_x, ρ_x')) dρ_x/dρ_y(f^-1(η)) exp(d_ℳ(f_* ρ_y, ρ_y')) ≤dρ_x/dρ_y(f^-1(η))= e^B(y, x, f^-1(η))≤ e^d(x,y)thuscosh(d(x', y')) ≤ e^d(x,y)as required.It follows easily that f̂ is locally 1/2-Holder:Since e^t ≤ 1 + 2t for 0 ≤ t ≤ 1, for x,y ∈ X, if d(x,y) ≤ 1 we have1 + d(f̂(x), f̂(y))^2/2 ≤cosh(d(f̂(x), f̂(y))) ≤ e^d(x,y)≤ 1 + 2d(x,y)henced(f̂(x), f̂(y)) ≤ 2 d(x,y)^1/2.♢Let X be a complete, simply connected Riemannian manifold with sectional curvatures K satisfying -b^2 ≤ K ≤ -1 for some b ≥ 1. For x ∈ X and ξ, η∈∂ X, let ∠ξ x η∈ [0, π] denote the Riemannian angle at x between the geodesic rays [x,ξ) and [x, η).We haveρ_x(ξ, η)^b ≤sin( ∠ξ x η/2) ≤ρ_x(ξ, η)Proof: Since the sectional curvature of X is bounded above and below by -1 and -b^2, we have∠^(-b^2)ξ x η≤∠ξ x η≤∠^(-1)ξ x ηhence by Lemma <ref>ρ_x(ξ, η)^b = sin( ∠^(-b^2)ξ x η/2) ≤sin( ∠ξ x η/2)≤sin( ∠^(-1)ξ x η/2) = ρ_x(ξ, η)♢Proof of Theorem <ref>: Let f : ∂ X →∂ Y be a Moebius homeomorphism between boundaries of complete, simply connected manifolds with sectional curvatures K satisfying -b^2 ≤ K ≤ -1.Let x ∈ X, and let y = f̂(x). Let M = d_ℳ(f_* ρ_x, ρ_y). Let K ⊂∂ Y be the set where d f_* ρ_x/dρ_y attains its maximum value e^M, and let η_1 ∈ K. Then by Proposition <ref>, there exists η_2 ∈ K such that ∠^(-1)η_1 y η_2 ≥π/2, so ρ_y(η_1, η_2) ≥ 1/√(2).Let ξ_i = f^-1(η_i) ∈∂ X, i = 1,2. Let η'_i ∈∂ Y be the unique point such that ρ_y(η_i, η'_i) = 1, i = 1,2. Then by Lemma <ref>, d f_* ρ_x/dρ_y attains its minimum value e^-M at η'_1, η'_2, and the points ξ'_i = f^-1(η'_i) satisfy ρ_x(ξ_i, ξ'_i) = 1, i = 1,2. The Geometric Mean Value Theorem givesρ_x(ξ_1, ξ_2) = e^M ρ_y(η_1, η_2), ρ_x(ξ'_1, ξ'_2) = e^-Mρ_y(η'_1, η'_2)Noting that ∠ξ_1 x ξ_2 = ∠ξ'_1 x ξ'_2 and ∠η_1 y η_2 = ∠η'_1 y η'_2, by Lemma <ref> we haveρ_x(ξ'_1, ξ'_2)≥sin( ∠ξ'_1 x ξ'_2/2)= sin( ∠ξ_1 x ξ_2/2) ≥ρ_x(ξ_1, ξ_2)^b andρ_y(η'_1, η'_1)≤(sin( ∠η'_1 y η'_2/2))^1/b = (sin( ∠η_1 y η_2/2))^1/b≤ρ_y(η_1, η_2)^1/b Using the above two inequalities in the equality ρ_x(ξ_1, ξ_2)/ρ_x(ξ'_1, ξ'_2) = e^2Mρ_y(η_1, η_2)/ρ_y(η'_1, η'_2) gives 1/ρ_x(ξ_1, ξ_2)^b-1≥ e^2Mρ_y(η_1, η_2)^1 - 1/b Thus 1≥ e^2Mρ_x(ξ_1, ξ_2)^b-1ρ_y(η_1, η_2)^1 - 1/b = e^2M e^(b-1)Mρ_y(η_1, η_2)^(b-1)+(1 - 1/b)≥e^(b+1)M/√(2)^b - 1/bhenceM ≤1/2b - 1/b/b+1log 2 = 1/2(1 - 1/b)log 2Thusd_ℳ(f_* ρ_x, ρ_f̂(x)) ≤1/2(1 - 1/b)log 2for all x ∈ X. Then for any x, y ∈ X,|d(f̂(x), f̂(y)) - d(x, y)| = |d_ℳ(ρ_f̂(x), ρ_f̂(y)) - d_ℳ(f_* ρ_x, f_* ρ_y)| ≤ d_ℳ(f_* ρ_x, ρ_f̂(x)) + d_ℳ(f_* ρ_y, ρ_f̂(y)) ≤ (1 - 1/b) log 2thus f̂ is a (1, (1 - 1/b)log 2)-quasi-isometry. As in <cit.> it is straightforward to show that the image of f̂ is 1/2(1 - 1/b)log 2-dense in Y and that the boundary map of f̂ equals f. ♢ alpha | http://arxiv.org/abs/1709.09110v2 | {
"authors": [
"Kingshook Biswas"
],
"categories": [
"math.DG"
],
"primary_category": "math.DG",
"published": "20170926161615",
"title": "Circumcenter extension of Moebius maps to CAT(-1) spaces"
} |
APS/123-QED [email protected] Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, Mé[email protected] Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, Mé[email protected] Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, México An effective toy model for an ideal one-dimensional nonstationary cavity is taken to be the starting point to derive a fitting markovian master equation for the corresponding leaky cavity. In the regime where the generation of photons via the dynamical Casimir effect is bounded, the master equation thus constructed allows us to investigate the effects of decoherence on the average number of Casimir photons and their quantum fluctuations through the second-order correlation function.Damped Casimir radiationand photon correlation measurements C. González-Gutiérrez December 30, 2023 =============================================================§ INTRODUCTION Cavity dynamical Casimir effect (DCE) is a fascinating quantum mechanical phenomenon in which real photons can be created out of vacuum fluctuations, via parametric amplification, as a consequence of nonadiabatic changes in the time-dependent electromagneticcavity boundary conditions <cit.>. Theoretically predicted by Moore in the70's <cit.>, the DCE was demonstrated experimentally, more than forty years later, by using asuperconducting Josephson metamaterial asa surrogate for a fast(a significant fraction of the speed of light)oscillating cavity mirror <cit.>. Besides being an important resultfrom the fundamental point of view of quantum field theory, the aforesaid phenomenon has also been investigated in a number of contexts such astrapped ions <cit.>, quantum refrigerators <cit.>, Kerr media <cit.>, and, still more recently, instochastic systems <cit.>; furthermore, it has found interesting applications in circuit quantum electrodynamics <cit.> where entangled artificial atoms <cit.> andGaussian boson samplers <cit.> can be realized. On the other hand, the issue of decoherence has also attracted a great deal of interest in recent times since, as known, a given physical system cannot be completely isolated from its surroundings; indeed, a more realistic scenario has to take into account the effects of loss of quantum coherence that stem from being oblivious of the environmental influences. In this regard, of particular interest to us is the fact that if one wants to detect and correlate Casimir photons, it is essential to consider the unavoidable interaction with their environment. Along this line of research, there have been several attempts to incorporate decoherence andenergy losses in nonstationary cavities including either amplitude <cit.> or phase <cit.> damping; and a rather cumbersome time dependent master equation has already been derived from first principles as asserted in Ref. <cit.>. However, despite all these efforts, a consensus about how to properly analyze the effects of dissipation on the DCE has not yet reached. This work is in keeping with the spirit of putting forward an algebraic toy model capable of describing, in an effective manner, decoherence effects in the process of creating and correlating photons in the DCE, a proposed model that is considered to be justified only under certain environmental and system conditions. More precisely, we restrict ourselves to a parameter regime in which the generation of photons remains bounded so as to be able to derive a fitting markovian master equation for the reduced density operator of the dynamical cavity viewed as an open quantum system. The optical scheme of the process it seeks to describe is sketched in Fig. <ref>. According to the figure, a nonstationary leaky cavity takes on the role of our central system from which Casimir photons are created and their quantum fluctuations are analyzed in terms of the second-order correlation (coherence) function by means of a standard Hanbury-Brown and Twiss intensity interferometer. The details of the effective model and the precise parameter settings within which we shall be focused on will be given below. The simplest effective Hamiltonian describing, in the Schrödingerpicture, the dynamics of an electromagnetic field inside a lossless one-dimensionalnonstationary single-mode cavity, whose instantaneous frequency ω(t)=ω_0[1+ϵsin(ν t)] follows from the time-dependent geometry of the system,is given by <cit.> (ħ=1):Ĥ_eff=(ϵω_0/4)(â^† 2+â^2)+(K/2)n̂,where ω_0 is the fundamental frequency of the cavity, and ϵ (ν) isthe amplitude (frequency) modulation.This time-independent Hamiltonian is an effective algebraic model obtained under quasi-resonant conditions, i.e.,ν=2ω_0+K, with K being a smallfrequency shift, and is written in a rotated reference frame in whichthe rotating-wave approximation (RWA) is alsoemployed <cit.>. The SU(1,1) symmetry of Ĥ_eff, generated by the set of operators {n̂, â^† 2, â^2}, enables us to obtain, by making use of standard Lie algebraic methods, the following expression for the expectation value of the number of photons generated from the vacuum state <cit.>:⟨n̂⟩=sinh^2(ηϵω_0t/2)/η^2,with η=√(1-(K/ϵω_0)^2). Hence, depending on whether the ratio K/ϵω_0 is smaller or greater than unity, one can identify a twofold behavior of photon generation ranging from the exponential growth (K/ϵω_0<1) to the bounded oscillatory regime (K/ϵω_0>1), in which case the argument of the hyperbolic function becomes imaginary so that the replacement η→η̃=√((K/ϵω_0)^2-1) follows; this crossover has recently been referred to as ametal-insulator phase transition <cit.>.Returning to the subject of considering the nonstationary cavity as an open system, an ansatz for the corresponding phenomenological master equation at zero temperature is considered to have the following structure <cit.>: dρ̂/dt =-i[Ĥ_eff,ρ̂]+κℒ[â]ρ̂, where κ is the decay rate, which is inversely proportional to the quality factor of the cavity, and the generator ℒ[x] is such that ℒ[x]ρ̂≡ 2xρ̂ x^†-x^† xρ̂-ρ̂ x^† x, with ρ̂ being the reduced system density operator. On the basis of this masterequation, we arrive at the modified version of (<ref>):⟨n̂⟩=-2⟨n̂⟩_st^phe^-2κ t[sinh^2(ηϵω_0 t/2)+1/2+(κ/ηϵω_0)sinh(ηϵω_0 t)]+ ⟨n̂⟩_st^ph,where ⟨n̂⟩_st^ph=1/2 [(2κ/ϵω_0)^2-η^2]^-1is the steady state average of the photon number provided that 2κ>ηϵω_0. In the resonant case, K=0, the result of Ref. <cit.> is recovered from (<ref>) and, again, a photon-number exponential growth is obtained as long as the amplitude modulation surpasses the rate at which the system decays. On the other hand, it was shown in <cit.> that dephasing effects themselves, ∝ℒ[n̂]ρ̂, tend to only slow down the photon generation rate. In this approach, however, the lack of knowledge about the explicit form of the system's steady state makes it difficult, for instance, to examine analytically the statistical behavior of the outgoing photons via the second order correlation function involving the use of the known quantum regression formula <cit.>. Further drawbacks of the phenomenological treatment have already been discussed in <cit.>.Section <ref> outlines the derivation of the microscopic master equation on the basis of the Born and Markov approximations, an approach that is suitable for the description of dissipation in the DCE evolving within the bounded regime of photon generation. Having determined the steady state limit of our system, we proceed, in section <ref>, to the description of the outcome of the proposed master equation reflected upon the average photon number and the intensity correlation of two created photons. And finally, some conclusions are given in section <ref>.§ MICROSCOPIC MASTER EQUATIONLet the time-independent effective Hamiltonian (<ref>) be our starting point. This Hamiltonian can easily be diagonalized by making use of the squeeze operator Ŝ(r) = exp[ r(â^2-â^† 2)/4 ] = e^ zâ^† 2/2 e^β/2(n̂+1/2)e^-z â^2/2 through the unitary transformation Ŝ(r)Ĥ_effŜ^†(r);here, the particular choice of the parameterr=1/2ln[(K+ϵω_0)/(K-ϵω_0)] guarantees the proper diagonalization process provided that the inequality K/ϵω_0>1 holds, and the remaining parameters z=-tanh (r/2) and β = ln (1-z^2) are stated by disentangling the exponential. So, via this reframed system, it is found that the corresponding eigenenergies and eigenstates of our effective Hamiltonian are, respectively, given by2E_n =ϵω_0√((K/ϵω_0)^2-1)(n+1/2)-K/2,|r ,n⟩ =Ŝ(r)|n⟩,where n=0,1,2,…, and the latter turn out to be the so-called squeezed number states <cit.>. Based upon the aforesaid unitary operator, we find it convenient to introduce the so-termed pseudo-annihilation (creation) operator, b̂ (b̂^†), firstly introduced by Yuen <cit.> in his work on two-photon coherent states, defined byb̂=Ŝ(r) âŜ^† (r)=cosh(r/2)â+sinh( r/2)â^†,b̂^†=Ŝ(r) â^†Ŝ^† (r)=cosh(r/2)â^†+sinh(r/2)â, which is nothing but a Bogoliubov transformation that generates b̂ and b̂^† from the standard operators â and â^†, thereby preserving the commutator [b̂,b̂^†]=1.Thus, in this representation, one is able to obtain the following diagonal form of the system Hamiltonian (<ref>)Ĥ_S= Ω(b̂^†b̂+1/2)-K/4,with the identificationΩ = η̃ϵω_0/2; note that the states given by (<ref>) are indeed eigenstates of the operator b̂^†b̂, also called quasi-number operator <cit.>. To be more precise, in this algebraic scheme, the action of b̂ and b̂^† upon the eigenstates of the system, the squeezed number states, is reminiscent of that of â and â^† upon the Fock states in such a way that b̂|r,n⟩=√(n) |r,n-1⟩ and b̂^† |r,n⟩ = √(n+1) |r,n+1⟩ <cit.>. I.e., such operators lower and raise one excitation by changing the number of quanta in ± 1, therebyconnecting transitions between adjacent energy levels within the squeezed-number-state basis. Indeed, these operators can also be regarded as the actual eigenoperators of the system Hamiltonian inthe sense that they obey the commutation relationships:[Ĥ_S, b̂ ]=-Ωb̂,[Ĥ_S, b̂^†]=Ωb̂^†.This fact motivates us to consider the possibility of deriving a fitting, albeit approximate, master equation, in the weak-coupling and Markovian regimes, in order to explore the damped dynamics of the system described by the effectiveHamiltonian (<ref>) but from a different algebraic point of view. That is, in such a representation, given by the pseudo-harmonic oscillator outlined above, we should be able to establish the proper master equation in a way such that the dissipative part of the evolution be modeled in terms of the actual system's eigenoperators. To this end, let the Hamiltonian describing our quadratic oscillator as an open system be structured as follows:Ĥ=Ĥ_S+Ĥ_E+ Ĥ_SE, where Ĥ_S corresponds to the unperturbed central system (S), Ĥ_E is the free Hamiltonian of the environment (E), and Ĥ_SE represents the interplay between them. Proceeding in the customary fashion, let the environment of the central system be modeled as a bath of harmonic oscillators, i.e., Ĥ_E= ∑_kω_kB̂_k^†B̂_k, with ω_k being the frequency of the k-th oscillator, and their interaction be governed by the following Hamiltonian in the Schrödinger pictureĤ_SE = (â+â^†)∑_kg_k(B̂_k+B̂_k^†),which is taken to be linear in both the cavity field and the environment amplitudes, B̂_k (B̂^†_k) is the annihilation (creation) operator within the k-th mode, and the g_k's are the coupling parameters;incidentally, this kind of system-environment interaction resembles the multimode coupling Hamiltonian model proposed in Ref. <cit.>, where the authors attempt to describe a leaky-cavity configuration in which a dispersive mirror is inserted into a larger ideal nonstationary cavity that, in turn, plays the role of a reservoir. So, Hamiltonian (<ref>), written in terms of the natural variables of the field, â and â^†, can be recast in the pseudo-harmonic-oscillator representation by inversion of (<ref>) and (<ref>). So, in the interaction picture generated by H_S+H_E, with H_S being taken to be (<ref>), we getH̃_SE(t) =(b̂e^-iΩ t+b̂^†e^iΩ t)∑_kg_k(r) (B̂_ke^-iω_kt+B̂_k^†e^iω_kt),where the coupling constants are now construed as being dependent on the squeezing parameter, that is to say,g_k(r)=e^-r/2g_k. Based upon the Born and Markov approximations, and within the framework of the master equation approach, one is able to establish the following equation for the reduced density operator associated with the system at hand which makes no restriction on the precise interaction between the latter and its surroundings<cit.>ρ̇̃̇(t) = -∫_0^∞ dτ Tr_E{[ H̃_SE(t),[H̃_SE(t-τ),ρ̃(t) ⊗ρ_E]] },where the tilde over the density operator means that it is in the interaction picture, Tr_E indicates the trace over the environment variables, and ρ_E represents the state of the environment which, according to the Born approximation, is taken to be constant in its evolution (ρ_E(t)≈ρ_E(0)=ρ_E) and determined by the Boltzman distribution ρ_E = e^-Ĥ_E/k_BT/Tr {e^-Ĥ_E/k_BT}. Substitution of (<ref>) into (<ref>), application of the RWA to the equation thus obtained, and going back to the Shrödinger representation, leads us, after some algebra, to a Lindblad master equation describing the damped dynamics of the system interacting with a bath of harmonic oscillators in thermal equilibrium at T temperature:dρ̂/dt=-i [Ĥ_S,ρ̂]+ γ_r ( N_Ω+1 ) ℒ [b̂]ρ̂ +γ_r N_Ωℒ [b̂^†] ρ̂ .Here, γ_r=e^-rγ is the overall damping rate, where we have let γ=π h(Ω) |g(Ω)|^2 be, approximately, a constant quantity, provided of course that a spectrally flat environment is taken into consideration, with h(Ω) and g(Ω) being, respectively, the density of states and the system-environment coupling at Ω; and N_Ω=1/(e^ħΩ/k_BT-1) is the average number of thermal photons in the reservoir at the aforesaid frequency. The dependency of the resulting master equation upon the squeezing parameter r is apparent from the viewpoint of the algebraic scheme we are working on: each set of parameters {K, ϵ, ω_0}, in terms of which the squeezing one is set down, is thought of as defining, correspondingly, a different oscillator system described by the algebraic Hamiltonian (<ref>), whose energy spectrum, albeit equally spaced for a given value of the parameters involved (see Eq. (<ref>)), displays an overall dependency on the ratio K/ϵω_0 with a tendency to merge at K/ϵω_0=1. So, for each value of this ratio, we have specific pseudo operators {b̂, b̂^†} describing the allowed transitions induced by the environment that take place at the time scales γ_r^-1 and at the specific transition frequency Ω between the energy levels involved in accord with the commutation relations (<ref>).It is also worth commenting that taking advantage of the algebraic scheme in terms of which it is written down, Eq. (<ref>) can be solved by using, for example, the standard technique based upon superoperators (see, for instance, Ref. <cit.>) allowing us to confirm that in the asymptotic limit t→∞, the reduced density operator approaches that of the squeezed thermal state, namely, lim_t→∞ρ̂=(1+ N_Ω)^-1∑_n( N_Ω/1+ N_Ω)^n |r,n ⟩⟨ r, n|, and, therefore, the steady state at zero temperature (N_Ω=0) becomes precisely the squeezed vacuum state, ρ̂(t→∞) → |r,0⟩⟨ r,0|, which clearly corresponds to the state of minimum energy (the ground state) of the system (see Eq. (<ref>)). We note in passing that (<ref>), written in terms of â and â^†, is somewhat similar to the one obtained by considering a bosonic system coupled to a phase-sensitive reservoir <cit.> and, additionally, bears some resemblance to the master equation derived in Ref. <cit.>.§ GENERATION AND CORRELATION OF PHOTONSLet us now discuss some results concerning the damped evolution of the average photon number and the second-order intensity correlation function. Firstly, to evaluate the expectation value of the number operator in the current representation, namely, ⟨n̂⟩=cosh(r) ⟨b̂^†b̂⟩-1/2sinh(r)[⟨b̂^† 2⟩+ ⟨b̂^2⟩]+sinh^2( r/2 ), we find it convenient to establish the following equations of motion for the its constituents: d⟨b̂^†b̂⟩/ dt =-2 γ_r⟨b̂^†b̂⟩+2 γ_r N_Ω,d⟨b̂^2⟩ /dt =-(2iΩ+2γ_r) ⟨b̂^2⟩, which follows from applying ⟨Ô̇⟩=Tr {ρ̇̂̇Ô} with the help of (<ref>). The solution to these equations is quite straightforward and a closed-form expression for the average photon number can be found when the state of the system at the initial moment of time is the vacuum, ρ̂(0)=|0⟩⟨ 0|, represented in the squeezed-number-state eigenbasis (<ref>). Thus, the sought result turns out to be⟨n̂ (t) ⟩= e^-2γ_rtsin^2(η̃ϵω_0 t/2 )/η̃^2 +(1-e^-2γ_rt)⟨n̂⟩_st,in which one is able to distinguish clear-cut limits being represented in Fig. <ref>: (i) on the one hand, we identify the undamped case (γ_r=0) that corresponds to the well-known result regarding the oscillatory regime of photon generation; (ii) on the other hand, once the transient evolution has elapsed, it is found that the steady state limit, lim_t →∞⟨n̂(t) ⟩ =⟨n̂⟩_st= ⟨n̂⟩_st,0(1+2N_Ω)+ N_Ω, corresponds to the mean photon number for the squeezed thermal state (blue line), with ⟨n̂⟩_st, 0 = 1/2[(1-(ϵω_0/K)^2)^-1/2-1 ] being the corresponding average without thermal photons, N_Ω = 0 (black line). Note that, in contradistinction to the phenomenological master equation's prediction, ⟨n̂⟩_st is independent of the rate at which the central system decays. (<ref>), which is one of the main results of this paper, could in principle be testedexperimentally by measuring the probability distribution of the outgoing cavity photons at different time intervals with the use of just one of the photo-detectors sketched in Fig. <ref>.As far as the intensity-intensity correlation function is concerned, we calculate the standard normalizedexpression of it given by g^(2)(τ) = ⟨â^†(0)â^†(τ) â(τ) â(0) ⟩/⟨â^†â⟩^2_st. This correlation function, which is intrinsically time-symmetric, represents a relative measure of the joint probability of detecting two photons separated by a time delay τ and allows us to discern whether two detection processes are correlated or independent of each other. According to the optical scheme displayed in Fig. <ref>, in order to obtain the value of g^(2)(τ), the outgoing generated photons are passed through a 50/50 beamsplitter, the transmitted signal registered by detector D_1 is multiplied with the reflected one registered by D_2 and, in turn, they are averaged over all the detected values <cit.>. Applying the quantum regression formula <cit.>, together with (<ref>), enables us to arrive at the expression g^(2)(τ) = 1+e^-2γ_rτ [ C_1+C_2cos(2Ωτ)],where we have set the constant terms C_1={ (1+2⟨n̂⟩_st,0)[ ⟨n̂⟩_st- N_Ω (⟨n̂⟩_st+⟨n̂⟩_st,0+2 ) ] +(1+2⟨n̂⟩_st,0)^2(2 N_Ω^2+ N_Ω) }/⟨n̂⟩_st^2, C_2=N_Ω( N_Ω+1)/⟨n̂⟩^2_stη̃^2. (<ref>) behaves differently, as a function of the photo-counting delay τ, as shown in Fig. <ref>, depending on whether or not thermal photons come into play. At zero temperature (black line), we see that the correlation function reveals a noticeable bunching effect for short time delays; particularly, g^(2)(0)|_N_Ω=0=3+⟨n̂⟩_st,0^-1 reveals a super-thermal photon statistics behavior that is in accord with the fact that, in the DCE, photons are created in pairs <cit.>. In addition to this, when T≠ 0 (blue line) the oscillatory fingerprint of (<ref>) comes about in a series of beats at the frequency 2Ω that, according to (<ref>), are bolstered by the thermal photons from the environment via the quadratic nature of the central system in a way such that both features reinforce each other. § CONCLUSIONSAn effective algebraic model, viewed as an open quantum system, has been proposed as a step towards a better understanding of decoherence effects on detecting and correlating photons in a nonstationary leaky cavity in which the DCE takes place. For the model to be applicable, we have restricted ourselves to the regime within which the photon generation is bounded for a given set of system parameters, i.e., when the inequality K/ϵω_0>1 is satisfied. Based upon the markovian master equation derived under this condition, it is found that the steady state limitof the system corresponds to the squeezed thermal state at finite temperature and, thus, to the squeezed vacuum state at zero temperature; this result, as opposed to the phenomenological treatment, allows for an explicit analysis of the outgoing photons in terms of the second-order correlation function. This last feature, besides reveling a conspicuous bunching effect, turns out to display a beating behavior that is fostered by merging the environmental thermal photons and the intrinsic quadratic character of the system. The procedure outlined in this letter can also be applied, and suitable, to pursue the investigation of another measurements of phase-dependent quantum fluctuations in the context of Casimir ratiadion, such as the time-dependent physical spectrum of light <cit.>, the spectrum of squeezing <cit.>, and the amplitude-intensity correlation function <cit.>; the last one, unlike the g^(2)(τ) function, is a wave-particle correlation that can exhibit large time asymmetries <cit.>.§ ACKNOWLEDGMENTSR.R.-A. and C.G.-G. thank CONACYT, Mexico, forfinancial support under Scholarships Nos.379732 and 385108, respectively, and DGAPA-UNAM, Mexico, for support under Project No. IN113016. O. de los S.-S wants to thank Professor J. Récamier for his hospitality at Instituto de Ciencias Físicas, UNAM.10DodonovReview V. V. Dodonov,Physica Scripta, vol. 82, no. 3, p. 038105, 2010.moore1970 G. T. Moore J. Math. Phys., vol. 11, p. 2679, 1970.Lahteenmaki12032013 P. Lähteenmäki, G. S. Paraoanu, J. Hassel, and P. J. Hakonen,Proceedings of the National Academy of Sciences, vol. 110, no. 11, pp. 4234–4238, 2013.Nils N. Trautmann and P. Hauke, New Journal of Physics, vol. 18, no. 4, p. 043029, 2016.JPaz N. Freitas and J. P. Paz,Phys. Rev. E, vol. 95, p. 012146, Jan 2017.ricardo2017 R. Román-Ancheyta, C. González-Gutiérrez, and J. Récamier, J. Opt. Soc. Am. B, vol. 34, pp. 1170–1176, Jun 2017.ricardoguias R. Román-Ancheyta, I. Ramos-Prieto, A. Perez-Leija, K. Busch, and R. d. J. León-Montiel, Phys. Rev. A, vol. 96, p. 032501, Sep 2017.RevModPhysFNori P. D. Nation, J. R. Johansson, M. P. Blencowe, and F. Nori, Rev. Mod. Phys., vol. 84, pp. 1–24, Jan 2012.Solano1 S. Felicetti, M. Sanz, L. Lamata, G. Romero, G. Johansson, P. Delsing, and E. Solano,Phys. Rev. Lett., vol. 113, p. 093602, Aug 2014.Borja1 B. Peropadre, J. Huh, and C. Sabín,arXiv preprint arXiv:1610.07777, 2016.Dodonovlosses V. V. Dodonov,Phys. Rev. A, vol. 58, pp. 4147–4152, Nov 1998.Ralfdecohe R. Schtzhold and M. Tiersch, Journal of Optics B: Quantum and Semiclassical Optics, vol. 7, no. 3, p. S120, 2005.Soffleaky G. Schaller, R. Schützhold, G. Plunien, and G. Soff,Phys. Rev. A, vol. 66, p. 023812, Aug 2002.Law1 C. K. Law,Phys. Rev. A, vol. 49, pp. 433–437, Jan 1994.Dalvit2011 D. A. R. Dalvit, P. A. M. Neto, and F. D. Mazzitelli, Fluctuations, Dissipation and the Dynamical Casimir Effect, pp. 419–457. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011.DodonovOneAtom A. V. Dodonov and V. V. Dodonov,Phys. Rev. A, vol. 85, p. 015805, Jan 2012.DodonovTwoAtoms A. V. Dodonov and V. V. Dodonov, Phys. Rev. A, vol. 85, p. 055805, May 2012.Ban93 M. Ban,J. Opt. Soc. Am. B, vol. 10, pp. 1347–1359, Aug 1993.carmichael H. Carmichael, An open systems approach to quantum optics, vol. 18. Springer Science & Business Media, 1993.nieto M. M. Nieto,Physics Letters A, vol. 229, no. 3, pp. 135 – 143, 1997.yuen H. P. Yuen,Phys. Rev. A, vol. 13, pp. 2226–2243, Jun 1976.mandel L. Mandel and E. Wolf, Optical coherence and quantum optics. Cambridge university press, 1995.gardiner C. W. Gardiner and H. Haken, Quantum noise, vol. 26. Springer Berlin, 1991.moya H. Moya-Cessa,Physics Reports, vol. 432, no. 1, pp. 1 – 41, 2006.ekert A. K. Ekert and P. L. Knight,Phys. Rev. A, vol. 42, pp. 487–493, Jul 1990.ficekbook Z. Ficek and M. R. Wahiddin, Quantum optics for beginners. CRC Press, 2014.Eberly77 J. H. Eberly and K. Wódkiewicz,J. Opt. Soc. Am., vol. 67, pp. 1252–1261, Sep 1977.Collett M. Collett, D. Walls, and P. Zoller,Optics Communications, vol. 52, no. 2, pp. 145 – 149, 1984.castro2016 H. M. Castro-Beltrán, R. Román-Ancheyta, and L. Gutiérrez, Phys. Rev. A, vol. 93, p. 033801, Mar 2016.castro2017 L. Gutiérrez, H. Castro-Beltrán, R. Román-Ancheyta, and L. Horvath, arXiv preprint arXiv:1706.03027, 2017. | http://arxiv.org/abs/1709.09685v1 | {
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"O. de los Santos-Sánchez",
"C. González-Gutiérrez"
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"primary_category": "quant-ph",
"published": "20170927181644",
"title": "Damped Casimir radiation and photon correlation measurements"
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Thin-shell wormholes from compact stellar objectsPeter K. F. Kuhfittig*[E-mail: [email protected]]Department of Mathematics, Milwaukee School of Engineering,Milwaukee, Wisconsin 53202-3109, USA===========================================================================================================================================================This paper introduces a new type of thin-shell wormhole constructed from a special class of compact stellar objects rather than black holes.The construction and concomitant investigation of the stability to linearized radial perturbations commences with an extended version of a regular Hayward black hole.Given the equation of state 𝒫=ωσ, ω <0, for the exotic matter on the thin shell, it is shown that whenever the value of the Hayward parameter is below its critical value, no stable solutions can exist.If the Hayward parameter is allowed to exceed its critical value, stable solutions can be found for moderately sized thin shells. Not only are the underlying structures ordinary compact objects, rather than black holes, the results are consistent with the properties of neutron stars, as well as other compact stellar objects.Keywords: thin-shell wormholes; compact stellar objects; Hayward black holes § INTRODUCTION A highly effective method for describing or mathematically constructing a class of spherically symmetric wormholes using the now standard cut-and-paste technique was proposed by Visser in 1989 <cit.>.The construction calls for grafting two black-hole spacetimes together, resulting in a thin-shell wormhole.By starting with a regular Hayward black hole, it is proposed in this paper that the construction can be extended to massive compact objects.Apart from a number of forerunners, the concept of a traversable wormhole suitable for interstellar travel was first proposed by Morris and Thorne <cit.>.It turned out that a wormhole could only be held open by violating the null energy condition, defined as follows:T_αβk^αk^β≥ 0for all null vectors k^α.Matter that violates this condition came to be called “exotic."In particular, for the radial outgoing null vector (1,1,0,0), the violation takes on the form T_αβk^αk^β =ρ +p<0.(Here T_00=ρ, the energy density, and T_11=p, the radial pressure.)We will assume in this paper that the exotic matter on the shell satisfies a certain equation of state (EoS).In an earlier paper, Eiroa <cit.> assumed the generalized Chaplygin EoS 𝒫=A/|σ|^α, where σ is the energy density of the shell and 𝒫 is the surface pressure. Kuhfittig <cit.> investigated the possible stability of thin-shell wormholes constructed from several spacetimes using the simpler EoS 𝒫=ωσ, ω <0.Since it resembles the EoS of a perfect fluid, this choice seems more natural and will therefore be employed in this paper.As will be seen below, the energy density σ of the thin shell is negative. Moreover, given that the shell is assumed to be infinitely thin, the radial pressure p is zero.So ρ +p=σ +0<0, so that the null energy condition is automatically violated.Our final goal in this paper is to determine criteria for making this new type of wormhole stable to linearized radial perturbations.§ THIN-SHELL WORMHOLE CONSTRUCTION Consider the line elementds^2 = -f(r) dt^2 + [f(r)]^-1dr^2 + h(r)(dθ^2+sin^2θ dϕ^2),where f(r) is a positive function of r.As in Ref. <cit.>, the construction begins with two copies of a black-hole spacetime and removing from each the four-dimensional regionΩ^± = {r≤ a | a>r_h},where r=r_h is the (outer) event horizon of the black hole.Now identify (in the sense of topology) the time-like hypersurfaces∂Ω^± ={r=a | a>r_h}.The resulting manifold is geodesically complete and possesses two asymptotically flat regions connected by a throat.Next, we use the Lanczos equations <cit.>S^i_ij=-1/8π([K^i_ij]-δ^i_ij[K]),where S^i_ij is the surface stress-energy tensor, K^i_ij is the extrinsic curvature tensor, [K_ij]=K^+_ij-K^-_ij and [K] is the trace of K^i_ij.In terms of the surface energy density σ and the surface pressure 𝒫, S^i_ij=diag(-σ, 𝒫, 𝒫).The Lanczos equations now yieldσ=-1/4π[K^θ_θθ]and𝒫=1/8π([K^τ_ττ] +[K^θ_θθ]). A dynamic analysis can be obtained by letting the radius r=a be a function of time <cit.>.As a result,σ = - 1/2π a√(f(a) + ȧ^2)and𝒫 =-1/2σ + 1/8π2ä + f^'(a) /√(f(a) + ȧ^2).Here the overdots denote the derivatives with respect to proper time τ.It is easy to check that σ and 𝒫 obey the conservation equationd/dτ(σ a^2)+𝒫d/dτ(a^2)=0,which can also be written in the formdσ/da + 2/a(σ+𝒫) = 0. To perform a stability analysis, we must first note that for a static configuration of radius a, we have ȧ=0 and ä=0.We must also consider linearized fluctuations around a static solution characterized by the constants a_0, σ_0, and 𝒫_0.Now, given the EoS 𝒫=ωσ, Eq. (<ref>) can be solved by separation of variables to yield|σ(a)|=|σ_0|(a_0/a)^2(ω+1),where σ_0=σ(a_0).The solution can therefore be written asσ(a)=σ_0(a_0/a)^2(ω+1),σ_0=σ(a_0). The next step is to rearrange Eq. (<ref>) to obtain the “equation of motion"ȧ^2 + V(a)= 0,where V(a) is the potential definedasV(a) =f(a) - [2π a σ(a)]^2.Taylor-expanding V(a) around a=a_0, we obtainV(a)= V(a_0) + V^'(a_0) ( a - a_0) + 1/2 V^''(a_0) ( a - a_0)^2 +higher-order terms.Since we are linearizing around a=a_0, we require that V(a_0)=0 and V'(a_0)=0.Since the higher-order terms are considered negligible, the configuration is in stable equilibrium if V”(a_0)>0. § THE REGULAR HAYWARD BLACK HOLE The following line element describes a spherically symmetric regular (nonsingular) black hole:ds^2=-(1-2mr^2/r^3+2ml^2)dt^2+(1-2mr^2/r^3+2ml^2)^-1dr^2 +r^2(dθ^2+sin^2θdϕ^2),Introduced by Hayward <cit.>, this is referred to as a Hayward black hole in Refs. <cit.>. Line element (<ref>) contains two free parameters, l and m; l is called the Hayward parameter, while m will be interpreted as the mass of the black hole.Thin-shell wormholes from Hayward black holes are discussed in Refs. <cit.>.It is apparent that for large r, the Hayward black hole becomes a Schwarzschild spacetime. To study the behavior for small r, we first rewrite line element (<ref>) in a form that is particularly convenient for later analysis:ds^2=-(1-2(r/m)^2/(r/m)^3+2(l/m)^2)dt^2+(1-2(r/m)^2/(r/m)^3+2(l/m)^2)^-1dr^2+r^2(dθ^2+sin^2θdϕ^2). Writing the g_tt-term in the form 1-(r/m)^2/(l/m)^2+(r/m/l/m)^51/2l/m/1+1/2(r/m)^3/(l/m)^2,it follows that for small r,f(r)=1-(r/m)^2/(l/m)^2 +𝒪(r/m/l/m)^5.The point is that for small r, the Hayward black hole has the form of a de Sitter black hole, immediately raising the question whether the resulting thin-shell wormhole could ever be stable to linearized radial perturbations.Ref. <cit.> discusses the de Sitter case f(r)= 1-2m/r-1/3Λ r^2, where Λ >0 is the cosmological constant.A stable solution requires that 1/3Λ m^2<1/27. Now, Refs. <cit.> note that for a Hayward black hole, there is a critical value (l/m)^2= 16/27 corresponding to a regular extremal black hole.The smaller value (l/m)^2<16/27 admits a double horizon.If (l/m)^2>16/27, there is no event horizon at all, a point that will be addressed later.For now, it is sufficient to observe that the values for 1/(l/m)^2 in Eq. (<ref>) are much larger than the allowed value of 1/27 from Ref. <cit.>.So assuming the EoS 𝒫=ωσ, there are no stable solutions for the de Sitter case. (Since the Hayward black hole becomes a Schwarzschild spacetime, nor are there stable solutions for large r <cit.>.) § EXTENDING THE HAYWARD BLACK HOLE At this point we need to return to Sec. <ref>.In particular, making use of Eqs. (<ref>), (<ref>), and (<ref>), we obtainV(a)=1-2(a/m)^2/(a/m)^3+2(l/m)^2-[1-2(a_0/m)^2/(a_0/m)^3+2(l/m)^2](a_0/a)^2+4ω.Evidently, V(a_0)=0.To meet the condition V'(a_0)=0, we differentiate V(a) in Eq. (<ref>) and solve for ω:ω =-1/2+1/4[(a_0/m)^3+2(l/m)^2](4)(a_0/m)^2-6(a_0/m)^5/[(a_0/m)^3+2(l/m)^2][(a_0/m)^3+2(l/m)^2-2(a_0/m)^2]. Since our spacetime is approximately de Sitter only for small r, we will confine ourselves to relatively small values of the shell radius.As already noted, values of the Hayward parameter below the critical value yield a black hole with two event horizons, but the resulting thin-shell wormholes are unstable.As we will see in the next section, however, the calculations using the Hayward line element do not require any particular restriction on l/m.In fact, stable solutions can be obtained for values above the critical value.The implication is that even though the line element retains its original form, we are no longer dealing with a black hole but instead with an ordinary, possibly compact, stellar object.More precisely, line element (<ref>) becomesds^2=-(1-2M(r)/r)dt^2+(1-2M(r)/r)^-1dr^2 +r^2(dθ^2+sin^2θdϕ^2),whereM(r)=r^3m/r^3+2ml^2,M(0)=0,showing that Eq. (<ref>) represents an ordinary mass such as a particle or a star <cit.>.(We will see below that we are dealing with a special class of compact stellar objects.)We are going to obtain some stable solutions in the next section. § STABLE SOLUTIONS; COMPACT STELLAR OBJECTS To illustrate the type calculation required, suppose we choose a_0/m=2 and l/m= √(16/27), the critical value.Then from Eq. (<ref>), we get ω=-1.534274 and 2+ω =-4.13710.Eq. (<ref>) then givesV(a)=1-2(a/m)^2/(a/m)^3+2(16/27)-[2(2^2)/2^3+2(16/27)](2/a/m)^-4.13710. The resulting graph in Fig. 1 is concavedown around a_0/m=2, showing that the wormhole is unstable.As another example, if a_0/m=3.5 and l/m=1.7, we obtain 2(l/m)^2=5.78, 2+4ω =-0.6528, andV(a)=1-2(a/m)^2/(a/m)^3+5.78-[1-2(3.5)^2/3.5^3+5.78](3.5/a/m)^-0.6528.The graph, shown in Fig. 2, is concave uparound a_0/m=3.5.So the wormhole is stable.Table 1 provides an overview of the results for various values of a_0/m, starting with a_0/m=1.5 by showing the dependence on the Hayward parameter: to ensure stability, a larger value of a_0/m requires a larger value of l/m.(For a_0/m≤ 1, we get only unstable solutions.) Since we are dealing with compact objects rather than black holes, we need to check the physical plausibility.Consider a typical neutron star having a radius ranging from 11 km to 11.5 km and a mass of 2M_⊙≈ 3 km. Then a_0/m ranges from 3.67 to 3.83, which are well within the values listed in Table 1. In fact, it is theoretically possible to have a stable thin shell right above the surface with a_0/m=4 and l/m=2.0 according to Table 1.Other compact objects such as quark stars or strange stars could allow even smaller values for a_0/m. § CONCLUSIONThis paper introduces a new type of thin-shell wormhole constructed from a special class of compact stellar objects rather than black holes.This finding follows from a discussion of the stability to linearized radial perturbations of an extended version of a regular Hayward black hole.Assuming the equation of state 𝒫=ωσ, ω <0, for the exotic matter on the thin shell, it is shown that whenever the value of the Hayward parameter is below its critical value, no stable solutions can exist.Stable solutions are obtained, however, if l/m is allowed to exceed the critical value, thereby eliminating the event horizon of the black hole.The resulting underlying structure supporting the thin-shell wormhole is a compact object rather than a black hole.The findings are consistent with the properties of neutron stars, as well as other compact stellar objects. 20mV89M. Visser, “Traversable wormholes fromsurgically modified Schwarzchild spacetimes,"Nucl. Phys. B 328, 203 (1989). MT88M.S. Morris and K.S. Thorne, “Wormholesin spacetime and their use for interstellar travel:A tool for teaching general relativity,"Amer. J. Phys.56, 395 (1988). eE09E.F. Eiroa, “Thin-shell wormholes witha generalized Chaplygin gas," Phys. Rev. D 80,044033 (2009). pK12P.K.F. Kuhfittig, “The stability ofthin-shell wormholes with a phantom-like equation ofstate," Acta Phys. Polon. B 41, 2017 (2010). PV95E. Poisson and M. Visser, “Thin-shellwormholes: Linearized stability," Phys. Rev. D 52, 7318 (1996). LC04F.S.N. Lobo and P. Crawford, “Linearizedstability analysis of thin-shell wormholes with acosmological constant," Class. Quant. Grav. 21,391 (2004). ER04E.F. Eiroa and G.E. Romero, “Linearized stability of charged thin-shell wormholes," Gen. Rel.Grav. 36, 651 (2004). TSE06M. Thibeault, C. Simeone, and E.F. Eiroa,“Thin-shell wormholes in Einstein-Maxwell theorywith a Gauss-Bonnet term," Gen. Rel. Grav.38, 1593 (2006). RKC06F. Rahaman, M. Kalam, and S. Chakraborty,“Thin shell wormholes in higher dimensionalEinstein-Maxwell theory," Gen. Rel. Grav.38, 1687 (2006). RKC07F. Rahaman, M.Kalam, and S. Chakraborty,“Gravitational lensing by a stable C-fieldwormhole," Chin. J. Phys. 45, 518 (2007). RS07M.G. Richarte and C. Simeone,“Thin-shell wormholes supported by ordinarymatter in Einstein-Gauss-Bonnet gravity,"Phys.Rev. D 76, 087502 (2007). LL08J.P.S. Lemos and F.S.N. Lobo, “Planesymmetric thin-shell wormholes: Solutions andstability," Phys. Rev D 78, 044030 (2008). sH06S.A. Hayward, “Formation and evaporationof nonsingular black holes," Phys. Rev. Lett.96, 031103 (2006). HOHM. Halilsoy, A. Ovgun, and S. Habib Mazharimousavi,“Thin-shell wormholes from regular Hayward black hole,"Eur. Phys. J. C 74, 2796 (2014). SMM. Sharif and S. Mumtaz, “Stabilty of regular Hayward thin-shell wormholes," Ad. High Energy Phys.2016, 2868750, (2016). MTWC.W. Misner, K.S.Thorne, and J.A. Wheeler,Gravitation. W. Freeman and Company, New York (1973)p. 608. | http://arxiv.org/abs/1709.09142v3 | {
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Department of Mathematics Princeton University Princeton, NJ 08544Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139D.K. was partially supported by an NSF Postdoctoral Research fellowship as well as ERC-2011-StG-278940. Y.L. was partially supported by NSF grant DMS-1711053. [email protected] [email protected] On the existence of unstable minimal Heegaard surfaces Yevgeny Liokumovich====================================================== We prove that for generic metrics on a 3-sphere, the minimal surface obtained from the min-max procedure of Simon-Smith has index 1.We prove an analogous result for minimal surfaces arising from strongly irreducible Heegaard sweepouts in 3-manifolds. We also confirm a conjecture of Pitts-Rubinstein that a strongly irreducible Heegaard splitting in a hyperbolic three-manifold can either be isotoped to a minimal surface of index at most 1 or else after a neck-pinch is isotopic to a one-sided minimal Heegaard surface. ℝ ℝℙ ℕ ℤ ℂ ℳ ℋ̋ Łℒ𝒱 Σ Γ𝐱 𝐳 γ 𝐤̨ 𝐯̌ |̊x̊|̊ |∂_r^⊤| |∂_r^| |∂_r| vol dmnRe HessRic areadℋ^n ḏℋ̱^̱ṉ-̱1̱ ḍịṿ dist ℋ^2 ℋ^1spt Proj łloc *conjConjecture *thmaTheorem A *thmbTheorem B *thmcTheorem C *thmdTheorem D thmTheorem[section] lemma[thm]Lemma cor[thm]Corollary prop[thm]Proposition remark *rmkRemark example[thm]Example definition defi[thm]Definition § INTRODUCTIONThe min-max theory was introduced by Almgren in the 60s and then later completed by Pitts in the 80s to construct embedded minimal hypersurfaces in Riemannian manifolds. Roughly speaking, one considers sweepouts of a manifold and the longest slice in a tightened sweepout that “pulls over" the entire manifold gives a minimal surface.Recently, the min-max theory has led to proofs of long-standing problems, for instance the proof of the Willmore Conjecture by Marques and Neves <cit.>.Almgren-Pitts' approach considers very general sweepouts and it is difficult to control the topology of the minimal surface obtained. In the 80s by Simon-Simon refined Pitts' arguments to allow one to consider sweepouts of a 3-manifold by surfaces of a fixed topology.For instance, given 𝕊^3 one can consider sweepouts by embedded two-spheres.Simon and Smith proved that one can work in this restricted class of sweepouts and still obtain a closed embedded minimal surface but with control on the topology.It was proved in <cit.> that the topology of the limiting minimal surfaces is achieved roughly speaking after finitely many neck-pinches.A basic question is to understand how the Morse index of the minimal surface obtained in either approach is related to the number of parameters used in the construction.Roughly speaking, a k parameter family should produce an index k critical point. Suprisingly, the question of estimating the Morse index had been left open since Pitts' original work. In the Almgren-Pitts setting, recently Marques-Neves <cit.> made the first advance on this problem byproving that when the ambient metric is generic (i.e., bumpy in the sense of White <cit.>) and contains no non-orientable embedded minimal surfaces,that the support of the minimal surface obtained has index 1 when one considers one-parameter sweepouts.In other words, precisely one component is unstable with index 1 and the other components are all stable.Under the same hypotheses, we prove in this paper that in the Simon-Smith setting, when running a min-max procedure with two-spheres in 𝕊^3, the support of the min-max limit has index 1.The work of Marques-Neves <cit.> involves three components: an upper index bound, a lower index bound, and the fact that the unstable component is achieved with multiplicity 1.While the first of these generalizes easily to the Simon-Smith setting, the second and third require new interpolation results. The main technical contribution of this paper is an interpolation result that rules out convergence of a min-max sequence to a stable minimal surface.In this paper, we also confirm a long-standing conjecture of Pitts-Rubinstein: namely to show that in a hyperbolic manifold, if a Heegaard surface is strongly irreducible then it can be isotoped to be an index 1 minimal surface (or else after neck-pinch to the boundary of a twisted interval bundle over a one-sided Heegaard surface).See Theorem <ref> for a precise statement.Let us now state our results.For this we need a number of definitions.Given a Heegaard splitting H of M, a sweepout by Heegaard surfaces or sweepout is a one parameter family of closed sets {Σ_t}_t∈ [0,1] continuous in the Hausdorff topology such that* Σ_t is an embedded smooth surface isotopic to H for t∈(0,1)* Σ_t varies smoothly for t∈ (0,1)* Σ_0 and Σ_1 are 1-d graphs, each one a spine of one of the handlebodies determined by the splitting surface H. If Λ is a collection of sweepouts, we say that the set Λ is saturated if given a map ϕ∈ C^∞(I× M,M) such that ϕ(t,-)∈Diff_0 M for all t∈ I, and a family{Σ_t}_t∈ I∈Λ, we have{ϕ(t,Σ_t) }_t∈ I∈Λ.Given a Heegaard splitting H, let Λ_H denote the set of all sweepoutsby Heegaard surfaces {_t }, such that the corresponding family of mod 2 flat 2-cycles is not contractible relative to ∂ [0,1] = {0,1}. Λ_H is a saturated family of sweepouts.The width associated to Λ_H is defined to beW(M,Λ_H)=inf_{Σ_t}∈Λsup_t∈ Iℋ^2(Σ_t),where ℋ^2 denotes 2-dimensional Hausdorff measure.It follows by an easy argument using the isoperimetric inequality (Proposition 1.4 in <cit.>) that W_H>0.This expresses the non-triviality of the sweepout.A minimizing sequence is a sequence of families {Σ^n_t}∈Λ_H such thatlim_n→∞sup_t∈[0,1]ℋ^2(Σ^n_t)=W(M,Λ_H).A min-max sequence is then a sequence of slices Σ^n_t_n, t_n∈ (0,1) such thatℋ^2(Σ^n_t_n)→ W(M,Λ_H). The main result due to Simon-Smith is that some min-max sequence converges to a smooth minimal surface realizing the width, whose genus is controlled.Some genus bounds were proved by Simon-Smith, but the optimal ones quoted below were proved in <cit.>: Let M be a closed oriented Riemannian 3-manifold admitting a Heegaard surface H of genus g. Then some min-max sequence Σ_t_i^i of surfaces isotopic to H converges as varifolds to ∑_j=1^k n_j Γ_j, where Γ_j are smooth embedded pairwise disjoint minimal surfaces and where n_j are positive integers.Moreover,W(M, Λ_H)=∑_j=1^k n_jℋ^2(Γ_j).The genus of the limiting minimal surface can be controlled as follows:∑_i∈ O n_ig(Γ_i) + 1/2∑_i∈ N n_i(g(Γ_i)-1)≤ g,where O denotes the set of i such that Γ_i is orientable, and N the set of i such that Γ_i is non-orientable, and g(Γ) denotes the genus of Γ.The genus of a non-orientable surface is the number of cross-caps one must attach to a two-sphere to obtain a homeomorphic surface.In particularBy sweeping out a Riemannian three-sphere by two-spheres, we obtain the existence of a family {Γ_1,...,Γ_k} of pairwise disjoint smooth embedded minimal two-spheres.Marques-Neves <cit.> recently obtained upper index bounds for the min-max minimal surface obtained in Theorem <ref> and <ref>:In the setting of Theorem <ref> suppose in addition that the metric is bumpy, there holds∑_i=1^kindex(Γ_i) ≤ 1. Recall that a metric is bumpy if no immersed minimal surface contains a non-trivial Jacobi field.White proved <cit.> that bumpiness is a generic property for metrics.In particular, any metric can be perturbed slightly to be bumpy.Our main result is the following equality in the case of spheres:In the setting of Theorem <ref>, suppose in addition that the metric is bumpy.Then the min-max limit satisfies:∑_i=1^kindex(Γ_i) = 1. If the metric is not assumed to be bumpy then we obtain the existence of a minimal surface satisfying (<ref>) and ∑_i=1^kindex(Γ_i) ≤1 ≤∑_i=1^kindex(Γ_i)+∑_i=1^knullity(Γ_i)The index bounds (<ref>) and (<ref>) were conjectured explicitly by Pitts-Rubinstein <cit.> in 1986.In particular we have the following:Let M be a Riemannian 3-manifold diffeomorphic to 𝕊^3 endowed with a bumpy metric.Then M contains an embedded index 1 minimal two-sphere. A long-standing problem is to prove that in a Riemannian three-sphere M, there are at least four embedded minimal two-spheres.This is the analog of Lusternick Schnirelman's result about the existence of three closed geodesics on two-spheres. If M contains a stable two-sphere, then Theorem <ref> implies the following (by considering the three-balls on each side of this two-sphere): Let M be a Riemannian three-sphere containing a stable embedded two-sphere.Then M contains at least two index one minimal two-spheres.Thus M contains at least three minimal two-spheres. For results in the case when M contains no stable two-spheres, see <cit.>.For strongly irreducible Heegaard splittings, we can use an iterated min-max procedure to obtain:Let Σ be a strongly irreducible Heegaard splitting surface in a manifold endowed with a bumpy metric.Then from an iterated min-max procedure we obtain the existence of a family of pairwise disjoint minimal surfaces {Γ_1,...,Γ_k} obtained from Σ after neck-pinch surgeries,so that ∑_i=1^kindex(Γ_i) = 1. Using Theorem <ref> together with the Catenoid Estimate <cit.>, we obtain: Let M be a Riemannian 3-manifold diffeomorphic to ℝℙ^3 endowed with a bumpy metric.Then M contains a minimal index 1 two-sphere or minimal index 1 torus.In the special case that ℝℙ^3 is endowed with a metric of positive Ricci curvature, it was proved in <cit.> that it contains a minimal index 1 torus.In this paper, we also confirm a long-standing conjecture of Pitts-Rubinstein <cit.> in hyperbolic manifolds: Let M be a hyperbolic 3-manifold and Σ a strongly irreducible Heegaard surface.Then either* Σ is isotopic to a minimal surface of index 1 or 0 or* after a neck-pinch performed on Σ, the resulting surface is isotopic to the boundary of a tubular neighborhood of a stable one sided Heegaard surface.If M is endowed with a bumpy metric, in case (1) we can assume the index of Σ is 1.Recall that a one-sided Heegaard surface Σ embedded in M is a non-orientable surface such that M∖Σ is an open handlebody.An example is ℝℙ^2⊂ℝℙ^3 as ℝℙ^3∖ℝℙ^2 is a three-ball.A Heegaard splitting is strongly irreducible if every curve on Σ bounding an essential disk in H_1 intersects every such curve bounding an essential disk in H_2.Strongly irreducible Heegaard splittings were first introduced by Casson-Gordon <cit.>, who proved that in non-Haken 3-manifolds, any splitting can be reduced until it is strongly irreducible.Thus lowest genus Heegaard splittings in any spherical space form are strongly irreducible.Even though they are not hyperbolic manifolds, we still obtainLet L(p,q) be a lens space other than ℝℙ^3. Then L(p,q) contains an index 1 or 0 Heegaard torus.If the metric is assumed bumpy, then the index can be assumed to be 1.As sketched by Rubinstein <cit.>, Theorem <ref> gives a minimal surface proof of Waldhausen's conjecture:Let M be a non-Haken hyperbolic 3-manifold.Then M contains finitely many irreducible Heegaard splittings of any given genus g. Waldhausen's conjecture was proved by Tao Li (<cit.>,<cit.>, <cit.>) using the combinatorial analog of index 1 minimal surfaces – almost normal surfaces.For effective versions of Theorem <ref>, see <cit.>, <cit.>. The organization of this paper is as follows. In Section <ref> we explain the main ideas and difficulties in our Interpolation Theorem, which roughly speaking allows us to interpolate between a surface Γ close as varifolds to a union of strictly stable minimal surface with integer multiplicities (Σ=∑ n_iΣ_i) and something canonical. In Section <ref> we consider the case that Σ is connected, and show how to isotope Γ to a union of several normal graphs over Σ joined by necks.In Section <ref> we describe the notion of “root sliding" which is useful for global deformations. In Section <ref> we introduce the Light Bulb Theorem and its generalizations which enable us to find necks to further reduce the number of graphs of Γ over Σ.In Section <ref> we generalize to the setting when Σ consists of several components. In Section <ref> we apply our interpolation result to obtain the lower index bound.In Section <ref> we use the index bounds, together with some observations regarding nested minimal surfaces and a characterization of minimal surfaces bounding small volumes to prove the conjecture of Pitts-Rubinstein. During the preparation of this article Antoine Song <cit.> obtained some related results. Acknowledgements: We would like to thank Andre Neves and Fernando C. Marques for their encouragement and several discussions.We thank Francesco Lin for some topological advice.D.K. would like to thank Toby Colding for suggesting this problem.We are grateful to Dave Gabai for several conversations. § INTERPOLATION In the proof of the lower index bound (Theorem <ref>) to rule out obtaining a stable surface with multiplicity, we must deform slices of a sweepout that come near such a configuration. To that end, the main technical tool is to deform a sequence close in the flat topology to a stable minimal surface with multiplicity to something canonical. §.§ Marques-Neves squeezing map. Let ⊂ M be a smooth two-sided surface and let exp_: Σ× [-h,h] → M denote the normal exponential map.Let N_ε(S) = exp_(Σ× [-ε,ε]) denote an open ε-tubular neighbourhood of submanifold S ⊂ M.It will be convenient for the purposes of this paper to foliate an open neighbourhood ofnot by level sets of the distance function, but rather by hypersurfaces with mean curvature vector pointing towards , which arise as graphsof the first eigenfunction of the stability operator over .Such a foliation gives rise to a diffeomorphism ϕ: × (-1,1) →Ω_1 ⊂ N_h(Σ), a collection of open neighbourhoods Ω_r = ϕ(Σ× (-r,r)) and squeezing maps P_t(ϕ(x,s)) = ϕ(x,(1-t)s). Let P: Ω_1 →Σ denote the projection map P(ϕ(x,s)) = x. We refer to <cit.> for the details of this construction. We summarize properties of the map P_t: * P_0(x) = x for all x ∈Ω_1 and P_t(x)=x for all x ∈Σ and 0 ≤ t< 1;* There exists h_0>0, such that N_h_0⊂Ω_1 and for all positive h<h_0 there exists t(h) ∈ (0,1)withP_t(h)(N_h_0)⊂ N_h;*For any surface S ⊂Ω_1 and for allt ∈ [0,1) we haveArea(P_t(S)) ≤ Area(S) with equality holding if and only if S ⊂Σ; *Let U ⊂Σ be an open set, f:U →ℝ be a smooth function with absolute value bounded by h_0 and let S = {ϕ(x,f(x)): x ∈ U }. Then we have a graphical smooth convergence of P_t(S) to U asas t → 1. Property (3) is proved in<cit.>. All other properties follow from the definition.The importance of the above is that we can use the squeezing map to pusha surface S in a small tubular neighbourhood of Σ towards Σ while simultaneously decreasing its area.In the rest of the paper we will say that a surface S is graphical if it satisfies S = {ϕ(x,f(x)): x ∈ U } for some function f and asubset U ⊂Σ.§.§ The case of connected stable minimal surface. The following is a special case of our main interpolation result.Setting g=0 in the statement of the proposition and assuming Γ is connected, one can interpret it as a quantitative form of Alexander's Theorem.Yet another way to interpret it is as a kind of Mean Curvature Flow performed “by hand."Letbe a smooth connected orientable surface of genus g, with a map P satisfying (1) and (2) above. Let ⊂ N_h_0() be a smooth embeddedsurface, such that each connected component ofhas genus at most g. For every δ>0there exists an isotopy _t ⊂N_h_0() with* _0 =* For each connected component 'of Γ_1 either '= ϕ (Σ× t) for some t ∈ [-1,1]or ' is contained in a ballof radius less than δ * Area(_t) ≤ Area()+δ for all t. The reason that the δ-constraint is important in (3) is that we will be gluing this isotopy into sweepouts with maximal area approaching the width W and we want the maximal area of the resulting sweepout to still be W.It follows from Alexander's theorem that any embedded two-sphere in N_ε(Σ)≅ S^2× [0,1] can be isotoped to either a round point or else to Σ itself.The difficulty is to obtain such an isotopy obeying the area constraint (<ref>).It is instructive to consider the analogous question in ℝ^3 to that addressed inProposition <ref>.Suppose one is given two embeddings Σ_0 and Σ_1 of two-spheres into ℝ^3. We can ask whether for any δ>0 there exists an isotopy Σ_t from Σ_0 and Σ_1 obeying the constraint (assuming |Σ_1|>|Σ_0|):|Σ_t|≤ |Σ_1|+δ t. It is easy to see that the answer is “yes."Namely, one can even do better and find an isotopy satisfying |Σ_t|≤ |Σ_1|t. To see this, one can first enclose Σ_0 and Σ_1 in a large ball about the origin B_R. By Alexander's theorem there is an isotopy ϕ_t between Σ_0 and Σ_1 increasing area by a factor at most A along the way. First shrink B_R into B_R/A, then perform the shrunken isotopy (1/A)ϕ_t on B_R/A, and then rescale back to unit size.Of course, in 3-manifolds that we must deal with in Proposition <ref> are 𝕊^2× [0,1] in which one does not have good global radial isotopies to exploit.However, the same idea of shrinking still applies if we first work locally in small balls to “straighten" our surface.We can also use the squeezing map to repeatedly press our surface closer to Σ in the flat topology while only decreasing area.Let us explain the ideas in our proof of Proposition <ref> in more detail. There are two main steps.In the first, we introduce a new local area-nonincreasing deformation process in balls.The end result of applying this process in multiple balls centered around Σ is to produce an isotopic surface Γ consisting of k parallel graphical sheets to Σ joined by (potentially very nastily) nested, knotted and linked tubes. The local deformation we introduce exploits the fact that in balls, we can using Shrinking Isotopies to “straighten" the surface while obeying the area constraint (a similar idea was used by Colding-De Lellis <cit.> in proving the regularity of 1/j-minimizing sequences). Our deformation process is a kind of discrete area minimizing procedure, somewhat akin to Birkhoff's curve shortening process.In the process, it“opens up" any folds or unknotted necks that are contained in a single ball. However,at this stage we can not open necks like on Figure <ref>.After the first stage of the process, we are left with k parallel graphical sheets arranged about Σ joined by potentially very complicated necks.If k is 1 or 0, the proposition is proved. If not, the second step is to use a global deformation to deform the surface through sliding of necks to one in which two parallel sheets are joined by a neck contained in a single ball.Then we go back to Step 1 to open these necks.After iterating, eventually k is1 or 0.The second stage is complicated by the fact that the necks joining the various sheets can be nastily nested, knotted or linked. We we need generalizations of the Light Bulb Theorem in topology to untangle this morass of cables and find a neck to open. The version of the Light Bulb Theorem that will be most useful to us is the following (see Theorem <ref>). Given a 3-manifold M and two arcs, α and β, with boundary points in ∂ M assume that one of the boundary points of α lies in the boundary component of M diffeomorphic to a sphere.Then α and β are isotopic as free boundary curves if and only if they are homotopic as free boundary curves. We will apply this theorem in the situation when α is a core arc of a “cable", a collection of (partially) nested necks in the tubular neighbourhood of α. In section <ref> we define cables and prove some auxiliarylemmas which allows us to treat these collections of tubes almost as if it was an arc attached to the surface. §.§ The case of multiple connected components. Proposition <ref> deals with the situation when surfaceis contained in a tubular neighbourhood of a connectedstable minimal surface . In general, we need to consider a situation when clustersaround a minimal surfacethat has multiple connected components. This is illustrated on Figure <ref>. Surfaceis mostly contained in the tubular neighbourhood of minimal surfaces _1 and _2, while the part ofoutside of N_h(_1 ∪_2) looks like a collection of thin tubes that can link with each other and knot around handles of _1. In this setting we prove the following proposition.Let _1, ..., _k be pairwise disjoint embedded two-sided stable minimal surfaces in a 3-manifold M and denote Σ:=∪_i_i.There exist ε_0 >0 and h_0>0 such that for all ε∈ (0, ε_0) and h ∈ (0,h_0) the following holds. Ifsatisfies a) Area (∖ N_h())< ε;b) genus(N_h(_i) ∩) ≤ genus(_i) for each i;then for every δ>0there exists an isotopy _t with* _0 =* Area(_1 ∖ N_h()) < δ * Area(_t) ≤ Area()+δ for all t. * Γ_1 is a surface consisting of some subcollection of the set , joined by thin necks.We describe ideas involved in the proof of Proposition <ref>. First, we can use a version of Almgren's pull-tight flow together with maximum principle for stationary varifolds to make the area ofoutside of N_h() arbitrarily small. We are grateful to Andre Neves from whom we learnedof arguments of this type. For each connected componentwe can intersectwith the tubular neighborhood N_h() and glue in small discsto the boundary components of N_h() ∩ so as to obtain a closed (possibly disconnected) surface. Then we can apply Proposition <ref> to deform this surface into disjoint graphical copies of _i. Of course, we are not allowed to actuallydo any surgeries on . Instead, we perform deformations ofProposition <ref> while simultaneously moving thin necks attached to the surface to preserve continuity.After the surface has been deformed into a canonical form in the neighborhood of each connected component _i, we need a global argument, showing that one can always finda neck that can be unknotted, using Generalized Light Bulb Theorem, and slide into the neighborhood of one of the _i's. This process terminates only when for each i surfaceeither avoids the neighbourhood of _i or looks like a single copy of _i with thin necks attached.Note that unlike in the setting considered by Marques-Neves (Appendix A of <cit.>), it is very important that we keep track of the part of the pulled-tight surface outside of the tubular neighborhood, as the neck we may ultimately need to find may pass through the complement of the tubular neighborhood.See Figure <ref> for an illustration of a case where this is necessary.§ DEFORMATION IN THE NEIGHBORHOOD OF A CONNECTED STABLE MINIMAL SURFACE §.§ Stacked surface. Let ⊂ N_h_0(). Given δ>0 we will say thatis a (δ,k)-stacked surface if there exists a decomposition = D ⊔ Y ⊔ Xwith the following properties:a) D= _i=1^m D_i, where eachD_i ⊂D_i, and {D_i} is a collection of disjoint graphs over , D_i= {ϕ(x,f_i(x))|x ∈}, satisfyingArea(D_i ) < Area() + δ/10k;b) Area(D_i ∖ D_i) < δ/10;c) each connected component of Y is a the boundary of a small tubular neighbourhood ofan embedded graph, and their total area is at most δ/10;d) X is a disjoint union of closed surfaces of total area less than δ/10, each contained in a ball of radius less than √(δ)/10.We can order the punctured surfaces D_i to have descending height relative to a fixed unit normal on Γ, with D_1 the top-most.Let us call D_i the ith sheet.Let us call Y the thin part of. Letbe a strictly stable two-sided connected minimal surface and ⊂ N_h_0() be a (not necessarily connected) smooth surface. For any δ>0 there exits k>0 and an isotopy {_t} with _0 =, _1 is (δ,k)-stacked and Area(_t) ≤ Area() + δ for all t.§.§ Choice of radius r and open neighbourhood Ω_h Consider the projected current P(Γ) supported on Σ.There exists r_1>0 so thatfor any r≤ r_1 the mass of P(Γ) in any ball B_r(x) (with x∈Σ) is less than δ/200. By continuity, we can choose t_1 close enough to 1 so thatfor every t ∈ [t_1,1) the mass of P_t(Γ) ⊂Ω_1 - t_1 in any ball B_r(x) is at most δ/100 for any r<r_1 and x∈Σ. We replace Γ with P_t_1(Γ) (but do not relabel it).Let h ∈ (0, 1 - t_1), so that Γ⊂Ω_h. Now we pick r= r(, , δ)>0, satisfying the following properties:1) r is smaller than the minimum of the convexityradii of M and ;2) r < r_1, that is, for every x ∈ and a ball B_r(x) of radius r we have that Area(∩ B_r(x)) < 1/100δ;3) for every x ∈ and a ball B_r(x) of radius r we have that the exponential map exp: B^Eucl_r(0) → B_r(x) satisfies 0.99 < |d exp_y| < 1.01 for all y ∈ B_r(x). §.§ Choice of triangulation and constant c. Fix a triangulation of , so that for each 2-simplex S_i, 1 ≤ i ≤ m, in the triangulation there exists a point p_i ∈ S_i withS_i ⊂ B_r/3(p_i). Assume that the number is chosen so that S_i+1 and S_i share an edge. We cover Ω_h() by a collection of cells {Δ_i = ϕ (S_i × [-(1-t_1),(1-t_1)]) }. The interiors of Δ_i's are disjoint and each Δ_i is contained in a ballB_r/2(p_i). Let C_i =ϕ ( ∂ S_i × [-h,h]) ⊂∂Δ_i. Let c = min{ Area(∩Δ_i)}. By applying squeezing map P_t we may assume that Ω_h ⊂ N_r/10(Σ) andis contained in the union of Δ_i.We will first need to prove a local version of Lemma <ref>. Namely, we will show thatcan be deformed into certain canonical form in each cell Δ_i.We introduce several definitions.§.§ Essential multiplicity.Suppose r' ∈ (r/2, r), ⊂Ω_h() and assumeintersects ∂ B_r/2(p_i) and ∂ B_r(p_i) transversally. Let 𝒮(,p_i, r') denote the set of surfacesS ⊂Ω_h(),such that S intersects ∂ B_r/2(p_i) transversally andthere exists an isotopy from S tothrough surfaces S' such that S' ∖ int(B_r'(p_i)) = ∖ int(B_r'(p_i)). Let k(S) denote the number of connected components of S ∩∂ B_r/2(p_i), which are not contractible in ∂ B_r/2(p_i) ∩Ω_h(). We define the essential multiplicity ofin B_r/2(p_i) to bek_ess(,i,r') = inf{k(S)| S ∈𝒮(,p_i,r') }.We have the following lemma.Area(∩ B_r'(p_i)) ≥ k_ess(,p_i,r') Area(∩ B_r'(p_i)) - O(h).By coarea inequality Area(∩ B_r'(p_i)) ≥∫_ρ=0^r' L(∂ B_ρ(p_i) ∩) d ρ. For almost every ρ we have that the number ofconnected components of ∩∂ B_ρ(p_i), which are not contractible in ∂ B_ρ(p_i) ∩Ω_h() is at least k_ess(,i,r'). Indeed, otherwise we could radially isotopto obtain a surfacewith fewer non-contractible components of ∩∂ B_r/2(p_i), contradicting the definition of k_ess(,i,r').We have L(∂ B_ρ(p_i) ∩) ≥ L(∂ B_ρ(p_i) ∩) - O(h) and the lemma follows. Note that it may happen that the relative map (∩ B_r'(p_i), ∩∂ B_r'(p_i)) → (B_r'(p_i),∂ B_r'(p_i))is null-homotopic (but not null-isotopic) and yet k_ess(,i) ≠ 0 (see Fig. <ref>).§.§ Surfaces stacked in a cell.An embedded surface S is (δ,k)-stacked in a cell Δ_i if there exists a decomposition S ∩Δ_i = D ⊔ Y ⊔ Xwith the following properties:a) D= _i=1^m D_i, where eachD_i ⊂D_i, and {D_i} is a collection of disjoint graphs over S_i, D_i= {ϕ(x,f_i(x))|x ∈ S_i }, satisfyingArea(D_i ) < Area(S_i) + δ/10k;b) Area(D_i ∖ D_i) < δ/10;c) each connected component of Y is a boundary of a small tubular neighbourhood ofan embedded graph, and their total area is at most δ/10;d) X is a disjoint union of closed surfaces of total area less than δ/10, each contained in a ball of radius less than √(δ)/10.We can order the punctured surfaces D_i to have descending height relative to a fixed unit normal on Γ, with D_1 the top-most.Let us call D_i the ith sheet.Let us call Y the thin part of S ∩Δ_i. §.§ Key lemmas used in the proof of Proposition <ref> .The following is the blow down - blow up lemma from <cit.>.Suppose B_r'(x) is a ball of radius r' ≤ r and _tbe an isotopy with _t ∖ B_r'(x)= _0 ∖ B_r'(x). Then there exists an isotopy _t, such that:(a) _0=_0 and _1=_1;(b) _t ∖ B_r'(x) =_0 ∖ B_r'(x);(c) Area(_t) ≤max{Area(_0), Area(_1) }+2 r' L(_0 ∩∂ B_r'(x)) for t ∈ [0,1]. For the proof see radial deformation constructionin <cit.>, Step 2 in the proof of Lemma 7.6.Letand Δ_i be as defined above.There exists an admissible family {_t} and k ≤ k_ess(,i,3r/4), such that:a) _0 = and _t ∖ int(B_r(p_i)) = ∖ int(B_r(p_i)) for all t;b) _1 is (δ,k)-stacked in Δ_i and Area(_1)< k Area(∩Δ_i) + δ/2;c) Area(_t)< Area() + δ for t ∈ [0,1];d) ifis (δ',k')-stacked in a cell Δ_j, j = i-1 or i+1,then either k=k' and _1 is (δ',k)-stacked in Δ_jor k_ess(_1,j, 3r/4)< k'.After applying the squeezing map P we may assume thatis contained in Ω_h(), where h sufficiently small, so that: - Area( ϕ(,h') ∩ B_r(p_i)) ≤ Area(B_r(p_i) ∩) + δ/20k for all h' ∈ [-h,h];- Area(∂ B_r'(p) ∩Ω_h()) < δ r' /100 forall r' ∈ (0,r). By coarea inequality and the definition of r (<ref>) there exists a radius r' ∈ [3r/4, r] withL( ∩∂ B_r'(p_i)) ≤δ/10 r Let k = k_ess(, p_i, r'). It follows from the definition that k ≤ k_ess(, p_i, 3r/4).We will show that there exists anisotopy {_t} ofthat does not changeoutside of the interior of B_r'(p_i) and deforms it to a surface _1 with the following properties:(a) _1 is (δ,k)-stacked in Δ_i(b) Area(_1 ∩ B_r'(p_i)) ≤ Area(_0 ∩ B_r'(p_i)) + δ/10.Then by Lemma <ref> and (<ref>) we may assume thatthe isotopy {_t } also satisfies Area(_t) ≤ Area() + δ for t∈ [0,1]. In other words, in the construction below we do not need to control the areas of the intermediatesurfaces.We start by deforming all connected components ofwhich are closed surfaces in the interior of B_r'(p_i), so that they lie in a small ball and have total area less than δ/100. From now on, without any loss of generality, we may assume that every connected component of∩ B_r'(p_i) intersects ∂ B_r'(p_i).By definition of k_ess(,i, r') we can deforminto a surface S, such thatexaclty k connected components ofS ∩∂ B_r/2(p_i) are not contractible in ∂ B_r/2(p_i) ∩Ω_h().Choose a collection of embedded mutually disjoint closedcurves {γ_j }, γ_j ⊂ S, such thatconnected components of S ∖∪γ_j are discs, annuli or pairs of pants. Moreover, collection of curves {γ_j } can be chosen sothat it includes all connected components of S ∩∂ B_r/2(p_i). We will say that a curve γ⊂ S ∩∂ B_r/2(p_i) (resp.γ⊂ S ∩∂ B_r'(p_i)) is essential if it is non-contractible in ∂ B_r/2(p_i) ∩Ω_h() (resp. ∂ B_r'(p_i) ∩Ω_h()).We may assume that S has been deformed in such a way that 1) every essential γ⊂ S ∩∂ B_r/2(p_i)is a latitudinal circle, that is γ = ∂ B_r/2(p_i) ∩ϕ (× t) for some t ∈ (-h,h); 2) every non-essential γ⊂ S ∩∂ B_r/2(p_i) is of the form γ = ∂ B_ρ(γ)(x(γ)) ∩∂ B_r/2(p_i) with the total sum of the areas of all B_ρ(γ)(x(γ)) ∩∂ B_r/2(p_i) less thanδ/100. Let S' be a connected component of S ∖∪γ_j that lies in B_r/2(p_i). If S' is a disc with a non-essential boundaryin ∂ B_r/2(p_i) we can isotop it to a small cap near its boundary and push it out of B_r/2(p_i). Similarly, if S' is an annulus or a pair of pants with non-essential boundary components we can isotop it to a surface given by the boundary of a tubular neighbourhood of a curve or a Y graph with the area at most2 ∑_l Area(B_ρ(γ_j_l)(x(γ_j_l)) ∩∂ B_r/2(p_i)), where γ_j_l are boundary components of S'.If S' is a disc with an essential boundary curve we isotop it to B_r/2(p_i) ∩ϕ (× t). Similarly, we isotop an annulus or a pair of pants with m=1,2 or 3 essential boundary components to a surface given by m stacked discs with holes connected by narrow tubes or boundaries of a tubular neighborhood of a graph.Ambient isotopy theorem guarantees that these deformations can be done so that different connected componentsdo not intersect each other. As a result we obtain that the new surface _1 is (k, δ)-stacked in Δ_i ⊂ B_r/2(p_i). We would like to deform connected components of S ∖∪γ_j that lie inB_r'(p_i) ∖ B_r/2(p_i) in a way that will guarantee theupper bound on the area and propertyd) in the statement of the Lemma.The main issue is that our deformation is not allowed to change the boundary S ∩∂ B_r'(p_i), which can be very wiggly. However, for some sufficiently small positive δ'< r' -r/2we can deform S so that S ∩∂ B_r'-δ'(p_i) satisfies the same properties 1)-2) as S ∩∂ B_r/2(p_i), while controlling the area of S ∩ (B_r'(p_i) ∖ B_r'-δ'(p_i)) in terms of h. We choose δ' sufficiently small, so that the distance function to p_i restricted to S ∩ (B_r'(p_i) ∖ B_r'-δ'(p_i)) is non-degenerate. First, we deform the collars of non-essential curves γ⊂ S ∩∂ B_r'(p_i), so that their intersection with ∂ B_r'-δ'(p_i) satisfies condition analogous to 2) above. This can be done in a way so that the area of the deformed part ofS ∩ (B_r'(p_i) ∖ B_r'-δ'(p_i)) is bounded by the area of the disc γ bounds in ∂ B_r'(p_i).Now we would like to straighten the essential curves. Let γ denote the highest (with respect to signed distance from ) essential curve in S ∩∂ B_r'(p_i). Choose t ≤ h, so that the latitudinal curve ϕ (× t) ∩∂ B_r'-δ'(p_i) lies above γ. We isotop the small non-essential necks in the neighbourhood of ∂ B_r'-δ'(p_i) so that their intersection with ∂ B_r'-δ'(p_i) lies either above ϕ(× t) or below γ. After this deformation the subset of∂ B_r'-δ'(p_i) ∖ S that lies between ϕ(× t) ∩∂ B_r'-δ'(p_i)and γ is homeomorphic to a cylinder. This implies that there exists an isotopy of S sliding the essential intersection γ toϕ(× t) ∩∂ B_r'-δ'(p_i). We iterate this procedure for every essential curve in S ∩∂ B_r'-δ'(p_i) deforming them into latitudinal curves. The isotopies done in this way have the property that the area of the deformed part ofS ∩ (B_r'(p_i) ∖ B_r'-δ'(p_i)) is bounded by the area of ∂ B_r'-δ'(p_i) ∩Ω_h().We conclude that total area of S ∩ (B_r'(p_i) ∖ B_r'-δ'(p_i)) after the deformation goes to 0 as h → 0.Suppose the collection of curves {γ_j } is chosen sothat it includes all connected components of S ∩∂ B_r' - δ'(p_i). Suppose S' is a connected component of S ∖∪γ_j that lies in( B_r'-δ'(p_i) ∖ B_r/2(p_i)).If all boundary components of S' are non-essential, we can deform itso that it is a boundary of a tubular neighbourhood of a curve or a Y graph.If S' is an annulus and one of its boundary components is essential then the second boundary component must also be essential (this follows by examining the homomorphism of fundamentalgroups induced by inclusion). Observe that if both boundary components lie in∂ B_r/2(p_i) we obtain a contradiction with the definition of k_ess. If both lie in ∂ B_r'-δ'(p_i) we push S' very close to ∂ B_r'-δ'(p_i), so that its area is at most Area(∂ B_r'-δ'(p_i)∩Ω_h()). If one component of ∂ S' lies in ∂ B_r'-δ'(p_i) and another component lies in ∂ B_r/2(p_i) we can isotop S' so that it is a graphical sheet of area at most Area(∩ (B_r'-δ'(p_i) ∖ B_r/2(p_i))) + O(h).Suppose now that S' a pair of pants. It follows by examining the homomorphism from π_1(S') = ℤ * ℤ to π_1((B_r'-δ'(p_i) ∖ B_r/2(p_i)) ∩Ω_h()) = ℤ_1 that S can have either 0 or 2 essential boundary components. In both cases we can deform it similarly tothe case of an annulus, but with a narrow tube attached.By Lemma <ref> we have that the area bound c) is satisfied for h sufficiently small.It is straightforward to check that the above deformations can be done so that if Δ_j ∩ B_r'(p_i) ≠∅ andwas (k',δ')-stacked in Δ_j for k' = k, then it will be(k,δ)-stacked after the deformations.Supposewas (k',δ')-stacked in Δ_j for k' > k. Then after the deformation there will be some open subset U ⊂ S_j, such that for every x ∈ S_j we have P^-1(x) ∩_1 has less than k points. It follows that k_ess(_1,j, 3/4)< k'.§.§ Proof of Proposition <ref>. Fix δ>0.First we construct a deformation ofto a surface that is (δ/10,k)-stacked in each cell Δ_i for some integer k, while increasing its area by at most δ/2.Recall the definition of c from (<ref>). Let δ_i = min{c/2, 1/2^iδ/100}. We will construct asequence of surfaces ^0,...,^N, such that1*) ^1= and^N is (δ/10,k)-stacked in each cell Δ_i.2*) Area (^i+1) ≤ Area(^i)+ δ_i and for every p_j,Area (^i+1∩ B_r(p_j)) ≤ Area(^i∩ B_r(p_j))+ δ_i 3*) There exists an isotopy {_t^i} with _t^i ⊂ N_h_0(), such that _0^i=^i, _1^i=^i+1 andArea(_t^i) < Area(^i) + δ/2 for t ∈ [0,1].The process consists of a finite number of iterations. The l'th iteration will consist of m_l ≤ m steps. Let m̃(l) = ∑_l' ≤ l m_l'. For j=1,..,m_l wedeform^m̃(l-1)+j-1 into ^m̃(l-1)+j. At the j'th step of l'th iteration we apply Lemma <ref> to ^m̃(l-1)+j-1 to construct an isotopy to ^m̃(l-1)+j, which is (k_m̃(l-1)+j, δ_m̃(l-1)+j)-stacked in the cell Δ_j.Now by induction and Lemma <ref> d) we have two possibilities:1) ^m̃(l-1)+j is (δ/10, k_m̃(l-1)+j) stacked incells Δ_1, ..., Δ_j; 2) ^m̃(l-1)+j is (k', δ/10)-stacked in Δ_j-1 and k_ess(^m̃(l-1)+j, j-1, 3r/4)< k'.In the second case we apply Lemma<ref> to ^m̃(l-1)+j inthe cell Δ_j-1. This deformation (preceded by an application of a squeezing map P if necessary) will, by Lemma <ref>, reduce the area of the surface by at least c - δ_m̃(l-1)+j> c/2. Since the area of _n can not be negative, we must have that eventually it is stacked in every cell. The total area increase after all the deformations is at most ∑δ_n < δ/10. This concludes the proof ofProposition <ref>.§ TUBES, CABLES AND ROOT SLIDING(Definition of a tube.) Let γ:[0,1] → N be an embeddedcurve and exp_γ: [0,1] × D^2 be the normal exponential map and supposeexp_γ is a diffeomorphism onto its image for v ∈ D^2 with |v| ≤ 2 ε. We will say that T = {exp_γ(t,v): |v| = ε} is an ε-tube with core curve γ.In this paper we will often need to isotopically deform parts of a surface so that it looks like a disjoint union of long tubes. We will then need to move these tubes around ina controlled way. Here we collect several definitions and lemmas related to this procedure.Letbe an embedded surface in M. 𝒞 = { (A_i, γ_i, ε_i)}_i=1^k will be called a cable of thickness ε>0 with root balls B_1 and B_2 and necks A = ⋃_i=1^k A_i , where(1) { A_i ⊂} is a collection of disjointε_i-tubes, ε_i ≤ε with core curves γ_i;(2) B_1 and B_2 are disjoint opens balls of radius r> ε. For j=1,2 we have that B_j ∩ = ⊔_i=1^k Ã_j,i, where each Ã_j,i is homeomorphic to an annulus with boundary circles c̃_j,i^1 and c̃_j,i^2,satisfying c̃_j,i^1 ⊂ A_i and c̃_j,i^2 ⊂ M ∖ A andwith γ_i(0) ⊂ B_1 and γ_i(1) ⊂ B_2;(3) let N_i denote the solid cylinder bounded by A_i,N_i={exp_γ_i(t,v): |v| ≤ε_i }, then N_i ⊂ N_1 for all i. In the following Lemma <ref> we observe that if a cable hassufficiently small thickness then we can squeeze it towards the core curve γ_1 to make the total area of necks arbitrarily small.There exists a constant C_sq>0, such that for all sufficiently small ε>0 the following holds. If_0 is a surface witha cable 𝒞 = { (A_i, γ_i, ε_i)}_i=1^kof thickness ε>0 with root balls B_r(p_1) and B_r(p_2) thenthere exists an isotopy _t, t ∈ [0,1) such that:(1) Area(_t) ≤ Area(_0) + C_sq k ε^2 for all t ∈ [0,1);(2) _t is a surface witha cable 𝒞_t = { (A_i^t, γ_i^t, ε_i^t)}_i=1^kof thickness ε_t = (1-t)ε with necks A_t and root balls B_r_t(p_1) and B_r_t(p_2) of radius r_t;(3) ε_i^t, r_t andArea(A_t)are monotone decreasing functions of t with A_t → 0, ε_i^t → 0and r_t → 0 as t → 1;(4) γ_1^t = γ_1 for all t ∈ [0,1).Before proving Lemma <ref> we state the following auxiliary result. Let B_1(0) be a ball in ^3 and ⊂ B_1(0) be a surface with ∂⊂∂ B_1(0). Let γ_t be an isotopy of curves in ∂ B_1(0) with γ_0 = ∂ and l(γ_t) < L. Then there exists an isotopy _t with _0 =, ∂_t = γ_t and Area(_t) < Area(_0) + L.The result follows by the blow down - blow up trick from <cit.> as inthe other parts of this paper. Let ϕ_t^1: N_1 → N_1 be a map given by ϕ_t^1(exp_1(v)) = exp_1 (tv) for v ∈ N γ_1.Choose monotone decreasing functionsf_i:[0,1] → [0,1], 2 ≤ i ≤ k, so that the map ϕ_t^i: N_i → M defined by ϕ_t^i(v) = ϕ_t (exp_1^-1 (exp_i(f_i(t) v ))) is a diffeomorphism onto its image andA_i^t = ϕ_t^i(A_i) are all disjoint.This defines the desired isotopy outside of the root balls B_r(p_1) and B_r(p_2). We extend the istopy inside the balls using Lemma <ref>. This finishes the proof of Lemma <ref>.Given a surfacewith a cable we definea new surface obtained by sliding the root ball B_1as illustrated on Fig. <ref>.Let _0 be a surfacewith cable 𝒞 = { (A_i, γ_i, ε_i)}_i=1^kof thickness ε>0 with root balls B_1 and B_2. Let c̃_j,i^1 and c̃_j,i^2 be as in Definition <ref>. Let B_3 be a ball intersecting _0 in a disc and α⊂ (_0 ∖ (B_1 ∪ B_2 ∪ B_3) an arc with endpoints α(0) ⊂c̃^2_1,1 and α(1) ⊂∂ B_3.Let β denote an arc in ∂ B_1 connecting the endpoint of α(0) to the endpointof γ_1 ∖ B_1. Let γ̃_a = α∪β∪ (γ_1 ∖ B_1). Perturb γ̃_a in the direction normal to _0, so that it does not intersect _0 ∖∪ A except at the endpoints.Let γ̃_b denote an arc obtained by perturbing α to the other side of _0 from γ̃_a.We will say that _1 is obtained from _0 by ε-cable sliding if the following holds:(a) _1 has a cable𝒞_1 ={ (A_i^1, γ_i^1, ε_i^1)}_i=1^k with ε_i^1 ≤ε_1, γ_1^1 = γ̃_a and root balls B_3 and B_2;(b) _1 has a cable𝒞_1 ={ (A_i^2, γ_i^2, ε_i^2)}_i=2^k with ε_i^1 ≤ε_1, γ_2^2 = γ̃_b and root balls B_1 and B_3;(c) Setting A^1 = ∪_1^k A_i^1 and A^2 = ∪_2^k A_i^2 we have_1 ∖ (A_1 ∪ A^2 ∪ B_1 ∪ B_3)= _0 ∖ (A ∪ B_1 ∪ B_3). The following lemma allows us to slide the root of a cable along a curve contained in while increasing its area by in a controlled way.For every ε_0>0 there exists ε>0 sufficiently small, so that if _1 is obtained from _0 by ε-cable sliding then there exists an isotopy _t, t ∈ [0,1], such that Area(_t) ≤ Area() + ε_0 for all t ∈ [0,1]. By Lemma <ref> we may assume that the thickness ε and the radius of the root balls r are as small as we like.Let I_L = { (x,0,0) | 0 ≤ x ≤ L }⊂^3 and S_xy denote the xy-plane in ^3. Let L= l(α). For every c>0 there exists ε_1>0 and a diffeomorphism Φ: N_ε_1(α) → N_ε_1(I_L) ⊂^3, such that(a) Φ(α) = I_L;(b) Φ(S ∩ N_ε_1)= S_xy∩ N_ε(I_L);(c) 1-c≤||D Φ|| ≤ 1+c.Fix c<1/10 to be chosen later (depending on ε_0) andassume 20 ε < ε_1 and 2r < ε_1. Let q_a= = Φ(α(0)) and q_b = Φ(α(1)). By our choice of c we have Φ(B_1) ⊂ B_ε_1/2(q_a) It is straightforward to construct a 1-parameter familyof diffeomorphisms ϕ_t: N_ε(I_L) → N_ε(I_L), ϕ_0 = id, and generated by a 1-parameter family of compactly supported vector fieldsξ_t with the following properties:(i) ϕ_t(B_ε_1/2(q_a)) is an isometric copy of B_ε_1/2(q_a) translated distance tL along thex axis;(ii) ξ_t(p) lies in S_xy for every p ∈ S_xy;(iii)ξ_t is supported in N_2 ε_1/3(α) for all t;(iv) l(ϕ_t(Φ(γ_i)) )< 10 l(γ_i) for all t. Composing with Φ^-1 we obtain a 1-parameterfamily of diffeomorphisms ϕ̃_t: M → M. Observe that by condition (ii) the restriction of ϕ̃_t to S is a diffeomorphism of S, in particular,Area (ϕ̃_t(S)) = Area(S). By compactness we can choose ε∈ (0, ε_1/20)sufficiently small, so that Area(ϕ̃_t(S)) ≤1/10 kε_0 for t ∈ [0,1].Moreover, we can isotop each surface ϕ̃_1(A_i) so thatit coincides with ∂ N_ε_i(ϕ̃_1(γ_i)).This finishes the construction of the desired isotopy.For every ε>0 there exists a δ>0 with the following property.Suppose a surfaceis in a canonical form, in particular it is (δ,m)-stacked in every ball B_r(x_j). There exists an isotopy of, increasing the area ofby at most ε, so that the thin part T = ⊔ A_i, where each A_i is homeomorphic to an annulus.If every connected component of T has 2 boundary components then we are done.Suppose T_1 is a connected component with k ≥ 3 boundary components. We describe how to use the root sliding lemma to deform T_1 into two disjoint thin subsets, each having a smaller number of boundary components.Since the surface is in a canonical form, there exists a cell Δ_j with γ = ∂Δ_j ∩ T_1 non-empty. Let γ_1 denote an inner most closed curve of γ. Let D_1 denote the small disc γ_1 bounds in∂Δ_j. We have that interior of D_1 does not intersectT_1. (Note, however, that there could beconnected components of (T ∖ T_1) ∩∂Δ_j intersecting D_1).We consider two possibilities:1) γ_1 bounds a disc D ⊂ T_1. Then we can find a ball B⊂ N_h_0() with ∂ B = D_1 ∪ D. Let B̃ denote a small ballwith D_1 ⊂∂B̃and int(B̃) ∩ int(B) = ∅. There exists a diffeotopy Φ_t of N_h_0(), such thatΦ_1(B) ⊂B̃. It is straightforward to checkthat using repeated applicationof the blow down - blow up trick Lemma <ref> we can make sure that the areas of Φ_t() do not increase by more than O(δ). In the end, we obtain that the number ofconnected components of T_1 ∩∂Δ_j has decreased by one.2) γ_1 separatesconnected components of ∂ T_1. Let A denote thecomponent of T_1 ∖γ_1, which has more than 2 boundary components. Let α be a path in A from γ_1 to a different boundary component of A and into the thick part of . Let γ̃ denote all connectedcomponents of T that are contained inside a small disc bounded by γ_1 (including γ_1). Let T̃ denote a small neighbourhood of γ̃ in T. In a small neighbourhood of D_i we can isotop T̃ so that it satisfiesthe properties of a cable with two roots. We can then use Lemma <ref> to move one of the roots into the thick part of . As a result we reduced the number of boundary components of T_1.Suppose the first possibility occurs. Then we have decreased the number components of T_1 ∩∂Δ_j by 1. We choose an inner most connectedcomponent once again. Eventuallywe will encounter possibility 2. Then we split T_1 into two connected components with a strictly smaller number of boundary components. § OPENING LONG NECKSIn this section, we prove the following:Let Σ be a strictly stable minimal two-sphere and let Γ⊂ N_h_0(Γ) be a two-sphere in (δ,k) canonical form for some k>1.Then there exists an isotopy Γ_t beginning at Γ_0=Γ through surfaces with areas increasing by at most δ so that in some cell, the number of essential components of Γ_1 is fewer than k. The difficulty in Proposition <ref> is that while the surface Γ is in canonical form, there can be many wildly knotted, linked and nested arcs comprising the set of tubes.In order to untangle this morass of tubes to obtain a vertical handle supported in a single ball requires the Light Bulb Theorem in topology, which we recall:Let α(t) be an embedded arc in𝕊^2× [0,1] so that α(0)={x}×{0} and α(1)={y}×{1} for some x,y∈𝕊^2.Then there is an isotopy ϕ_t of α so that* ϕ_0(α)=α * ϕ_1(α) is the vertical arc {x}× [0,1].The Light Bulb theorem can be interpreted physically as that one can untangle a lightbulb cord hanging from the ceiling and attached to a lightbulb by passing the cord around the bulb many times.The simplest nontrivial case of Proposition <ref> consists of two parallel spheres joined by a very knotted neck.Here the Light Bulb Theorem <ref> allows us to untangle this neck so that it is vertical and contained in one of the balls B_i. Thus the resulting surface is no longer in canonical form and we can iterate Step 1.We will in fact need the following generalization of the light bulb theorem (cf. Proposition 4 in<cit.>): Let M be a 3-manifold and α an arc with one boundary point on a sphere component Γ of ∂ M and the other on a different boundary component.Let β be a different arc with the same end points as α.Then if α and β are homotopic, then they are isotopic.Moreover, if γ is an arc freely homotopic to α (i.e. joined through a homotopy where the boundary points are allowed to slide in the homotopy along ∂ M), then they are freely isotopic (i.e., they are joined by an isotopy with the same property). Sketch of Proof:The homotopy between α and β can be realized by a family of arcs α_t so that α_t is embedded or has a single double point for each t∈ [0,1].If α_t_0 contains a double point, the curve α_t_0 consists of three consecutive sub-arcs [0,a], [a,b], and [b,1] so that without loss of generality [0,a] connects to the two sphere Γ.We can pull the arc α_t_0([b-ε,b+ε]) transverse to α_t_0([0,a]) along the arc α_t_0([0,a]) and then pull it over the two sphere Γ, and then reverse the process.We can then glue this deformation smoothly in the family α_t for t near t_0 to obtain the desired isotopy.The proof is illustrated in Figure <ref>.Let us now prove Proposition <ref>:Since k>1, we can find a cell Δ_k so that Γ∩∂Δ_k contains several small non-essential circles.Let C denote an innermost such circle.By squeezing a small collar around C we obtain a neck with two roots.There are two connected components of Γ∖ C.Let us denote them A and B.There must be two consecutive sheets S_1 and S_2 comprising Σ so that S_1 is contained in A and S_2 is contained in B.Thus we can move the roots of the collar about C using Lemma <ref> so that they are stacked on top of each other, one in S_1 and the other in S_2.By Proposition <ref> we can isotope the neck to then be a vertical neck contained in a single cell.Thus the number of essential components has gone down by at least 1 in this cell. §.§ Proof of Proposition <ref>. First we apply Proposition <ref> to deformso that it is (δ,k)-stacked. We consider two cases.Case 1.is not homeomorphic to a sphere. By assumption we have that the genus of every connected component ofis less or equal to the genus of . It follows that each connected component ofeither coincides with a graphical sheet over(as more than one graphical sheet would imply that the genus ofis greater than that of ) or is contained in a small ball of radius less than δ.Case 2.is homeomorphic to a sphere. Supposehas a connected component which intersects more than one sheet D_i. We apply Proposition <ref> reducing the essential multiplicity ofin some cell Δ_i. Then we can apply Lemma <ref> to reduce the area of the surface by at least c/2. We iterate this procedure. Eventually, every connected component ofwill either be graphical or contained in a small ball. This finishes the proof ofProposition <ref>. § CONVERGENCE TO A SURFACEWITH MULTIPLE CONNECTED COMPONENTSIn this section, we generalize Proposition <ref> to the situation whereis disconnected and the area ofoutside of N_h() is small. Letbe an orientable connected surface and let p: N_h() → be the projection map. Given a positive integer m we will say that a surfacehas ε-multiplicity m if there exists a subset U ⊂ with Area(U) < ε and for almost every x ∈∖ U the set { p^-1(x) ∩} has exactly m points. Similarly, we will say that a surfacehas ε-even(resp. ε-odd) multiplicity in N_h() if there exists a subset U ⊂ with Area(U) < ε and for almost every x ∈∖ U the set { p^-1(x) ∩} has an even (resp. odd) number of points.It is straightforward to check that for all sufficiently small h> 0 and ε>0, if L(∩∂ N_h ()) ≤1/100√(ε), thenis either ε-even or ε-odd inN_h().Let = ⊔_k, where each _k is a smooth strictly stable two-sided connected minimal surface.There exists h_0, ε_0 >0, such that for all h ∈ (0, h_0) and ε∈ (0, ε_0) the following holds. Supposesatisfies (a) Area (∖ N_h())< ε;(b) genus(N_h(_i) ∩) ≤ genus(_i) for each i;(c) L(∩∂ N_h ()) ≤1/100√(ε).Let {_k_j} denote the subset of minimal surfaces for whichis ε-odd in N_h(_k_j). Then for every δ>0 there exists an isotopy _t, such that:(i) Area(_t) < Area() + δ(ii) _1 is the union of {_k_j} connected by thin necks.(iii) If {_k_j} is empty then _1 can be chosen to be a closed surface in a ball of arbitrarily small radius.Note that assumption (b) is satisfied wheneveris a strongly irreducible Heegaard splitting.Proposition <ref> follows immediately from Proposition <ref>.§.§ Reducing the area outside of the tubular neighbourhood ofThe following lemma is useful for reducing the area of thin hair (see <cit.> for an analogue lemma in the context of Almgren-Pitts theory). Let S be a surface in M then for all sufficiently smallh>0 the following holds. There exists ε(M, S, h)>0 with the following property.For every δ>0 and every surfacewithArea(∖ N_h(S))< ε their exists a smooth isotopy _t with(1) _0 =, (2) Area(_t) ≤ Area()+ δ,(3) Area(_1 ∖ N_2h(S))< δ,We recall the following “bounded path"version of the γ-reduction of <cit.>used in the min-max setting of Simon-Smith(see <cit.>).Let Σ be an embedded surface in M, and U be an open set included in M. Let ℑ𝔰(U) be the set of isotopies of M fixing M\ U, with parameter in [0,1]. For δ>0 defineℑ𝔰_δ(U) = {ψ∈ℑ𝔰(U); ℋ^2(ψ(τ,Σ))≤ℋ^2(Σ̃) + δ for all τ∈[0,1]}.An element of the above set is called a δ-isotopy. Suppose that the sequence {ψ^k}⊂ℑ𝔰_δ(U) is such that lim_k→∞ℋ^2(ψ^k(1,Σ)) = inf_ψ∈ℑ𝔰_δ(U)ℋ^2(ψ(1,Σ)).Such a sequence is called minimizing. Then in U, ψ^k(1,Σ) subsequently converges in the varifold sense to a smooth minimal surface Σ̂.Let us apply this γ-reduction with constraint to U:=M\N̅_̅h̅(S). Let {ψ^k}⊂ℑ𝔰_δ(U) be a minimizing sequence. Then by the monotonicity formula for minimal surfaces, the area of (M\N_2h(S))∩ψ^k(1,Σ) goes to zero so in particular for a k' large enough, this area is smaller than δ. The lemma is proved by taking Σ_t = ψ^k'(t,Σ). The following lemma allows us to isotopically pushdiscs into the neighbourhood of a surface with almost no increase in the area.There exists ε(M, , h)>0 with the following property.Suppose D ⊂ M ∖ N_h()is an embedded disc withArea(D)< ε, l(∂ D) < √(ε) and ∂ D ⊂∂N_h()There exists a smooth isotopy D_t with(1) D_0 =, (2) Area(D_t) ≤ Area(D) + δ,(3) D_1 ⊂N_2h().ε < 1/100 r^2. Fix a triangulation of M ∖ N_h() with each 3-simplex Δ_j contained in a ball of radius r/2 centered at the barycenter of the simplex p_j.Let C(2) denote the union of the interiors of 2-simplices in the triangulation. After an initial deformation we may assume that D intersects C(2) in a collection of small circles and does not intersect the 1-skeleton or 0-skeleton of the triangulation. Indeed, by coarea formula we can find a radius r_j ∈ (r,2r), so that ∂ B_r_j(p_j) ∩ D is a finite union of circles of total length less than 2 ϵ/r. Using blow down - blow upLemma <ref> we can deform D to D' so thatD' intersects each 2-face of Delta_j in a collection of circles and does not intersect 1-skeleton or 0-skeleton of Delta_j, and so that the area of D' is less than or equal to the area of D. We perform a similar deformation for every simplex in the triangulation. We claim that there exists a 3-simplex Δ', so that Δ' ∩ D contains a connected component D' diffeomorphic to a disc. Given this claim we retract D by inductively reducing the number of connected components of D ∩ C(2). Namely, we push out the disc outside of Δ' together withall connected components D ∩Δ' contained between D' and the 2-simplex of Δ' that contains ∂ D'.To prove the claim, consider a tree where each vertex corresponds to aconnected component of D ∖ C(2), and two vertices are connectedby an edge if the corresponding connected components have a common boundary circle. A terminal vertex of the tree must correspond to a disc.Letbe connected and suppose ∩∂ N_h() is a collection of small disjoint circles. Let C() denote a closed surface in N_h() obtainedfrom ∩ N_h() by capping each connected component of ∩∂ N_h() with a small disc and perturbing to remove self-intersections.For every sufficiently small δ>0 the following holds. Suppose L(∩∂ N_h()) ≤δ and let {_t} be an isotopy of closed surfaces with_0 = C(). Then there exists an isotopy {'_t }, such that(a) '_0 =;(b) '_t ∖ N_h() = ∖ N_h();(c) ('_t ∩ N_h()) ∖_t consists ofa union of disjoint necks of total area O(δ^2).Let γ_1 be an outermost connected component of ∩∂ N_h(). Let D_1 and D_2 denote the two discs in C() and ∂ N_h() respectively, corresponding tothe surgery along γ_1. Let α_t be an embdedded arc withendpoints p in D_1 and q in D_2. It is a consequence of standard topological theorems (in particular, Cerf's theorem)that there exists an isotopy of embedded arcs α_t, which does not intersect _t, except at the endpoint p_t ∈_t, with α_0 = α and the other endpoint equal to q ∈ D_2 ⊂∂ N_h().If follows by compactness that a sufficiently small tube around α_t will be disjoint from _t except at the root. We glue in this family of necks to obtain a new isotopy of surfaces. We proceed by induction on the number of connected components of ∩∂ N_h().We need one more lemma before we can prove Proposition <ref>. Supposesatisfies the assumptions of Proposition <ref>.Then the conclusions of Lemma <ref> hold and, moreover, we may also assume that (4) for each i, there exists an integerm_i such that _1 has ε-multiplicity m_i;(5) if genus(_i) ≥ 1 thenm_i = 0 or 1; (6) if ' ⊂ N_h(_i) is a connected component of_1 ∩N_h(_i), such that m_i>1 and γ⊂∂' is an inner most circle in ∂ N_h(_i), then ∂' = γ.By applyingProposition <ref> together withLemma <ref> we may assume that in the tubular neighborhood of each connected component _i surface ∩ N_h(_i) looks like m_i disjoint copies of _i with thin necks that go into the boundary ∂ N_h(_i). This proves (4). By assumption(b) of Proposition <ref>if _i has genus greater than or equal to 1 then m_i = 0 or 1. This proves (5).Finally, we can apply Lemma <ref> to slide tubes attaching to an inner most sphere, so that condition (6) is satisfied. §.§ Proof of Proposition <ref>.Assume that the part of the surface outside of small tubular neighborhood ofsatisfies conclusions (1)-(6) of Lemmas <ref>and <ref>. By Proposition <ref> andLemma <ref> we may assume that in the tubular neighborhood of each connected component _i surface ∩ N_h(_i) looks like k disjoint copies of _i with thin necks that go into the boundary ∂ N_h(_i). Also, if _i has genus greater than or equal to1 then ∩ N_h(_i) is either empty or has one connected component (which look like _i with necksescaping into the boundary ofN_h(_i)). Hence, we need to deal with multiple connected component only in tubular neighbourhood of spherical components of . Suppose ∩ N_h(_i) has more than one connected component.Choose an inner most closed curve γ⊂∂ N_h(_i) ∩. By squeezing a small collar that contains γ we obtain a neck with two roots. There are two connected components Γ∖γ.Let us denote them A and B.Moreover, since we chose γ to be innermost, by Lemma <ref> (6), exactly one sheet of ∩ N_h(_i) is contained in A and all other sheets are contained in B.Hence, we can move the roots using Lemma <ref> so that they are stacked on top of each other in the neighborhood of component Σ_i andapply Proposition <ref>, so that in some cell on Σ_i, the number of essential components has gone down.Thus we can reducethe number of sheets in N_h(Σ_i) by at least two. We proceed this way for as long as ∂ N_h(_i) ∩ consists of more than one connected component for any i.Since the even/odd parity is preserved in this process eventually we obtain that we have exactly one connected component in the neighborhood for each _k_j wherehad ε-odd multiplicity.§.§ Gluing two families of isotopies and interpolation. Proposition <ref> implies that we can deformthe surface converging to a minimal surface with multiplicity so that it has form = _thin⊔_thick, where _thick is a disjoint union of embedded stablesurfaces _i with δ-size discs removedand _thin is contained in the boundary ofa tubular neighbourhood of an embedded graph.Given two isotopic surfaces with such a decomposition, and assuming some additional topological conditions, we would like to find an isotopy between them that fixes _thick.Suppose _1 and _2 are two isotopic embedded surfaces in M satisfying assumptions of Proposition <ref> and, moreover, suppose the set of stable connected components _i, where _j, j=1,2, is ϵ-odd coincides. Assume, in addition, that M = S^3 or is a lens space and _j is a strongly irreducibleHeegaard splitting of M diffeomorphic to a torus or 2-sphere.Then for every δ>0 there exists an isotopy between _1 and _2 of area at most max{Area(_1), Area(_2) }+δ.By Proposition <ref>, _j can be deformed into a union of stable spheres connected by thin necks. Suppose first that _j is a 2-sphere. Then by results of sections 4 and 5 any two such configurations are isotopicwith arbitrarily small area increase.Suppose that _j is a 2-torus. If _i has a torus component, then by applying Proposition <ref> we can find an isotopy between thin parts of _1 and _2 through surfaces of small area. If the thick parts of _1 and _2consist of spheres, we can use Lemma <ref> to deform _j so that it consists of stable spheres connected consecutively by thin loops with the last sphere connected to a thin torus T_j (that is, boundary of a δ-neighbourhood of an embedded loop). By strong irreducibility assumption each sphere _i bounds a ball on one of the sides. In particular, thin tori T_1 and T_2 are isotopic in the complement of _i's. The desired isotopy can then be obtained by repeated application of Proposition <ref>.We conjecture that the analogue of this Proposition holds when _1 and _2 are strongly irreducible Heegaard splittings of a 3-manifold M.§ DEFORMATION THEOREMS AND INDEX BOUNDS In this section we apply our Interpolation result to obtain the Deformation Theorem.Then we show how the Deformation Theorem easily implies the index bound Theorem <ref>. First we recall the following lemma from Marques-Neves (Corollary 5.8 in <cit.>) expressing the fact that strictly stable surfaces are isolated from other stationary integral varifolds:Suppose Σ is strictly stable and two-sided.Then there exists ε_0:=ε_0(Σ) so that every stationary integral varifold V∈𝒱_n(M) in B^F_ε_0(Σ) coincides with Σ.Note that the assumption of stability is essential in Lemma <ref>.For example, in round 𝕊^3 there exist a sequence of minimal surfaces Σ_g converging to twice the Clifford torus C.Thus in any small neighborhood of 2C are many stationary integral varifolds (<cit.>, <cit.>). The proof of the lemma follows from the application of the squeezing map P_t (seesection <ref>).Let us record some further properties of the map P_t (and sometimes we will write P_t(x) as P(t,x)).There exists r_0>0 such that P: N_r_0× [0,1]→ N_r_0 satisfies: * P(x,0)=x for all x∈ N_r_0, P(x,t)=x for all x∈ S and 0≤ t≤ 1* P(N_r_0,t)⊂ N_r for all 0≤ t≤ 1 and r≤ r_0, and P(Σ_r_0,1)=S* the map P_t is a diffeomorphism onto its image for all 0≤ t <1 * for all varifolds, and every connected component N of N_r_0, the function t→ ||P_t(V)||_N has strictly negative derivative unless V is supported on S, in which case it is constant. In the following, we consider a minimizing sequence Φ_i of sweepouts which means thatsup_t∈ [0,1]ℋ^2(Φ_i)→ W. We have the following result (similar to Deformation Theorem C in Section 5.6 in <cit.>) which allows us to deform the sweep-out Φ_i away from stable surfaces with multiplicities in such a way that no new stationary integral varifolds of mass W arise as limits of min-max sequences:Suppose M = S^3 or a lens space.Suppose Φ_i is a pulled tight sequence of sweepouts by spheres or tori (so that any min-max sequence obtained from it has a stationary limit). Let Σ be a stationary varifold such that* The support of Σ is a strictly stable two-sided closed embedded minimal surface S (potentially disconnected) so that Σ=∑_i=1^k n_i S_i,where n_i are positive integers* W = |Σ|.Then there exist η>0 and j_0⊂ℕ so that for all i≥ j_0, one can find a non-trivial sweepout Ψ_iso that * lim sup_i→∞sup_t∈ [0,1](||Ψ_i(t)||)≤ W* Any min-max sequence obtained from Ψ_i converges to either (a) the limit of a min-max sequence from Φ_i and is disjoint from an η-ball in the 𝐅-metric aroundor else (b) disjoint from an η-ball in 𝐅-metric about any stationary integral varifold of mass W.Let α>0 be a small number to be specified later.If α is small enough, we can find finitely many disjoint intervals V'_i,α⊂ [0,1] so that if t∈ [0,1]∖ V'_i,α then 𝐅(Φ_i(t),Σ)≥α and for t∈ V'_i,α, we have 𝐅(Φ_i(t),Σ)≤ 2α.Moreover, for t∈∂ V'_i,α, we obtain α≤𝐅(Φ_i(t),Σ)≤ 2α.Note that for α small enough, it cannot happen that V'_i,α=[0,1] since Φ_i([0,1] is a non-trivial sweepout of M. Moreover, it cannot happen that V'_i,α is empty (as otherwise the deformation theorem would already be proved). Thus the set V'_i,α consists of several sub-intervals of [0,1].Let V_i,α denote one connected component of V'_i,α.We will amend the sweepout Φ_i in the interval V_i,α and since V'_i,α consists of disjoint such intervals, we can apply the analagous alteration in each connected component. We can choose α so small so that if 𝐅(Φ_i(t),Σ)<α then the area of Φ(t) outside of N(Σ) is smaller than the ε in the statement of Lemma <ref>, the ε_0 in the statement of Proposition <ref>, and less than 3ε_0 (where ε_0 is from Lemma <ref>).Let us consider one such interval V_i,α, which is [t_a,t_b]. For each i, Lemma <ref> furnishes an isotopy H^1_i(t)_t∈[0,1] so that * H^1_i(0)=Φ_i(t_a)* H^1_i(t)≤ |Φ_i(t_a)|+1/i* H^1_i(1) is a surface with |H^1_i(t)∩ N()|<1/i We can further let H^2_i(t) = P_t(H^1_i(1)) where t is parameterized from t=0 to t=1-q_i (where q_i will be a sequence approaching zero as i→ 0 to be chosen later). Furthermore, when i is large enough it follows that from Proposition <ref> there exists an isotopy H_i^3(t)_t∈[0,1] so that* H^3_i(0)=H^2_i(1-q_i)* H^3_i(t)≤ |H^3_i(1)|+1/i* H^3_i(1)=⋃_i∈𝒞S'_i∪⋃_k T^1_k In (3), 𝒞 denotes the subcollection of {1,2,...k} such that S_i is ε-odd. S'_i denotes the surface S_i with several disks removed, and the set {T^1_i} consists of thin tubes connecting to the the various S'_i at the circles where the disks from S_i have been removed so that ⋃_i∈𝒞S'_i∪⋃_k T^1_k is a closed surface. Let us define the isotopy H^4_i(t)_t∈[0,1] by concatenatingH^1_i(t)_t∈[0,1], H^2_i(t)_t∈[0,1-q_i] and H^3_i(t)_t∈[0,1]Thus H^4_i(t)_t∈[0,1] furnishes an isotopy between Φ_i(t_a) and the closed surface ⋃_i∈𝒞S'_i∪⋃_k T^1_k.Similarly, we can obtain an isotopy H^5_i(t)_t∈[0,1] beginning at Φ_i(t_b) when t=0 and terminating at ⋃_i∈𝒞S'_i∪⋃_k T^2_k satisfying the hypothesis above where the thin tubes T^1_k are replaced by potentially different T^2_k.By Proposition <ref> there is an isotopy connecting ⋃_i∈𝒞S'_i∪⋃_k T^2_k and ⋃_i∈𝒞S'_i∪⋃_k T^1_k that increases area at most 1/i along the way.Denote the isotopy I_i(t)_t∈ [0,1] that begins at t=0 at ⋃_i∈𝒞S'_i∪⋃_k T^1_k and terminates at t=1 at ⋃_i∈𝒞S'_i∪⋃_k T^2_k. Thus we can concatenate H^4_i(t)_t∈[0,1], together with I_i(t)_t∈ [0,1] and then H^5_i(1-t)_t∈[0,1] to obtain an isotopy H^6_i(t)_t∈[0,1] beginning at t=0 atΦ_i(t_a) and terminating at t=1 atΦ_i(t_b). Let us replace by the sweepout Φ_i(t) in the interval [t_a,t_b] by the isotopy H^6_i(t)_t∈[0,1].In this way we obtain a new sweep-out Φ'_i(t).We claim that Φ'_i(t) satisfies the conclusion of the theorem. It follows from the properties of H^1_i, H^2_i, H^3_i and their mirror images H^5_i thatsup_t∈ [0,1] |Φ'_i(t)| → W i→∞We need to show that no min-max sequence of Φ'_i(t) obtained from the intervals V_i,ε converges to anything in an ε-ball about the space 𝒮 of stationary integral varifolds with mass W.Let us denote this latter space B_ε(𝒮). Claim 1: There exists an ε_1>0 so that for i large enough,H^1_i(t)∩ B_ε_1(𝒮)=∅ for all t∈ [0,1].Suppose toward a contradiction that there is a subsequence (not relabelled) i→∞ as well as a sequence of stationary integral varifolds V_i as well as t_i∈ [0,1] so that 𝐅(V_i, H^1_i(t_i))→ 0 as i→∞.We can assume |V_i|=W and that V_i converge to a stationary integral varifold V.Note that for i large enough, 𝐅(H^1_i(t_i),Σ)<𝐅(H^1_i(t_i),Φ_i(t_a))+𝐅(Φ_i(t_a),Σ) ≤ 1/i+2ε <ε_0 because of item (3) in the list of properties that H^1_i satisfies and because the 𝐅 metric between two surfaces is bounded from above by the area of the symmetric difference of the surfaces. Thus V_i∈ B_ε_0^𝐅(Σ) for i large and so by Lemma <ref> we have that for i large enough, V=V_i=Σ. Thus we have 𝐅(H^1_t(t_i),Σ)→ 0.From this we can easily deduce that Φ_i(t_a)→Σ, which contradicts the fact that 𝐅(Φ_i(t_a),Σ)=α by definition.Indeed, by construction the limit L of Φ_i(t_a) and L' of H^1_t(t_i) coincide in N(Σ) with total mass W, and moreover L can have no support outside of N(Σ) as Φ_i are a minimizing sequence with maximal area approaching W as i→∞.Claim 2:At least one of the integers {n_i}_i=1^k is greater than 1.Moreover,H^2_i(t)∩ B_ε_2(𝒮)=∅for all t∈ [0,1] and some ε_2>0.Suppose toward a contradiction that n_1=n_2=...=n_k=1. Consider the sequence of surface H^1_i(1), and their limit A which is an integral varifold supported in the tubular neighborhood N(Σ) that is homologous to Σ.We can also consider Ã=P_1(A).Note that |A|≤ W.It follows from the properites of the projection mapping P_t that |Ã|≤|A|.But W≤ |Ã| since à is an integral varifold supported on Σ with multiplicity at least one everywhere.Thus we obtain |Ã|=|A|=W which implies by item (4) in Lemma <ref> that A=Σ.Thus H^1_i(1)→Σ.Since 𝐅(H^1_i(1),Φ_i(t_a))→ 0, we obtain as in Claim 1 thatF(Φ_i(t_a), Σ)→ 0.But this contradicts the definition of t_a as in Claim 1. For the second part of the claim, suppose toward a contradiction that there is a subsequence (not relabelled) i→∞ as well as a sequence of stationary integral varifolds V_i as well as t_i∈ [0,1] so that 𝐅(V_i, H^2_i(t_i))→ 0 as i→∞.We can assume |V_i|=W and that V_i converge to a stationary integral varifold V and that t_i→ t.Note that V is supported on S by Property (4) of the projection map and the fact that the area of H^2_i(t_i) outside of N(Σ) is approaching 0 as i→∞.Letting A denote the limit of the surfaces H^1_i(1), we have |A|≤ W.We have from the definition of H^2_i it follows that V=P_t(A) and |V|≤ |A| as the map P_t is area non-increasing. Thus we obtain |V|=|A|=W from which we deduce that H^1_i(1)=V, contradicting Claim 1.Claim 3:There exists ε_2>0 so that for i large enough, and a choice of q_i we have that for some connected component S_1 of S, there holds|H^2_i(1-q_i)∩ N(S_1)| < |Σ| -ε_2.We will in fact show that |P_1(H^1_i(1))∩ N(S_1)| ≤ |Σ|-ε_2, from which Claim 3 follows immediately by choosing q_i small enough.If (<ref>) were to fail we would have a subsquence of i so that for all k, lim sup_i→∞ |P^k_1(H^1_i(1))|≥ |Σ|(N(S_k)) Let us assume P_1(H^1_i(1)) converges as varifolds to a varifold V with |V|=W and V is supported in S.Arguing as in the previous claims, we conclude that H^1_i(1)→ V and moreover that Φ_i(t_a)→ V.By (<ref>), the fact that |V|=W, and the Constancy Theorem, we obtain that V is equal to Σ. So we have Φ_i(t_a)→Σ contradicting the fact that 𝐅(Φ_t(t_a),Σ)=α.Claim 4:There exists an ε_2>0 so that for i large enough,H^3_i(t)∩ B_ε(𝒮)=∅ for all t∈ [0,1]. As in Claim 1), suppose toward a contradiction that there is a subsequence (not relabelled) i→∞ as well as a sequence of stationary integral varifolds V_i as well as t_i∈ [0,1] so that 𝐅(V_i, H^3_i(t_i))→ 0 as i→∞.We can assume |V_i|=W and by Allard's compactness theorem V_i converge to a stationary integral varifold V.Thus also H^3_i(t_i) converge to V.The only stationary integral varifolds supported in N(S) are the S_i with some integer multiplicities.It follows that V=∑ k_i S_i for some non-negative integers k_i. Note that since |V|=W, and by claim 3), V≠Σ, it follows that at least one of the k_i is less than n_i, and at least one of the k_i is is greater than n_i (say k_r).Thus H^3_i(t_i)∩ N(S_r) converges to k_r S_r.But this as before implies Φ_i(t_a)∩ N(S_r) converges to k_r S_r, which contradicts the fact that 𝐅(Φ_i(t_a), Σ)≥α.Since F(Φ'(t),Σ)≥α for t∈ [0,1]∖ V'_i,α, by setting η:=min(ε_1,ε_2,α), Claim (3) and Claim (1) together imply the Deformation theorem. Finally, we observe that family Psi_i forms a non-contractible loop in the space of flat cycles. This follows by Interpolation results in Appendix Aof <cit.>. In particular, the sweepout thatwe obtained is non-trivial.Given the Deformation Theorem, together with Deformation Theorem A of Marques-Neves <cit.> it is easy to prove the Index Bound in the Simon-Smith setting which we restate: Let M be a Riemannian three-sphere with bumpy metric. Then a min-max limit of minimal two-spheres {Γ_1, ...Γ_k} satisfies:∑_i=1^kindex(Γ_i) = 1.If the metric is not assumed to be bumpy, then we obtain∑_i=1^kindex(Γ_i) ≤1 ≤∑_i=1^kindex(Γ_i)+∑_i=1^knullity(Γ_i).Suppose {Φ_i} is a pulled-tight minimizng sequence.Since the metric is bumpy, there are only finitely many embedded index 0 orientable minimal surfaces with area at most W. Thus there are only finitely many stationary integral varifold supported on a strictly stable minimal surface with total mass W.Denote this set 𝒲_0.Applying the Deformation Theorem <ref> we obtain a new minimizing family {Φ'_i} so that no element of 𝒲_0 is in the critical set of {Φ'_i}, while no new stationary integral varifolds are in the critical set.After pulling this family tight, we can apply Theorem 6.1 (<cit.>) we can find a minimal surface in the critical set of {Φ”_i} with index at most 1.Thus the Theorem is proved. If the metric g not bumpy, then we can take a sequence of metrics g_i converging to g. The min-max limit Λ_i for each g_i can be chosen to have index 1 by the above.From convergence of eigenvalues of the stability operator we have that, if the convergence is with multiplicity 1,the stable components can converge to a stable minimal surface in g with or without nullity, and the unstable component may converge to either a) an unstable minimal surface or else b) a stable minimal surface with nullity.This establishes the bounds (<ref>) in these cases.If the convergence is instead with multiplicity for some component, the limit is automatically stable as it has a positive Jacobi field, establishing (<ref>).Finally note that in the case of collapse with multiplicity the genus bounds (<ref>) are preserved. We will also need to consider the situation of widths of manifolds with boundary.To that end, let M be a three-manifold and Σ a strongly irreducible splitting so that M∖Σ=H_1∪ H_2.Suppose Σ^1_k<...<Σ^1_0=Σ is a chain of compressions on the H_1 side and Σ^2_j<...<Σ^2_0=Σ is a chain of compressions on the H_2 side.Remove from M the handlebodies bounded by the components of Σ^1_k and Σ^2_j to obtain a manifold M̃ with boundary B_1 in H_1 and B_2 in H_2. Suppose B_1 and B_2 are strictly stable minimal surfaces. Consider sweepouts {Σ_t} beginning at B_1 together with arcs joining the components,and terminating at B_2 together with arcs.We moreover demand that Σ_t is isotopic to Σ for 0<t<1. Let W denote the width for this min-max problem.Then we have the following proposition, which allows us to apply min-max theory:W>sup_C∈ B_1∪ B_2 |C|.Thus by the index bound (<ref>) we obtain in the interior of M̃ an index 1 minimal surface. Suppose toward a contradiction W=sup_C∈ B_1∪ B_2 |C|.Suppose without loss of generality that the supremum is realized by B_1. Let us consider a pulled-tight minimizing sequence {Σ^i_t}_i=1^∞ so thatsup_t∈ [0,1] |Σ^i_t|→ W i→∞ Let N:=N_r_0 denote the tubular neighborhood about B_1 on which P_t is defined via Lemma <ref>.For all ε>0 there exists δ>0 so that if 𝐅(X,B_1)<δ then ℋ^2(X∖ N))<ε.Choose ε_0 to be that provided by Lemma <ref>.Since Σ^i_t converges as varifolds to B_1 as t→ 0, for each i, we can choose t_i so that F(Σ^i_t_i,B_1)=δ/2 and ℋ^2(Σ^i_t_i∖ N)<ε_0.Thus from Lemma <ref> we can isotope Σ^i_t_i to a surface satisfying ℋ^2(Σ̃^i_t_i∖ N)<δ_i where δ_i is any sequence approaching 0 as i→∞.Since δ_i is approaching zero, we can perform neck-pinches on ∂ N∩Σ̃^i_t_i (c.f. Proposition 2.3 in <cit.>) o obtain a sequence of surfaces Σ^i_t_i entirely contained in N so that|Σ^i_t_i|≤ |Σ̃^i_t_i|≤ |Σ^i_t_i|.Let A denote a subsequential limit of Σ^i_t_i as i→∞.Note that |A|≤ W by (<ref>) and the fact that Σ^i are a minimizing sequence. Let Ã=P_1(A).We have W≤ |Ã| since à is an integral varifold supported on B_1 with multiplicity at least one everywhere.By the properties of P (Lemma <ref>) we have |Ã|≤ |A|.Thus putting this together we obtain W≤ |Ã|≤ |A|≤ W.In other words, Ã=A again by the properties of the projection map (Lemma <ref>).It follows that A is supported on B_1 and thus must consist of the surfaces comprising B_1 with multiplicity 1 by the Constancy Theorem since A is stationary (as the limit of a min-max sequence from a pulled-tight minimizing sequence).Thus we have Σ^i_t_i→ B_1 with multiplicity 1.But this contradicts the fact that F(Σ^i_t_i,B_1)=δ/2 for all i. § APPLICATIONSIn this section, let us prove the claim of Pitts-Rubinstein (Theorem <ref>), which we restate: Let M be a hyperbolic 3-manifold and Σ a strongly irreducible Heegaard surface.Then either* Σ is isotopic to a minimal surface of index 1 or 0 or* after a neck-pinch performed on Σ, the resulting surface is isotopic to the boundary of a tubular neighborhood of a stable one sided Heegaard surface.If M is endowed with a bumpy metric, in case (1) we can assume the index of Σ is 1. We alsoprove Theorem <ref> under the assumption M is a lens space not equal to ℝℙ^3:If M≠ℝℙ^3 is a lens space, then M contains a minimal index 1 or 0 torus.We will use repeatedly the following essential estimate due to Schoen <cit.>:Let M ba three-manifold.Then there exists C>0 (depending only on M) so that if Σ is a stable minimal surface embedded in M, thensup_x∈Σ |A|^2≤ C. We will need the following non-collapsing lemma:Let M be a closed Riemannian three-manifold.For all C>0 there exists ε(C,M)>0 so that if Σ⊂ M is a closed embedded two-sided minimal surface bounding a region H that is not a twisted I-bundle over a non-orientable surface and satisfyingsup_x∈Σ |A|≤ C,then the volume of H is at least ε(C,M). Note that the assumption on the topology of H is essential.In ℝℙ^3 one can easily find metrics in which a sequence of stable two-spheres converges smoothly with bounded curvature to ℝℙ^2 with multiplicity 2. These stable two spheres bound a twisted interval bundle about ℝℙ^2 with volumes approaching zero. Moreover, the assumption of bounded curvature is essential, as the doublings of the Clifford torus give an example of a sequence of minimal surfaces bounding volumes approaching zero.Suppose the lemma fails.Thus for some C>0 there is a sequence Σ_i of embedded minimal surfaces satisfyingsup_i∈ℕsup_x∈Σ_i |A|≤ C,where R_i is a handlebody bounded by Σ_i with (R_i)→ 0.If the area of Σ_i are uniformly bounded, then from the curvature bound we obtain that Σ_i converges with multiplicity one to a closed embedded minimal surface Σ or else with multiplicity 2 to a non-orientable surface Γ that is a one-sided Heegaard splitting. In the first case, since Σ bounds a definite volume on both sides, the smooth convergence implies that Σ_i do as well.In the second case we still have (R_i)→(M∖Γ)>0.Thus we can assume that the areas of Σ_i are unbounded.Again because of the curvature bound (<ref>), upon passing to a subsequence, we can assume Σ_i converges to a smooth minimal lamination ℒ.It follows that one can find a covering {B_j} of M with the property that in any ball, there's a diffeomorphism ϕ_j:B_j→ D^2× [0,1] so that ϕ_j(ℒ) consists of D^2× K, where K is a closed subset of [0,1].For each i, we can consider ϕ_j(Σ_i) which consist of a disjoint union of several graphs G^i,j_1,..., G^i,j_r_i (where, potentially r_i→∞ as i→∞).Let A^i,j_k denote the region between G^i,j_k and G^i,j_k+1.Note that A^i,j_k⊂ R_i for all k odd.Moreover, since (R_i)→ 0, it follows that (A^i,j_k)→ 0 for k odd.Thus we obtain that|G^i,j_k-G^i,j_k+1|→ 0i→∞.We can find a graph G^i,j_k+1/2 between G^i,j_k and G^i,j_k+1 and thus in B_i we can retract Σ_i to smoothly with multiplicity 2, to the graph G^i,j_k+1. Since we have such retractions in all balls B_i, by gluing these together we obtain a retraction of Σ_i onto a closed embedded surface Γ. But a connected surface cannot retract to a closed surface Γ smoothly with multiplicity two unless Γ is non-orientable.Thus we obtain that R_i is homeomorphic to a twisted interval bundle over Γ, contradicting the assumption on R_i. This is a contradiction. We have the following finiteness statement for nested minimal surfaces:Let M be a three manifold with boundary X, a stable, minimal genus g surface.Suppose M is not homeomorphic to a twisted interval bundle over a non-orientable surface. Let {X_i}_i∈ℕ be a sequence of stable minimal surfaces isotopic to X so that each X_i bounds X× [0,1] on one side and H_i:=M∖ (X× [0,1]) on the other.Suppose {X_i}_i∈ℐ are nested in the sense that H_i⊊ H_j i>j.Then the areas of X_i are uniformly bounded and thus some subsequence obtained from {X_i}_i∈𝐍 converges to a minimal surface of genus g with a non-trivial Jacobi field.If M is endowed with a bumpy metric, no such infinite sequence {X_i}_i∈𝐍 can exist.The nestedness assumption is key in Proposition <ref>.Colding-Minicozzi have constructed <cit.> examples of stable tori without any area bound (later B. Dean <cit.> found examples of any genus greater than 1, and J. Kramer <cit.> found examples of stable spheres). The proof of Proposition <ref> is related to an idea due to M. Freedman and S.T. Yau toward proving the Poincare conjecture.If one had a counterexample to the Poincare conjecture, the sketch was to endow it with a bumpy metric and then using an (as yet conjectural) min-max process to produce an infinite sequence of minimal embedded nested two-spheres.Proposition <ref> then leads to a contradiction. Since X_i do not bound a twisted I-bundle over a non-orientable surface, it follows from Lemma <ref> that there exists ε_0>0 so that (H_i)≥ε_0>0 for all i.For each i≥ 1, denote R_i=H_i+1∖ H_i. Note that as ⋃_i=1^∞ R_i⊂ H_1,is a disjoint decomposition, it follows that(R_i)→ 0r→ 0. For each i≥ 1, because X_i and X_i+1 are nested we can define the lamination ℒ_i to consist of two leafs, X_i and X_i+1. By the curvature bounds for stable minimal surface (<ref>)it follows that ℒ_i subconverge to a lamination ℒ.From the definition of lamination, we can cover M by finitely many open balls {B_i}_i=1^p so that on any ball B_i, after applying a diffeomorphism ϕ^i:B_i→ D^2× [0,1] ⊂ℝ^3, where D^2 is the unit disk in the xy-plane.For i large, in each ball B_k,ℒ_i|_B_k consists of several parallel graphs {G^i,k_j}_j=1^r, where r depends on i and k.The region R_i restricted to B_k consists of several slab regions between consecutive graphs from {G^i,k_j}_j=1^r. Let R_i^k(s,s+1) denotesuch a slab region between G^i,k_s and G^i,k_s+1 in B_k.Slab regions come in three possible types: (a) both boundary walls G^i,k_s and G^i,k_s+1 are in X_i, (b) one boundary wall is in X_i and the other in X_i+1, or (c) both boundaries in X_i+1. For each slab R_i^k(s,s+1) of type (b) there's a well defined retraction of R_i^k(s,s+1) onto the wall in X_i.For a slab G^i,k_s of type (c), we can retract the slab onto a graph G̃ in between G^i,k_s and G^i,k_s+1. Note that there must be some points of X_i+1 which are retracted to X_i (otherwise, all points of X_i+1 are of type (c) and thus we obtain that X_i+1 bounds a set of tiny volume, contradicting the previous lemma).In this way, we obtain a smooth retraction of X_i+1 onto X'_i∪X̃, where X̃ are the union of graphs arising in case (c) and X'_i denotes a subset of X_i.But since the retraction is smooth, in fact X̃ is empty and X'_i=X_i.Thus we obtain that X_i+1 is a normal graph over X_i for i large.In fact, the same argument shows that X_j is a normal graph over X_i for any j>i.It follows that the areas of X_i are uniformly bounded.We can now prove the following Proposition:Let H be a Riemannian handlebody with strictly stable boundary, endowed with a bumpy metric.Then there exists a minimal surface Σ of index 1 or 0 obtained after finitely many neck-pinch surgeries performed on ∂ B. Thus the genus of Σ is at most the genus of ∂ H.We consider sweep-outs of H beginning at ∂ H and ending at the one-dimensional spine of H.We then consider the associated min-max problem and its width.By Proposition <ref>, since ∂ H is strictly stable we obtain that the width of the handlebody H is strictly larger than the area of ∂ H. If ∂ H is diffeomorphic to S^2, then by the Deformation Theorem <ref> the min-max limit can be realized by an unstable surface in the interior of H. Otherwise, by the genus bounds, the minimal surface realizing the width cannot be equal to the boundary surface ∂ H obtained with some multiplicity.Moreover, by the upper index estimates of Marques-Neves (Deformation Theorem A), we can guarantee that the min-max limit has index at most 1.If we are in the case of obtaining a stable minimal surface Σ in Proposition <ref>, then we can apply Proposition <ref> iteratively to the handlebody bounded by Σ inside of H.By the Nesting Proposition <ref>, there can only be finitely many iterations and thus we obtain the following improvement on Proposition <ref> which rules out obtaining a stable minimal surface: Let H be a Riemannian handlebody with strictly stable boundary, endowed with a bumpy metric.Then there exists a minimal surface Σ of index 1 obtained from ∂ H after finitely many neck-pinch surgeries performed. Thus the genus of Σ is at most the genus of ∂ H. Note that Proposition <ref> implies the Index Bound Theorem <ref> as if we obtain a minimal surface that is strictly stable, Proposition <ref> implies the existence of an index 1 surface inside each handlebody. §.§ Strong Irreducibility Let us recall some basic fact about strongly irreducible Heegaard splittings Σ.Let Σ be a Heegaard surface in M.By definition, this means M∖Σ consists of two open handlebodies, H_1, and H_2.Let γ be a simple closed curve on Σ.We say γ is a compressing curve if it bounds an embedded disk D with ∂ D=γ whose interior is contained in H_1 or H_2. We call D the compressing disk bounded by γ. There are three types of compression curves: * γ bounds no disk on Σ and bounds a disk in H_1* γ bounds no disk on Σ andbounds a disk in H_2* γ bounds a disk in Σ isotopic to its compressing disk in either H_1 or H_2 In the third case, let us say that γ bounds an inessential disk and the compression is inessential. In the first and second cases, let us say γ bounds an essential disk in H_1 or H_2, respectively..A Heegaard surface is strongly irreducible if every essential compressing disk in H_1 intersects every essential compressing disk in H_2. Given a compressing curve on Σ, we can perform a neck-pinch surgery on Σ along γ to produce a new surface, Σ'.Let us write Σ'<Σ in this case.Note that we have (Σ')≤(Σ). In fact, whenever Σ is the boundary of a handlebody and Σ'<Σ precisely one of the following holds: * (Σ')=(Σ)-1 * (Σ')=(Σ) and the number of connected components of Σ' is one more than Σ. Because of strongly irreducibility, every compressing curve bounding a disk in H_1 intersects every such curve in H_2.It follows that if we perform an H_i-compression (for i=1 or i=2) on Σ to obtain a surface Σ', then any further compression on Σ' must be performed on the same side, except for inessential compressions which can happen on either side.Suppose we perform an essential neck-pinch on a strongly irreducible Heegaard splitting of genus g, Σ_g to obtain Σ_g'.It is possible that the genus of Σ'_g is one less than Σ_g, and, while Σ'_g bounds on one side a genus g-1 handlebody, on the other side it bounds a twisted interval bundle over a non-orientable surface Γ.If such a neck-pinch exists, then Γ is an incompressible non-orientable minimal surface.Moreover, Γ is known as a one-sided Heegaard splitting, since M∖Γ is a handlebody of genus g-1.The simplest example of this phenomenon arisees from the genus 1 Heegaard splitting of ℝℙ^3.After a neck-pinch performed on a Heegaard torus, one obtains a three ball that bounds an interval bundle over ℝℙ^2. Let us summarize this discussion in the following dichotomy for surgeries performed on a strongly irreducible splitting.Suppose Σ_k<Σ_k-1<...< Σ_0 where Σ_0 is strongly irreducible.Then one of the following holds: * For each j∈{0,,,,k-1}, Σ_j is obtained from an essential surgery on Σ_j-1 performed on the H_1 side or an inessential surgery splitting off a two-sphere* For each j∈{0,,,,k-1}, Σ_j is obtained from an essential surgery on Σ_j-1 performed on the H_2 side or an inessential surgery splitting off a two-sphereWe have the following further dichotomy:* Σ_1 bounds a twisted interval bundle over an incompressible one sided Heegaard surface and Σ_j for j>1 are obtained from Σ_j-1 by inessential compressions or else* for any j∈{1,2,...,k}each non-sphere component of Σ_j is incompressible in the manifold M∖(Σ_j), where (Σ_j) denotes the interior of the handlebody determined by Σ_j. For any sphere component Σ_j, the infimal area of a surface isotopic to Σ_j in M∖(Σ_j) is positive.If we are in case (a) we can minimize area in the isotopy class of the one sided Heegaard surface by a theorem of Meeks-Simon-Yau (<cit.>) to obtain a stable embedded non-orientable minimal surface. Proof of Theorems <ref> and <ref>: Let us first assume that M is endowed with a bumpy metric.So we are assuming M is either hyperbolic or a lens space.Let Σ be a strongly irreducible Heegaard splitting.Suppose that after performing a neck-pinch on H, one obtains a surface isotopic to the boundary of a tubular neighborhood of a one-sided Heegaard splitting surface Γ.Then we are in case (2) of the Theorem and by (a) above we can minimize in the isotopy class of Γ to obtain a stable one-sided Heegaard splitting of M.Let us therefore assume without loss of generality no such neck-pinch is possible. If M is a hyperbolic manifold, there are no minimal tori or spheres and moreover, if Σ_g is a minimal surface of genus g, then|Σ_g|< 4π(1-g). Consider the min-max limit Γ_0 obtained relative to H.Thus we haveΓ_0=∑_i=1^k n_iΛ_i,where Λ_i are closed embedded minimal surfaces and n_i as positive integers.Moreover, if n_i>1 then Λ_i is a two-sphere. Also we have∑_i=1^k n_i(Λ_i)≤ g.From the Index bound (<ref>) we obtain ∑_i=1^k(Λ_i)=1 Suppose that Λ_1 is the unique unstable component of Γ_0. If Λ_1 has genus g, we are done.Assume without loss of generality that the genus g' of Λ_1 is less than g.Thus from case (b) Λ_1 bounds a handlebody Y_i.Let us remove from the manifold M the set Y_i to obtain a new manifold with boundary M'=M∖ Y_i.Since Λ_1 is unstable, we can minimize area in its isotopy class in M' to obtain a closed embedded strictly stable minimal surface Λ'_1.If Λ_1 has positive genus, then it is incompressible and Λ'_1 is a genus g' strictly stable minimal surface in the isotopy class of Λ_1 together (potentially) with some minimal two-spheres.If Λ_1 is a sphere, we obtain that Λ_1' is a collection of minimal two-spheres. If Λ_1 has positive genus, let us define M” to be obtained from M' be removing the collar neighborhood between the positive genus component of Λ_1' and Λ_1.If Λ_1 has genus zero, then one of the components of Λ_1' is homologous to Λ_1, and thus we can form M” be removing the cylindrical region between this component and Λ_1. In the end, we obtain a manifold M” with strictly stable boundary consisting of a single strictly stable minimal surface of genus 0≤ g'<g.Let us assume toward a contradiction that M contains no index 1 minimal surface isotopic to Σ. We will now describe an iteration process.Beginning with N_0=M”, we obtain an infinite sequence of manifolds {N_i}_i=0^∞ so that * N_i has stable boundary ∂ N_i whose components are partitioned into two sets: B^1_i, and B^2_i, so thatfor j∈{1,2} each surface in B^j_i is obtained from a sequence of H^i-surgeries on Σ. * for each i and j∈{1,2} there holds∑_C∈ B^j_i(C)≤ g * N_i⊊ N_j when i>jNote first that N_0 satisfies (1) and (2).Given the surface N_i, let us describe how to construct N_i+1.We can consider sweep-outs {Σ_t}_t=1^1 of N_i with the following properties * Σ_0 = B^1_i ∪{ B^1_t}* Σ_t for 0<t<1 is a surface isotopic to the strongly irreducible Heegaard surface Σ * Σ_1 = B^2_i ∪{ B^2_t}* Σ_t varies smoothly for 0<t<1 and in the Hausdorff topology for t→ 0 and t→ 0 We can consider the saturation of all such sweepouts Π_Σ_t and width W_i for the corresponding min-max problem.Since the components of B^2_i∪ B^1_i are strictly stable, it follows from Proposition <ref> that W_i>sup_C∈ B^2_i∪ B^1_i |C|. Thus the min-max limit associated to the min-max problem consists of a varifoldΛ=∑_i=1^kΛ_i. If M is hyperbolic, then there are no minimal two-spheres and thus each component of Λ arises with multiplicity 1.It then follows from Proposition <ref> that (at least) one component Λ_1 of Λ is contained in the interior of the manifold.If instead M is a lens space, then the Deformation Theorem similarly implies that a connected component of Λ is contained in the interior of M. If Λ_1 is isotopic to one of the components of∂ N_i,then we remove from N_i the collar region diffeomorphic to Λ_1× [0,1] to obtain a new manifold N_i'. If Λ_1 bounds.a handlebody in N_i, then we remove this handlebody to obtain N_i' (note in the process we may remove some components of ∂ N_i contained in this handlebody). Since N'_i now has one unstable component, Λ_1, we can minimize area <cit.> in the isotopy class of Λ_1 in N'_i to obtain a stable minimal surface Λ̃_1.If Λ_1 has positive genus, then Λ̃_1 and Λ_1 are isotopic and we remove the collar region between them to obtain N_i+1.If Λ_1 is a two-sphere, then it follows from our interpolation theorem that in fact Λ̃_1 consists of several two spheres with multiplicity 1, one Λ̃'_1 of which is homologous to Λ_1.We remove the collar region from N_i between Λ̃'_1 and Λ_1 to obtain N_i+1.This completes the construction of the iteration process to obtain an infinite sequence N_1,N_2,...N_k... of manifolds satisfying (a), (b) and (c) above under the assumption that there is no index 1 minimal surface isotopic to Σ. We will now deduce a contradiction.Since by (b) the genus of the surfaces ∂ N_i are bounded by g, if the areas of ∂ N_i were bounded, we obtain a Jacobi field, contradicting the bumpiness of the metric. In the case that M is hyperbolic, since ∂ N_i have bounded genus, it follows from the area bound (<ref>) that the areas are in fact bounded, violating the bumpiness of the ambient metric. This is a contradiction and completes the proof in the hyperbolic case. Thus henceforth we may assume that the areas of ∂ N_i are an unbounded sequence of positive numbers and the ambient manifold is a lens space. Suppose for some subsequence of the ∂ N_i, at least one component ∂ N^*_i of ∂ N_i that is distinct from ∂ N_i-1 has genus 0<g'<g.In this case, we can assume that ∂ N_i are a nested sequence of genus g' surfaces. By Proposition <ref> it follows that this case cannot happen as the metric is bumpy.Suppose instead for some subsequence of the ∂ N_i, at least one component ∂ N^*_i of ∂ N_i distinct from∂ N_i-1 is a sphere.We claim that upon passing to a subsequence, we can assume that these spheres ∂ N_i are nested.If not, by Lemma <ref>, each two-sphere bounds a definite volume .If the two-spheres ∂ N_i are non-nested, then we obtain an impossibility:(M)≥∑_i=1^∞((∂ N^*_i))≥∑_i=1^∞=∞,where (∂ N^*_i) denotes the three-ball in M enclosed by the two-sphere ∂ N^*_i.Thus we can assume the two spheres ∂ N^*_i are nested.By Proposition <ref>, we obtain a contradiction. Since all cases result in contradictions, it follows that the sequence N_0, N_1, N_2... must terminate at a finite stage so that N_k' is a surface isotopic to Σ with Morse index 1.If the metric g is not bumpy, we can consider a sequence of metrics g_i→ g.By the above, there is an index 1 minimal surface Σ_i isotopic to Σ.We can consider the limit Σ_i→Σ_∞. Since Σ_i have bounded area and bounded genus, Σ_∞ is a smooth connected minimal surface obtained with some multiplicity. By strong irreducibility, the converge is multiplicity 1 (as the only other alternative is for it converge with multiplicity two to one sided Heegaard splitting surface).Since the convergence is multiplicity 1 it is smooth everywhere, and thus the genus of Σ is the same as Σ_i and moreover, Σ is isotopic to Σ_i. The index of Σ_∞ is either 0 or 1 because under smooth convergence, the eigenvalues of the stability operator vary smoothly.Finally, let us prove Theorem <ref>, which we restate:Let M be a Riemannian 3-manifold diffeomorphic to ℝℙ^3 endowed with a bumpy metric.Then M contains a minimal index 1 two-sphere or minimal index 1 torus.If the minimal surface realizing the width of ℝℙ^3 contains a two-sphere or torus, then we are done.Otherwise, the width is realized by kΓ, where Γ is an embedded ℝℙ^2 and k is an even integer.Let us pass to a double cover M̃ of M so that Γ lifts to Γ̃ which a two-sphere and M̃ is a three-sphere. If Γ̃ is strictly stable, then by our Index Bound, each ball B̃_1, B̃_2 bounded by Γ̃ contains an unstable two-sphere S̃_1 and S̃_2, respectively.Since S̃_1 is contained in a fundamental domain of the deck group, it follows that S̃_1 descends to a minimal two-sphere in M.If instead Γ̃ is unstable, then we can push off Γ̃ to one side of Γ̃ using the lowest eigenfunction of the stability operator.Using a by-now standard argument (c.f. Lemma 3.5 in <cit.>) the sweepout can either be extended to the rest of B̃_1 with all areas below Γ̃ or else we obtain an unstable two-sphere in B̃_1, which means the theorem is proved as the unstable two-sphere in B̃ descends to M.Thus we can assume we have realized Γ̃ as a minimal surface in an optimal foliation of M.By the catenoid estimate <cit.> we can easily construct a sweep-out of M by tori with area less than 2|Γ|.Thus the width of M could not have been realized by kΓ.This generalizes in a straightforward way to the case of strongly irreducible Heegaard splittings:Let M be a hyperbolic three-manifold and Σ a strongly irreducible Heegaard splitting of genus g.Then M contains an orientable index 1 minimal surface of genus at most g.The proof is identical to the case of ℝℙ^3 above.By Pitts-Rubinstein claim, either M contains a genus g minimal surface isotopic to Σ, or M contains a one sided minimal Heegaard splitting Γ. 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Applications of minimax to minimal surfaces and the topology of 3-manifolds, Miniconference on geometry and partial differential equations, 2, Canberra, (1986), 137–170; Proc. Centre Math. Anal. Austral. Nat. Univ.[R]R D. Rolfson. Knots and links, Publish or Perish Press, 1976.[R1]R1 J.H. Rubinstein, Minimal surfaces in geometric three-manifolds, preprint available at http://www.ms.unimelb.edu.au/ rubin/publications/minimalsurfacenotes8.pdf.[R2]R2 J.H. Rubinstein, One sided Heegaard splittings of three-manifolds, Pacific J. Math. vol. 26, no. 1, 1978.[S]sR. Schoen.Estimates for stable minimal surfaces in three-dimensional manifolds. In Seminar on Minimal Submanifolds, vol 103 of Ann. of Math. Stud. p. 111–126, Princeton University Press, Princeton, NJ 1983.[So]So A. Song. Local min-max surfaces and strongly irreducible minimal Heegaard splittings, preprint available at arXiv:1706.01037. [SS]ss F. Smith. On the existence of embedded minimal two spheres in the three sphere, endowed with an arbitrary riemannian metric. PhD thesis, Supervisor: Leon Simon, University of Melbourne, 1982.[W]W B. White. The space of minimal surfaces for varying Riemannian metrics, Indiana Math. Journal 40 (1991), no.1, 161–200. | http://arxiv.org/abs/1709.09744v2 | {
"authors": [
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tinyhmargin=2.5cm, vmargin=2cm changemargin[2] #1 #2*+0.15in+0.15in 30pt 8.1in 6in 1.3.05in theoremTheorem plain acknowledgementAcknowledgement algorithmAlgorithm axiomAxiom caseCase claimClaim assumptionAssumption conclusionConclusion conditionCondition conjectureConjecture corollaryCorollary criterionCriterion definitionDefinition *definition*Definition exampleExample exerciseExercise lemmaLemma notationNotation problemProblem propositionProposition remarkRemark solutionSolution summarySummary pilotPilot continued[1][continued]#1 varproof[1][Proof]#1#1#2#1#22mu#1#2 equationsection Sharp bounds for the Roy model]Sharp bounds and testability of a Roy model of STEM major choices University of Toronto The Pennsylvania State UniversityMunich Center for the Economics of Ageing at the Max-Planck Insitute for Social Law and Social PolicyWe analyze the empirical content of the Roy model, stripped down to its essential features, namely sector specific unobserved heterogeneity and self-selection on the basis of potential outcomes. We characterize sharp bounds on the joint distribution of potential outcomes and testable implications of the Roy self-selection model under an instrumental constraint on the joint distribution of potential outcomes we call stochastically monotone instrumental variable (SMIV). We show that testing the Roy model selection is equivalent to testing stochastic monotonicity of observed outcomes relative to the instrument. We apply our sharp bounds to the derivation of a measure of departure from Roy self-selection to identify values of observable characteristics that induce the most costly misallocation of talent and sector and are therefore prime targets for intervention. Special emphasis is put on the case of binary outcomes, which has received little attention in the literature to date. For richer sets of outcomes, we emphasize the distinction between pointwise sharp bounds and functional sharp bounds, and its importance, when constructing sharp bounds on functional features, such as inequality measures. We analyze a Roy model of college major choice in Canada and Germany within this framework, and we take a new look at the under-representation of women in STEM.[ Romuald Méango The first version is of 22 April 2012. The present version is of December 30, 2023. This research was supported by SSHRC Grants 410-2010-242, 435-2013-0292 and 435-2018-1273, NSERC Grant 356491-2013, and Leibniz Association Grant SAW-2012-ifo-3. The research was conducted in part, while Marc Henry was visiting the University of Tokyo and Isma^̂22el Mourifié was visiting Penn State and the University of Chicago. The authors thank their respective hosts for their hospitality and support. They also thank Désiré Kédagni, Lixiong Li, Karim N'Chare, Idrissa Ouili and particularly Thomas Russell and Sara Hossain for excellent research assistance. Helpful discussions with Laurent Davezies, James Heckman, Hidehiko Ichimura, Koen Jochmans, Essie Maasoumi, Chuck Manski, Ulrich M^̂22uller, Aureo de Paula, Azeem Shaikh and very helpful and detailed comments from five anonymous referees, from numerous seminar audiencesand the 2018 Canadian senate open caucus on women and girls in STEM are also gratefully acknowledged. Correspondence address: Department of Economics, Max Gluskin House, University of Toronto, 150 St. George St., Toronto, Ontario M5S 3G7, Canada =================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Keywords: Roy model, partial identification, stochastic monotonicity, functional sharp bounds, inequality, college major, gender profiling, STEM, SMIV.JEL subject classification: C31, C34, C35, I21, J24§ INTRODUCTION In a seminal contribution that is now part of the folklore of economics, <cit.> proposed a model of earnings with sorting on sector specific skills. Roy's objective was to provide a channel by which skills translate into earnings and to capture the idea that favorable sorting reduces earnings inequality.The simplicity of this mechanism and the richness of its implications turned the Roy model into one of the most successful tools in the analysis of environments, where skills and choices interact: they include the <cit.>-<cit.> labor supply model, the unionization model of <cit.>, the model of education self-selection proposed by <cit.>, sector selection in <cit.>, and the <cit.> immigration model. More recently, <cit.> revisited the issue of inequality in the unionization model, <cit.> used the Roy model to shed light on the recent evolution of the gender gap, <cit.> to analyze the choice of surgical procedures and <cit.> to analyze benefits and costs of educational choices. The list is, of course, far from complete, but quite sufficient to show the enormous success of the Roy model.In the original model, skills are jointly log normal and <cit.> show that under this assumption, the joint distribution of skills and the marginal distributions of potential earnings are identified. <cit.> further show that self-selection does indeed reduce aggregate inequality when skills are log normal and within sector inequality when skills have a log concave distribution. Naturally, the effect of self-selection on outcome inequality remains empirically relevant when skills do not have a log concave distribution. However, the analysis of the nonparametric version of the Roy model, stripped down to the self-selection mechanism, has long been hampered by (lack of) identification issues. The <cit.> and <cit.> comments on non-identifiability of competing risks imply that any continuous outcome distribution could be rationalized with independent sector-specific skills, so that the <cit.> notion of skill hierarchies loses empirical content.One way to resolve this lack of identification issue, pioneered by <cit.>, <cit.>, is to bring in additional information to achieve identification, such as repeated cross sections, panel data, factor structure, exclusion restrictions and large support assumptions within restricted specifications of the model. A vast literature, both theoretical and empirical, followed this lead (see for instance <cit.>, <cit.>, <cit.> or <cit.> for recent accounts). In particular, recent developments in nonparametric inference in Roy and competing risks models can be found in <cit.>, <cit.>, <cit.>, and <cit.>. Another way to approach the issue, which was pioneered by <cit.> and which we follow here, is to recognize that, despite the identification failure because of self-selection, the Roy model is not devoid of empirical content.The object of the present article is to characterize this empirical content, with special emphasis on the joint distribution of potential outcomes and testability of the Roy selection mechanism. This implies considering distributional features of outcomes, which are important if one is to evaluate the effect of self-selection on wage inequality, as Roy initially intended; see <cit.> for a discussion. It further implies considering joint distributional aspects. As <cit.> noted, information on the joint distribution is necessary to evaluate welfare implications of policy changes that affect the relative price of skills in both sectors. Correlation between outcomes can be important to policy evaluation, as discussed in <cit.>, as can the difference between potential outcomes or the distribution of outcomes conditional on the chosen sector. In all such cases, the joint distribution of potential outcomes is the relevant object to characterize. We refer to <cit.>, <cit.>, and <cit.> for in-depth discussions of this issue.We devote a considerable amount of attention to the case with binary outcomes, which we call the binary outcomes Roy model. The reason is twofold. First, the identification failure is starker with binary outcomes, and the characterization of the joint distribution is easier to derive and explain in the binary case, before it is extended to the more general cases of discrete, continuous or mixed discrete and continuous outcomes. Second, the case of discrete outcomes has received very little attention in the Roy model literature, <cit.> and <cit.> being notable exceptions. Most of this literature concerns the case of continuous outcomes and many applications, where outcomes are discrete, fall outside its scope. They include analysis of the effects of different training programs on the ability to secure employment, of competing medical treatments or surgical procedures on survival, and of competing policies on schooling decisions in developing countries among numerous others. The Roy model is still highly relevant to those applications, but very little is known of its empirical content in such cases.We derive sharp bounds for the joint distribution of (Y_0,Y_1), using techniques from <cit.> and <cit.> -see also <cit.> and <cit.>.[<cit.>, and later <cit.>, <cit.> and <cit.> look at more general treatment effects models from a partial identification point of view and use rearrangement inequalities to derive bounds on the distribution of treatments effects, a feature of the joint distribution of potential outcomes, under conditions, where the marginal distributions of potential outcomes are identified.] Bounds do not cross, and the model is not testable, unless we observe variables that have a restricted effect on potential outcomes. A special case is that of selection shifters that are statistically independent of potential outcomes. Such variables have two major drawbacks in this framework. First, they are very elusive in important areas of application of this methodology. To take one classical example in the literature on returns to education, parental education, measures of school quality, and fees may be correlated with unobserved cognitive and non cognitive parental investments and is therefore unlikely to be independent of potential outcomes. Second, within the Roy model, given the sector selection mechanism, a variable that is independent of potential outcomes can only affect sector selection if potential outcomes are equal in both sectors, which severely restricts the extent of resulting variation in sector selection. To resolve both of these issues, we introduce stochastically monotone instrumental variables. They are selection shifters that are restricted to affect potential outcomes monotonically. For instance, parental education may not be independent of potential wages because of unobserved parental cognitive and non cognitive investments. However, it is unlikely that such additional investments will negatively affect potential future wages. Moreover, allowing for monotonic effects on potential outcomes resolves the second issue, since stochastically monotone instrumental variables may shift selection even when potential outcomes are different. Our stochastically monotone instrumental variable assumption is stronger than the <cit.> monotone instrumental variable assumption, which only requires mean potential outcomes to be monotonic in the instrument, rather than the whole distribution. It is different from Constraint (9) in<cit.>, which is a restriction on the univariate distribution of realized wages, as opposed to the joint distribution of potential outcomes. This difference is crucial, when deriving bounds on joint distributional features in a sector selection model.We derive the identified set for the joint distribution of potential outcomes in the binary outcomes Roy model under this assumption of stochastically monotone instrumental variable (hereafter SMIV), and we show that stochastic monotonicity of observed outcomes in the instrument summarizes all observable implications of the model. Hence a test of Roy selection behavior boils down to a test of stochastic monotonicity,and can therefore be conducted with existing inference methods, as in <cit.>, <cit.> and <cit.>. This provides a fully nonparametric alternative to tests of the Roy model proposed since <cit.> (see also <cit.>, with multiple treatments and repeated cross-sections). Intuition about the relation between stochastic monotonicity and Roy selection can be gained from the following hypothetical scenario. Suppose two identical young women have higher economic prospects in non STEM fields than in STEM fields. One of them is induced to choose a STEM degree by a larger proportion of women on the STEM faculty in her region, whereas the other, who lives in a region with lower proportion of women on the STEM faculty, chooses a non-STEM field to maximize economic prospects. It will then appear that an increase in the proportion of women on the STEM faculty produces a decrease in observed outcomes, hence a rejection of monotonicity.To alleviate the concern that rejection of Roy selection behavior may in fact be down to a rejection of the assumption that individuals are perfectly informed of their potential future outcomes at the time of sector selection, we derive sharp bounds for the joint distribution of potential outcomes for an imperfect foresight binary outcomes Roy model. In the latter, agents select the sector that maximizes the expectation of their outcome with respect to their information set at the time of decision. Since the model is rejected if and only if this identified set is empty, it allows us to summarize all observable implications of that version of the model as well. We also provide bounds for a measure of departure from Roy selection, which is constructed from the difference between the maximum potential outcome and the realized outcome (both being equal under Roy selection), and which, again, requires bounding the joint distribution of potential outcomes, as we do here, rather than marginal average outcomes, as is customary in the literature. These measures of departure from the Roy selection model serve to identify values of observable characteristics that induce the most costly misallocation of talent and field of study and are therefore prime targets for intervention.We extend the test of Roy self-sorting and the measure of departure from Roy to more general discrete and continuous outcomes. When extending the analysis of the bounds on potential outcomes, distributional issues come to the fore. The classic <cit.> bounds are sharp for ℙ(Y_0≤ y) and ℙ(Y_1≤ y), for each quantile y, but, as noted by <cit.>, they do not incorporate monotonicity and right-continuity restrictions on distribution functions. Hence they entail loss of information, when the object of interest involves densities, such as hazard rates, or functionals of the distribution, such as inequality measures.We provide a general characterization of the joint distribution and bounds on the marginal distributions of potential outcomes that are functionally sharp, in the sense that they incorporate slope restrictions. In this, we follow <cit.>, although the model specification, hence the bounds, are different. As in <cit.>, <cit.> and <cit.> for different models, we apply the latter bounds to derive sharp bounds on inequality measures, which we show are more informative than would have been obtained from pointwise sharp bounds on the distributions of potential outcomes, such as Peterson bounds.In the tradition of <cit.>, <cit.> and, more recently, <cit.>, we analyze returns to education through the lens of the Roy selection model. It is well documented, since at least <cit.>, that major choice is an important determinant of labor market outcomes. The account in <cit.> shows that the literature on the determinants of and the returns to major choice is now substantial, including most notably <cit.>, <cit.> and <cit.>. The STEM versus non-STEM classification has come to dominate the debate. We therefore analyze a Roy model of choice of field of study, between STEM and non-STEM degrees, based on data from nationally representative surveys of university graduates in Canada and Germany. Following the recent literature on the subject, surveyed in <cit.>, we focus on mathematics intensive fields, including economics, but excluding life sciences. We consider a Roy model of major choice, where the target labor market outcomes are wage, obtaining a permanent job by the year of the survey, or holding a job related to the field of study, respectively. Our main objective is to shed some insight onto the under-representation of women in STEM education and even more so in STEM jobs, the gender gap in STEM labor market outcomes, and the contribution of the STEM economy to rising wage inequality. If choices conform to the Roy self-sorting mechanism, only policies directed at ex-post wage discrimination arelikely to be effective in reducing inefficiencies, not policies directed at reducing gender profiling in major choice. Hence, in our investigation of the determinants of under-representation of women in STEM fields, we give prominence to testing Roy selection behavior. As the latter is not testable without covariate restrictions, we consider variation induced in major choices by parental education. There are reasons to doubt the validity of parental education according to Assumption <ref> below, which requires independence of the instrument and the vector of potential outcomes. It is more reasonable to assume parental education level has a monotonic effect on potential outcomes, as prescribed by Assumption <ref> below, which requires stochastic monotonicity of the distribution of potential outcomes, conditionally on the instrument. When testing whether women graduate choices conform to the Roy self-sorting mechanism, we also use the fraction of women in the faculty of STEM programs in the region and at the time of major choice as a SMIV instrument, based on the assumption that role models may not negatively affect future prospects for women graduates. We also perform our test of Roy self-sorting based on additional vectors of instruments incorporating local labor market conditions at the time of choice for robustness purposes, and find very little variation in test results.Our tests of the Roy model for different choices of instruments and different employment related outcome variables reveal significant gender, racial and regional differences.We find a pattern of rejections of Roy self-sorting based on outcomes for white women in the former Federal Republic of Germany and the rest of Canada, and a lack of rejections for visible minorities and for white women from Québec, white men from all of Canada and the former German Democratic Republic. Confidence intervals for measures of departure from Roy behavior reveal that in the case of white women from the former Federal Republic, for instance, rejection of Roy behavior seems to be driven by lower income women with high school educated mothers and middle income women with postgraduate educated mothers. Among groups, where Roy self-sorting is not rejected, comparisons of interquartile ranges for observed and counterfactual income distributions are inconclusive except in the case of white women in the former Democratic Republic and white men in Québec, where self-sorting is found not to increase inequality, complementing the result for log-concave talent distributions in <cit.>.The pattern of rejections of Roy self-sorting in major choice points to non labor market related determinants of choice. For instance, our results are consistent with a story involving gender profiling pushing white men in the West of Germany into STEM fields and white women in the West of Germany and in Canada out of STEM fields. They are also consistent with gender profiling being less prevalent in the former communist Germany. However, the results are also consistent with a story involving non pecuniary field preferences driving major choices of more privileged groups in more affluent regions, but not the choices of the more financially constrained.§.§ OutlineThe remainder of the paper is organized as follows. Section <ref> details the general frame of analysis. Section <ref> concerns sharp bounds and testability of the binary outcomes Roy model, and its version with imperfect foresight, under the assumption of stochastic monotonicity of potential outcomes relative to an instrument. Section <ref> derives functional sharp bounds for the Roy model with mixed discrete-continuous outcomes, and their implications for testability of Roy behavior and effects of endogenous sector selection on functional features such as inequality measures.Section <ref> applies the derived bounds to the analysis of major choice in Canada and Germany and the under-representation of women in STEM.The last section concludes. Proofs of the main results are collected in the appendix. § ANALYTICAL FRAMEWORKWe adopt the framework of the potential outcomes model Y=Y_1D+Y_0(1-D), where Y is an observed scalar outcome, D is an observed selection indicator, which takes value 1 if Sector 1 is chosen, and 0 if Sector 0 is chosen, and Y_1, Y_0, are unobserved potential outcomes, with common lower bound b on their support (b is usually 0 or -∞). <cit.> trace the genealogy of this model and we refer to them for terminology and attribution. The object of interest is the joint distribution of (Y_0,Y_1) and features thereof. Since Y and D are observed, the joint distribution of (Y,D) is directly identified from the data.We strip the model down to its self-selection mechanism, where agents are perfectly informed of the joint distribution of their potential outcomes (Y_0,Y_1) in both sectors and choose the sector that maximizes outcomes, so that D=1 when Y_1>Y_0 and D=0 if Y_1<Y_0. The model is silent on the tie-breaking mechanism agents use in case Y_1=Y_0. As is customary in such frameworks, the assumption that agents are perfectly informed is intended to reflect, within a simple static model, the result of dynamic adjustments and learning on the one hand, and to put in stark relief the difference between the agents' and the analyst's information sets, on the other hand. We summarize the model with the following assumptions. [Potential outcomes]Observed outcomes are the realizations of a random variable Y satisfying Y=Y_1D+Y_0(1-D), where (Y_0,Y_1) is a pair of possibly dependent unobserved random variables and D is an observed indicator variable.[Selection]The selection indicator satisfies Y_1>Y_0⇒ D=1, Y_1<Y_0⇒ D=0.Individuals choose the sector that yields higher outcome, when Y_1 Y_0. Their choice criterion is unspecified if Y_1=Y_0. When outcomes are discrete, the possibility of ties has to be considered. More generally, in a Roy model of earnings, the possibility of equal earnings in both sectors has to be entertained, if wage setters propose contracts that pool different skill levels, for instance. If the probability of ties is non zero, the Roy model specification described here is different from the specification of the competing risks model with non zero probability of ties in <cit.>. The model we consider here is tuned to economic applications, where the sector selection is unknown, when both sectors yield the same outcome. Hence, we identify ℙ(Y_d≤ y,D=d), for d=0,1, but not ℙ(Y_d≤ y,Y_d>Y_1-d). All we know is that ℙ(Y_d≤ y,Y_d>Y_1-d)≤ℙ(Y_d≤ y,D=d)≤ℙ(Y_d≤ y,Y_d≥ Y_1-d). In the competing risks analysis of <cit.>, on the other hand, ℙ(Y_1≤ y,Y_1>Y_0), ℙ(Y_0≤ y,Y_1<Y_0) and ℙ(Y_1≤ y,Y_1=Y_0) are all assumed identified, so that one observes when both components of the system fail simultaneously. Our analysis can be extended to the case, where Y, Y_0 and Y_1, take values inan ordered subset of a Euclidean space, such as ℝ^2 endowed with the lexicographic order ≾_lex, for instance. In the latter case, with outcome variable Y=(W,T) ordered lexicographically, Assumption <ref>, would read [ W_d>W_1-d [ W_d=W_1-d T_d > T_1-d ] ] ⇒ D=d.Take the case of university STEM major choice for instance. Lexicographic Roy preferences based on relatedness and income implies that prospective students choose STEM if they anticipate only STEM degrees will provide them with employment in their field of study, or if both STEM and non STEM provide them with employment in their field but they anticipate higher earnings in STEM.All results below would relate to the probability distributions of outcomes and potential outcomes relative to the chosen order, not the multivariate probability distributions. For instance, in the lexicographic example, the probability distribution is defined as ℙ(Y≾_lexy)=ℙ((W,T)≾_lex(w,t))=ℙ(W<w[W=wT≤ t]).The whole analysis, model, distributional assumptions and theoretical results, are understood to be conditional on a set of observed covariates, which will be omitted from the notation, unless they are involved in identifying assumptions.§ BINARY OUTCOME ROY MODELA great deal of the intuition for the characterization that we propose for the Roy model can be developed with the simplest version, whereY_0 and Y_1 are both binary outcomes. It models success or failure in securing a desired outcome, and the way it depends on a binary choice of treatment.In the case of college major choice, considered in Section <ref>, Y_1 will model the ability to secure permanent employment at the time of the survey interview, if the degree or the major is classified as STEM, whereas Y_0 will model the ability to secure employment, with a non-STEM degree or major. A model satisfying Assumptions <ref> and <ref>, with Y_0,Y_1∈{0,1}, is called binary outcome Roy model. An alternative way of defining a binary outcomes model, which shares the main features of the Roy model, i.e., self-selection on unobserved heterogeneity, involves latent potential outcomes. It is identical to the Roy model, except that potential outcomes are censored. Observed outcomes are the realizations of a random variable Y satisfying Y=Y_1D+Y_0(1-D), where* potential outcomes satisfy Y_d=1{Y_d^∗>0}, for d=0,1, for a pair of possibly dependent unobserved random variables (Y_0^∗,Y_1^∗),* D is an observed indicator variable, satisfying Y_1^∗>Y_0^∗⇒ D=1, Y_1^∗<Y_0^∗⇒ D=0. The alternative binary Roy model of Definition <ref> can be interpreted in two ways. First, it is equivalent to a model with Y=1{Y^∗>0}, where Y^∗ satisfies a Roy model. Hence, it can be interpreted as a censored Roy model. The latent variables may be continuous variables, such as wages, and the analyst only observes whether or not they fall above or below a threshold. Other examples include examination grades, which are unobserved, except for the pass or fail outcome. Second, the actual outcome may be binary and be the result of a two-stage decision by the agent. In a first stage, they choose the sector of activity, with their choice of college major, for instance. In a second stage, they decide whether or not to work. The labor supply decision hinges on the difference between wage and reservation wage in the chosen sector. Then, Y_d^∗ can be interpreted as the difference between wage in Sector d and reservation wage in Sector d. If reservation wages are equal in both sectors, the model still conforms to the simple Roy incentive mechanism, where wages are the only determinant of sector choice. If reservation wages differ in both sectors, however, the model no longer conforms to the simple Roy incentive mechanism, as sector selection internalizes possibly non-pecuniary costs and benefits of each sector, as in the recent analysis of the generalized Roy model in <cit.>.Despite their distinct interpretations, it will be shown that sharp bounds for the joint distribution of (Y_0,Y_1) are identical in both models, so that both models carry exactly the same informationon the joint distribution of censored potential outcomes. They also share the reduced form implication𝔼(Y_d-Y_1-d| D=d)≥0,which can be interpreted as a condition of chosen sector advantage or as nonnegative average treatment effect on the treated (where choice of Sector 1 corresponds to treatment). However, we show below that the reduced form condition (<ref>) contains less information on the joint distribution of potential outcomes than the structural models of Definitions <ref> and <ref> do. In particular, constraint (<ref>) is also shared by a binary outcome Roy model with imperfect foresight, identical to the binary outcome Roy model of Definition <ref>, except that the selection equation of Assumption <ref> is replaced with the following: (Imperfect foresight)The selection indicator satisfies𝔼[Y_1-Y_0|ℐ]>0⇒ D=1, 𝔼[Y_0-Y_1|ℐ]>0⇒ D=0, where ℐ is the sigma-algebra characterizing the agent's information set at the time of sector choice.A model satisfying Assumptions <ref> and <ref>, with Y_0,Y_1∈{0,1}, is called binary outcome Roy model with imperfect foresight. Our results in the next section characterize sharp bounds on the joint distributions of potential outcomes and highlight the difference in empirical content between perfect and imperfect foresight Roy models.§.§ Sharp bounds for the binary outcome Roy model In the binary outcomes Roy model, the lack of point identification comes from the fact that the mapping from observed sector and success to unobserved skills is not single valued. We know that when success in Sector 1 is observed, potential outcomes can be either (Y_0=1,Y_1=1), i.e., success in both sectors, or (Y_0=0,Y_1=1), i.e., success in Sector 1 only. Hence the identified probability that a random individual in the population chooses Sector 1 and succeeds will not be sufficient to identify the probability of succeeding in Sector 1. What we do know, however, is thatY=0 is observed if and only if the individual has neither the skills to succeed in Sector 0 nor in Sector 1. Hence, ℙ(Y_0=0,Y_1=0)= ℙ(Y=0). Moreover, if the individual has the skills to succeed in Sector 0, but not in Sector 1, then, success in Sector 0 will be observed, so that ℙ(Y_0=1,Y_1=0)≤ℙ(Y=1,D=0). Symmetrically, if the individual has the skills to succeed in Sector 1, but not in Sector 0, then, success in Sector 1 will be observed, so that ℙ(Y_0=0,Y_1=1)≤ℙ(Y=1,D=1).The discussion above shows that the expressions hold. Showing sharpness of these bounds is more involved, and the proof of the Proposition <ref> is given in the appendix, together with a more fastidious statement of the theorem, with a rigorous and unambiguous definition of sharp bounds in this context. Note that the bounds can take the form of an equality in case upper and lower bounds coincide. The following equality and inequalities provide a set of sharp bounds for the joint distribution of potential outcomes (Y_0,Y_1) in the binary outcomes Roy model (Definition <ref>) and the alternative binary Roy model (Definition <ref>).[ ℙ(Y_0=1,Y_1=0)≤ℙ(Y=1,D=0),; ℙ(Y_0=0,Y_1=1)≤ℙ(Y=1,D=1),; ℙ(Y_0=0,Y_1=0)=ℙ(Y=0). ]The bounds in Proposition <ref> summarize all the information in the (alternative) binary outcome Roy model about the joint distribution of potential outcomes. From these bounds, sharp bounds on the marginals, which are akin to traditional bounds on average treatment outcomes, can be recovered. Combining the equality and inequalities of (<ref>), we obtain traditional bounds on the marginals (see for example <cit.>, Section 7.5).ℙ(Y=1,D=0)≤𝔼Y_0 ≤ℙ(Y=1)ℙ(Y=1,D=1)≤𝔼Y_1 ≤ℙ(Y=1).If the means of marginal potential outcomes are the objects of interest, as in <cit.>, the bounds above are sharp without additional restrictions. Here, we take bounds on the joint distribution of potential outcomes as the object of interest. It is easy to see that (<ref>) and ℙ(Y_0=0,Y_1=0)=ℙ(Y=0) are jointly equivalent to (<ref>). However, from (<ref>) alone, (<ref>) cannot be recovered, so that information on the joint distribution is lost. The bounds on the average sector difference are-ℙ(Y=1,D=0)≤𝔼(Y_1-Y_0)≤ℙ(Y=1,D=1).The sharp bounds of Proposition <ref> emphasize two important facts: * On the one hand, despite the literature on non identification of competing risks, starting with <cit.> and <cit.>, the Roy model does in fact contain non trivial information about the joint distribution of potential outcomes, hence of skills, or more generally, of sector specific unobserved heterogeneity. * On the other hand, the sharp bounds of Proposition <ref> can be very wide and they do not cross. For any joint distribution for (Y,D), there exists a joint distribution for (Y_0,Y_1) that fits the binary outcome Roy model, so that the latter is not falsifiable in the absence of additional constraints.Since the Roy model imposes strong restrictions on behavior, the lack of testability is particularly vexing. We shall consider exclusion and monotonicity restrictions that allow us to recover testability of behavior characterized by Roy sector selection. In the case of college major choice, considered in Section <ref>, one of our main concerns will be with explanations of the under representation of women in STEM. One candidate is wage discrimination in STEM, which is compatible with a Roy model of behavior. Another is gender profiling in major choice, which is not. Hence the ability to test Roy maximizing behavior in major selection is paramount.§.§ Stochastically monotone instrumental variables (SMIV) In order to allow falsifiability of the Roy model, we now investigate the implications of exclusion restrictions. Such exclusions are of two types: sector-specific variables, i.e., variables affecting only one outcome equation, but not the other (Assumption <ref> in Appendix <ref>), and variables that shift sector selection, but shift potential outcomes either not at all (Assumption <ref> below), or only in one direction (Assumption <ref> below). To sharpen the focus and save space, we discuss the conceptually relatively straightforward implications of sector specific variables in Appendix <ref>, and consider mostly the effect of vectors Z of variables that affect sector selection, but have restricted impact on potential outcomes. We shall comment on the way in which sector specific exclusions modify the expressions and leave details to Appendix <ref> (As mentioned before, conditioning on remaining observed covariates is implicit in all the paper). We start the discussion with variables that shift selection, but not potential outcomes. There exists a vector Z of observable random variables, such that (Y_0,Y_1)⊥⊥ Z.Such variables are akin to typical instrumental variables, and examples within Roy models in the existing literature include parental education in <cit.>, distance to a college in <cit.> and attendance in a Catholic high school in <cit.>. Local aggregate labor market variables at the time of sector selection are also often used, as in <cit.> and references therein.First, it is important to emphasize, that, unlike the generalized Roy model used in the contributions cited in the previous paragraph, the pure Roy selection mechanism imposes D=1 when Y_1>Y_0 and D=0 when Y_1<Y_0. Hence, a selection shifter Z satisfying Assumption <ref> can only affect the model in case of ties Y_1=Y_0. The model is lexicographic, in the sense that agents care only about outcomes when choosing their sector of activity, unless the outcomes are equal in the two sectors, at which point other considerations guide their decision. As a result, Y is independent of Z, but (Y,D) is not jointly independent of Z, so that the bounds in Proposition <ref> can be sharpened using variation in ℙ(Y=1,D=1| Z) and in ℙ(Y=1,D=0| Z). Taking the expressions in (<ref>) conditionally on Z and using Assumption <ref> to remove conditioning in the left-hand sides yields the bounds [ℙ(Y_0=1,Y_1=0) ≤ inf_zℙ(Y=1,D=0| Z=z),; ;ℙ(Y_0=0,Y_1=1) ≤ inf_zℙ(Y=1,D=1| Z=z),; ;ℙ(Y_0=0,Y_1=0) = ℙ(Y=0) = ℙ(Y=0| Z=z). ] The tightened bounds are proven to be sharp in Appendix <ref> and illustrated in Figure <ref>. They are intersection bounds, and inference can be carried out with the method proposed in <cit.>.The third expression in (<ref>) gives a testable implication, since the binary outcomes Roy model under Assumptions <ref> implies Y⊥⊥ Z. We now argue that the latter summarizes all possible testable implications of the model. Indeed, for any joint distribution of (Y,D,Z) on {0,1}^2×ℝ satisfying Y⊥⊥ Z, we can always define the pair of potential outcomes (Y_0,Y_1) by Y_0=Y_1:=Y and satisfy the constraints of the binary outcome Roy model under Assumption <ref>.However, rejection of Y⊥⊥ Z cannot be attributed to a violation of the Roy selection assumption (Assumption <ref>) if the validity of the instrument is under question.In the case of college major choice, considered in Section <ref>, one of the proposed instrument is parental education. Unfortunately, the validity of this instrument is doubtful, as parental education level may be correlated with unobserved individual productivity in one or both of the sectors, hence affect potential outcomes directly. Indeed, <cit.> argue that cognitive and non-cognitive unobserved skills are determined in great part by parental environment and investment, which in turn is highly correlated with parental education. Distance to college is a similarly tainted instrument for returns to education, as discussed in <cit.>, since parental location preferences are correlated with unobserved cognitive and non cognitive parental investments. The same applies to local labor market conditions, which may drive endogenous location choices. Moreover, <cit.> derive sharp testable implications of (Y_0,Y_1)⊥⊥ Z within a binary potential outcomes model (without the Roy selection assumption) and their test tends to reject validity of parental education as an instrument, including in our data. More generally, instruments are elusive in the study of returns to education. The rest of this section is concerned with a weakening of Assumption <ref> and a discussion of its validity, in order to recover testable implications of the Roy selection assumption.Our objective now is to bring covariate information to bear and restore falsifiability of the Roy selection mechanism without relying on strong independence assumptions that are hard to substantiate. Joint independence of potential labor market outcomes and parental education is indeed hard to substantiate, as unobserved benefits of parental education can raise productivity. However, it is natural to assume that increasing parental education cannot worsen potential labor market outcomes (see the discussion after the statement of Assumption <ref>). Similarly, local aggregate labor market variables, such as the average wage in STEM for an individual socio-economic category at the time of college major decision, are also likely to be correlated with ex-post job market outcomes, but higher local average wages in STEM at the time of major decision are unlikely to produce lower wages in STEM at the time of graduation, barring complex general equilibrium adjustments. Measures of school quality, merit based scholarships, and distance to college also fall in the category of useful variation shifters that are typically not independent of potential outcomes, but may shift them in only one direction.The following weakening of Assumption <ref> formalizes this insight. We adopt the following notion of monotonicity for the instrument. For details, refer to <cit.>, Section 6B. When comparing vectors, “≥” denotes the componentwise partial order.(First Order Stochastic Dominance) A distribution F_1 on ℝ^k is said to be first order stochastically dominated by a distribution F_2 if there exists random vectors Y_1 with distribution F_1 and Y_2 with distribution F_2 such that ℙ(Y_2≥ Y_1)=1. By extension, a random vector with distribution F_2 is also said to stochastically dominate a random vector with distribution F_1. (SMIV)For any pair z_2≥ z_1 in the support of a vector of observable variables Z, the conditional distribution of (Y_0,Y_1) given Z=z_2 first order stochastically dominates the distribution of (Y_0,Y_1) given Z=z_1 (denoted (Y_0,Y_1)| Z=z_2≿_FSD(Y_0,Y_1)| Z=z_1). Assumption <ref> is inspired by the monotone instrumental variable (hereafter MIV) of <cit.>. Assumption <ref> is stronger than MIV, which, adapted to our setting, would only constrain the means of potential outcomes, but not the marginal distributions or any feature of the joint distribution, which are crucial to the sharp bounds and the testing procedure we develop.[Note, however, that observed variable ν (researcher measured ability), presented as an example of MIV in Lemma 3.1 of <cit.>, can be shown to actually satisfy our SMIV Assumption <ref> under the assumptions of the lemma.] Assumption <ref> is different from Constraint (9) in <cit.>, which restricts the endogenous realized wage distribution, whereas our assumption is meant to operate on the vector of skills. As we shall discuss below, in our framework, Constraint (9) in BGIM is a testable implication of wage maximization behavior under Assumption <ref>. Note also that Assumption <ref> can hold with respect to a vector of instruments, which can increase the tightness of bounds on parameters of interest.We now discuss the economic content and validity of Assumption <ref> within the context of a sector selection model. Assume, as in the skill formation technology of <cit.> <cit.> and <cit.>, that the vector of potential outcomes, such as the potential wages in both sectors, is determined by (Y_0,Y_1)=f(θ,p,η)=(f_0(θ,p_0,η_0),f_1(θ,p_1,η_1)), where θ=(θ_C,θ_N) is a vector of cognitive and non cognitive skills (or abilities), p_d, d=0,1, is a vector of prices of cognitive and non cognitive skills in Sector d, η_d, d=0,1, is a shock in Sector d, and f_d, d=0,1, is a scalar function (see Equations (7) and (9) of <cit.> for a special case). Suppose the function f is increasing in θ, which is a reasonable assumption, even if different sectors value cognitive and non cognitive skills differently. If (θ_C,θ_N) is stochastically monotone with respect to a (vector of) determinant(s) Z of skill investment, and Z is independent of prices and shocks, then, the vector of potential outcomes (Y_0,Y_1) will inherit from (θ_C,θ_N) stochastic monotonicity with respect to Z, and Assumption <ref> will hold. §.§ Sharp bounds and testability of the binary outcomes Roy model under SMIV An important distinction between the roles of the independence assumption (Assumption <ref>) and the stochastic monotonicity assumption (Assumption <ref>) is that, under the former, the instrument Z can only shift sector selection when Y_0=Y_1, whereas under the latter, Z is no longer required to be independent of potential outcomes (Y_0,Y_1) and can therefore induce variation in D, even when Y_0 Y_1.To see how the stochastic monotonicity assumption (Assumption <ref>) combines with the Roy selection mechanism (Assumption <ref>), start from the sharp bounds of Proposition <ref> in the equivalent representation[ ℙ(Y=1,D=0| Z=z) ≤ ℙ(Y_0=1| Z=z) ≤ℙ(Y=1| Z=z),; ℙ(Y=1,D=1| Z=z) ≤ ℙ(Y_1=1| Z=z) ≤ℙ(Y=1| Z=z),; ℙ(Y_0=Y_1=0| Z=z) =ℙ(Y=0| Z=z). ] The statement (Y_0,Y_1)| Z=z_2≿_FSD(Y_0,Y_1)| Z=z_1 is equivalent to ℙ((Y_0,Y_1)∈ U| Z=z_2)≥ℙ((Y_0,Y_1)∈ U| Z=z_1), for all upper sets U (Theorem 6.B.1 of <cit.>, Section 6B). (Upper Sets)A subset U of a partially ordered set (𝒮,≥) is called an upper set if y∈ U implies ỹ∈ U for all ỹ≥ y. The non trivial upper subsets of{0,1}^2 are{(1,1)}, {(0,1),(1,1)}, {(1,0),(1,1)}, {(0,1),(1,0),(1,1)}.Consider, for instance, the upper set {(0,1),(1,1)}. Stochastic monotonicity of (Y_0,Y_1) in z implies that ℙ((Y_0,Y_1)∈{(0,1),(1,1)}| Z=z)≤ℙ((Y_0,Y_1)∈{(0,1),(1,1)}| Z=z̃) for all z̃≥ z, or equivalently ℙ(Y_1=1| Z=z)≤ℙ(Y_1=1| Z=z̃). Since the latter is smaller than or equal to ℙ(Y=1| Z=z̃) by Assumptions <ref> and <ref>, we obtain ℙ(Y_0=1| Z=z)≤ℙ(Y=1| Z=z̃) for all z̃≥ z in the domain of Z. Proceeding similarly with all upper subsets of {0,1}, we obtain the following sharp bounds for the joint distribution of potential outcomes under Assumptions <ref>, <ref> and <ref>: for all z in the domain of Z, [ sup_z̃≤ zℙ(Y=1,D=0| Z=z̃) ≤ ℙ(Y_0=1| Z=z) ≤inf_z̃≥ zℙ(Y=1| Z=z̃),; sup_z̃≤ zℙ(Y=1,D=1| Z=z̃) ≤ ℙ(Y_1=1| Z=z) ≤inf_z̃≥ zℙ(Y=1| Z=z̃),; sup_z̃≥ zℙ(Y=0| Z=z̃) ≤ ℙ(Y_0=Y_1=0| Z=z) ≤inf_z̃≤ zℙ(Y=0| Z=z̃). ] The third line of the display in (<ref>) combines identification of ℙ(Y_0=Y_1=0| Z=z), which is equal to ℙ(Y=0| Z=z), and the testable implications sup_z̃≥ zℙ(Y=0| Z=z̃)≤inf_z̃≤ zℙ(Y=0| Z=z̃) for all z in the domain of Z. The latter is equivalent to stochastic monotonicity of Y in z, which turns out to summarize all testable implications of Roy under Assumption <ref> as formalized in the following theorem. 2pt2pt* The display in (<ref>) characterizes the identified set for the joint distribution of potential outcomes in the binary outcomes Roy model under Assumptions <ref>, <ref> and <ref>.* Under Assumptions <ref>, <ref> and <ref>, the following holds: (∗) For any pair z_2≥ z_1 in the support of the vector of observable variables Z, Y| Z=z_2≿_FSDY| Z=z_1.* If (Y,Z) satisfies (∗), then there is a pair (Y_0,Y_1) such that Assumptions <ref>, <ref> and <ref> hold.When Y is stochastically monotone in z, inf_z̃≥ zℙ(Y=1| Z=z̃) is equal to ℙ(Y=1| Z=z), which, by the third line of (<ref>) is equal to 1-ℙ(Y_0=Y_1=0| Z=z). Hence the right-hand side inequalities in the first two lines of (<ref>) are redundant, and the identified set for the joint distribution of potential outcomes is characterized by two inequalities and one equality. The proof of Theorem <ref>(1) is given in Appendix <ref> and the identified set is represented graphicallyon the right-hand-side panel of Figure <ref>.The proof of Theorem <ref>(2,3) is straightforward. Indeed, under Assumptions <ref> and <ref>, we have ℙ(Y_0≤ y,Y_1≤ y| Z)=ℙ(Y≤ y| Z), for all y, since counterfactual outcomes cannot be larger than realized ones. Hence stochastic monotonicity of (Y_0,Y_1) immediately implies stochastic monotonicity of Y. We now argue that it constitutes a sharp testable implication of the Roy selection mechanism. Indeed, given any joint distribution of observable variables (Y,D,Z) on ℝ×{0,1}×ℝ^d, with Y| Z=z_2≿_FSDY| Z=z_1 for each z_2≥ z_1, the pair of potential outcomes (Y_0,Y_1) can always be chosen in such a way that Assumptions <ref>, <ref> and <ref> are satisfied. For example, setting Y_0=Y_1=Y would satisfy all the constraints.Theorem <ref> shows that testing the Roy selection mechanism simply boils down to testing stochastic monotonicity of observed outcomes with respect to the monotone instrumental variable, which can be performed with existing inference methods in <cit.>, <cit.> and <cit.>. Statements (2) and (3) of Theorem <ref> make no mention of the binary outcomes Roy model, since they are valid without restrictions on the domain of the outcome variables. Theorem <ref> also sheds new light on Assumption (9) in <cit.>, which is identical to our testable implication of the Roy model, when the outcome of interest is wage. Hence, the stochastic monotonicity constraint of <cit.> can be seen as an implication of wage maximization behavior in the sector selection stage. §.§ Imperfect foresight To address the concern that rejection of the Roy selection mechanism may be down to rejecting the assumption that agents are perfectly informed of their future potential outcomes at the time of sector selection, we also derive testable implications of the binary outcome Roy model with imperfect foresight of Definition <ref>. The latter is identical to the binary outcome Roy model, except that the Roy selection assumption,Assumption <ref>, is replaced by imperfect foresight, namely Assumption <ref>. Under the potential outcomes model, i.e., Assumption <ref>, only, we still know that an individual with the skill to succeed in Sector 0, but not in Sector 1, will be observed as having succeeded in Sector 0 or as having failed in Sector 1 (the latter was ruled out under the Roy selection rule of Assumption <ref>). Hence ℙ(Y_0=1,Y_1=0| Z)≤ℙ(Y=1,D=0| Z)+ℙ(Y=0,D=1| Z) and symmetrically for ℙ(Y_0=0,Y_1=1| Z). An individual without the skills to succeed in either sector will be observed to fail, so that ℙ(Y_0=0,Y_1=0| Z=z)≤ℙ(Y=0| Z=z). Under stochastic monotonicity in Z (Assumption <ref>), the latter yields ℙ(Y_0=0,Y_1=0| Z=z)≤inf_z̃≤ zℙ(Y=0| Z=z̃).In addition, observing success in Sector d necessary implies that the agent has the skills required for Sector d, hence ℙ(Y_d=1| Z)≥ℙ(Y=1,D=d| Z). Under Assumption <ref>, the latter yields ℙ(Y_d=1| Z=z)≥sup_z̃≤ zℙ(Y=1,D=d| Z=z̃), d∈{0,1}. We now add selection information according to Assumption <ref>. The latter is equivalent toY=Y_d⇒𝔼[Y|ℐ]=𝔼[Y_d|ℐ]≥𝔼[Y_1-d|ℐ], d=0,1.This yields 𝔼(Y|ℐ)=max{𝔼(Y_0|ℐ),𝔼(Y_1|ℐ)}. Under Assumption <ref>, the latter yields monotonicity of 𝔼(Y| Z=z) in z. Note that the same testable implications would be obtained had Assumption <ref> been replaced with a more general sector selection rule based on the comparison of expected utilities, namely the rule 𝔼[u(Y_d)|ℐ]>𝔼[u(Y_1-d)|ℐ]⇒ D=d, for d=0,1.Putting it all together yields the following sharp bounds on the joint distribution of potential outcomes under Assumptions <ref>, <ref> and <ref>: [ ℙ(Y_0=1,Y_1=0| Z=z) ≤ ℙ(Y=1,D=0| Z=z) +ℙ(Y=0,D=1| Z=z),; ℙ(Y_0=0,Y_1=1| Z=z) ≤ ℙ(Y=0,D=0| Z=z) +ℙ(Y=1,D=1| Z=z),; ℙ(Y_0=0,Y_1=0| Z=z) ≤1-𝔼(Y| Z=z), ]and[sup_z̃≤ zℙ(Y=1,D=0| Z=z̃)≤ 𝔼(Y_0| Z=z),;sup_z̃≤ zℙ(Y=1,D=1| Z=z̃)≤ 𝔼(Y_1| Z=z),; max{𝔼(Y_0| Z=z),𝔼(Y_1| Z=z)}= 𝔼[Y| Z=z], ]for all z in the support of Z.The inequalities above define the identified set for the joint distribution of potential outcomes. Testable implications of the Roy model with imperfect foresight include monotonicity of 𝔼[Y| Z=z] in z as derived above, which proves Theorem <ref>(2) below. It can be easily shown that in the binary case, this monotonicity summarizes the empirical content of the Roy selection assumption with imperfect foresight, as stated in Theorem <ref>. Indeed, for any given vector (Y,D,Z) such that 𝔼[Y| Z=z] is non decreasing in z, setting Y_0=Y_1=Y satisfies the assumptions, which proves Theorem <ref>(3) below.2pt2pt* The displays in (<ref>) and (<ref>) jointly characterize the identified set for the joint distribution of potential outcomes in the binary outcomes Roy model with imperfect foresight under Assumptions <ref>, <ref> and <ref>.* If Assumptions <ref>, <ref>, <ref> hold with ℐ-measurable Z, then 𝔼[Y| Z=z] is non decreasing in z.* For any distribution G on {0,1}^2×Supp(Z), such that 𝔼[Y| Z=z] is non decreasing in z, there exists a random vector (Y_0,Y_1,D,Z)∈{0,1}^3×Supp(Z) such that (Y_1D+Y_0(1-D),D,Z) has distribution G and Assumptions <ref> and <ref> are satisfied with ℐ=σ(Z).We can therefore test Roy with imperfect foresight under Assumption <ref> simply by testing monotonicity of 𝔼[Y| Z=z] in z, using existing inference methods in <cit.> or <cit.>. We can also verify that stochastic monotonicity of Y (the testable implication of Roy selection as shown in Theorem <ref>) does indeed imply monotonicity of 𝔼[Y| Z=z] in z, which is consistent with the fact that Assumption <ref> implies Assumption <ref>. Moreover, in the binary outcomes case, the testable implications of Roy behavior under SMIV and those of imperfect foresight Roy under SMIV are identical, since when Y is binary, stochastic monotonicity of Y| Z=z and monotonicity of 𝔼[Y| Z=z] are equivalent. Hence, rejection of Roy selection behavior under SMIV implies rejection of Roy with imperfect foresight as well. However, the identified set for the joint distribution of potential outcomes in Theorem <ref>(1) is nested in, and weakly tighter than the identified set of Theorem <ref>(1), since the combination of Assumptions <ref> and <ref> contains more information on the joint distribution of potential outcomes than the combination of Assumptions <ref> and <ref>.§.§ Bounds on departures from Roy selection In case of rejection of the Roy selection mechanism, the methodology developed here, and particularly the information on the joint distribution of potential outcomes, allows us to quantify departures from the Roy sector selection rule (Assumption <ref>). If agents are believed to be expected outcome maximizers, i.e., to behave according to the binary outcome Roy model with imperfect foresight, this measure of departure can be interpreted as a measure of the cost of imperfect foresight. If, on the other hand, departures from the Roy model with imperfect foresight are entertained, then the measure of departure we propose also captures the extent to which considerations other than potential outcome maximization enter in the decision. This may be the result of maximization of a utility function that depends on aspects beyond the chosen outcomes. It may also reveal a bias in decision making. This would be the case, in our application to major choice, if gender profiling discouraged women from choosing STEM majors. Departure from Roy sector selection, which we therefore interpret as inefficiency of sector choice, can be measured as the difference between maximum potential outcome and realized outcome, the two being equal by definition in the case of Roy selection according to Assumption <ref>.Efficiency loss from Roy selection departures is defined for each z∈ Supp(Z) as el(z):=ℙ(max(Y_0,Y_1)=1| Z=z)-ℙ(Y=1| Z=z) in the binary outcomes case, and, for each y∈ Supp(Y), as el(y,z):=ℙ(Y≤ y| Z=z)-ℙ(max(Y_0,Y_1)≤ y| Z=z), otherwise. We have ℙ(max(Y_0,Y_1)=1| Z=z)-ℙ(Y=1| Z=z)=ℙ(Y=0| Z=z)-ℙ(Y_0=Y_1=0| Z=z). Since in the binary outcomes Roy model, ℙ(Y_0=Y_1=0| Z=z) is identified as ℙ(Y=0| Z=z), efficiency loss is zero, which justifies the interpretation as a departure from Roy selection (Assumption <ref>). When Assumption <ref> is dropped, efficiency loss is non negative. Since ℙ(Y=0| Z=z) is identified, bounds on efficiency loss or departure from Roy will be obtained from bounds on ℙ(Y_0=Y_1=0| Z=z) under Assumptions <ref> and <ref> only. Since ℙ(Y_0=Y_1=0| Z=z) involves the joint distribution of potential outcomes, sharp bounds on marginal distributions alone cannot deliver the desired bounds on efficiency loss. This feature is shared by other policy relevant parameters such as ex-post regret, skill correlation, sector effect conditional on the chosen sector and the proportion who benefit from a given sector, all of which can also be bounded using this methodology.Note that the efficiency loss criterion is instrument-dependent. First, it is a function of the value z of the instrument Z, which is part of the appeal, since it allows to identify regions of the support of the instrument that are most susceptible to departures from wage maximization. Second, after taking the expectation over Z, the parameter no longer depends on the choice of instrument, however, the bounds do. If there are two instruments and the vector also satisfies SMIV, then the whole analysis can be carried out with respect to the vector of instruments to tighten the bounds.Under Assumptions <ref> and <ref>, the identified set for the joint distribution of potential outcomes is obtained in a similar fashion to (<ref>)-(<ref>), except that we cannot rely on selection information, so that the upper bounds in (<ref>) are obtained from the fact that talent for Sector d only precludes observing failure in Sector d. Bounds (<ref>)-(<ref>) are therefore replaced by [ℙ(Y_0=0,Y_1=0| Z=z)≤inf_z̃≤ zℙ(Y=0| Z=z̃);ℙ(Y_0=0,Y_1=1| Z=z)≤ℙ(Y=0,D=0| Z=z)+ ℙ(Y=1,D=1| Z=z),;ℙ(Y_0=1,Y_1=0| Z=z)≤ℙ(Y=1,D=0| Z=z)+ ℙ(Y=0,D=1| Z=z),;ℙ(Y_0=1,Y_1=1| Z=z)≤ inf_z̃≥ zℙ(Y=1| Z=z̃),]and[sup_z̃≤ zℙ(Y=1,D=0| Z=z̃)≤ℙ(Y_0=1| Z=z)≤1-sup_z̃≥ zℙ(Y=0,D=0| Z=z̃); ;sup_z̃≤ zℙ(Y=1,D=1| Z=z̃)≤ℙ(Y_1=1| Z=z)≤ 1-sup_z̃≥ zℙ(Y=0,D=1| Z=z̃), ]for all z in the support of Z. From (<ref>), we obtain immediately an upper bound on ℙ(Y_0=Y_1=0| z), namely inf_z̃≤ zℙ(Y=0| Z=z̃). Sharp bounds are obtained by projecting (<ref>)-(<ref>) onto component ℙ(Y_0=Y_1=0| z) in the 3-simplex. 1pt 1pt * The displays in (<ref>) and (<ref>) jointly characterize the identified set for the joint distribution of potential outcomes under Assumptions <ref> and <ref>, with Y∈{0,1}.* Under Assumptions <ref> and <ref> with Y∈{0,1}, efficiency loss due to departures from Roy selection satisfies, for each z∈ Supp(Z), el(z) ≥ ℙ(Y=0| Z=z)-inf_z̃≤ zℙ(Y=0| Z=z̃). The results on efficiency loss allow us to identify values of observable characteristics that induce the most costly misallocation of talent and field of study and are therefore prime targets for intervention. § ROY MODEL WITH DISCRETE-CONTINUOUS OUTCOMESExtending the analysis to richer sets of outcomes, including mixed discrete and continuous potential outcomes does not remove the lack of identification issue in the Roy model (and the related competing risks model). The range of observables is richer, but so is the object of interest, i.e., the joint distribution of potential outcomes. Given partial observability and endogenous sector selection, the Roy model is essentially partially identified. Results obtained in the form of sharp bounds on the joint distribution of potential outcomes and the methods used to derive them are analogous to the corresponding results and methods in the binary outcome case, except in one important respect. When considering distributional aspects, such as inequality, the distinction between pointwise bounds and functional bounds is crucial as described below. After a discussion of the latter point, we proceed to analyze testability and quantifying departures from the Roy selection mechanism along the same lines as in the binary outcomes case in Section <ref>. §.§ Functionally sharp bounds for the Roy model Consider the Roy model of Section <ref>, under Assumptions <ref> and <ref>. Bounds on the marginal distributions of potential outcomes can be derived very easily as follows.For any real number y, ℙ(Y_d≤ y)=ℙ(Y_d≤ y, D=d)+ℙ(Y_d≤ y, D=1-d). The first term on the right-hand-side is identified. The second term on the right-hand-side is bounded below by ℙ(Y_1-d≤ y, D=1-d), and above by ℙ(D=1-d). The resulting bounds were shown by <cit.> to be pointwise sharp for the marginal distributions of potential outcomes, in the sense that any pair of distributions of potential outcomes that satisfy the bounds for a given fixed y, can be obtained from some joint distribution of observable variables (Y,D) under the assumptions of the Roy model. However, as <cit.> pointed out, there are additional non redundant cross quantile restrictions, namely, for all y≥ x, ℙ(x<Y_d≤ y)≥ℙ(x< Y_d≤ y, D=d). If the object of interest involves densities, such as the hazard rate, or functional features, such as inequality measures, the difference between the latter bounds and pointwise bounds can be considerable. Indeed, combining Peterson boundsinvolves an additional term -ℙ(D=1-d) in the lower bound. This difference arises because the monotonicity of the distribution function is not factored in. Graphically, the difference between pointwise bounds and functional bounds can be highlighted on Figure <ref>. A candidate distribution function for Y_d that is drawn through the two points (ỹ_1,q_1) and (ỹ_2,q_2) can lie between the curves ℙ(Y≤ y) and ℙ(Y≤ y,D=d)+ℙ(D=1-d). Hence it satisfies pointwise bounds. However, its slope is lower in some regions than the slope of the curve ℙ(Y≤ y,D=d), so that it fails to satisfy the functional bounds.Turning to the joint distribution function of potential outcomes, pointwise bounds can also be derived very easily. Indeed, we immediately have[ ℙ(Y≤min(y_0,y_1))5pt≤5pt ℙ(Y_0≤ y_0,Y_1≤ y_1) 5pt≤5ptℙ(Y≤ y_0,D=0)+ℙ(Y≤ y_1,D=1). ]Corollary 1 of <cit.>shows that the bounds (<ref>) can be attained under their competing risks specification. However, once again, these bounds fail to incorporate monotonicity conditions, and they can entail loss of information, when describing functional features of potential outcomes.The object of interest is the joint distribution (Y_0,Y_1), the information on which we wish to characterize using the identified joint distribution of observable variables (Y,D). Take any subset A of ℝ^2 and consider bounding the probability of (Y_0,Y_1)∈ A. If A contains points (y_0,y) such that y_0≤ y, it can give rise to observation (Y=y,D=1), and if A contains points (y,y_1) such that y_1≤ y, it can give rise to observation (Y=y,D=0). Hence, observation (Y=y,D=d) such that y∈ U_A,0 below, and only those, can be rationalized by elements of A. Similarly, to derive the lower bound, notice that (Y,D)=(y,1) can arise for any (Y_0,Y_1)∈[b,y]×{y}, so that (Y_0,Y_1) mass could be concentrated outside A unless the whole of [b,y]×{y} is contained in A (where b is the common lower bound of the supports of Y_0 and Y_1).For any Borel set A in ℝ^2, define the sets U_A,0, U_A,1 and L_A,0, L_A,1 as[ U_A,0={y∈ℝ | {y}×[b,y]∩ A∅},L_A,0={y∈ℝ | {y}×[b,y]⊆ A},; U_A,1={y∈ℝ | [b,y]×{y}∩ A∅},L_A,1={y∈ℝ | [b,y]×{y}⊆ A}. ] We shall formally show that the upper bound is ℙ(Y∈ U_A,0,D=0)+ℙ(Y∈ U_A,1,D=1). Similarly, the lower bound will be shown to be ℙ(Y∈ L_A,0,D=0)+ℙ(Y∈ L_A,1,D=1). In the case, where A is an upper set (Definition <ref>), the bounding sets of Definition <ref> take a very simple form, and we derive the constraints associated with Assumption <ref> (SMIV) accordingly. 1pt 1pt * Let the distribution of observable variables (Y,D) on ℝ×{0,1} be given. Under Assumptions <ref> and <ref>, the distribution of (Y_0,Y_1) on ℝ^2 satisfies, for all Borel subset A of ℝ^2, ℙ(Y∈ L_A,0,D=0)+ℙ(Y∈ L_A,1,D=1) 50pt≤ ℙ((Y_0,Y_1)∈ A) 100pt≤ℙ(Y∈ U_A,0,D=0)+ℙ(Y∈ U_A,1,D=1).* Conversely, for any joint distribution satisfying the bounds above, there exists a pair (Y_0,Y_1) with that distribution, which satisfies Assumptions <ref> and <ref>.* If Assumption <ref> also holds, then the distribution of (Y_0,Y_1) also satisfies, for all upper set A of ℝ^2, all z∈ Supp(Z), sup_z̃≤ z[ℙ(Y≥y_0^A,D=0| Z=z̃)+ℙ(Y≥y_1^A,D=1| Z=z̃)] 150pt≤ ℙ((Y_0,Y_1)∈ A| Z=z) ≤ inf_z̃≥ zℙ(Y≥y^A| Z=z̃),with y^A:=inf{y:(y,y)∈ A}, y_0^A:=inf{y: (y,+∞)×ℝ⊆ A} and y_1^A:=inf{y: ℝ×(y,+∞)⊆ A}. Theorem <ref>(1) allows us to easily recover Peterson bounds with suitable choices of A. Choosing A=[b,y]×ℝ yields Peterson bounds on the marginal distribution of Y_0. Choosing A=[b,y_0]×[b,y_1] yields Peterson bounds on the joint distribution of (Y_0,Y_1).Finally, applying Theorem <ref> to sets of the form (y_1,y_2]×ℝ and ℝ×(y_1,y_2] yields the following bounds on the marginal distributions of Y_d, for d=1,0: ℙ(y_1<Y≤ y_2,D=d)+ℙ(Y≤ y_2, D=1-d)1{y_1≤b} 50pt≤ℙ(y_1<Y_d≤ y_2)100pt≤ℙ(y_1<Y≤ y_2,D=d)+ℙ(y_1<Y, D=1-d). The upper bound (<ref>) is redundant. Indeed, it can be recovered from lower bounds on ℙ(y_2<Y_d≤∞) and ℙ(b<Y_d≤ y_1). We shall show that the class of sets of the form (y_1,y_2]×ℝ and ℝ×(y_1,y_2] suffice to characterize the marginal potential distributions and that the lower bounds are functionally sharp, as formulated in Corollary <ref> below. The bounds are similar, though not identical, to the bounds in Theorem 1 of <cit.> for a related competing risks model, discussed in the paragraph below Assumption <ref>. The result is proved in the appendix, with a more rigorous statement and formal definition of functional sharp bounds.Under Assumptions <ref> and <ref>, the bounds ℙ(y_1<Y_d≤ y_2)≥ℙ(y_1<Y≤ y_2,D=d)+ℙ(Y≤ y_2, D=1-d)1{y_1≤b} for all y_1, y_2∈ℝ∪{±∞}, y_1<y_2, and d=0,1,are functional sharp bounds.Corollary <ref> tells us that intervals are sufficient to characterize all the information we have on the marginal distribution of potential outcomes (Y_1,Y_0). They form a core determining class of sets, in the terminology of <cit.>, <cit.>. This has several advantages. It allows the incorporation of exclusion restrictions and lends itself to the partial identified inference of <cit.> and <cit.>. The characterization of Corollary <ref> allows us to derive sharp bounds on functional features such as measures of inequality. §.§ Testing Roy and bounding departures from RoyAs in the binary outcome case of Section <ref>, the Roy model defined by Assumptions <ref> and <ref>, is not falsifiable without additional information. Indeed, for any joint distribution (Y,D), potential outcomes (Y_0,Y_1) can be chosen, for instance with Y_0=Y_1=Y, such that Assumptions <ref> and <ref> hold. Given the unavailability of an instrument that satisfies Assumption <ref>, we examine falsifiability of the model under the stochastically monotone instrumental variable assumption (Assumption <ref>). Theorem <ref>(2,3) shows that stochastic monotonicity of observed outcomes with respect to the instrument summarizes all observable implications of the Roy selection mechanism under SMIV. Hence Roy selection behavior can be tested using existing inferential methods to test stochastic monotonicity. As concerns falsifiability of the Roy model with imperfect foresight, Theorem <ref>(2) shows that a testable implication is monotonicity of 𝔼[Y| Z=z] in z, which can also be tested using existing inference methods on regression monotonicity. However, Theorem <ref>(3) only holds in the binary outcomes case, since monotonicity of 𝔼[Y| Z=z] does not otherwise imply stochastic monotonicity of Y| Z=z in z, and therefore does not summarize the empirical content of Roy with imperfect foresight under Assumption <ref>.According to Definition <ref>, departure from Roy selection behavior or inefficiency of sector choice can be measured with the difference ℙ(Y≤ y| Z=z)-ℙ(max(Y_0,Y_1)≤ y| Z=z). The latter is zero under Assumption <ref> (Roy selection mechanism). Otherwise, ℙ(Y≤ y| Z=z)-ℙ(max(Y_0,Y_1)≤ y| Z=z)≥ℙ(Y≤ y| Z=z)-inf_z̃≤ zℙ(Y≤ y| Z=z̃), under Assumption <ref>.Under Assumptions <ref> and <ref>, efficiency loss of Definition <ref> satisfies el(y,z) ≥ ℙ(Y≤ y| Z=z)-inf_z̃≤ zℙ(Y≤ y| Z=z̃),for all (y,z) in the support of (Y,Z).As the binary case, the lower bound on efficiency loss is zero under Assumption <ref> and can serve to construct a test statistic for a test of Roy selection behavior.§.§ Functional features of potential distributionsThe original motivation of the Roy model was to analyze the effect of self-selection on wage distributions, and particularly on wage inequality. <cit.> show that self-selection reduces aggregate inequality when skills are log normal and within sector inequality when skills have a log concave distribution. One of the purposes of functional sharp bounds derived in the previous section is to analyze the effect of self-selection on inequality of potential outcomes in the specification of the Roy model we consider here, where the Roy model structure is stripped down to the self-selection mechanism. Functional sharp bounds on the potential outcome distributions allow us to derive sharp bounds on inequality measures. In this section, we concentrate on the interquantile range, although the same reasoning applies to other functionals from the vast literature on distributional inequality.Consider two quantiles q_1 and q_2 with q_2>q_1, as illustrated on Figure <ref>. The most commonly used range is the interquartile range, where q_1=1-q_2=1/4, but other cases, such as q_1=1-q_2=0.1, are also of great empirical relevance. Peterson bounds on the distribution of Y_d impose ℙ(Y_d≤ y_1)≤ℙ(Y≤ y_1,D=d)+ℙ(D=1-d)=q_1 and ℙ(Y_d≤ y_2)≥ℙ(Y≤ y_2)=q_2. Hence, the upper bound on the interquantile range based on pointwise sharp bounds for the distribution of Y_d is y_2-y_1. However, functional sharp bounds of Corollary <ref> are violated, since q_2-q_1<ℙ(y_1<Y≤ y_2,D=d). On Figure 4, we exhibit another pair of points, namely (ỹ_1,q_1) and (ỹ_2,q_2) such that a distribution for potential Y_d cannot cross these two points and satisfy the functional sharp bounds of Corollary <ref>. We now show how to derive sharp bounds for the interquantile range. For ease of notation throughout this section, for d=0,1, and for each y∈ℝ, denote F(y):=ℙ(Y≤ y), F_d(y):=ℙ(Y_d≤ y), F_d(y):=ℙ(Y≤ y, D=d), F̅_d(y):=ℙ(Y≤ y,D=d)+ℙ(D=1-d), and f^-1 the generalized inverse of f, i.e., f^-1(q)=inf{y:f(y)>q}. Start from any y within the pointwise quantile bounds F̅_d^-1(q_1)≤ y≤ F^-1(q_2). From y, the largest interquantile range obtains in either of the following two cases: * when F(y) is hit first, in which case the interquantile range is F^-1(q_2)-y,* when the potential distribution F_d follows the slope of F_d starting from the point with coordinates (y,q_1), in which case the interquantile range isỹ-y, where ỹ achieves sup{ỹ: q_2≥q_1+ F_d(ỹ)- F_d(y)}. Hence, the interquantile range starting from quantile y is:(y)=min( F^-1(q_2)-y,F_d^-1(q_2-q_1+ F_d(y))-y). Finally, maximizing IQR(y) over admissible y's yields the upper bound on the interquantile range. Hence, under Assumptions <ref> and <ref>, the sharp bounds on the interquantile (q_1,q_2) range are given by:[ max(0,F̅_d^-1(q_2)-F^-1(q_1)) 10pt≤10pt(q_1,q_2) 10pt≤10pt; ; 75ptmax_F̅_d^-1(q_1)≤ y≤ F^-1(q_1)( min( F^-1(q_2)-y,F_d^-1(q_2-q_1+ F_d(y))-y) ). ]Under Assumption <ref> (SMIV), bounds on the interquantile range would be constructed in the same way, conditionally on Z, with F(y) replaced with sup_z̃≤ zℙ(Y≤ y| Z=z̃) and F̅_d(y) replaced with inf_z̃≤ z[ℙ(Y≤ y,D=d| Z=z̃)+ℙ(D=1-d| Z=z̃)].From the viewpoint of the interquantile range, we can now consider the effect of self-selection into the sector of activity (or treatment) on inequality, both within sector and in the aggregate. We compare outcome distributions resulting from self-selection, hereafter called outcome distributions in the self-selection economy, to distributions of outcomes that would result from random assignment of individuals to sectors of activity, hereafter called outcome distributions in the random assignment economy, as in <cit.>, <cit.> and <cit.>. In Sector d, the distribution of outcomes in the random assignment economy is the distribution of potential outcome Y_d, while the distribution of outcomes of the self-selection economy is ℙ(Y≤ y| D=d). In the aggregate population, the distribution of outcomes of the random assignment economy is ℙ(Y_0≤ y)ℙ(D=0)+ℙ(Y_1≤ y)ℙ(D=1), whereas the distribution of outcomes of the self-selection economy is simply the distribution of observable outcomes Y. These cases are collected in Table <ref>. In Sector d, the interquantile range between quantiles q_1 and q_2 of the distribution of outcomes in the random assignment economy is bounded above by (<ref>). In the self-selection economy, it is identified as the interquantile range of the distribution ℙ(Y≤ y| D=d). The following proposition shows how they compare.* If the distribution of outcomes Y conditional on D=d first order stochastically dominates the distribution of outcomes Y conditional on D=1-d, i.e., ℙ(Y≤ y| D=d)≤ℙ(Y≤ y| D=1-d) for all y∈ℝ, then, for any pair of quantiles, the interquantile range of the distribution of outcomes in Sector d in the self-selected economy is lower than the upper bound of the interquantile range of the distribution of outcomes in Sector d in the random assignment economy.* If the stochastic dominance relation of (1) does not hold, then there exists distributions for the pair (Y,D) such that the interquantile range of the distribution of outcomes in Sector d in the self-selected economy is larger than the upper bound of the interquantile range of the distribution of outcomes in Sector d in the random assignment economy. Proposition <ref> tells us two things. On the one hand, if Sector d is unambiguously more profitable in the self-selected economy, it is possible for inequality in Sector d, as measured by the interquantile range, to decrease with self-selection, relative to an economy with random assignment of individuals to sectors. On the other hand, if neither sector dominates the other in the self-selection economy, then there are joint distributions of observables under which we know that self-selection unambiguously increases inequality in Sector d. In case no sector stochastically dominates the other, the hypothesis that self-selection increases inequality is testable based on the bounds of (<ref>), in the sense that one can test the hypothesis that the interquantile range in the self-selected economy is larger than the upper bound of the interquantile range in the randomized economy.§ ROY MODEL OF COLLEGE MAJOR CHOICE IN CANADA AND GERMANYSince <cit.> pointed out that major choice mattered more to labor market outcomes than college choice, the literature on returns to college education has placed some focus on the determinants of major choice and the effects on labor market outcomes. The salient classification that has come to dominate the debate is between STEM and non-STEM degrees, and there is ample evidence of the labor market advantages conferred on male graduates by STEM degrees: <cit.> and <cit.> for the US, <cit.> for Ireland, <cit.> for the UK, <cit.> for several EU countries. The wage benefits of STEM degrees have been found to be a significant but not sole determinant of major choice. <cit.> finds that high ability students view education as a consumption good in the US. <cit.> find elasticity of major choice to expected income to be significant, but less important as a determinant of major choice in France than heterogeneity in preferences for the subject matter. We revisit the issue using our nonparametric bounds methodology on Canadian and German data. We examine whether the data is consistent with a Roy selection of students into the two sectors based on anticipated labor market outcomes only. We study how the answers depend on visible minority status and residency in Québec and the former German Democratic Republic. The picture is rather different for women. The labor market advantages, if present, are not so clear-cut, as noted by <cit.> and <cit.>, and women are severely under-represented in STEM education and even more so in STEM jobs. The evidence is summarized in <cit.>. Two dominant explanations for the under-representation of women in STEM education and in STEM careers are discrimination, which lowers expected wages for women in STEM, and gender profiling, which keeps young women away from STEM education. The former is compatible with a Roy model of career choice, assuming wage discrimination is anticipated, and can be addressed by policies fighting lower labor market outcomes for women in STEM. The latter involves non pecuniary considerations in major choice, therefore requires generalized Roy modeling and can be addressed by policies aimed at encouraging young women into STEM education. However, differential costs between STEM and non STEM majors are nonexistent in Germany and Canada, so that generalized Roy models based on differential costs are not directly applicable here. In any event, given the divergence in policy implications of the two channels above, it is important to investigate which of the two is the dominant effect.The under-representation of women in STEM jobs is often cited as a major contributor to the gender wage gap, as in <cit.>. More generally, there is a large amount of informal discussion, although, to the best of our knowledge, little formal investigation, of the contribution of the STEM economy to rising wage inequality; see, for instance, <cit.>, who attribute rising inequality to skill-based technological change. Our methodology allows us to address this issue by comparing inequality in STEM wages to inequality in non-STEM wages in a counterfactual economy, where sector allocation is random. We can also investigate the effect of self-selection on sectoral and aggregate wage inequality. §.§ Data Our empirical analysis relies on Canadian and German nationally representative graduates surveys. Both countries have a tradition of running extensive surveys on graduate education, and they differ substantially on the proportions of graduates choosing STEM fields (53% of our German sample versus 20% of our Canadian sample), so that analyzing both simultaneously allows us to present a more robust picture of the differences in choice of men and women in graduate education. Both data sets contain detailed information on a representative sample of recent university graduates in their respective countries. The German data are collected by the German Centre for Higher Education Research and Science Studies (DZHW) as part of the DZHW Graduate Survey Series. Data and methodology are described in <cit.>. In Germany, the wave we consider includes graduates who obtained their highest degree during the academic year 2008-2009. The Canadian data is drawn from the National Graduate Survey of Statistics Canada. In Canada, the wave we consider includes graduates who obtained their highest degree during the academic year 2009-2010. We also examine data from earlier waves, namely 1997, 2001, 2005 for Germany and 2000, 2005 for Canada. In the case of the earlier Canadian waves, we rely on publicly available data, which has fewer variables than the data we use for the 2009-2010 wave, and which, unlike the latter, only provides interval censored income information. Graduates were interviewed 1 year and 5 years after graduation in the German survey and 3 years after graduation in the Canadian survey. At that point, extensive information was collected on their educational experience, employment history, including wages and hours worked, along with detailed socio-economic variables. Geographical information is more precise in the German data, with 38 regions, as opposed to 13 in the Canadian data. The German data also contains information on talent, with results at the Abitur (high school final exam), whereas the Canadian survey only provides a self-assessed measure of ability.Both data sets allow us to observe whether employment is permanent or temporary and whether it is related to the specific field of study. In both data sets, fields of study are recorded at a high level of disaggregation, which allows us to discriminate subjects that require mathematics from those that don't. We then merge the fields of study into two categories. We call STEM the mathematics intensive category, which consists mostly of mathematic, physical, economic and computer sciences, as well as engineering and related fields, although other STEM definitions often include life sciences and exclude economics. The remaining majors are merged in the non-STEM-degree category. In Canada and in Germany, the choice of field of study is made prior to enrolment in the program. In both countries, we only consider graduates from institutions in the country of the survey, who are active on their respective country's labor market at the time of the interview.We consider a selection of outcome variables: the ability to secure a permanent employment, the ability to secure employment within the field of study, and annual wage and average hourly wage during the year prior to the time of the interview. Given the high correlation between wage and hourly wage measures, we report only results for wage. Annual wage is non censored in the Canadian data and reported in 1,000 euro bins in the German data.The potential instruments we consider are the education level of both parents (the surveys report parental education in discrete categories, which we translate into years of education following <cit.>), the proportion of women among STEM faculty members (which we call rate of feminization of the STEM faculty) in universities in the individual's region of residence at the time of choice.The German version of this variable is drawn from data on gender distribution of faculty by field and by federal State provided by the Federal Statistical Office of Germany (DESTATIS). The Canadian version of this variable is drawn from Statistics Canada, University and College Academic Staff System (UCASS). There is a very high level of assortative matching in parent's education both in Canada and Germany, so we only report results using the mother's education and the rate of feminization of STEM faculty as stochastically monotone instrumental variables. We also use local labor market conditions at the time of choice as instruments for robustness purposes, although their validity relies on neglecting general equilibrium effects.We compare results for gender and visible minority status. In the Canadian survey, visible minority status is self-reported. In Germany, we construct this variable from the country of birth, and we assign an individual in the survey the status of visible minority if they were born in a country with a non-white majority population. This unfortunately excludes a large number of graduates of Turkish descent, whom we are unable to track. We also distinguish German graduates from institutions in the former German Democratic Republic and Canadian graduates from institutions in Québec.Our study focuses on the latest cohort. The raw sample from the German survey consists of 10,494 individuals. From the raw sample, we exclude all respondents who are still in education, have never worked or are currently inactive, unemployed, in part-time employment or self-employed. This leaves 9,202 observations. We keep only graduates who hold a “Bachelor”, “Magister” or “Diplom”, excluding those with “Staatsexamen” and “Lehramt” degrees, which are specific tracks mainly for teachers. This leaves us with 7,729 observations. Finally, we divide the population between those who completed the Abitur (high school final exam) in the former Federal Republic and in the former Democratic Republic and exclude those for which we do not have this information or obtained their Abitur abroad (107 individuals). Most of our econometric analysis is based on the sample of individual with complete information on gender, degree, migration background, year and place of Abitur completion, mother's education, and income or job characteristics, that is between 4,559 and 4,890 observations.The raw sample from the Canadian survey consists of 28,715 observations who participated in the survey. From the raw sample, we exclude all respondents who have completed trade, vocational, college and CEGEP diploma or certificate at the time of their 2009/2010 graduation and 2013 interview. We keep only those individuals who have “university diploma or certificate below Bachelor level”, “Bachelor’s degree or first professional degree”, “university diploma/certificate above the Bachelor’s level but below the Master’s level”, “Master’s degree” and “Doctorate”. We also exclude all respondents who are still in education, self-employed, working in family business without pay and live in the U.S. as primary residence. We further filter the data set to include respondents who are in the labor force, employed, work full-time and have age below 40. Since the econometric analysis is based on the sample with complete information on gender, minority status, income, degree, related job, permanent job, mother’s education and father’s education, the sample size ranges between 4,361 and 10,150 observations. §.§ Descriptive statistics Income distributions in Germany and Canada show a clear STEM advantage for both men and women and a clear gender gap. In Figure <ref>, distributions appear to be stochastically ordered. In both Canada and Germany, based on quartiles only, the distribution of male STEM wages dominates the distribution of female STEM wages, which dominates male non STEM wages, which dominates female non STEM wages.A similar pattern emerges from Table <ref>, where we see that men with STEM degrees are more likely to hold permanent employment in a field related to their studies, than men with non STEM degrees and women in both categories. More precisely, in Germany, 41% of men with STEM degrees obtain permanent employment one year after graduation, 35%in a field related to their studies and 6% in other fields. For women with STEM degrees, the proportion is only 36%, with 29% in their field of study, and for men and women with non STEM degrees, the proportion falls to 23%, with 16% in their field of study. In Canada, 90% of men with STEM degrees obtain permanent employment three years after graduation, 84%in a field related to their studies and 6% in other fields. For women with STEM degrees, the proportion is 82%, with 73% in their field of study, and for men and women with non STEM degrees, the proportion is 81%, with 68% in their field of study. Since the proportion of men with STEM degrees is larger, the overall proportion of women with a permanent employment after 1 year in Germany is lower (27%) than for men (36%) and the proportion of women with a permanent employment after 3 years in Canada is lower (80%) than for men (86%). Table <ref> shows the degree of under representation of women in STEM degrees in both Germany and Canada, which tallies with the overwhelming evidence from previous studies in different contexts. In Germany, 37% of women's degrees are in STEM, as opposed to 75% for men. The difference is somewhat less pronounced for minorities, where 48% of women's degrees are in STEM, as opposed to 80% for men. In Canada, 8% of women's degrees are in STEM, as opposed to 35% for men. The difference is, again, less pronounced for minorities, where 15% of women's degrees are in STEM, as opposed to 45% for men. We examine the variation in sector choice induced by the instruments and illustrate it in the case of white women in affluent regions in Figure <ref>, where the brown line is the point estimator and the grey lines are the 95% confidence bands. There is some indication of a hump-shaped response of STEM choices in mother's education.The humped-shaped response to mother's education may be due to a larger involvement in major choices for parents with a bachelor's degree and a more laissez-faire approach beyond that. The effect of the proportion of women on the STEM faculty on women's choices is increasing for low proportions, as we would expect, then levels for larger proportions. §.§ Discussion of the SMIV instruments We contend that female mentors and maternal educational attainment are two determinants of skill investment that affect the vector of cognitive and non cognitive skills monotonically. Take maternal educational attainment first. A large literature on the intergenerational transmission of human capital suggests that cognitive skills are positively impacted by parental education (see for instance <cit.>). More recent evidence points to the same conclusion about non cognitive skills (see for instance <cit.> and <cit.>). Combined with evidence of complementarity between cognitive and non cognitive skills (<cit.>, <cit.> and references therein), this tends to support stochastic monotonicityof the vector of cognitive and non cognitive skills with respect to maternal educational attainment, hence the validity of Assumption <ref> for the latter variable. Consider now the presence of female mentors on the faculty, or more precisely the proportion of women on the STEM faculty. <cit.> report that “theory and evidence suggest that female instructors may be instrumental in encouraging women to enroll and excel in subjects in which they are underrepresented.” Hence, we expect that female instructors will help female students improve and adapt their cognitive and non cognitive skills to the demands of the market, hence increasing the vector of potential outcomes, so that Assumption <ref> is satisfied. §.§ Methodology and results From the survey samples, we first construct sub-samples based on gender, visible minority status, and the broad region of residence at the time of the interview (former East and West Germany, Québec and the rest of Canada). We are interested in comparing behavior by gender and by race or immigrant status, as well as socio-economic background. As far as the latter determinant is concerned, since we have no data on socio-economic background of respondents, we use a coarse subdivision in regions for each of the countries, differentiating the relatively poorer former East Germany and Québec. The latter division has the added benefit of distinguishing very different cultural spheres, the role of which we can also investigate.We test monotonicity of the conditional mean for each binary outcome, and both mean monotonicity and stochastic monotonicity of the non-binary discrete and continuous outcomes with respect to the instruments. We implement the stochastic monotonicity test proposed in <cit.>.[We thank Yu-Chin Hsu andChu-An Liu for sharing their code.] The sensitivity of inference results to the generalized moment selection procedure is usually the major concern with this type of procedure, see for instance <cit.>[The generalized moment selection procedure, originally introduced in <cit.>, <cit.> and <cit.>, increases the power of moment inequality tests, while controlling size, by pre-selecting inequalities that are close to binding. In the specific implementation of moment inequality testing in <cit.>, the threshold according to which moment inequalities are pre-selected depends on the user-chosen quantities κ_n and B_n.]. We choose the recommended values for the user-chosen parameters governing the generalized moment selection in <cit.>, namely B_n=0.85ln n/lnln n and κ_n=0.15ln n. To investigate robustness of the inference results to variations around this choice, we ran the tests in the case of the mother's education as an instrument for all pairs of values in {B_n/2,B_n,2B_n}×{κ_n/2,κ_n,2κ_n}. Of the 48 test results in the Canadian portions of Tables <ref> and <ref>, we see variation in the rejection level in one case only, related to Québec.In the German portions of Tables <ref> and <ref>, we see variation in rejection levels in four cases and reversal of the test results in three cases, related to East Germany.Table <ref> collects results of the test of the Roy model with imperfect foresight using the mother's education as an instrument satisfying Assumption <ref> (SMIV).The hypotheses that white men and women in the former Federal Republic choose their major to maximize expected income or the probability of a permanent employment a year after graduation are both rejected at the 1% level. The hypothesis that white women in the former Democratic Republic choose their major to maximize the probability of a permanent employment a year after graduation is also rejected at the 10% level. No other rejection of imperfect foresight Roy selection are found for residents of the former Democratic Republic or for minorities.The hypotheses that white women in the rest of Canada choose their major to maximize expected income or the probability of a permanent employment three years after graduation are both rejected at the 5% level. The hypothesis that white women in Québec choose their major to maximize the probability of securing employment related to their field of study is also rejected at the 5% level. For men, Roy self-sorting is never rejected, which again shows a significantly different behavior for men and women. As in Germany, we find no rejections for visible minority men or women. A notable feature of the results presented in Table <ref> is that the hypothesis that white Canadian women's choices are driven by expected income or the probability of securing permanent employment is rejected for the rest of Canada, but not in Québec, whereas the hypothesis that choices are driven by the probability of securing employment related to the field of study is rejected in Québec, but not in the rest of Canada. This is consistent with the interpretation that labor market outcomes are stronger determinants of choices for women in Québec, whereas field preferences are stronger determinants of choice for women in the rest of Canada.[To investigate this issue further, we tested a Roy model of self-sorting based on a variable equal to 1 when the applicant says they obtained the employment they were hoping for, and zero otherwise. This variable is available in Canada and the test result are identical to those obtained for the test of Roy self-sorting based on the relatedness of employment with field of specialization at university. Interpretation of this result, however, would hinge on a correct interpretation of the variable itself, which we do not have at this point.]Table <ref> reports results of the test of pure Roy self-sorting behavior based on three outcome variables, namely income, the degree to which employment is related to the field of study, and the vector (permanent, related) with lexicographically ordered components. We no longer include the ability to secure permanent employment, since it is a binary variable, and the tests of pure and imperfect foresight models are identical. As we see in Table <ref>, the same conclusions hold for the pure Roy selection model, except that the hypothesis that white women in the former Democratic Republic choose their major to maximize expected income a year after graduation is now also rejected at the 1% level, and the hypothesis that white women in Québec choose their major to maximize the probability of securing permanent employment three years after graduation (and in case of ties decide based on relatedness of the employment) is now also rejected at the 10% level. Again, there are no rejections of the pure Roy selection model for minorities anywhere, or for men anywhere in Canada.Tables <ref> and <ref> collect similar results to those in Tables <ref> and <ref>, except that Assumption <ref> (SMIV) holds for the vector of instruments combining mother's education and the proportion of women on the STEM faculty in the individual's region at the time of choice. Hence, only results for women are presented, since the proportion of women on the STEM faculty is conceived as a valid SMIV for women only. Again, there are no rejections of either the imperfect or the perfect foresight Roy models for minorities. The hypotheses that white women in the former Federal Republic choose their major to maximize expected income or the probability of a permanent employment a year after graduation are both rejected at the 1% level.The hypothesis that white women in the former Democratic Republic choose their major to maximize the their expected income (resp. probability of a permanent employment) a year after graduation is also rejected at the 1% level. The same results hold for the test of perfect foresight Roy self-sorting. Looking at the Canadian portion of Tables <ref> and <ref> reveals only slight discrepancies with test results with only the mother's education as the SMIV. For comparison, we look at older cohorts, based on the mother's education as an instrument (the proportion of women on the STEM being unavailable for these cohorts). We find rejection of Roy behavior based on income after one year (both perfect and imperfect foresight) for men from the former Federal Republic who obtained their degrees in 2005, but not for women from the former Federal Republic or either gender from the former Democratic Republic.One initially surprising result in the Canadian portion of Table <ref> is the fact that Roy self-sorting behavior for white women based on income is no longer rejected when the test is based on the vector of instruments, whereas it was rejected based on mother’s education only. Although the theoretical bounds are tighter, an increased number of redundant moment inequalities reduces the power of the inference procedure.Our results show a prevalence of rejections of Roy major selection behavior, possibly in favor of non pecuniary considerations, for categories that are generally considered privileged, particularly women, i.e., white women in Canada and white men and women from the former West German Federal Republic. We tend not to reject Roy major selection behavior for all other categories. This is borne out by the differences in responses to a survey question on the importance of labor market considerations on major choice. Table <ref> shows that minorities and residents of Québec and the former Democratic Republic of Germany tend to weigh labor market considerations more than their counterparts.To further investigate rejections of the Roy self-sorting behavior, we compute confidence intervals for the measure of departure from Roy (also called “efficiency loss”) provided in Section <ref> and <ref>. We report the confidence lower bounds for white men and women from the former West Germany, for whom the Roy self-sorting behavior was rejected. For each of these categories, we plot the lower confidence bound as a function of income and the mother's education to identify regions of values (of income and mother's education) that are responsible for the rejection of Roy self-sorting. For white men in the former West Germany, we find that rejections are mostly driven by individuals, whose mothers earned postgraduate degrees. For white women in the former West Germany, we find rejections are driven by lower income women with high school educated mothers and median income women, whose mothers earned a high school degree only or a postgraduate degree.Finally, we investigate the impact of Roy self-selection on income inequality in the case of individuals for whom the hypothesis of Roy self-sorting is not rejected, i.e., minorities of both genders in Germany and Canada, white women in the former East Germany and Canadian white men. Inference on the bounds from Propositions <ref> and <ref> on the efficiency loss from non maximizing behavior and on the bounds (<ref>) and (<ref>) on the interquartile range in the randomized economy, is carried out with the STATA package clrbounds implementing <cit.>. In Figure <ref>, we report confidence intervals for the partially identified interquartile range of potential non STEM income Y_0, potential STEM income Y_1, and aggregate income in an economy where individuals are randomized into sectors, next to the interquartile range for observed distributions of STEM, non STEM and aggregate income distributions. Most results are inconclusive, in the sense that realized interquartile ranges are well within the bounds for potential distributions, except in the case of white men in Québec and white women in the former East Germany, where observed STEM interquartile range coincides with the lower bound on potential interquartile range . § CONCLUSION In this paper, we analyzed the Roy model of self-sorting into economic activities on the basis of anticipated outcomes. We stripped the model down to its essential features: we assumed that heterogeneous agents are characterized by a pair of potential outcomes, one for each sector of activity, and that they choose the sector that gives them a strictly higher outcome, leaving choice undetermined in case of ties. We characterized the restrictions this mechanism imposes on the joint distribution of potential outcomes.This characterization showed, on the one hand, that the Roy self-sorting mechanism puts non trivial restrictions on joint distributional features of potential outcomes, but, on the other hand, that the identified set is never empty, so that the Roy self-sorting mechanism described is not testable. Testability can be restored using selection shifters that are jointly independent of potential outcomes. However, such shifters are difficult to find in applications, and their usefulness is severely restricted by the Roy self-sorting mechanism, which only lets them affect selection in case potential outcomes are equal. We therefore introduced an extension of the notion of monotone instrumental variable, designed to constrain the joint distribution of potential outcomes, the quantity of interest. We considered (vectors of) variables that affect the vector of potential outcomes monotonically, in the sense of multivariate first order stochastic dominance, and called such (vectors of) variables stochastically monotone instrumental variables (SMIV). We repeated the characterization of the identified set for the joint distribution of potential outcomes under the SMIV assumption, and showed that testing the Roy self-sorting mechanism is equivalent to testing stochastic monotonicity of observed outcomes in the instrument. To alleviate the concern that rejections are due to the assumption that agents are perfectly informed of their future outcomes, we repeated the exercise with an imperfect foresight version of the model, where agents select sectors based on expectations. Beyond testing the Roy self-sorting mechanism and providing measures of departure from outcome-based decisions, we highlighted another important application of our characterization of the identified set of joint potential outcome distributions, namely the derivation of sharp bounds on the interquantile range of potential outcome distributions to revisit the effects of self selection on inequality in employment outcomes.We applied our methodology to the analysis of major choices made by graduates of Canadian and German universities based on the national graduate surveys of each of these two countries. We analyzed selection of mathematics-intensive versus other fields of study by graduates within the framework of the Roy model with employment based outcomes that include income 1 and 3 years after graduation, the ability to secure permanent employment by the time of the survey and the extent to which employment secured is related to the field of study. The data supports previous evidence of a labor market advantage of mathematics-intensive fields (STEM), severe under-representation of women in STEM, over-representation of visible minorities in STEM and male labor market advantage in both sectors. We investigated whether selection behavior is consistent with Roy self-sorting on outcomes for categories of graduates by gender, visible minority status and region of residence (former East and former West Germany, Québec and the rest of Canada). To test Roy self-sorting based on employment outcomes, we used parental education level and the proportion of women on the faculty of STEM programs in the region and at the time of choice as stochastically monotone instruments. We found a pattern of rejections of Roy self-sorting based on outcomes for white women in the former Federal Republic of Germany and the rest of Canada, and a lack of rejections for visible minorities and for white women from Québec and white men from all of Canada and the former German Democratic Republic. Confidence intervals for measures of departure from Roy behavior revealed that in the case of white women from the former Federal Republic, for instance, rejection of Roy behavior seems to be driven by lower income women with high school educated mothers and middle income women with postgraduate educated mothers. Among groups, where Roy self-sorting is not rejected, comparisons of interquartile ranges for observed and counterfactual income distributions are inconclusive except in the cases of women in the former Democratic Republic and white men from Québec, where self-sorting is found not to increase inequality.The pattern of rejections of Roy self-sorting in major choice points to non labor market related determinants of choice. For instance, our results are consistent with a story involving gender profiling pushing white men in the West of Germany into STEM fields and white women in the West of Germany and in Canada out of STEM fields. They are also consistent with gender profiling being less prevalent in the former communist Germany. However, the results are also consistent with a story involving non pecuniary field preferences driving major choices of more privileged groups in more affluent regions, but not the choices of the more financially constrained. The methodology proposed here should then be construed as a tool for exploratory analysis of the determinants of major choice prior to a fully structural generalized Roy modeling of preferences, à la <cit.>, <cit.>, in a context where, unlike the analysis of returns to college, there is no clear cost differential between different choices. Non rejections of Roy self-selection based on labor market outcomes, on the other hand, are a warning that policies directly aimed at increasing the share of women in STEM majors at university may have a (possibly short term) negative effect on the gender gap and wage inequality, and that both upstream (early childhood) and downstream (labor market) interventions are required. § PROOFS AND ADDITIONAL RESULTS RELATING TO BINARY OUTCOMES §.§ Sharp bounds for the binary outcome Roy model §.§.§ Statement of Proposition <ref>Fix the pair of binary random variables (Y,D) with probability mass function (q_00,q_01,q_10,q_11), with q_ij:=ℙ(Y=i,D=j). The following two statements hold. (1) If the non negative vector (p_00,p_01,p_10,p_11)∈ℝ^4 satisfies p_00+p_01+p_10+p_11=1, p_10≤ q_10, p_01≤ q_11 and p_00=q_00+q_01, then there exists a pair of binary random variables (Y_0,Y_1) such that Assumptions <ref> and <ref> are satisfied and ℙ(Y_0=0,Y_1=0)=p_00, ℙ(Y_0=0,Y_1=1)=p_01, ℙ(Y_0=1,Y_1=0)=p_10 and ℙ(Y_0=1,Y_1=1)=p_11. (2) Conversely, if the pair of binary random variables (Y_0,Y_1) satisfies Assumptions <ref> and <ref>, then ℙ(Y_0=1,Y_1=0)≤ q_10, ℙ(Y_0=0,Y_1=1)≤ q_11 and ℙ(Y_0=0,Y_1=0)=q_00+q_01. §.§.§ Proof of Proposition <ref>Write p_ij:=ℙ(Y_0=i,Y_1=j) for each i,j=0,1. The binary outcomes Roy model of Definition <ref> can be equivalently defined as a correspondence G betweenvalues of observables (y,d)∈𝒜:={(0,0),(0,1),(1,0),(1,1)} and values of unobservables (y_0,y_1)∈𝒜. The correspondence is defined by its values G(y,d) for each (y,d)∈𝒜, namely G(1,1):={(1,1),(0,1)}, G(1,0):={(1,1),(1,0)}, G(0,1):={(0,0)} and G(0,0):={(0,0)}. By Theorem 1 of <cit.>, the 14 inequalities ℙ((Y_0,Y_1)∈ A)≤ℙ(G(Y,D)∩ A∅) for each A⊂𝒜 provide a collection of sharp bounds for the model defined by the correspondence G. For instance, A={(0,0)} yields the inequality p_00≤ q_00+q_01 and A={(1,1),(0,1)} yields the inequality p_11+p_01≤ q_11+q_10. To prove the result, it suffices to show that all 14 inequalities are implied by 0≤ p_10≤ q_10, 0≤ p_01≤ q_11 and p_00=q_00+q_01. The 14 inequalities are listed below. Singleton A's yield[ p_11≤q_11+q_10; p_10≤ q_10; p_01≤ q_11; p_00≤ q_01+q_00. ]Pairs yield[ p_11+p_10 ≤ q_11+q_10; p_11+p_01 ≤ q_11+q_10; p_11+p_00 ≤ 1; p_10+p_01 ≤ q_11+q_10; p_10+p_00 ≤q_10+q_01+q_00; p_01+p_00 ≤ q_11+q_01+q_00. ]Finally, triplets yield[ p_11+p_10+p_01≤q_11+q_10; p_11+p_10+p_00≤1; p_11+p_01+p_00≤1; p_10+p_01+p_00≤ 1. ]The first four inequalities in (<ref>) are implied by the first inequality in (<ref>). The last two are implied by (<ref>). All inequalities in (<ref>) are therefore redundant. Since p_11=1-p_00-p_01-p_10, all four inequalities in (<ref>) are implied by 0≤ p_10≤ q_10, 0≤ p_01≤ q_11 and p_00=q_00+q_01. Finally, since p_11+p_10+p_01=1-p_00, the first inequality in (<ref>) is implied by p_00=q_00+q_01 and the result follows.§.§.§ Extension to the alternative binary Roy model(1) First, we show that ℙ(Y_0=1,Y_1=0)≤ q_10, ℙ(Y_0=0,Y_1=1)≤ q_11 and ℙ(Y_0=0,Y_1=0)=q_00+q_01 hold if (Y,D,Y_0,Y_1) satisfy the assumptions of Definition <ref>. Under the specification of Definition <ref>, Y_0=1 and Y_1=0 jointly imply that Y_0^∗>Y_1^∗, which in turn implies D=0 and Y=1, so that the first inequality holds. The second holds by the same reasoning and the roles of Y_0 and Y_1 reversed. Finally, Y_0=Y_1=0 implies Y=0, and Y=1 implies that Y_0=1 or Y_1=1, so the equality holds as well.(2) Second, the binary outcomes Roy model specification of Definition <ref> is nested in the alternative binary Roy model specification of Definition <ref>. Indeed, the former can be obtained by restricting (Y_0^∗,Y_1^∗) to be binary. Hence, sharpness of the bounds for the binary outcomes Roy model implies sharpness for the alternative binary Roy model. The result follows.§.§.§ Representation of the bounds on the 2-simplexWe continue to denote ℙ(Y=i,D=j)=q_ij and ℙ(Y_0=i,Y_1=j)=p_ij. According to Proposition <ref>, p_00=q_01+q_00. Hence, the remaining three probabilities, namely p_10, p_01 and p_11=q_11+q_10-p_10-p_01 can be represented in barycentric coordinates in the rescaled 2-simplex of Figure <ref>, where the three vertices correspond to the cases, where p_11=q_11+q_10, p_10=q_11+q_10 and p_01=q_11+q_10 respectively. §.§ Covariate restrictions§.§.§ Statement of Theorem <ref>(1)Fix the joint distribution (Y,D,Z) and denote the conditional probability mass function (q_00(z),q_01(z),q_10(z),q_11(z)), where q_ij(z)=ℙ(Y=i,D=j| Z=z). The following two statements hold. (1) If for each z∈ Supp(Z), the non-negative vector (p_00(z),p_01(z),p_10(z),p_11(z))∈ℝ^4 satisfies p_00(z)+p_01(z)+p_10(z)+p_11(z)=1, sup_z̃≤ zq_10(z̃)≤ p_01(z)+p_11(z), sup_z̃≤ zq_11(z̃)≤ p_10(z)+p_11(z) and p_00(z)=q_01(z)+q_00(z), then there exists a pair of binary random variables (Y_0,Y_1) such that Assumptions <ref>, <ref> and <ref> are satisfied and ℙ(Y_0=0,Y_1=0| Z=z)=p_00(z), ℙ(Y_0=0,Y_1=1| Z=z)=p_01(z) and ℙ(Y_0=1,Y_1=0| Z=z)=p_10(z), for each z∈ Supp(Z). (2) Conversely, if the random vector (Y,D,Z) satisfies Assumptions <ref>, <ref> and <ref> for some the pair of binary random variables (Y_0,Y_1), then (<ref>) holds.§.§.§ Proof of Theorem <ref>(1)From the proof of Proposition <ref>, the identified set under Assumptions <ref> and <ref> is characterized by q_10(z)≤ p_10(z)+p_11(z)≤ q_10(z)+q_11(z), q_11(z)≤ p_01(z)+p_11(z)≤ q_10(z)+q_11(z) and p_00(z)=q_01(z)+q_00(z) for all z∈ Supp(Z). Assumption <ref> is equivalent to ℙ((Y_0,Y_1)∈ U| Z=z_1)≤ℙ((Y_0,Y_1)∈ U| Z=z_2) for all z_1≤ z_2 and all upper set U in {0,1}^2. The upper sets are {(1,1)}, {(1,1),(1,0)}, {(1,1),(0,1)} and {(1,1),(1,0),(0,1)}. Hence, Assumption <ref> is equivalent to sup_z̃≤ zp_11(z̃)≤ p_11(z)≤inf_z̃≥ zp_11(z̃), sup_z̃≤ z[p_11(z̃)+p_10(z̃)]≤ p_11(z)+p_10(z)≤inf_z̃≥ z[p_11(z̃)+p_10(z̃)], sup_z̃≤ z[p_11(z̃)+p_01(z̃)]≤ p_11(z)+p_01(z)≤inf_z̃≥ z[p_11(z̃)+p_01(z̃)], and sup_z̃≤ z[1-p_00(z̃)]≤ 1-p_00(z)≤inf_z̃≥ z[1-p_00(z̃)] for all z∈ Supp(Z). Combining the two sets of inequalities yields the result.§.§.§ Sector specific exclusionsWe denote by X_d the vector of observable variables (if any) that enter in the equation determining potential outcome Y_d, but not Y_1-d. Since there is some ambiguity in notation, it is worth stressing the fact that both vectors X_0 and X_1 are observed, irrespective of the chosen sector, unlike Y_0, which is only observed if D=0 and Y_1, when D=1. [Sector specific exclusions]The random vectors X_0 and X_1 denote vectors of observed variables (when they exist) such that Y_d⊥⊥ X_1-d|X_d, for d=0 and 1.The exclusions of Assumption <ref> are conditional on a set of additional observed covariates, as noted before. Excluded variables X_d are variables that change the price of skills relevant for one sector without affecting the price of skills in the other, as discussed in <cit.>. Typical examples would include sector specific shifters of labor market conditions, as in <cit.>, <cit.>. In the case of college major choice, considered in Section <ref>, in a narrow partial equilibrium sense, exogenous and unanticipated variation (at the time of college major choice) in the gross number of STEM jobs could be thought to affect only conditions for success in securing employment with a STEM degree, without affecting success in securing employment with a non STEM degree.The classical way to derive bounds under an exclusion restriction is to observe that 𝔼(Y_d| X_d,X_1-d)=𝔼(Y_d| X_d) under Assumption <ref>, so that the bounds (<ref>) hold for all values of X_1-d. We contribute to the literature here, in showing sharpness of these bounds for the binary (and alternative binary) Roy model. Conditioning on all non excluded variables remains implicit throughout.For any (x_0,x_1) in the support of (X_0,X_1), the identified set for the parameter vector (𝔼(Y_0| X_0=x_0),𝔼(Y_1| X_1=x_1)) in the binary (and alternative binary) Roy model is characterized by:[ℙ(Y=1,D=0| X_0=x_0,X_1=x̃_1) ≤ 𝔼(Y_0| X_0=x_0) ≤ℙ(Y=1|X_0=x_0,X_1=x̃_1),; ; ℙ(Y=1,D=1| X_0=x̃_0, X_1=x_1) ≤ 𝔼(Y_1| X_1=x_1) ≤ℙ(Y=1| X_0=x̃_0, X_1=x_1), ]for almost all x̃_1∈ Supp(X_1| X_0=x_0), and x̃_0∈ Supp(X_0| X_1=x_1).The bounds define the identified set for the vector (𝔼(Y_0| X_0=x_0),𝔼(Y_1| X_1=x_1)), namely, any value of that vector satisfying the bounds can be achieved as a solution of the model for some distribution of the observable variables (Y,D) conditional on (X_0=x_0,X_1=x_1). In other words, no value for the pair (𝔼(Y_0| X_0=x_0),𝔼(Y_1| X_1=x_1)) that satisfies both equations can be rejected solely on the basis of the model specification. The bounds are well-known, but the joint sharpness result is new. As before, the bounds of Proposition <ref> are intersection bounds, so that inference can be carried out with the method proposed in <cit.>.A salient consequence of Proposition <ref> is the fact that the binary outcomes Roy model can be rejected when the bounds cross, i.e., when there is a value x_1 in the support of X_1 and two values x_0^1 and x_0^2 in the support of X_0 conditional on X_1=x_1, such that ℙ(Y=1,D=1| X_0=x_0^1,X_1=x_1)>ℙ(Y=1| X_0=x_0^2,X_1=x_1) or a value x_0 in the support of X_0 and two values x_1^1 and x_1^2 in the support of X_1 conditional on X_0=x_0, such that ℙ(Y=1,D=0| X_0=x_0,X_1=x_1^1)>ℙ(Y=1| X_0=x_0,X_1=x_1^2).Identification of the pair (𝔼(Y_0| X_0=x_0),𝔼(Y_1| X_1=x_1)) can be achieved as a simple implication of the previous result if there is x̃_1∈ Supp(X_1| X_0=x_0) such that ℙ(Y=1,D=1| X_0=x_0,X_1=x̃_1)=0 and x̃_0∈ Supp(X_0| X_1=x_1) such that ℙ(Y=1,D=0| X_0=x̃_0,X_1=x_1)=0, in which case lower and upper bounds coincide in Proposition <ref>. This identification result is akin to the identification at infinity of <cit.>.§.§.§ Proof of Proposition <ref>Validity of the bounds was shown above. For sharpness, fix (x_0,x_1) in the Support of (X_0,X_1). For a given random vector (Y,D) of binary random variables, denote by q_ij(x̃_0,x̃_1) the conditional probability ℙ(Y=i,D=j| X_0=x̃_0,X_1=x̃_1) for any (x̃_0,x̃_1) in the Support of (X_0,X_1). Consider any pair (a(x_0),b(x_1)) satisfying q_10(x_0,x̃_1)≤ a(x_0)≤ q_11(x_0,x̃_1)+q_10(x_0,x̃_1)for almost all x̃_1∈ Supp(X_1| X_0=x_0), and q_11(x̃_0,x_1)≤ b(x_1)≤ q_11(x̃_0,x_1)+q_10(x̃_0,x_1) for almost all x̃_0∈ Supp(X_0| X_1=x_1). We exhibit a pair of binary random variables (Y_0,Y_1) with joint distribution p_ij:=ℙ(Y_0=i,Y_1=j| X_0=x_0,X_1=x_1), such that Assumptions <ref>, <ref> and <ref> are satisfied, and such thatp_11(x_0,x_1)+p_10(x_0,x_1)=a(x_0)p_11(x_0,x_1)+p_01(x_0,x_1)=b(x_1).Here is our proposed distribution.p_00(x_0,x_1) = q_00(x_0,x_1) + q_01(x_0,x_1), p_11(x_0,x_1) = b(x_1) + a(x_0)-q_10(x_0,x_1) - q_11(x_0,x_1),p_10(x_0,x_1) = q_10(x_0,x_1) +q_11(x_0,x_1)-b(x_1),p_01(x_0,x_1) = q_10(x_0,x_1) +q_11(x_0,x_1)-a(x_0).Note that (<ref>) is verified by construction. We also verify that p_00(x_0,x_1)+p_01(x_0,x_1)+p_10(x_0,x_1)+p_11(x_0,x_1)=1 and thatp_00, p_10,and p_01 are nonnegative. From (<ref>) and (<ref>), q_10(x_1,x_0) + q_11(x_1,x_0)≤a(x_0) + b(x_1),which implies that p_11(x_0,x_1) is also nonnegative.Assumption <ref> is implied by (<ref>) irrespective of the construction of (Y_0,Y_1). We now construct a pair (Y_0,Y_1) with conditional distribution p_ij(x_0,x_1) such that Assumptions <ref> and <ref> are both satisfied. First construct a random variable U with uniform distribution on [0,1] in the following way. Set U∈[0,q_00(x_0,x_1)+q_01(x_0,x_1)] if and only if Y=0. Set U∈(q_00(x_0,x_1)+q_01(x_0,x_1),q_00(x_0,x_1)+q_01(x_0,x_1)+q_10(x_0,x_1)] if and only if (Y,D)=(1,0). Finally, set U∈(q_00(x_0,x_1)+q_01(x_0,x_1)+q_10(x_0,x_1),1] if and only if (Y,D)=(1,1).Now set (Y_0,Y_1)=(0,0) if and only if U≤ q_00(x_0,x_1)+q_01(x_0,x_1), (Y_0,Y_1)=(1,0) if and only if U∈(q_00(x_0,x_1)+q_01(x_0,x_1),q_00(x_0,x_1)+q_01(x_0,x_1)+p_10(x_0,x_1)], (Y_0,Y_1)=(1,1) if and only if U∈(q_00(x_0,x_1)+q_01(x_0,x_1)+p_10(x_0,x_1),1-p_01(x_0,x_1)], and (Y_0,Y_1)=(0,1) if and only if U∈(1-p_01(x_0,x_1),1]. By construction, (Y_0,Y_1) has probability mass distribution p_ij(x_0,x_1) and satisfies Assumptions <ref> and <ref>. This completes the proof.§.§.§ Statement of Theorem <ref>(1)Fix the joint distribution (Y,D,Z) and denote the conditional probability mass function (q_00(z),q_01(z),q_10(z),q_11(z)), where q_ij(z)=ℙ(Y=i,D=j| Z=z). The following two statements hold. (1) If for each z∈ Supp(Z), the non-negative vector (p_00(z),p_01(z),p_10(z),p_11(z))∈ℝ^4 satisfies p_00(z)+p_01(z)+p_10(z)+p_11(z)=1, sup_z̃≤ zq_10(z̃)≤ p_10(z)+p_11(z)≤inf_z̃≥ z[q_11(z̃)+q_10(z̃)], sup_z̃≤ zq_11(z̃)≤ p_01(z)+p_11(z)≤inf_z̃≥ z[q_11(z̃)+q_10(z̃)], p_00(z)≤inf_z̃≤ z[q_01(z̃)+q_00(z̃)], p_10(z)≤ q_01(z)+q_10(z), and p_01(z)≤ q_00(z)+q_11(z), then there exists a pair of binary random variables (Y_0,Y_1) such that Assumptions <ref>, <ref> and <ref> are satisfied and ℙ(Y_0=0,Y_1=0| Z=z)=p_00(z), ℙ(Y_0=0,Y_1=1| Z=z)=p_01(z) and ℙ(Y_0=1,Y_1=0| Z=z)=p_10(z), for each z∈ Supp(Z). (2) Conversely, if the random vector (Y,D,Z) satisfies Assumptions <ref>, <ref> and <ref> for some the pair of binary random variables (Y_0,Y_1), then (<ref>) and (<ref>) hold.§.§.§ Statement of Proposition <ref>(1)Fix the joint distribution (Y,D,Z) and denote the conditional probability mass function (q_00(z),q_01(z),q_10(z),q_11(z)), where q_ij(z)=ℙ(Y=i,D=j| Z=z). The following two statements hold. (1) If for each z∈ Supp(Z), the non-negative vector (p_00(z),p_01(z),p_10(z),p_11(z))∈ℝ^4 satisfies p_00(z)+p_01(z)+p_10(z)+p_11(z)=1, sup_z̃≤ zq_10(z̃)≤ p_10(z)+p_11(z)≤1-sup_z̃≥ zq_00(z̃), sup_z̃≤ zq_11(z̃)≤ p_01(z)+p_11(z)≤1-sup_z̃≥ zq_01(z̃), p_11(z)≤inf_z̃≥ z[q_10(z̃)+q_11(z̃)], p_00(z)≤inf_z̃≤ z[q_01(z̃)+q_00(z̃)], p_10(z)≤ q_01(z)+q_10(z), and p_01(z)≤ q_00(z)+q_11(z), then there exists a pair of binary random variables (Y_0,Y_1) such that Assumptions <ref> and <ref> are satisfied and ℙ(Y_0=0,Y_1=0| Z=z)=p_00(z), ℙ(Y_0=0,Y_1=1| Z=z)=p_01(z) and ℙ(Y_0=1,Y_1=0| Z=z)=p_10(z), for each z∈ Supp(Z). (2) Conversely, if the random vector (Y,D,Z) satisfies Assumptions <ref> and <ref> for some the pair of binary random variables (Y_0,Y_1), then (<ref>) and (<ref>) hold. §.§.§ Proof of Proposition <ref>(1)For each z in the support of Z, the binary outcomes model under Assumption <ref> can be equivalently defined as a correspondence G betweenvalues of observables (y,d)∈𝒜:={(0,0),(0,1),(1,0),(1,1)} and values of unobservables (y_0,y_1)∈𝒜. The correspondence is defined by its values G(y,d) for each (y,d)∈𝒜, namely G(1,1):={(1,1),(0,1)}, G(1,0):={(1,1),(1,0)}, G(0,1):={(1,0),(0,0)} and G(0,0):={(0,1),(0,0)}. By Theorem 1 of <cit.>, the 14 inequalities ℙ((Y_0,Y_1)∈ A| Z=z)≤ℙ(G(Y,D)∩ A∅| Z=z) for each A⊂𝒜 provide a collection of sharp bounds for the model defined by the correspondence G. The 14 inequalities are listed below. Singleton A's yield[p_11(z)≤q_11(z)+q_10(z);p_10(z)≤q_10(z)+q_01(z);p_01(z)≤q_11(z)+q_00(z);p_00(z)≤ q_01(z)+q_00(z). ]Pairs yield[p_11(z)+p_10(z)≤q_11(z)+q_10(z)+q_01(z);p_11(z)+p_01(z)≤q_11(z)+q_10(z)+q_00(z);p_11(z)+p_00(z)≤1;p_10(z)+p_01(z)≤1;p_10(z)+p_00(z)≤q_10(z)+q_01(z)+q_00(z);p_01(z)+p_00(z)≤ q_11(z)+q_01(z)+q_00(z). ]Finally, triplets yield only trivial inequalities of the form p_11(z)+p_10(z)+p_01(z)≤ 1.All non trivial inequalities in (<ref>) are equivalent to q_10(z)≤ p_10(z)+p_11(z)≤ 1-q_00(z) andq_11(z)≤ p_01(z)+p_11(z)≤ 1-q_01(z).Combining with Assumption <ref> as in the proof of Theorem <ref>(1) and removing redundant inequalities yields the result.§.§.§ Proof of Theorem <ref>(1)Assumption <ref> is equivalent to Y=Y_d⇒𝔼[Y|ℐ]=𝔼[Y|ℐ]≥𝔼[Y_1-d|ℐ] for d=0,1. The latter statement is true for some σ-algebra that contains σ(Z) if and only if Y=Y_d⇒𝔼[Y| Z]=𝔼[Y_d| Z]≥𝔼[Y_1-d| Z] for d=0,1. The latter is equivalent to max{p_01(z)+p_11(z),p_10(z)+p_11(z)}≤ q_11(z)+q_10(z) for all z∈ Supp(Z). Combining with (<ref>) and (<ref>) and removing redundant inequalities, yields q_10(z)≤ p_01(z)+p_11(z)≤ q_11(z)+q_10(z), q_11(z)≤ p_10(z)+p_11(z)≤ q_11(z)+q_10(z), p_00(z)≤ q_01(z)+q_00(z), p_10(z)≤ q_01(z)+q_10(z), and p_01(z)≤ q_00(z)+q_11(z) for all z∈ Supp(Z). Combining with Assumption <ref> as in the proof of Theorem <ref>(1) yields the result.§.§.§ Combined sector-specific and SMIV instrumentSuppose Z satisfies Assumption <ref> (SMIV) and Y_0⊥⊥ Z, so that Z is both a stochastically monotone instrument and a sector specific variable in the sense that it does not directly affect potential outcomes in the non STEM sector. Then, the joint distribution of potential outcomes in the binary outcomes Roy model (Assumptions <ref> and <ref>) satisfies [sup_zℙ(Y=1,D=0| Z=z) ≤ ℙ(Y_0=1)≤inf_zℙ(Y=1| Z=z); ; sup_z̃≤ zℙ(Y=1,D=1| Z=z̃) ≤ ℙ(Y_1=1| Z=z); ; ℙ(Y_0=Y_1=0| Z=z) =ℙ(Y=0| Z=z). ] Testable implications are stochastic monotonicity of Y relative to Z and sup_zℙ(Y=1,D=0| Z=z)≤inf_zℙ(Y=1| Z=z). § PROOFS AND ADDITIONAL RESULTS RELATING TO MIXED DISCRETE-CONTINUOUS OUTCOMES§.§ Functionally sharp bounds for the Roy model We first illustrate functional sharpness by showing improvements over Peterson bounds. Combining Peterson bounds (<ref>) and assuming y_12>y_01 and y_02>y_11 yields the following upper bound.ℙ(y_01<Y_0≤ y_02,y_11<Y_1≤ y_12) = ℙ(Y_0≤ y_02,Y_1≤ y_12)-ℙ(Y_0≤ y_02,Y_1≤ y_11) -ℙ(Y_0≤ y_01,Y_1≤ y_12)+ℙ(Y_0≤ y_01,Y_1≤ y_11)≤ ℙ(Y≤ y_02,D=0)+ℙ(Y≤ y_12,D=1) -ℙ(Y≤min(y_02,y_11)) -ℙ(Y≤min(y_01,y_12)) +ℙ(Y≤ y_01,D=0)+ℙ(Y≤ y_11,D=1)= ℙ(y_11<Y≤ y_02,D=0)+ℙ(y_01<Y≤ y_12,D=1).The latter bounds are not sharp. Indeed:ℙ(y_01<Y_0≤ y_02,y_11<Y_1≤ y_12) = ℙ(y_01<Y_0≤ y_02,y_11<Y_1≤ y_12,Y_1≤ Y_0) 25pt+10ptℙ(y_01<Y_0≤ y_02,y_11<Y_1≤ y_12,Y_1>Y_0)≤ ℙ(max(y_01,y_11)<Y≤ y_02,D=0)25pt+10ptℙ(max(y_01,y_11)<Y≤ y_12,D=1),obtained from Theorem <ref> (or directly), are sharper unless y_01=y_11.§.§.§ Proof of Theorem <ref>(1,2)The Roy model defined by Assumptions <ref> and <ref> can be equivalently recast as a correspondence G:ℝ×{0,1}⇉ℝ^2 defined as follows, with the order convention (Y_0,Y_1) for the pair of unobserved variables. For all y∈ℝ, [ G(y,0)={y}×[b,y]; ; G(y,1)= [b,y]×{y}. ]Indeed, if D=0, by Assumption <ref>, Y_0=Y. By Assumption <ref>, Y_1≤ Y. Hence the set of values compatible with the Roy model specification is (Y_0,Y_1)∈{Y}×[b,Y], as in the definition of G. Similarly, if D=1, by Assumption <ref>, Y_1=Y. By Assumption <ref>, Y_0≤ Y. Hence the set of values compatible with the Roy model specification is (Y_0,Y_1)∈[b,Y]×{Y}.The collection (μ,G,ν), where μ is the joint distribution of the vector (Y,D) of observable variables and ν is the joint distribution of the vector (Y_1,Y_0) of unobservable variables, forms a structure in the terminology of <cit.> extended by <cit.>. The correspondence G is non-empty valued and measurable, in the sense that for any open set 𝒪⊆ℝ^2, G^-1(𝒪):={(y,d)∈ℝ×{0,1} |G(y,d)∩𝒪∅} is a Borel subset of ℝ×{0,1}. Hence Theorem 1 of <cit.> applies and the collection of inequalitiesμ(A)≤ν [G^-1(A)],A⊆ℝ^2define sharp bounds for the joint distribution ν of the unobservable variables (Y_1,Y_0).For any Borel A⊆ℝ^2, G^-1(A) = {(y,d)∈ℝ×{0,1} |G(y,d)∩ A∅}= {(y,0) |y∈ U_A,0}∪{(y,1) |y∈ U_A,1}.Hence, μ(A)≤ν [G^-1(A)] is equivalent to the second inequality in the display of Proposition <ref>. The first inequality in that same display is obtained by complementation as follows.μ(A^c)≤ν[G^-1(A^c)] 15pt ⇒15pt μ (A)≥ ν[{(y,d)∈ℝ×{0,1} |G(y,d)⊆ A)}] = ν[{(y,0) |y∈ L_A,0}∪{(y,1) |y∈ L_A,1}]as required.§.§.§ Proof of Theorem <ref>(3)Let A be an upper set in ℝ^2. By Assumption <ref>, ℙ((Y_0,Y_1)∈ A| Z=z)≤inf_z̃≥ zℙ((Y_0,Y_1)∈ A| Z=z̃). As shown in the proof of Theorem <ref>(1,2), ℙ((Y_0,Y_1)∈ A| Z=z̃)≤ℙ(Y∈ U_A,0,D=0| Z=z̃)+ℙ(Y∈ U_A,1,D=1| Z=z̃). Since A is an upper set, Y∈ U_A,d implies (Y,Y)∈ A and the upper bound follows. Similarly, we have ℙ((Y_0,Y_1)∈ A| Z=z)≥ℙ(Y∈ L_A,0,D=0| Z=z̃)+ℙ(Y∈ L_A,1,D=1| Z=z̃) for any z̃≤ z. By definition of L_A,0, Y∈ L_A,0 implies Y×[b,Y]⊆ A. Since A is an upper set, this in turn implies that [Y,∞)×ℝ⊆ A, hence that Y≥ y^A_0. Reasoning identically for L_A,1 yields the lower bound and the result follows. §.§.§ Statement of Corollary <ref>* Let (Y_0,Y_1) be an arbitrary pair of random variables. Let Y and D satisfy Assumptions <ref> and <ref>. Then the distribution functions F_1 and F_0 of Y_1 and Y_0 respectively, satisfyF_d(y_2)-F_d(y_1)≥ℙ(y_1<Y≤ y_2,D=d)+ℙ(Y≤ y_2,D=1-d)1{y_1≤b} for d=0,1, and for all y_1 and y_2 in ℝ∪{±∞}, such that y_1<y_2.* Let Y be an arbitrary random variable and D be a binary random variable. Let F_1 and F_0 be cumulative distribution functions satisfying (<ref>). Then there exists a pair (Y_1,Y_0) with cdfs F_1 and F_0 respectively, such that Assumptions <ref> and <ref> are satisfied. §.§.§ Proof of Corollary <ref>(1) Validity of the bounds: As shown in the main text, Proposition <ref> yields bounds (<ref>)-(<ref>) and (<ref>) is redundant. Hence the result.(2) Sharpness of the bounds: Let Y and D be given. Let F_1 and F_0 be cdfs satisfying (<ref>). We shall construct a pair (Y_0,Y_1) with cdfs F_0 and F_1 respectively, such that Assumptions <ref> and <ref> are satisfied. Define F_d with y↦ F_d(y)=ℙ(Y≤ y,D=d) for each y. Let F^-1 be the generalized inverse, defined as F^-1(u)=inf{y: F(y)≥ u}. Let U be a uniform random variable on [0,1] such that U< ℙ(D=1)⇔ D=1. Define Y_0 and Y_1 in the following way. When U<ℙ(D=1), let Y_1= F_1^-1(U) and Y_0=(F_0- F_0)^-1(U). The latter is well defined, since F_0≥F_0, and U remains in the range of F_0- F_0. Indeed, (<ref>) implies F_d(y)≥ F_d(y)+ F_1-d(y),y∈ℝ̅.Hence, F_d(y)- F_d(y)≥ℙ(Y≤ y,D=1-d), hence, in particular, (F_0- F_0)(+∞)≥ℙ(D=1). For U>ℙ(D=1), letY_0= F_0^-1(U-ℙ(D=1)) and Y_1=(F_1- F_1)^-1(U-ℙ(D=1)). The latter is well defined because, as before, (F_1- F_1)(+∞)≥ℙ(D=0).We first verify Assumption <ref>. Note first that Assumption <ref> is equivalent to D=d⇒ Y_d≥ Y_1-d for d=0,1. Hence, we need to show that U<ℙ(D=1)⇒ Y_1≥ Y_0 and U>ℙ(D=1)⇒ Y_1≤ Y_0. By symmetry, we only show the first implication. Suppose U<ℙ(D=1). If U is a continuity value of F_1, then U= F_1(Y_1). By (<ref>), F_1≤ F_0- F_0. Hence, U= F_1(Y_1)≤(F_0- F_0)(Y_1). So if we can show right-continuity and monotonicity of F_d- F_d, then Y_0=(F_0- F_0)^-1(U)≤ Y_1 as required. Now, monotonicity of F_d- F_d follows immediately from (<ref>) and right continuity of F_d- F_d from that of F_d and F_d. If the distribution of Y_1 has an atom at F_1^-1(U), then, by right-continuity of F_1, U≤ F_1(Y_1)≤ (F_0- F_0)(Y_1), so that, by right continuity and monotonicity of F_0- F_0, we have Y_0=(F_0- F_0)^-1(U)≤ Y_1 as required.We now verify Assumption <ref>. We need to show that for each d=1,0, ℙ(Y_d≤ y,D=d)= F_d(y). By symmetry, we only deal with Y_1. By monotonicity and right continuity of F_1, F_1^-1(U)≤ y⇔ U≤ F_1(y) (Proposition 1(5) in <cit.>). Hence, we have the following as required.ℙ(Y_1≤ y,D=1)=ℙ( F_1^-1(U)≤ y,U<ℙ(D=1))=ℙ(U≤ F_1(y),U<ℙ(D=1))= F_1(y). Finally, we need to verify that Y_1 and Y_0 do indeed have the announced distributions. We shall show that the cdf of Y_1 is indeed F_1. Reasoning as above, we have the following.ℙ(Y_1≤ y,D=0) = ℙ((F_1- F_1)^-1(U-ℙ(D=1))≤ y,U>ℙ(D=1))= ℙ(U≤ (F_1- F_1)(y)+ℙ(D=1),U<ℙ(D=1))= (F_1- F_1)(y).Therefore ℙ(Y_1≤ y)=ℙ(Y_1≤ y,D=1)+ℙ(Y_1≤ y, D=0)= F_1(y)+(F_1- F_1)(y)=F_1(y) as required.§.§ Functional features of potential outcomes §.§.§ Proof of Proposition <ref>Let 0<q_1<q_2<1 and let y_1 and y_2 be the q_1 and q_2 quantiles of the distribution of outcomes in Sector d for the self-selected economy. The following holds.ℙ(Y≤ y_2| D=d)-ℙ(Y≤ y_1| D=d)= 1/ℙ(D=d)(F_d(y_2)- F_d(y_1) ) ≥ F_d(y_2)- F_d(y_1).Hence, q_2-q_1≥ F_d(y_2)- F_d(y_1). In addition, for any y∈ℝ, ℙ(Y≤ y| D=d)= F_d(y)/ℙ(D=d)= F_d(y)+P(D=1-d) F_d(y)/ℙ(D=d)≤F̅_d(y).Finally, under the stochastic dominance condition, F(y_j)=ℙ(Y≤ y_j)≤ℙ(Y≤ y_j| D=d) for j=0,1.Since the sharp upper bound for the (q_1,q_2)-interquantile range of the distribution of Y_d is given by(q_1,q_2,F_d)=max{y_2-y_1|[ F(y_1)≤ q_1≤F̅_d(y_1),; F(y_2)≤ q_2≤F̅_d(y_2),; q_2-q_1≥ F_d(y_2)- F_d(y_1), ].}these three inequalities imply that the interquantile range y_2-y_1 satisfies the sharp bounds on the interquantile range for the distribution of Y_d.However, if we relax the first order stochastic dominance condition, we now show that there exist situations, where the interquantile range in Sector d under self-selection is strictly larger than the upper bound for the corresponding interquantile range of the distribution of potential outcomes under random assignment. Let ℙ(Y≤ y) be continuous. Let y_02 be defined by ℙ(Y≤ y_02,D=d)+ℙ(D=1-d)=q_2. Finally, suppose that D=1-d⇒ Y≤ y_02, so that ℙ(Y≤ y) coincides with ℙ(Y≤ y,D=d)+ℙ(D=1-d) on the right of y_02. Then the upper bound for the interquantile range of Y_d is F_d(q_2-ℙ(D=d))- F_d(q_1-ℙ(D=d)), which can be made lower than the interquantile range for Y| D=d, namely ℙ(D=d)( F_d(q_2)- F_d(q_1)), with a suitable choice of slope for ℙ(Y≤ y,D=d). plain | http://arxiv.org/abs/1709.09284v2 | {
"authors": [
"Ismael Mourifie",
"Marc Henry",
"Romuald Meango"
],
"categories": [
"econ.EM"
],
"primary_category": "econ.EM",
"published": "20170926232535",
"title": "Sharp bounds and testability of a Roy model of STEM major choices"
} |
3D Modeling of the Magnetization of Superconducting Rectangular-Based Bulks and Tape Stacks M. Kapolka, V. M. R. Zermeño, S. Zou, A. Morandi, P. L. Ribani, E. Pardo, F. Grilli Document written on September 19, 2017. M. Kapolka, E. Pardo are with the Slovak Academy of Sciences, Institute of Electrical Engineering, Bratislava, Slovakia. V. M. R. Zermeño, S. Zou and F. Grilli are with the Karlsruhe institute of Technology, Institute for Technical Physics, Karlsruhe, Germany. P. L. Ribani and A. Morandi are with the University of Bologna, Italy. M. Kapolka and E. Pardo acknowledge the use of computing resources provided by the project SIVVP, ITMS 26230120002 supported by the Research & Development Operational Programme funded by the ERDF, the financial support of the Grant Agency of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences (VEGA) under contract No. 2/0126/15. Corresponding author's email: [email protected] 30, 2023 ===============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================empty empty Controller synthesis techniques for continuous systems with respect to temporal logic specifications typically use a finite-state symbolic abstraction of the system.Constructing this abstraction for the entire system is computationally expensive, and does not exploit natural decompositions of many systems into interacting components.We have recently introduced a new relation, called (approximate) disturbance bisimulation for compositional symbolic abstractionto help scale controller synthesis for temporal logic to larger systems.In this paper, we extend the results to stochastic control systems modeled by stochastic differential equations.Given any stochastic control system satisfying a stochastic version of the incremental input-to-state stability property and a positive error bound, we show how to construct a finite-state transition system (if there exists one) which is disturbance bisimilar to the given stochastic control system. Given a network of stochastic control systems, we give conditions on the simultaneous existence of disturbance bisimilar abstractions to every component allowing for compositional abstraction of the network system. § INTRODUCTIONIn abstraction-based controller synthesis, a finite-state symbolic model of a continuous systemis used to synthesize a symbolic controller for a logical specification, and the controller is then refined to a controller for the original system.This technique has recently gained a lot of attention due totwo main advantages.First, it allows for fully automated controller synthesis for systems with continuous dynamicswhile handling complex specifications (given e.g. as ω-regular languages) inaddition to stability.Second, it naturally accounts for the complex interplay between discrete and continuous components within a control loop. The soundness of the abstraction-based synthesis technique relies on notions of behavioral closeness ofthe original system and its abstraction, which is formalized using system equivalence relations(see e.g. <cit.> and the references therein).Recently, abstraction-based controller synthesis has been extended to stochastic control systems <cit.>. In the stochastic setting, behavioral closeness of the original system and its abstraction is formalized using then-th moment of the trajectories.Despite its nice theoretical properties and applicability to many different system classes,abstraction based controller synthesis does not scale very well becauseboth the abstraction step and the controller synthesis step areexponential in the dimension of the continuous state space.This issue motivated us to propose disturbance bisimulation <cit.>, an equivalence relation that exploits the intrinsic compositionality of systems.Given a network of metric systems and their equivalently interconnected disturbance bisimilar abstractions, the main result of<cit.> shows that the overall network system is also disturbance bisimilar to the network of the abstractions. This result has an interesting consequence: given a construction of a disturbance bisimilar abstraction fora given system class, we can compositionally abstract a network of systems from this classwhenever all local abstractions are guaranteed to exist simultaneously. In <cit.> we exploited this fact for large networks of incremental input-to-statestable deterministic control systems and demonstrated the effectiveness of our approach in a case study.In this paper, we extend these results to the class ofstochastic control systems which satisfy a stochastic version of the incremental input-to-state stability condition. Our main contribution in this paper is to show how a stochastic control system, whichmay be connected to other components within a network,allows for the algorithmic computation of(1) a metric system capturing its time-sampled dynamics and(2) a metric system capturing its abstract symbolic dynamics, such that the two constructed systems are disturbance bisimilar.This construction allows us to use the results from <cit.>to provide a compositional abstraction-based controller synthesis technique for a given network of continuous and stochastic dynamical systems. Our results relate to recent results in <cit.> on compositional abstraction for stochastic systems.The main difference between our work and <cit.> is the type of abstraction:while we work with finite state symbolic abstractions, their abstractions are infinite state. There have been some efforts for improving scalability of abstraction techniques for stochastic systems in a different setting, where abstract models are Markov chains and the goal is to match distributions on states up to a fixed horizon.In <cit.>, the state space discretization is done adaptively and, in <cit.>, the abstract state space of a monolithic system is represented compositionally. The paper is organized as follows.After introducing preliminaries on stochastic control systems in <ref>,we define metric systems in <ref>, and present how the two particular metric systems discussed abovecan be obtained from a stochastic control system.Given these two metric systems, we give sufficient conditions for them to be disturbance bisimilarin <ref>, after recalling the notion of disturbance bisimulation from <cit.>.In <ref>, we invoke results from <cit.> toextend our result from a single stochastic control system to a network of such systems.All proofs can be found in the appendix. § STOCHASTSIC CONTROL SYSTEMS Most part of this section is adapted from <cit.> to systems with stochastic disturbance inputs. §.§ Notation We use the symbols , , _>0, _≥ 0 and ℤ to denote the set of natural,real, positive real, non-negative real numbers, and integers, respectively.The symbols I_n, 0_n, and 0_n×m denote the identity matrix, the zero vector, and the zero matrixin ^n×n, ^n, and ^n×m, respectively.Given a vector , we denote by x_i the i-th element of x and by x the infinity norm of x. A continuous function is said to belong to class 𝒦 if it is strictly increasing and ;γ is said to belong to class 𝒦_∞ifand .A continuous function is said to belong to class 𝒦ℒ if, for each fixed s, the function belongs to class 𝒦_∞ and, for each fixed r, the mapβ(r,·):ℝ_≥ 0→ℝ_≥ 0 is decreasing and . Let f:ℝ_≥ 0→ℝ^k be a measurable function. We define the (essential) supremum f of f asf := ess(sup)f(t)| t≥ 0.A function f is bounded if f < ∞.Given a square matrix M, we denote by Tr(M) the trace of M, and by λ_min(M) and λ_max(M) the minimum and maximum eigenvalue of M respectively. Given a matrix , we denote by M:=max_1≤i≤n∑_j=1^m|m_ij| the infinitynorm of M,and by M_F √(Tr(MM^T)) the Frobenius norm of M. We denote by Diag(a_1,…,a_n) the diagonal matrix with diagonal entries a_1,…,a_n∈. If a_1,…,a_n are matrices, then Diag(a_1,…,a_n) is a block diagonal matrix of appropriate dimension. §.§ Stochastic Control SystemWe fix the probability space for the whole paper as (Ω, , ), where Ω is a sample space,is a sigma algebra over Ω representing the set of events, andis a probability measure. Let (Ω, , ) admits a filtration = (_s)_s≥ 0 which is complete and right continuous <cit.>. Let (B_s)_s ≥ 0 be a r-dimensional -Brownian motion.A stochastic control system is a tuple Σ=(X,U,𝒰,W,𝒲,f,σ), where X = ℝ^n is the state space,U⊆^m is the input set that is assumed to be compact,𝒰 is a subset of set of all measurable, locally essentially bounded functions of time from _≥ 0 to U,W⊆^p is the disturbance input space that is assumed to be compact,𝒲 isa set of stochastic processes with elements ,f:X× U × W → X is a continuous function of its arguments representing the drift of Σ, σ:X→^n×r is a function representing the diffusion of Σ. A stochastic processis called a solution process of Σ if there exists μ∈𝒰 and ν∈𝒲 satisfying the following stochastic differential equation:ξ(t)= f(ξ(t),μ(t),ν(t)) t+σ(ξ(t)) B_t, -almost surely (-a.s.).For succinctness of representation, we use the notation ξ_a μν to denote a stochastic solution process of Σ from the initial condition ξ_a μν(0) = a -a.s., and under effect of input signal μ∈𝒰 and disturbance signal ν∈𝒲. Note that given any time instant t,ξ_a μν(t) represents a random variable from Ω to X measurable in _t.We make the following two assumptions on stochastic control systems to ensure a unique global continuous solution. [Lipschitz condition] There exist constants L_f,L_σ∈_≥ 0 such that the following inequalities hold ‖ f(x,u,w)-f(x',u',w')‖≤ L_f(‖ x-x'‖+‖ u-u'‖+ ‖ w-w'‖), and ‖σ(x)-σ(x')‖≤ L_σ‖x-x'‖, for all x,x'∈ X, u,u'∈ U and w,w'∈ W. [Linear growth] There exists a positive constant K such that for all x∈ X, u,∈ U and w∈ W, max(f(x,u,w)^2,σ(x)^2)≤ K(1+x^2). <ref> on Lipschitz continuity gives uniqueness and <ref> on linear growth gives global existence (<cit.>). The latter will also be used in <ref> (cf. <ref>) to provide an upper bound on the second moment of the solution process.In this paper the disturbances in the set 𝒲 are allowed to be stochastic. This is necessary because, as will be described later, in our setting the disturbances play the role of trajectories of other stochastic control systems after interconnection. For the results of this paper to hold, we require the process (ξ,ν):Ω×_≥ 0→ X× W to be an Itô process, i.e., (ξ,ν) has to be the solution of a possibly time-inhomogeneous Itô diffusion. §.§ δ-ISS-M_qWe now generalize the notion of incremental input-to-state stability in the q-the moment (δ-ISS-M_q) for stochastic control systems from <cit.> by considering disturbances. In the absence of noise, these notions correspond to δ-ISS for deterministic systems <cit.>.A stochastic control system Σ=(X, U ,𝒰,W,𝒲,f,σ) is stochastically incrementally input-to-state stable in the q-th moment (δ-ISS-M_q), if there exists a 𝒦ℒ function β andfunctions ρ_u and ρ_d such that for any t∈ℝ_≥ 0, any μ, μ'∈𝒰, any ν, ν'∈𝒲, and anyrandom variables a and a' that are measurable in _0, the following condition is satisfied:[‖ξ_aμν(t)-ξ_a'μ'ν'(t)‖ ^q] ≤β( [ ‖ a-a' ‖ ^q], t ) + ρ_u (μ - μ') + ρ_d ([ν - ν'^q] ).The δ-ISS-M_q property can be characterized in terms of the existence of stochastic incremental Lyapnuov functions. <cit.>Define the diagonal set Δ as. Consider a stochastic control system Σ=(X, U ,𝒰,W,𝒲,f,σ)and a continuous function V:X× X→ℝ_≥ 0 which issmooth on {^n×^n}\Δ.Function V is called aδ-ISS-M_q Lyapunov function for Σif there existfunctionsα, α, σ_u, σ_d, and a constant κ∈ℝ_> 0 such that (i) α is a convex function, and α and σ_d are concave functions;(ii) for any x,x'∈ X, α(‖ x-x'‖ ^q)≤V(x,x')≤α(‖ x-x'‖ ^q);(iii) for any x,x'∈ X, x≠ x', any u,u'∈ U, and any w,w'∈ W,ℒ^u,u',w,w' V(x, x') [∂_xV ∂_x'V] [f(x,u,w); f(x',u',w') ]+ 1/2Tr([σ(x); σ(x') ][σ^T(x) σ^T(x')] [ ∂_x,x V∂_x,x' V;∂_x',x V ∂_x',x' V ]) ≤-κ V(x,x')+σ_u(u-u')+σ_d(w-w'^q),where ℒ^u,u',w,w' is the infinitesimal generator (<cit.>) associated to the stochastic control system (<ref>),which depends on two separate controls u, u'∈ U and two separate disturbances w,w'∈ W. In this case we say that the stochastic control system Σ admits a δ-ISS-M_q Lyapunov function,witnessed by α, α, σ_u, σ_d, and κ∈ℝ_> 0. Note that condition (i) is not required in the context of deterministic control systems.Condition (ii) implies that the growth rates of the functions α and α are linear, as a concave function is supposed to dominate a convexone.These conditions are not restrictive provided we are interested in the dynamics of Σ on a compact subset D⊂^n, which is often the case in practice.It can be readily verified that the δ-ISS-M_q Lyapunov function in Definition <ref>is a stochastic bisimulation function between Σ and itself, as defined in <cit.>, Def. 5.The following theoremdescribes δ-ISS-M_q in terms of the existence of δ-ISS-M_q Lyapunov functions. It generalizes the corresponding theorem <cit.> in the presence of disturbances.A stochastic control system Σ is δ-ISS-M_q if it admits a δ-ISS-M_q Lyapunov function.In order to keep the notation simple, we present the results only for second momentin the rest of paper with the understanding that they can be generalized for other moments.The following lemma (compare <cit.>) provides a sufficient condition on a particularfunction V to be a δ-ISS-M_q Lyapunov function. Let Σ=(X, U ,𝒰,W,𝒲,f,σ) be a stochastic control system. Let P∈^n×n be a symmetric positive definite matrix. Consider the function V: ^n ×^n _≥ 0 defined as: V(x,x')1/2(x-x')^TP(x-x')and satisfying (x-x')^T P(f(x,u,w)-f(x',u,w)) + +1/2√(P)( σ(x) - σ(x')) _F^2 ≤-2κV(x,x'), or, if f is differentiable, satisfying (x-x')^T P ∂_x f(z,u,w)(x-x') + +1/2√(P)( σ(x) - σ(x')) _F^2 ≤-2κ V(x,x'),for all x,x',z∈ X, for all u ∈U, for all w∈ W, and for some constant κ∈_> 0.Then V is a δ-ISS-M_2 Lyapunov function for Σ.§.§ Noisy and Noise Free Trajectories In this section we provide an upper bound on the distance between a stochastic state trajectory and its associated noise-freetrajectory at any instant of time.This bound is a generalization of the bound in <cit.> to the case when there is disturbance in the system.The bound will be instrumental in proving closeness between the trajectories of a stochastic control system and its abstraction in <ref>. Consider a stochastic control system Σ=(X, U ,𝒰,W,𝒲,f,σ). Suppose there exists a δ-ISS-M_2 Lyapunov function V of Σ s.t. its Hessian matrix in ^2n× 2n satisfies 0≤∂_x,xV(x,x')≤ P, for some positive semi-definite matrix P∈^2n× 2n and for any x,x'∈^n. Define ξ_xμν as the solution of the ordinary differential equation (ODE) ξ̇xμν(t)=f(ξxμν(t),μ(t),ν(t)) starting from the initial condition x. Then for any x in a compact set D⊂^n, any μ∈𝒰 and any ν∈𝒲, we have [‖ξxμν(t)-ξxμν(t)‖^2] ≤ h(σ,t), where h(σ,t) := ^-1( 1/2√(P)^2· n·minn,r·e^-κ t· L_σ^2··∫_0^t [ β(_x∈ Dx^2,s) +ρ_u( _u∈ Uu).. + ρ_d( _w∈ Ww^2 ) ]ds). The non-negative valued function h tends to zero as t0, t∞, or as L_σ 0, where L_σ is the Lipschitz constant introduced in <ref>. (<ref>) gives a representation of the function h in terms of ρ_u and ρ_d. This representation of h can be translated into a form using σ_u and σ_d instead, asshown in <cit.>.§ FROM STOCHASTIC CONTROL SYSTEMS TO METRIC SYSTEMSWe now introduce (deterministic) metric systems and interpretstochastic control systems and their abstractions as metric systems.As in <cit.>,we consider metric systems that are time sampled w.r.t.a globally fixed time sampling parameter τ∈.Given the probability space Ω and a time sampling parameter τ∈,a stochastic metric system[ Often, metric systems are defined with an additional output space and an output map from states to the output space. We omit the output space for notational simplicity; for us, the state and the output space coincide, and the output map is the identity function.] S=(X,U,𝒰_τ,W,𝒲_τ,δ_τ) consists of a (possibly infinite) set of statesX, given by a set of random variables, and equipped with a metric d:X× X →, a set of piece-wise constant inputs 𝒰_τ of duration τ taking values in U⊆m, i.e.,𝒰_τ = μ:[0,]→ U |∀ t_1,t_2∈[0,] . μ(t_1) = μ(t_2),a set of disturbances 𝒲_τ taking values in W⊆p, i.e.,𝒲_τ⊆ν:Ω× [0,]→ W,and a transition function δ_τ: X×𝒰_τ×𝒲_τ→ 2^X. We write x x' if x'∈δ_τ(x,μ, ν), and we denote the unique value of μ∈𝒰 over [0,τ] byu_μ∈ U. A deterministic metric system is a special type of a stochastic metric system where the states are deterministic points (i.e. random variables with Dirac delta distributions), and disturbances are deterministic signals of the form [0,τ]→ W.If the metric system S is undisturbed, we define W=0. In this case we occasionally represent S by the tuple S = (X, U, 𝒰_τ, δ_τ) and use δ_τ:X×𝒰_τ→ 2^X with the understanding that x'∈δ_τ(x,μ, ν) holds for the zero trajectory :→0 whenever x'∈δ_τ(x,μ). By slightly abusing notation we write x'=δ_τ(x,μ, ν) as a short form when the set δ_τ(x,μ, ν) = x' is a singleton. If X, 𝒰_τ and 𝒲_τ are finite (resp. countable), S is called finite (resp. countable).We also assign to a transition x'=δ_τ(x,μ, ν)any continuous time evolution ξ:[0,τ]→ X ξ(0) = x and ξ(τ) = x'.In the following we introduce two approaches to capture an abstracted version of the dynamics of a stochastic control systems Σ by a metric system conforming to <ref>. The first approach results in a sampled time abstraction which we denote by Σ. Given a stochastic control system Σ = (X, U, 𝒰, W, 𝒲, f,σ),a time-sampling parameter τ∈,and a probability space (Ω, ,),the discrete-time stochastic metric system induced by Σ is defined by Σ = (X_𝗋, U, 𝒰_τ, W, 𝒲_τ, δ_τ) s.t.X_𝗋 is the set of all X-valued random variables, 𝒰_τ and 𝒲_τ are defined over U and W, respectively, as in (<ref>)-(<ref>) and δ_τ(x,μ,ν) = x' if x and x' are measurable in ℱ_t and ℱ_t+τ, respectively, for some t∈_≥ 0, and there exists a solution process ξ:Ω×_≥ 0→ℝ^n of Σ satisfying ξ(0) = x and ξ_xμν(τ) = x' ℙ-a.s.Since we allow any state to be initial, all states in X need to be measurable on ℱ_0.We equip X_𝗋 with the metric xx'( 𝔼[x-x'^2] )^1/2.Recall that the disturbances in the set 𝒲 are stochastic.Hence the above metric system must be constructed by looking at the sampled version of (ξ,ν), which is the solution process of the Itô diffusion associated with (x,w).The second approach additionally imposes a quantization of the state, input and disturbance spaces and results in a metric system denoted by Σ. Before defining this system formally we introduce notation for quantization. For any A⊆n and any vectorwith elements i>0, we defineA:=(a_1,…, a_n) ∈ A | a_i = 2ki, k∈ℤ, i=1,…, n. For x∈n and vector λ with elements λ_i>0, let 𝔹_λ(x) = x'∈n|x_i-x_i'≤λ_i denote the closed rectangle centered at x. Note that for any λ≥ (element-wise), the collection of sets 𝔹_λ(q) with q∈n is a cover of n, that is, n⊆∪𝔹_λ(q)| q∈n. We will use this insight to discretize the state and the input space of Σ using discretization parameters η and ω, respectively.Also we need to define a vector-valued metric for comparing two disturbance vectors.Let A_1,…, A_k be a finite set of metric spaces, where each A_i, i=1,…,k has a metric d_i:A_i× A_i →ℝ_≥ 0.Let A = ∏_i=1^k A_i.We construct the metric :A× A →ℝ_≥ 0^kas an extension ofthe metrics d_i on A_i: for any a=(a_1,…, a_k) and a'=(a'_1,…,a'_k), we define(a,a'):= (d_1(a_1,a_1'), …, d_k(a_k, a_k')).For the disturbance space W we allow the discretization of W to be predefined.We make the following general assumptions on the discretizaion of W which will be useful when wedeal with networks of stochastic control systems in <ref>. Let Σ = (X, U, 𝒰, W, 𝒲, f,σ) be a stochastic control system. Then we assume that there exists a countable set ⊆ W, that exists a vector ε̃∈ℝ_≥ 0^p, and a vector-valued metric :W× W→ℝ_≥ 0^p,s.t. for all w∈ W there exists a ∈ for which(w,)≤ε̃andw-≤(w,). Using this assumption we formally define the abstract metric system Σ induced by Σ as follows.Let Σ = (X, U, 𝒰, W, 𝒲, f,σ) be a stochastic control system for which <ref> holds.Given three constants τ∈, ∈, and ∈, the discrete-time discrete-space deterministic metric systeminduced by Σ is defined by Σ = (, U,, ,, ) = X,is defined over U, as in (<ref>), := ν:[0,τ]→|∀ t,k∈ [0,τ] . ν(t) = ν(k), and (x, μ, ν) = x'∈|ξxμν(τ) - x'≤, where ξxμν(·) are the noise free trajectories of Σ defined via (<ref>). We equipwith the metricnaturally inherited from X.We denote the unique value of ν∈ over [0,τ] byw_ν∈.Let us emphasize that even though Σ is a stochastic metric system and Σ is a deterministic metric system,since we are interested in studying the closeness of their trajectories in the next section,it is important that X_𝗋 andare part of the same state space. We interpretto be contained in X_𝗋, since a set of points can be associated with a set of random variables with Dirac delta distributions. § DISTURBANCE BISIMULATION This section contains the main contribution of the paper; after recalling the notion of disturbance bisimulation from <cit.> we present sufficient conditions under which the two metric systems Σ and Σ associated with a stochastic control system Σ are disturbance bisimilar.For this analysis, we restrict our attention to δ-ISS-M_2 stochastic control systems with f(0_n,0_m,0_p) =0_n and σ(0_n) = 0_n×r, whose evolution is restricted to a compact region D⊂ℝ^n. Let i=(X_i,U_i,𝒰_τ,i,W_i,𝒲_τ,i,δ_τ,i), i=1,2, be two metric systems, with state-spaces 1,2⊆ and disturbance sets W_1,W_2⊆ W⊆ℝ^p. Furthermore, letadmit the metric d:×→ℝ_≥ 0 and W admit the vector-valued metric :W× W→ℝ_≥ 0^p. A binary relation R ⊆ X_1× X_2 is a disturbance bisimulation with parameters (ε,ε̃) where ε∈ and ε̃∈ℝ_≥ 0^p, iff for each (x_1,x_2)∈ R: * d(x_1,x_2)≤ε; * for every 1∈𝒰_τ,1 there exists a 2∈𝒰_τ,2 such that for all 2∈𝒲_τ,2 and 1∈𝒲_τ,1 with (ν_1(0),ν_2(0))≤ε̃, we have that (δ_τ,1(x_1,1,1), δ_τ,2(x_2, 2, 2))∈ R; and* for every 2∈𝒰_τ,2 there exists a 1∈𝒰_τ,1 such that for all 1∈𝒲_τ,1 and 2∈𝒲_τ,2 with (ν_1(0),ν_2(0))≤ε̃, we have that (δ_τ,1(x_1,1,1), δ_τ,2(x_2, 2, 2))∈ R. 1 and 2 are said to be disturbance bisimilar with parameters (ε,ε̃) if there is a disturbance bisimulation relationR between 1 and 2 with parameters (ε,ε̃). In order to prove the existence of a disturbance bisimulation between Σ and Σ we require two additional assumptions. Let Σ be a stochastic control system admitting a δ-ISS-M_2 Lyapunov function V. There exists a 𝒦_∞ and concave function γ s.t. for any x,x',x”∈ X,|V(x,x')-V(x,x”)|≤γ(x'-x”).This assumption is not restrictive as we are interested in the dynamics of Σ on a compact subset D⊂^n. Let Σ be a stochastic control system with the associated metric systems Σ and Σ introduced in <ref>. Then there exists afunction ψ for all disturbance pairs ν∈𝒲_τ and ν̂∈𝒲_τηω with d(ν(0),ν̂(0)) ≤ε̃, the following holds for all t∈[0,τ]: d(ν(t),ν̂(t)) = ([ν̂(t)-ν(t)^2])^1/2≤ψ(t) + ε̃. Given <ref> and <ref>, we present our first main result in the following theorem. Let Σ be a stochastic control system admitting a δ-ISS-M_2 Lyapunov function witnessed by κ, , , , and , that satisfies <ref> withfunction . Fix >0 and ⊆ W s.t. (<ref>) holds and let Σ be the countable deterministic metric system associated wih Σ according to <ref> such that <ref> holds.If 0≤≤min{()^-1∘(ε^2),^-1[ (1-𝖾^-κτ)(ε^2) - 1/𝖾κσ_u(ω) - 1/𝖾κσ_d(ψ(τ) + ε̃) ] - (h(σ,τ))^1/2}, where h(σ,τ) is as in (<ref>), then the relation ={ (x̂,x)∈× X_𝗋 | 𝔼[(x̂,x)]≤(ε_^2) } is a disturbance bisimulation relation (in the second moment) with parameters (ε,ε̃) betweenand . Given any fixed τ and ε̃, one can always find sufficiently small η and ω s.t. (<ref>) and (<ref>) hold, as long as ε is lower bounded according to ε^2 > ^-1( 1/𝖾κ(ψ(τ) + ε̃) + γ( h(σ,τ)^1/2)/( 1- 𝖾^-κτ)). The lower bound on ε can be minimized by choosing an optimal Lyapunov function V for a given system Σ (see e.g. <cit.>). Note that, when the system does not experience any disturbance,(<ref>) reduces to <cit.>. § COMPOSITIONAL ABSTRACTIONLet us first summarize what we have presented so far. In <ref> we have introduced two different metric systems Σ and Σ associated with a given stochastic control system Σ. Recall that Σ is an infinite state system, whereas Σ is a finite state system under the assumption that the state space of Σ is restricted to a compact subset of ℝ^n. Then we gave sufficient conditions for these two abstractions to be disturbance bisimilar in <ref>.In this section, we consider a network of stochastic control systems Σ_i_i∈ I, and the respective local abstractions Σ_i_i∈ I and iΣ_i_i∈ I of Σ_i_i∈ I, s.t. for all i∈ I, Σ_i and iΣ_i are disturbance bisimilar with parameters (ε_i, ε̃_i). Then we adapt our result from <cit.>, and prove that the isomorphic networks of Σ_i_i∈ I and iΣ_i_i∈ I, which are isomorphic to the network of Σ_i_i∈ I as well, are again disturbance bisimilar. §.§ Network of Stochastic Control Systems We first formalize networks of stochastic control systems and their abstractions by locally treating state trajectories of neighboring systems as disturbances. Let I be an index set (e.g., I = 1,…, N for some natural number N) and let ⊆ I× I be a binary irreflexive connectivity relation on I.Furthermore, let I'⊆ I be a subset of systems with ' := (I'× I')∩. For i∈ I we define_ℐ(i) = j| (j,i)∈ and extend this notion to subsets of systems I'⊆ I as _ℐ(I') = j|∃ i∈ I'.j∈_ℐ∖ℐ'(i). Intuitively, a set of systems can be imagined to be the set of vertices 1,2,…,|I| of a directed graph 𝒢, and ℐ to be the corresponding adjacency relation. Given any vertex i of 𝒢, the set of incoming (resp. outgoing) edges are the inputs (resp. outputs) of a subsystem i, and _ℐ(i) is the set of neighboring vertices from which the incoming edges originate. Let Σ_i = (X_i, U_i, 𝒰_i, W_i, 𝒲_i, f_i,σ_i), for i∈ I, be a collection of stochastic control systems. We say that the set of stochastic control systems Σ_i_i∈ I is compatible for composition w.r.t. the interconnection relation ℐ, iffor each i∈ I, we have W_i = ∏_j∈_ℐ(i)X_j. By slightly abusing notation we write w_i=∏_j∈_ℐ(i)x_j for x_j∈ X_j and w_i∈ W_i as a short form for the single element of the set ∏_j∈_ℐ(i)x_j. We extend this notation to all sets with a single element. Let I'⊂ I be a subset of systems in the network. We divide the set of disturbances W_i for any i∈ I' into the sets of coupling and external disturbances, defined byW_i^c = ∏_j∈_ℐ'(i)X_j and W_i^e = ∏_j∈_∖'(i)X_j, respectively.If Σ_i_i∈ I is compatible, we define the composition of any subset I'⊆ I of systems as the stochastic control system Σ_i_i∈ I'= (X, U, 𝒰, W, 𝒲, f,σ) where X, U and W are defined as X = ∏_i∈ I'X_i, U = ∏_i∈ I'U_i, and W = ∏_j∈_ℐ(I')X_j. Furthermore, 𝒰 and 𝒲 are defined as the sets of functions μ : → U andν : Ω×→ W such that the projection μ_i of μ on to U_i (written μ_i=μ|_U_i) belongs to 𝒰_i, and the projection ν_i^e of ν on to W_i^e belongs to 𝒲_i^e. The composed drift is then defined as f(∏_i∈ I'x_i, ∏_i∈ I'u_i, ∏_i∈ I'w_i^e) = ∏_i∈ I'f_i(x_i, u_i, w_i^c × w_i^e), where w_i^c = ∏_j∈_ℐ'(i)x_j, and the composed diffusion is defined as σ(∏_i∈ I'x_i) = Diag( σ_1(x_1),…, σ_|I|(x_|I|) ).The Brownian motion of the overall system is defined as: dB_t = [ dB_1,tdB_|I|,t ]^T. If I'=I, then Σ is undisturbed, modeled by W:=0. It is easy to see that Σ_i_i∈ I' is again a stochastic control system in the sense of <ref>. Networks of discrete time stochastic metric systems(Σ_i) and of abstract metric systems (iΣ_i) are defined analogously. Note that we assume a nice structure of the network: the diffusion functions and the Brownian motions of the systems in a network are decoupled from the states of the other systems. This is explicitly induced via the SDE (<ref>) as the diffusion terms σ_i(·) are only functions of system's state and not the disturbance. However since the states of the systems are coupled through the drift functions, the respective random variables are implicitly dependent.§.§ Simultaneous Approximation Given I and I' ⊆ I, consider a set of compatible stochastic control systems Σ_i_i∈ I, the subset composition Σ_i_i∈ I'= (X, U, 𝒰, W, 𝒲, f,σ) anda global time-sampling parameter τ.Then we can apply <ref> and <ref> to each Σ_i to construct the correspondingmetric systems Σ_i and Σ_i. To be able to do that, we need to equip W_i with a vector-valued metric _i:W_i× W_i→|_ℐ(i)|_≥ 0 and define _i for all i∈ I s.t. Ass. <ref> holds.Intuitively, _i(w_i,w_i') is a vector with dimension |_ℐ(i)|, where the j^th entry measures the mismatch of the respective state vector of the j^th neighbor of i.We define _i as the product of state spaces of jΣ_j, i.e., the abstraction of its neighbors,_i:=∏_j ∈_ℐ(i) [X_j]_η_j. Let Σ_i = (X_i, U_i, 𝒰_i, W_i, 𝒲_i, f_i,σ_i),i∈ I, be a set of compatible stochastic control systems and the set of abstract metric systemsiΣ_i = ([X_i]_η_i,Ui,i, _i,i,i) are constructed according to <ref>, where W_i is equipped with metric (<ref>) and _i as defined in (<ref>). Select local quantization parameters i_i∈ I η_i≤ε_i. Then Ass. <ref> holds for every i∈ Iwith ε̃_i defined asε̃_i:=∏_j∈_ℐ(i)ε_j.Given <ref>, it immediately follows that the sets Σ_i_i∈ I' and Σ_i_i∈ I' of metric systems are again compatible. In order to guarantee the result of <ref> for the network, we have additionally used <ref> which essentially bounds the effect of the disturbances on the state evolution. Given the particular choice of disturbances in the network as state trajectories of neighboring systems, we can explicitly compute function ψ(t) in <ref> using the following proposition from <cit.>. Under <ref> the solution process ξ_aμν(·) satisfies the inequality 𝔼[ξ_aμν(t)-ξ_aμν(s)_2^2]≤ C|t-s|,∀ s,t∈[0,τ],for any τ>0, where ·_2 indicates the 2-norm of a vector. The constant C := 2(1+𝔼a_2^2)(τ+1)e^ατ with α:=K+2√(K) and K from <ref>. The next lemma followsfrom <ref>. Given a set of stochastic control systems Σ_i_i∈ I which is compatible for composition, let each system Σ_i satisfy <ref> with constant K_i in (<ref>). Then <ref> holds for each Σ_i withfunction ψ_i(t):= [t(t+1)∑_j∈_ℐ(i)β_j e^α_j t]^1/2,where α_j:=K_j+2√(K_j) and β_j = 2(1+ sup_x_j∈ X_jx_j_2^2).Lemmas <ref>-<ref> show that the assumptions of <ref> on disturbance sets of Σ_i hold after composition. Then the next theorem follows from <ref> which establishes simultaneous disturbance bisimilarity between abstractions of components in a network. In this theorem, using the results in <ref>, we give conditions on all local state, input, anddisturbance quantization parameters in a composed stochastic control system Σ_i_i∈ I which allow for a simultaneous construction of local abstractions Σ_i using <ref>such that they are disturbance bisimilar with parameters (ε_i,ε̃_i)to their respective discrete-time stochastic metric systems Σ_i. Let Σ_i_i∈ I be a set of compatible stochastic control systems, each admitting aδ-ISS-M_2 Lyapunov function i witnessed by κ_i, i, i, i, and i, and let i be afunction (<ref>) holds. Let Σ_i_i∈ I be the set of discrete-time stochastic metric systemsinduced by Σ_i_i∈ I and let Σ_i_i∈ I be the set of countable deterministic metric systems induced by Σ_i_i∈ I and _i as in (<ref>). If all local quantization parameters i,ε_i,i_i∈ I simultaneously fulfill i≤ε_i and 0≤i≤min{(i)^-1∘i(ε_i^2),i^-1[ (1-𝖾^-κ_iτ)i(ε_i^2) - 1/𝖾κ_iσ_u,i(ω_i) - 1/𝖾κ_iσ_d,i(ψ_i(τ) + ε̃_i) ] - (h_i(σ_i,τ))^1/2}, with ε̃_i_i∈ I defined as (<ref>), then the relation i ={ (x̂_i,x_i)∈[X_i]_η_i× X_i,𝗋 | 𝔼[i(x̂_i,x_i)]≤i(ε_i^2) } is a disturbance bisimulation relation in the second moment with parameters (ε_i,ε̃_i) between ii and i for all i∈ I.§.§ Composition of Approximations We have discussed in <ref> that the sets Σ_i_i∈ I and Σ_i_i∈ I of metric systems is compatible.We also established conditions on local quantization parameters under which the metric systems Σ_i and Σ_i are disturbance bisimilar for any i∈ I.We now use the fundamental property of disturbance bisimulation relation proved in <cit.> that disturbance bisimilarity is preserved under composition of components in a network.This property together with <ref> result in the following theorem that explicitly gives the disturbance bisimulation relation on the composed abstractions of components in a network. Given the preliminaries of <ref> and I'⊆ I, let Σ_i_i∈ I' and Σ_i_i∈ I' be systems with state spaces X_𝗋 and , composed from the sets Σ_i_i∈ I and Σ_i_i∈ I, respectively. Then the relation ={ (q̂^T_1…q̂^T_|I'|, q^T_1… q^T_|I'|)∈× X_𝗋 |(q̂_i,q_i)∈i, ∀ i∈ I' )} is a disturbance bisimulation relation between Σ_i_i∈ I' and Σ_i_i∈ I' with parameters ε= ∏_i∈ I'ε_i and ε̃=∏_j ∈_ℐ(I')ε_j. Note that in the special case I'=I the composed system replaces the overall network without extra external disturbances. In this case it is easy to see that the relation in <ref> simplifies to a usual bisimulation relation.Given the premises of <ref> and that I'=I, the relationin (<ref>) is an ε-approximate bisimulation relation between Σ_i_i∈ I and Σ_i_i∈ I.§ CONCLUSIONIn this paper, we extended our previous result on compositional abstraction based control for non-probabilistic control systems to stochastic control systems. We gave sufficient conditions s.t. a stochastic control system, admitting a δ-ISS-M_q Lyapunov function and subjected to small mismatch in the continuous and abstract disturbances, admits a disturbance bisimilar abstract system. Then we used the property of disturbance bisimulation to show that given a network of stochastic control systems, the abstract systems can be computed compositionally. One can then use this paper's claim for compositional synthesis of controllers for networks of stochastic control systems, as is done in <cit.> for network of deterministic systems.plain 10angeli D. Angeli. A Lyapunov approach to incremental stability properties. IEEE Transactions on Automatic Control, 47(3):410–21, 2002.jagtap2017automated P. Jagtap and M. Zamani. Automated synthesis of infinite dimensional stochastic hybrid systems. arXiv preprint arXiv:1704.03690, 2017.julius1 A. A. Julius and G. J. Pappas. Approximations of stochastic hybrid systems. 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Construction of approximations of stochastic control systems: A compositional approach. In Decision and Control (CDC), 2015 IEEE 54th Annual Conference on, pages 525–530. IEEE, 2015. §.§ Proof of <ref>eqn First observe that ⊂, hence the metric d onis also a metric on . Now we prove the three parts of <ref> separately. [(a)] * By definition ofin (<ref>), (x̂,x)∈ implies eqnsubsection.eqn d(x̂,x) = (𝔼[ x̂-x^2])^1/2≤(α^-1(𝔼[V(x̂,x)]))^1/2≤ε. We used the convexity assumption of α and the Jensen inequality <cit.> to show the inequalities in (<ref>). * Given a pair (x̂, x)∈, for any μ∈, observe that there exists a μ̂∈ s.t. u_μ̂-u_μ≤ holds. Given any ν̂∈ and ν∈ s.t. (w_ν̂,w_ν)≤ holds, observe that w_ν̂-w_ν≤(w_ν̂,w_ν)≤ from (<ref>). Now we can apply transitions δ_τ (x,μ,ν) = ξ_xμν(τ) = x', ξ_x̂μ̂ν̂(τ) = z, ξ_x̂μ̂ν̂(τ) = z, and observe that there exists a x̂' ∈ s.t. x̂'-z≤η, and hence we have δ_τηω(x̂,μ̂,ν̂) = x̂'. Now consider the following derivation: eqnsubsection.eqn 𝔼[V(x̂',x')]=𝔼[V(z,x')+V(x̂',x')-V(z,x')]=𝔼[V(z,x')]+𝔼[V(x̂',x')-V(z,x')]≤α(ε^2)𝖾^-κτ+1/𝖾κσ_u(ω) + 1/𝖾κσ_d(ψ_i(τ) + ε̃) +𝔼[γ( x̂'-z)]≤α(ε^2)𝖾^-κτ +1/𝖾κσ_u(ω) + 1/𝖾κσ_d(ψ_i(τ) + ε̃) +γ(𝔼[ x̂' -z + z -z])≤α(ε^2)𝖾^-κτ+ 1/𝖾κσ_u(ω) + 1/𝖾κσ_d(ψ_i(τ) + ε̃)+γ(𝔼[ z-z]+x̂'-z)≤α(ε^2)𝖾^-κτ+ 1/𝖾κσ_u(ω) + 1/𝖾κσ_d(ψ_i(τ) + ε̃) +γ((h(σ,τ))^1/2+η)≤α(ε^2). Hence by (<ref>), (x̂',x')∈. * Given a pair (x̂, x)∈, for any μ̂∈, observe that we can choose μ∈ s.t. μ = μ̂, i.e., u_μ̂-u_μ = 0. Given any ν∈ and ν̂∈ s.t. (w_ν̂,w_ν)≤, we have as before δ_τ (x,μ,ν) = ξ_xμν(τ) = x', ξ_x̂μ̂ν̂(τ) = z, ξ_x̂μ̂ν̂(τ) = z, and observe that there exists a x̂' ∈ s.t. x̂'-z≤η, and hence we have δ_τηω(x̂,μ̂,ν̂) = x̂'. With a very similar derivation as in (<ref>) it follows from (<ref>) that (x̂',x')∈. §.§ Proof of other statements The proof of Lemma <ref> can be obtained from the proof of 3.4 in <cit.> by replacing all instances of f(x,u), f(z,u), f(x',u) and f(x',u') with f(x,u,w), f(z,u,w) , f(x',u,w) and f(x'.u',w') respectively, and defining the positive constant κ = κ, thefunctions σ_u(r) = ( nL_u^2/κ)√(P)^2r^2 and σ_d(r) = ( nL_w^2/κ)√(P)^2r. The proof of Lemma <ref> follows closely the proof of 3.8 in <cit.> and hence is omitted. Pick any i∈ I, w_i∈ W_i and observe that w_i=∏_j ∈_ℐ(i)x_j. By the choice of j as X_jj we furthermore know that for any x_j there exists x̂_j x_j-x̂_j≤η_j≤ε_j. Now recall that_i=∏_j ∈_ℐ(i)j=∏_j ∈_ℐ(i)X_jj. Using the definition of ε̃_i in (<ref>) andin (<ref>) we therefore know that for any w_i∈ W_i there exists w̃_i∈_i s.t. (w_i,_i)=∏_j ∈_ℐ(i)x_j - x̂_j≤∏_j ∈_ℐ(i)ε_j=ε̃_i. Furthermore, w_i-_i = ∏_j∈_ℐ(i)x_j - x̂_j = ∏_j∈_ℐ(i)x_j - x̂_j = (w_i,_i). The proof follows from the following derivation: d(ν_i(t),ν̂_i(t)) =([ ν_i(t) - ν̂_i(t)^2])^1/2 =([ ν_i(t) - ν̂_i(0)^2 ])^1/2 =( [ ν_i(t) - ν_i(0) + ν_i(0) -ν̂_i(0)^2 ] )^1/2 ≤ ( [ ν_i(t) - ν_i(0)^2] )^1/2 + ( [ ν_i(0) -ν̂_i(0)^2 ] )^1/2 =( [ ∏_j∈_ℐ(i)ξ_j(t) - ξ_j(0)^2] )^1/2 + ε̃_i =( [ (sup_j∈_ℐ(i)ξ_j(t) - ξ_j(0))^2] )^1/2 + ε̃_i ≤ ( [ ∑_j∈_ℐ(i)ξ_j(t) - ξ_j(0)^2] )^1/2 + ε̃_i ≤ ( ∑_j∈_ℐ(i)[ξ_j(t) - ξ_j(0)_2^2] )^1/2 + ε̃_i ≤ (∑_j∈_ℐ(i)2t(1+𝔼ξ_j(0)_2^2)(t+1)e^α_j t)^1/2 + ε̃_i ≤ (∑_j∈_ℐ(i)2t(1+ sup_x_j∈ X_jx_j_2^2)(t+1)e^α_j t)^1/2 + ε̃_i where α_j = K_j+2√(K_j) and K_j is the constant K as given in <ref> for the j-th system, and the last step follows from <ref>. We define for system Σ_i thefunction ψ_i(t):= [t(t+1)∑_j∈_ℐ(i)β_j e^α_j t]^1/2, which concludes the proof. | http://arxiv.org/abs/1709.09546v1 | {
"authors": [
"Kaushik Mallik",
"Sadegh Esmaeil Zadeh Soudjani",
"Anne-Kathrin Schmuck",
"Rupak Majumdar"
],
"categories": [
"cs.SY"
],
"primary_category": "cs.SY",
"published": "20170927141908",
"title": "Compositional Construction of Finite State Abstractions for Stochastic Control Systems"
} |
firstpage–lastpage Near L-Edge Single and Multiple Photoionization of Singly Charged Iron Ions [ Accepted . Received; in original form===========================================================================We investigate the large scale matter distribution adopting QSOs as matter tracer. The quasar catalogue based on the SDSS DR7 is used. The void finding algorithm is presented and statistical properties of void sizes and shapes are determined. Number of large voids in the quasar distribution is greater than the number of the same size voids found in the random distribution.The largest voids with diameters exceeding 300 Mpc indicate an existence of comparable size areas of lower than the average matter density. No void-void space correlations have been detected, and no larger scale deviations from the uniform distribution are revealed. The average CMB temperature in the directions of the largest voids is lower than in the surrounding areas by 0.0046 ± 0.0028 mK. This figure is compared to the amplitude of the expected temperature depletion caused by the Integrated Sachs-Wolfe effect.Large-scale structure of universe – cosmic background radiation – quasars: general. § INTRODUCTION Statistical characteristics of matter distribution depend on a number of cosmological parameters. Albeit structures on various scales carry the information of cosmological relevance, matter agglomerations on the largest scales attract the greatest interest. This is simply because the large structures are rare, and partly because some `unusual' accumulations of matter could impose unique constraints on the precision cosmology ΛCDM model.A question of identifying structures is of statistical nature and has a long history. It was recently discussed by <cit.>. Here we examine statistics of large voids found in the SDSS DR7 quasar catalogue.Space distributions of individual matter components – luminous matter, diffuse baryonic matter and dark matter – are strongly correlated, but not identical <cit.>.Also individual types of galaxies do not follow one universal distribution pattern. It seems, however, that noticeable differences that show up at small scales, systematically disappear at large scales. In particular, it seems legitimate to assume that at scales of hundreds Mpc the distribution of baryonic matter follows that of the dark matter.Quasars are suitable to study the matter distribution at the largest scales for several reasons. Being the most luminous active galactic nuclei, samples of quasars cover usually huge volumes. Magnitude limited samples of quasars show lower than normal galaxies radial density gradients because of strong cosmic evolution. This allows to construct voluminous data sets with low observational selection bias <cit.>. Clustering properties of quasars and galaxies are not distinctly different at small and medium scales <cit.>, what assures us that at scales of several hundreds Mpc quasar distribution is representative for the luminous matter distribution. To be more specific, the relationship between the spatial distribution of galaxies and quasars is expected to be linear, i.e.amplitudes of the relative fluctuations of both components are equal.Numerous galaxy surveys reveal a variety of structures that span a very wide range of linear sizes. Early 3D maps show pronounced filaments and voids extending over 50 and more Mpc <cit.>. Still larger structures based on the SDSS have been reported; in particular, the Sloan Great Wall 420 Mpc long <cit.>, and the largest filamentary structure extending above 1 Gpc found in the DR7QSO catalogue <cit.>. However, statistical significance of the latter one was questioned by <cit.> on the grounds that group finding algorithm used by <cit.> was not sufficiently restrictive and structures formed by chance were recognized as physical quasar group.A question of distinction between `real' and `by chance' structures is crucial for statistical studies of the largest matter accumulations observed in the Universe. Long filaments may arise as a result of a coherent process that involves simultaneously adequately big amount of matter, or may be a product of chance alignment of separate `short' filaments <cit.>. Likewise, large volumes of low matter density may develop from a single large scale fluctuation, or be a close group of `normal' voids similar to those observed in the local Universe.In the present paper we address this last question. Distribution of quasars in the SDSS DR7 catalogue is investigated in respect of the number and size of empty regions (voids). Then, shapes of voids and void correlation is examined.All distances and linear dimensions are expressed in co-moving coordinates. To convert redshifts to the co-moving distances, we use the flat cosmological model with H_ o = 70 km s^-1Mpc^-1, Ω_ m = 0.30 and Ω_Λ = 0.70. We focus our study on the area between 3000 and 4500 Mpc, what corresponds approx. to the redshift range 0.8 - 1.6.The present investigation seems to belong to a broad field of galaxy distribution studies that adopt a void concept. However, our work is rather weakly related to this area. This is because of several reason. The most conspicuous are the void scale and definition. Here we examine voids with radii above 145 Mpc that are completely empty.i.e. with no objects inside.Such zero-one approach is usually dropped in the galaxy studies. In our geometrical attitude to void definition we ignore kinematic effects of deviations from the Hubble flow. The redshift – distance relationship is determined by the cosmological model. Also, questions on dynamics of the large scale inhomogeneitiesof thematter distribution cannot be addressed here at the present stage.Concentration of quasars is several orders of magnitude lower than the galaxy space density. Consequently, the area nominally covered by the DR7 QSO catalogue is sparsely populated with the average distance between neighbouring quasars of r_ n∼ 125 Mpc. Statistical relationship between the galaxy and quasar space distribution holds for scales considerably larger than r_ n, and onlyvoids covering several r_ n indicate areas of the actually low galaxy density and the low total matter density[One should note that the number of large voids depends strongly on the local average density of points (see Fig. <ref> in the Appendix <ref>). Obviously, this effect depends on the cosmic fluctuations of quasar concentration, as well as on depth and homogeneity of the survey.]. Because of that, our analysis of quasar voids properties is limited to the largest voids.An advancedmethod to investigate topology of continuous fields in cosmology, namely a watershed void finder (WVF), has been developed in recent years <cit.>. WVF is a particularly effective tool to study evolution of complex, hierarchical structures both in the real data and simulations <cit.>. The observational material is practically always represented by the discrete samples, and transformation of the discrete distribution into the continuous one constitutes the inherent element of the WVF method <cit.>. Here we will examine scales only a few times larger than the average quasar separations (see below).Voids in the present situation are defined by just a few objects, what does not allow for a credible construction of the continuous density field.N-body simulations show that in scales of few dozens MPC void properties evolve with time <cit.>.Because the present investigation concentrates on much larger structures than typical voids observed in galactic catalogues, and we concentrate on a single large volume contained within a relatively narrow range of redshifts (0.8 ≲ z ≲ 1.6), no cosmic evolution effects are considered in the paper.Voids detected in galaxy surveys span a wide range of sizes. For instance, <cit.> identify voids in the SDSS DR7 galaxy catalogue with radii in the range 6 - 24 h^-1 Mpc.Only relatively small and moderate size voids are almost completely devoid of galaxies <cit.>.Linear sizes of the present voids exceed 300 Mpc, and are comparable to the largest voids detected in the galaxy distribution <cit.>. Such large volumes are called voids because of distinctly lower than average number of galaxies.However, the amplitude of density fluctuation within large voids is not well determined <cit.>. Most information on this question is obtained from N-body large scale simulations <cit.>. Also the present investigation cannot provide direct information on the distribution of galaxies in the areas coinciding with the quasar voids. One of the objectives of our paper is to measure the amplitude of the Integrated Sachs-Wolfe signal generated by the largest voids. Potential correlation between the cosmic microwave background (CMB) temperature variationswith the quasar voids would confirm the physical nature of voids, and – in the future – could be used to measure amplitude of the large scale density fluctuations.The paper is organized as follows. In the next section we present geometric construction that is used to define voids. In Sec. <ref>a short description of the quasar sample used in the investigation is given. Statistical characteristics of voids found in the data are presented in Sec. <ref>. Number of voids, their sizes and shapes are parametrized by a sphere radius used in the void finding algorithm. Correlation of the sky position of the largest voids with the CMB temperature local minima is discussed in Sec. <ref>.Some peculiarities in void shapes are discussed in Sec. <ref>. Main results are summarized in Sec. <ref>. Statistics of voids in a random point distribution and details of the computer algorithm applied to findvoids are given in the Appendices.§ THE VOID – DEFINITION Large and roughly spherical areas devoid of galaxies together with galaxy clusters, walls and filaments constitute a complex structures of the galaxy distribution known as a cosmic web. Thus, the notion of cosmic voids was developed with the advent of the extensive galaxy surveys.The galaxy voids are easily identified visually in 3D surveys, but are also revealed in 2D massive counts such as the Lick galaxy counts.Space concentration of quasars is much lower than that of galaxies. Thus, the apparent void in the quasar distribution does not imply presence of the comparable size galaxy void. Nonetheless, assuming no bias in the large scale distribution of quasars and galaxies, large quasar voids found in the present study indicate areas of the lower concentration of galaxies.To analyze quantitatively even the basic void parameters, such as size and shape, one should replace the vague concept of void as a rounded empty volume by more rigorous void definition. Consequently, cosmic voids have been defined in diverse ways in the past. It has been established that most galaxy voids are not totally empty, and this observations were incorporated in various void investigations. In the present study the void is defined as a region completely devoid of objects.Figs. <ref> (a) and (b), which are taken from <cit.>, illustrate in 2D a relationship between: (a) –the common realization of 'void' notion, and (b) – the geometric construction representing the void.Let us consider the distribution of n objects in the selected volume V. This distribution contains a void of size R if one can insert into V a circle (sphere in 3D) of radius R with no objects inside.Shaded areas in Fig. <ref> (b) indicate the underlying constructions of n(R) = 6voids of size R.It is convenient to call each of these areas a 'void centre'.Rigorously, the shaded areas are the geometric places of all the points which are centres for empty circles (spheres in 3D) of radius R.One should note that shape of the void centre contains the information on the shape of the entire void. Both number of void centres and their shapes depend on the radius R, and –obviously – on the statistical characteristics of the distribution of points. <cit.> gives the formula for the expected number of voids (void centres), n(R), as a function of number of objects, N, volume, V, and R for the Poissonian distribution of points in 2D and 3D (see also Appendix <ref>).§ THE DATA The Sloan Digital Sky Survey Quasar Catalogueis used to study potential inhomogeneities in the matter distribution on very large scales, i.e. above ∼ 300 Mpc. The fifth edition of this catalogue is presented in <cit.>.The multi step procedure to identify quasars, collect photometry and spectroscopic redshifts is described in that paper.Also the references to papers presenting all the successive releases of the catalogue are given. Total number of quasars in the catalogue exceeds 105 000, of which more than 90 000 lie in the northern galactic hemisphere. The redshifts range from 0.065 to 5.46, and 80 per cent of them are between 0.55 and 2.8 (2050 Mpc and 6150 Mpc).Generally, the catalogue is magnitude limited what determines the overall distance distribution. Nevertheless, the distribution of objects in the redshift magnitude plane is complex what reflects multiple criteria applied to construct the final list of objects. In particular, quasar identification and redshift measurement depend on the position of emission lines in the SDSS photometric system and spectral band pass. Potentially this effect could introduce spurious variations of the redshift distribution for the catalogued objects. However the authors stated that this is not an issue for quasars in this catalogue. In the paper we investigate the quasar space distribution in the redshift range ∼0.8 ÷∼1.6. In this area the resultant selection (see Fig. 4 of ) apparently generates flat space density distribution of the catalogued objects.Void analysis, as other statistics dedicated to the space investigation, is hindered by the interference of the local effects with the cosmic data. Although large sections of the catalogue display a high degree of homogeneity, the surface distribution of objects in several regions of the celestial sphere is nonuniform and exhibits patches of distinctly different concentration of quasars. The observational material has been gathered for several years. Its homogeneity inevitably suffers from instrument-related biases as well as from the specifics of the data processing. In particular, selected areas are subject to different magnitude limits. In effect, most of the distinct features clearly visible in the surface distribution are not related to the cosmic signal. Our analysis is more sensitive to various selection effects (in most cases - unrecognized) that vary across the sky rather than to the radial bias that acts uniformly over the whole investigated area. Hence, in the present investigation we introduce an additional magnitude limit of 19.5 in the z band. Albeit, this z cut-off decreases the number of quasars in the northern hemisphere from above 90000 to 72067, it effectively reduces conspicuous surface structures, apparently of the local origin.The z band, with its average wavelength of 893 nm, is used to select a possibly homogeneous sample because it is least affected by interstellar extinction. Fig. <ref> shows the distribution of quasars in the north galactic hemisphere between 2750 and 4750 Mpc selected from the original SDSS catalogue brighter than z = 19.50. Despite its featureless appearance, the subsequent analysis will show that the space distribution of objects is not random.In Fig. <ref> the space density of the selected quasar sample is shown as a function of distance. Overall decline of the density with increasing distance results from an apparent magnitude selection.A wide plateau between 3000 and 4500 Mpc comes froma kind of interplay between the observational selection and quasar cosmic evolution. This flat density distribution is helpful for studies of large structures, including voids.In the paper we concentrate on this section of the data.Apparent depression in the distance range 4000-4300Mpc (centered at redshift of 1.4) is discussed below.§ VOIDS IN THE SDSS QUASAR DISTRIBUTIONThe void centres in the northern galactic hemisphere were searched for at distances between 3000 and 4500 Mpc. All the calculations were performed with the resolution of 1 Mpc in 3D. Te radial coordinates were determined from redshifts assuming the strict Hubble flow in the ΛCDM model. The computational details of the void finding algorithm are described in the Appendix <ref>. The radius of the largest void in the investigated area R = 193 Mpc.Fig. <ref> shows the distributions of void centres projected on the sky for the selected void radii between 180 and 145 Mpc. The number of void centres, n(R), grows as the radius decreases.Due to centre mergers the relationship is non-monotonic.However, within the radius range covered by the present computations, a rate at which `new' centres emerge with decreasing radius,exceeds the rate of mergers. The effect of void mergers and the percolation phenomenon is illustrated in Appendix <ref> where we show the analytic n(R) relationship over a wide range of radii for the random distribution of points. §.§ Space distribution of voidsThe void centres occupy a small fraction of the investigated volume, V = 3.58× 10^10Mpc^3, over the entire range of void radii presented in Fig. <ref>. At the radius R = 145 Mpc, the volume of all the void centres amounts to 1.96× 10^-3 of the V. The present radii are still distinctly greater than the percolation radius inthe random (Poissonian) distribution of approximately 110 Mpc. Thus, all the selected voids are situated at the 'large void' tail of the void distribution. In thisradius range, relatively small fluctuations of the space density of objects result in a strong variations of the void concentration (see Appendix <ref>). Consequently, possible asymmetry of the void distribution on both sides of the galactic longitude line of 70^∘ - 250^∘ (see bottom-right panel of Fig. <ref>), if real, may be a result of relatively small inhomogeneities of the survey rather than the true variations of the matter density on a Gpc scale.Some insight into this question is provided by exploring the void distribution along the line of sight. The area between 3000 and 4500 Mpc is divided into approximately equal volumes. The near field extends between 3000 and 3900 Mpc, end the far one stretches from 3900 to 4500 Mpc.The distribution of void centres found for the radius of 145 Mpc in both regions is shown in Fig. <ref>. The number of void centres in the more distant section is larger then in the near one by more then 26 percent. The difference of void concentration in both fields has been expected because the quasar density in the more distant field shows a wide minimum centered at the distance of ∼4100 Mpc, or the redshift z ≈ 1.4 (Fig. <ref>).Most likely this minimum is caused by the varying effectiveness of the line identification in quasar spectra <cit.>, and is unrelated to the intrinsic quasar distribution. It appears visually that void centre angular distributions in the celestial sphere in two distance bins exhibit overall similarities, what would indicate some inhomogeneities in the SDSS catalogue.However, it should be stressed that visually isolated differences of the distribution of void complexes, are not necessarily significant in the statistical terms.The whole population of void centres found at R = 145 Mpc (Fig. <ref>, bottom right) does not show clear deviations from uniformity - the void centres are apparently scattered randomly.To investigate this question in a quantitative way, the observed distribution of separations between the void centres is analyzed. We compare the distribution of centre pairs in the real and random data sets.According to the present definition, the void centre is a 3D object, often of a complex shape. Separation of two void centres is defined as a distance between their centres of mass.In the computations, the volume of the void centre is represented by a set of cubic cells 1 Mpc a side (see Appendix <ref>).Due to computer memory constraints, the void centre is identified and localized in space just by the cells distributed on the `centre surface'. Only these cells are kept for further analysis. Because of that, the position of the void centre is described by the centre of mass of the surface cells.Two schemes to generate mock catalogues were applied. In the first one, angular positions of all the centres in the simulated data are the same as for the real ones, and randomized are only distances to the voids. We use the bootstrap method, and draw the random distances from the true distribution. In this case, the randomized void centres preserve some statistical characteristics of the real data.In the second method, a population of void centres found in strictly randomly distributed points is used.We examine the distribution of centre pair separations for voids selected at radius R=145 Mpc. Let p_ d(S) and p_ r(S) denote the numbers of centre pairs with a separation S normalized to the total number of pairs for the real and randomized void samples, respectively.A ratio of both quantities shifted to 0 for the uncorrelated data represents a auto-correlation function (ACF) of void centres: ξ(S)= p_ d(S)/p_ r(S) - 1 . ∋For both randomization schemes no correlation signal was detected.∋Fig. <ref> shows results for the perfectly random data – within statistical fluctuations a ratio p_ d(S) / p_ r(S) is equal to 1. The pair separations in the random distribution are the average of 30 data sets generated using the Monte Carlo scheme. The error bars represent the rms scatter between the simulations.No significant deviations of p_ d(S) from the random case over a wide range of separations S indicates that voids are not arranged into larger structures.Consequently, the distribution of voids provides no evidences for the under-dense areas significantly larger than the individual void. §.§ Number of voids Featureless surface distribution of quasars in the SDSS catalogue (Fig. <ref>) and close to uniform radial distribution (Fig. <ref>), indicate that any potential deviations from the the homogeneous space distribution of quasars are small at large scales. Therefore, the main objective of this analysis is to assess to what extent the true quasar distribution in fact differs from the random one. In the present analysis, the void statistics is used to determine the maximum linear scale at which the data exhibit non-random characteristics. To facilitate interpretation of the statistical void parameters, mock catalogues of uniformlydistributed 'quasars' have been constructed. Objects in the mock catalogues are randomly distributed in 3Dusing pseudo random number generator within a volume covered by the SDSS. Space density is equal to the average density in the true catalogue between 2750 and 4750 Mpc to allow for voids extending beyond the reference distance range of 3000 - 4500 Mpc.A number of 30 data sets were created to gain understanding on statistical scatter of the measured parameters.The number of separate void centres in the true and simulated data as a function of radius is shown in Fig. <ref>. In order to account for the finite survey volume, the void centres are weighted according to their position relative to the area borders.The unit weight is given to all the centres that are completely contained in the analyzed volume. If the centre touches the boundary of the area at one face, the weight is reduce to 1/2, if it touches two faces its weight is assumed to be equal to 1/4, and for three faces - 1/8. Thus, the total number of voids generally is not an integer. The error bars represent the rms dispersion determined for 30 randomized catalogues.Over the whole range of void radii the real quasar data accommodate larger number of void centres, n_ d(R), than the random simulated distributions, n_ r(R). Although, the relativedifference, [n_ d(R) - n_ r(R)]/n_ r(R), fluctuates due to the stochastic nature of both distribution, the discrepancy between both quantities seems to decline steadily at the smaller radii. In the absolute numbers, the excess of true voids increases with diminishing radius, and stabilizes or begins to decrease below r ≈ 150 Mpc.The true void excess demonstrates that the void algorithm is an effective tool to investigate the large scale variations of quasar concentration and that the catalogued quasars are not distributed randomly in space. Apparently, the density of quasars is not perfectly constant in the investigated volume. Because of that, the number of large voids is greater than the number expected for the randomdistribution of objects populating the same volume with the density equal to the average density of quasars (see formula <ref> for the number of voids in the Poissonian distribution).It is likely that to some extent residual imperfections of the data selection procedures introduce some bias in the catalogue which is responsible for the the n_ d(R) - n_ r(R) difference. To assess, how strongly the cosmic signal contributes to this difference, other statistical properties of voids are investigated in the following sections.§ VOIDS VS. CMBFluctuations of matter distribution on very large scales influence the cosmic microwave background (CMB) what is known as the Integrated Sachs-Wolfe effect (ISW). CMB photons crossing mass agglomerations, as well as broad depressions of the mass density. gain or lose energy.Amplitude of the effect is defined by the net change of height of the potential hill (or depth of the well) during the photon travel time. In the matter dominated flat universe the time evolution of density fluctuations in the linear approximation is balancedby the matter dilution caused by the Hubble expansion and the effect is cancelled. In models with the cosmological constant the expansion of the universe is not matched by the evolution rate of density fluctuations and the ISW effect turns up. From the observation point of view the ISW effect is still debatable. Although, the large structures (both superclusters and supervoids) found in extensive galaxy surveys, such as SDSS DR6 <cit.>, seem to correlate with the CMB temperature variations, the reported amplitude of the signal (e.g.) is substantially larger than that expected for the ISW effect (e.g. ).Here, we explore the potential relationship between the voids found in the quasar catalogue and the CMB temperature fluctuations under the assumption that the correlation of both distributions is generated via the ISW effect. However, to confirm the existence of the under-dense regions associated with our voids, the nature of this correlation is not crucial.Obviously, if the observed void excess results from the catalogue deficiencies generated locally, one should expect no correlations of the detected voids with the CMB temperature. Moreover, the void excess can be generated by various kinds of fluctuations of the matter density that also do not introduce void - CMB correlation. For example, if the void excess is produced by variations of the density field on scales much larger than the present voids, the number of detected voids will exceed that for the Poissonian field, but a topology of individual voids will be defined just by geometrical structures unrelated to the local matter density. Only the physical connection of some voids with the true matter density depressions generates the sought correlation.Figure <ref> shows that the number of voids detected in the quasar distribution is greater than that expected in the perfect Poissonian case, but the relative difference of both quantities is rather modest. Thus, the majority of voids is not associated with the areas of lower than average matter density. Nevertheless, the systematic excess of the void number over a wide range of radii could indicate that some fraction of large voids is genetically related to the under-dense regions. We investigate in detail a possible coincidence of `cold spots' in the CMB temperature map with the population of voids found in the present investigation. We note the obvious fact that the number of voids selected at two void radii are correlated.This is illustrated in Fig. <ref>, where all the voids selected at one radius show up also in the voidmaps constructed at all smaller radii. To obtain independent sets of void centres for different radii, we apply the following procedure. First, a void of the largest radius is found.It is centred at galactic coordinates (l, b) = (216^∘, 89^∘) at the distance of 4139 Mpc. Then the centre of the next in size void is searched for, excluding the volume occupied by the first one.The minimum void centre separation equal to the radius of the larger void is introduced to eliminate the situation where several void centres are clustered in the small area. Such tight void group in fact represents potentially a single under-dense region. The second void with radius of 192 Mpc is centred at (l, b) = (309^∘, 64^∘) at the distance of 4234 Mpc. The procedure was repeated for all the voids down to r =145 Mpc, the smallest voids investigated here, and provided a list of 568 void centres.A sky distribution of a sample of 18 largest voids selected in this way in the radius range 193 - 176 Mpc is shown in Fig. <ref>.The number of the observed voids with smaller radii approaches that predicted for the Poissonian case. These voids result from the purely random quasar structures and do not produce the ISW effect. Thus, the expected ISW signal averaged over the whole void population is highly diluted.Moreover, the fluctuations of the CMB temperature in the WMAP data at all the interesting scales are much larger than the expected ISW amplitude created by voids, what strongly reduces the signal-to-noise (S/N) ratio of the present ISW signal detection.The CMB temperatures at direction of voids are calculated using the 9 year WMAP CMB maps available at LAMBDA[] which isa part of NASA's High Energy Astrophysics Science Archive Research Center. We usedWMAP's standard Res 9 HEALPix projection with ∼ 7 arcmin resolution.For each void the CMB temperature was determined. The WMAP data were averaged over a circular sky area positioned at the void centre. One can expect that the highest amplitude of the ISW signal coincides with with the void centre and decays outside.To maximize the S/N ratio, the radius of the area was delineated as θ_ CMB = ϰ θ_ void, with ϰ = 0.9, where θ_ void is the angular radius of a void (different values of ϰ give either the weaker signal or the stronger noise).The amplitude of the ISW signal was defined as the difference between the void temperature and the average CMB temperature in the surrounding area.The CMB map was smoothed using a spherical harmonic filter of degree l = 9.The low value of the degree l was chosen to produce sufficiently flat temperature map around each void. The reference temperature to calculate the ISW signal was assumed as the average temperature of the filtered CMB data in the annulus surrounding the void. The inner and outer annulus radii of θ_ CMB and 3 θ_ CMB were taken.The Δ T_ CMB plotted in Fig. <ref> show the differences between the void and reference temperatures in the set of 568 voids defined above. The data are divided into 5 bins according to the void radius:145 ≤ R < 148 Mpc, 148 ≤ R < 153 Mpc, 153 ≤ R < 159 Mpc, 159≤ R < 165 Mpc, and R ≥ 165 Mpc.Thebins contain respectively: 125, 155, 136, 69, and 83 voids. The cross shows the temperature difference for the merged two bins with largest radii (R ≥ 159 Mpc). The 1 σ uncertainties are estimated using the simulations. The void and reference temperatures were obtained for a set of 30 randomized catalogues. The data have been processed in the same way as the original material, and the rms scatter of the corresponding Δ T_ CMB distributions in the mock catalogues is shown as the error bars.Details of the radial distribution and radii of voids with R ≥ 165 Mpc are displayed in Fig. <ref>.The distribution of temperatures in large voids is not symmetric with respect to the Δ T = 0 line.The average negative Δ T_ CMB signal for voids with R ≥ 159 differs from zero by more than 1.64 σ, and in the highest bin of R ≥ 165 Mpc the net temperature in 50 voids is negative.Using the binomial distribution and assuming equal probabilities of temperatures below and above the average, a chance that at least 50 of 83 temperatures drawn at random will be negative amounts to 0.039. In statistical terms, significance of this asymmetry is not very high.Nevertheless, the data are consistent with the correlation between the void distribution and the CMB temperature depressions, and it is legitimate to assume that some voids indeed coincide with the areas of lower than average matter density.Unfortunately, large intrinsic scatter of the CMB temperature maps strongly impedes assessing the amplitude of the ISW signal.This in turn, prevents us from imposing restrictive constraints on the density distribution in voids. Consequently, our objective in the subsequent calculations, is to examine to what extent the present estimates of Δ T_ CMB are consistent with the existing data on the large scale matter distribution derived from N-body simulations.The relationship between the cosmic structures and the CMB temperature variations generated by the ISW effect has been broadly discussed in the past.In particular, a measurable correlation of the void distribution with depressions of the CMB temperature is expected in models with non-vanishing cosmological constant <cit.>, although the predicted ISW signal is typically much weaker than those reported in the literature, <cit.>.Amplitude of Δ T depletion produced by the void depends on the time evolution of the gravitational potential, Φ(r), along the CMB photon path.The distribution of Φ(r) in the vicinity of voids was investigated by <cit.> using the N-body simulations. They found that for large voids, Φ(r) scales in a simple way with the void radius and the average galaxy density contrast within a void, δ_g =ρ_ gv/ρ_ gl-1, where ρ_ gl and ρ_ gv are the global and void galaxy number densities, respectively. The approximate galaxy bias factor in voids is estimated at 2.In the following we assume that these scaling relationships apply to the present voids, albeit the <cit.> investigation concentrates on lower redshifts and smaller void sizes than those considered in the present paper.Thus, the subsequent calculations have only indicative character.To assess capabilities to measure the large scale inhomogeneities of the matter distribution using the present void algorithm we apply the linear formula for the ISW temperature change <cit.>: Δ T/T =-2 ∫_0^z_ LS a(z) [1 - f(z)] Φ dz , ∋where z_ LS is the redshift of the last scattering, a(z) –the cosmic scale factor, andf = d ln D/d ln a , ∋is the density fluctuation growth rate, where D(a) is the linear evolution of the matter density perturbation.One can expect that the highestΔ T_ CMB amplitudes are generated preferentially by the largest voids. In the present investigation, however, the estimates of the average temperature signal are strongly affected by sampling errors. Because of that we include in the calculations all the voids with R ≥ 159 Mpc.The average void radius in a sample of all the voids with R ≥ 159 Mpc is R = 167.7 Mpc, and the average void distance d = 3820 Mpc (z≈ 1.2). Using <cit.> Eqs (6) and (10) with scaling coefficients in their Table B2, we get the distribution of potential, Φ(r), associated with the our 'average' void for different amplitudes of the galaxy contrast δ_g. Then, the integral in Eq. <ref> is calculated in the ΛCDM cosmological model with parameters specified in the Introduction. The formulae for the linear growth rate, D(a) were taken from <cit.>.To compare the model with our Δ T_ CMB estimates, we note that only a fraction of the voids is actually associated with the low density areas, while majority of voids results from random quasar configurations. Therefore, we define the average temperature signal produced by a true fluctuation of the matter distribution as: Δ T_ u = η Δ T_ v· n_ v/n_ v - n_ rv , ∋where Δ T_ v and n_ v are the average temperature deficit in a void and the number of voids in the sample, and n_ rv denotes the number of voids expected for the random distribution. The coefficient η takes into account partial overlapping of voids, and is equal to the ratio of the total solid angle covered by voids to the summed up area of all the voids. For R ≥ 159and ϰ = 0.9 we have η = 0.753, Δ T_ v = -0.00465 ± 0,0028, and n_ v = 152. The number of voids expected in the random distribution n_ rv = 126.9 ± 10.2 is derived from 30 mock catalogues. Thus, the temperature depletion associated with the under-dense regions Δ T_ u = -0.022 ± 0.016 mK. According to Eq. <ref> the temperature depletion at the level of -0.022 mK is produced by a void with the average density contrast of ∼ -0.62. Although, the data on the galaxy distribution at redshifts under consideration are scarce, such high density contrasts seem unlikely. Also, the galaxy bias factor generally grows with redshift (e.g. ) and the corresponding galaxy concentration contrast would be even higher. Assuming large (as compared to the <cit.> assessments) matter density contrast of -0.3, the temperature depletion according to Eqs <ref> and <ref> from our `average' void Δ T_ u = -0.006 mK, andΔ T_ v = 0.0013 mK. In view of the large CMB temperature variance, the `dilution' of true underdense regions among the random quasar voidsdrastically reduces ability of the present method to investigate the ISW effect from voids.We stress, that all these estimates are based on the extrapolation of the <cit.> scaling relations for the relevant void parameters. Their distribution of void radii peaks at ∼ 40 Mpc with no voidsof R above 90 Mpc. <cit.> assess that in ΛCDM models ∼ 100 Mpc structures at z ≈ 0.5 generate the ISW signal of ≲ 0.002 mK. The hypothetical structures reported here exceed 300 Mpc, and proportional higher temperature effect is expected. Such structures are, however, extremely rare. In the volume of ∼ 3.6· 10^10 Mpc, the number of under-dense regions associated with voids is estimated at 27± 10. Thus, there is less than one such object in a cube of side 1100 Mpc.It underlines the potential role of large void searches in the quasar distributions for the investigation of matter density fluctuations at scales much larger than accessible in the galaxy catalogues.§ VOID SHAPESA shape of the void centre is defined by cells distributed in the centre surface, as described in the Sec. <ref>.Obviously, shapes evolve with the void radius R. One can expect that statistical investigation of void centre structures should give also some insight into the typical properties of voids understood as volumes of space free of quasars. A moment of inertia is a natural tool to study the basic geometric properties of 3D structures. For that purpose, the void centre is considered as a rigid body.Since the void centre is here represented only by cells in the outer layer of the spatially extended centre structure, the moment of inertia pertains just the `mass distribution' of the surface of the centre.Principal moments of inertia and principal axes (eigenvalues and eigenvectors of the moment of inertia tensor) of void centres were calculated for the real and randomized data. Gross features of the centres, such as sizes, ratios of the principal moments, and directions of principal axes in both data sets were investigated. No conspicuous distinctions between the sets have been found, albeit some marginally significant differences in the centres shapes are present.The distribution of quasars is a Poissonian stochastic process, and voids in the real data not necessarily reflect statistically significant large scale depression of the matter distribution. One can expect that preferably only the largest voids that are found occasionally in the real catalogue indicate areas of lower matter density. Among voids selected at R=145 Mpc, 17 have `surface' (as defined above) containing more that 1.5· 10^5 cubic 1 Mpc cells.In the 30 sets of the randomized distributions, only 242 such large voids have been found. Fig. <ref> shows the cumulative distribution functions, CDF, of the principal moments of inertia ratios for the real data (17 step functions) and the randomized data. A two sample K-S test applied to the distributions of the principal moments of inertia, I_ max, I_ med, I_ min, reveals possible differences of void centre shapes in the data and in the simulations. In Fig <ref> the cumulative distribution functions (CDF) of I_ min/I_ max, I_ min/I_ med, and I_ med/I_ max for the true and random voids are shown. The distributions differ at a significance level of 0.039, 0.032 and 0.006, respectively. Relationships between three principal moments are also visualized in Fig. <ref>, where the distribution of I_ min/I_ max vs. I_ min/I_ med is shown for the void centres in the quasar data and in the mock catalogues. To compare both distributions we apply two-dimensional version of the K-S test developed by <cit.>. The test indicates that real and random data differ at significance level of 0.0073. The signal in the Peacock's test is produced by the apparent excess of void centres in the top-left quadrant of Fig. <ref>. Judging fromthis feature, it implies that the shapes of centres are generally less elongated (or more spherical) as compared to the random case. One should keep in mind, however, that the actual shapes of void centres of the large voids are highly different from the `regular' shapes of triaxial ellipsoids.§ CONCLUDING REMARKSWe study the space distribution of quasars in the SDSS to assess the matter distribution on very large scales. A suitable quasar void finding algorithm allowed for the investigation of void sizes and shapes. An existence of structures involving groups of voids were also examined. The SDSS quasar catalogue, taken as a whole, is subject to various selection biases. A section of the catalogue that covers the redshift range of roughly 0.8 - 1.6 was used in the paper. Although, this area seems the best fitted for this kind of investigation, the data also suffered from imperfections, what limited the present study. Potential inhomogeneities of the SDSS could affected overall space distribution of voids.Because of that we concentrate on void characteristics that are fairly immune to the survey deficiencies.It is shown that the distribution of void sizes is inconsistent with the random distribution of quasars. The excess of the number of voids is observed fordiameters above ∼300 Mpc.We examine the largest voids, since they most likely coincide with the underlying large scale low matter density (baryonic and dark) areas. To relate the present voids with the true depressions of the matter density, we investigate the angular correlation between the voids and the CMB temperature distribution. Such correlation is expected due to the ISW effect. It is found that the average temperature in the direction of large voids is lower then in the surrounding areas by a few μK in rough agreement with the ISW mechanism in the ΛCDM model. However, the statistical significance of the detection is too low to perform a quantitative analysis of the ISW effect. Excess of voids with the negative temperature deviation over those with the positive one among 152 voids with R ≥ 159 Mpc is significant at 0.039 level. We conclude that the low significance of both statistics results from the fact that extraneous variations of the CMB temperature are much larger that the measured amplitude of the ISW signal.The space autocorrelation function of void centres is determined.The ACF amplitude is consistent with no correlation signal over a whole accessible range of void separations. However, this conclusion is not highly restrictive due to large uncertainties of the ACF estimate. To improve the voids statistics, data covering wider redshift range would be required.In statistical terms, the shapes of the quasar void centres define space structures of under-dense areas. Thus, investigation of quasar voids could provide valuable information on the large scale matter distribution. The observed void centre shapes and those found in the random distributions are statistically different, although the differences are not high. Shapes here are defined solely by the ratios of principal moments of inertia I_ min / I_ max and I_ med / I_ max.Assuming that distributions of these parameters describe a population of triaxial ellipsoids, real objects tend to be more spherical as compared to the simulated ones.However, the true shapes of the largest void centres in Fig. <ref> (bottom right panel) strongly differ from ellipsoids, and conclusions based on such approximation should be treated cautiously. We plan to extend the investigation of voids using quasars in other redshift ranges. Broader observational basis should help to clarify also this point of the present paper. § ACKNOWLEDGEMENTSI thank the anonymous reviewer for valuable recommendations that greatly helped me to improve the material content of the paper.§ VOIDS IN POISSONIAN DISTRIBUTIONIn Appendices we use terms `void' and `void centre' as defined in Sec. <ref>.Although the space distribution of quasars listed in the SDSS catalogue is different from the random (Poissonian) one, it is instructive to compare the void centres characteristics of the real data with this idealized case. Additionally, analytic formula for the number of voids in the Poissonian case derived in <cit.> allows us to check quality of the void finder computer algorithm.A number of random mock catalogues in the SDSS area were generated using the Monte Carlo scheme. Then, the void centres were found applying the same method as for the real data (see below).Results averaged over 30 mock catalogues are shown in Fig. <ref> with dots. The error bars represent the rms dispersion of 30 data sets around the average number of centres n(R), divided by the square root of the number of sets.The volume searched for void centres, V, and the space density of points, λ were chosen to match corresponding quantities in the the real catalogue. In our case V = 3.58·10^10 Mpc^3, and λ = 5.15·10^-7 Mpc^-3.The total number of points within V, N=Vλ≈ 18400. To account for the edge effects the mock catalogues were generated over the larger area stretching out 250 Mpc around V.The solid curve gives the expected number of centres according to the analytic formula <cit.>: n(R) = V λ e^-4/3 R^3 λ (^4 R^6 λ^2/6 - 4R^3 λ + 1) , ∋for parameters V and λ given above. Equation <ref> was derived under the assumption that for given R, the void centres are not considerably larger than the sphere of radius R. Thus, the formula breaks down at radii just a couple percent greater than the percolation radius, R_ p, i.e. the radius at which the individual void centres merge and create an infinite web. In the present case R_ p≈ 110 Mpc.Even small variations of the local average matter density have a substantial impact on space concentration of voids in the relevant range of radii. To illustrate a steep dependence of the void probability on the quasar density in the limit of large voids, we use the Eq. <ref>.The dotted curve in Fig. <ref> shows the expected number of voids in the same volume for 10 percent lower density, i.e. for λ = 4.64·10^-7 Mpc^-3. At radius R ≈ 171 Mpc the number of voids as compared to the original λ is higher by a factor of 2 and the ratio increases at greater R.§ VOIDS – NUMERICAL ALGORITHMSpatial resolution of all the computations is set to 1 Mpc. To search the whole volume of ∼3.6· 10^10 Mpc for the void centres, a several step procedure is applied. First, the volume is divided into ∼8.7· 10^6 cubic `domains' of a = 16 Mpc a side. For given void radius R, a distance d between the domain centre and each quasar is examined. If d < R - a√(3) / 2, the domain is eliminated from further computations.Number of accepted domains, n_ ad(R), rises rapidly with the decreasing R.So, for R equal to 193, 190, 155, and 145 Mpc the corresponding numbers of domains are: 1, 35, 13544, and 45720. Then, space distribution of accepted domains is inspected, and domains are segregated into clusters, where a cluster is defined asa group of mutually contiguous domains. Each cluster hosts one or more void centres.In the next step, each accepted domain is divided into 16^3 1 Mpc `cells'.For the each cell the distance from the cell centre to the nearest quasar is determined. If it is smaller than R, the cell is removed.Obviously, the total number of saved cells raises with the decreasing R proportionally to n_ ad(R), and quickly becomes unmanageable. To facilitate computations over the entire interesting range of R, only the cells distributed on the surface of the void centre are saved for further processing.A removal of the `interior' cells allows for the effective analysis of void structures, although, the numbers of kept cells are still quite big. The largest void found at radius R = 145 Mpc contains more that 400 000 `surface cells'. Space arrangement of cells determines the distribution and shape of void centres.Cells are split into clumps of `neighbours'. An isolated clump of cells defines a single void centre.Two cells are assumed to be neighbours if their separation in each coordinate does not exceed 2 Mpc.This particular separation was chosen to balance a discrete character of cell selection, despite the fact that the void centre is a continuous body.∋Very good agreement between the number of voids detected in simulations and given by the Eq. <ref> assures us that the present void finding algorithm not only generates correct void numbers, but delivers accurate information on all the remaining void characteristics, as shape details, mutual relationships and orientation relative to the coordinate system. | http://arxiv.org/abs/1709.09086v1 | {
"authors": [
"A. M. Sołtan"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170926152122",
"title": "Topology of large scale under-dense regions"
} |
APC, Université Paris Diderot,CNRS, CEA, Observatoire de Paris, Sorbonne Paris Cité 10, rue Alice Domon et Léonie Duquet, F-75205 Paris CEDEX 13, France. Starting from the second post-Keplerian (2PK) Hamiltonian describing the conservative part of the two-body dynamics in massless scalar-tensor (ST) theories, we build an effective-one-body (EOB) Hamiltonian which is a ν-deformation (where ν= 0 is the test mass limit) of the analytically known ST Hamiltonian of a test particle. This ST-EOB Hamiltonian leads to a simple (yet canonically equivalent) formulation of the conservative 2PK two-body problem, but also defines a resummation of the dynamics which is well-suited to ST regimes that depart strongly from general relativity (GR) and whichmay provide information on the strong-field dynamics, in particular, the ST innermost stable circular orbit (ISCO) location and associated orbital frequency. Results will be compared and contrasted with those deduced from the ST-deformation of the (5PN) GR-EOB Hamiltonian previoulsy obtained in [Phys. Rev. D95, 124054 (2017)]. Reducing the two-body problem in scalar-tensor theoriesto the motion of a test particle :a scalar-tensor effective-one-body approachFélix-Louis Julié September 28, 2017 =========================================================================================================================================§ INTRODUCTIONBuilding libraries of accurate gravitational waveform templates is essential for detecting the coalescence of compact binary systems. To this aim, the effective-one-body (EOB) approach has proven to be a very powerful framework to analytically encompass and combine the post-Newtonian (PN) and numerical descriptions of the inspiral, merger, as well as “ring-down" phases of the dynamics of binary systems of comparable masses in general relativity, see, e.g., <cit.>. Matching and comparing gravitational wave templates to the present and future data from the LIGO-Virgo and forthcoming interferometers will bring the opportunity to test GR at high PN order and in the strong field regime of a merger. A next step to test gravity in this regime is to match gravitational wave data with templates predicted in the framework of modified gravities. In this context, scalar-tensor (ST) theories with a single massless scalar field have been the most thoroughly studied. For instance, the corresponding dynamics of binary systems is known at 2.5PN order <cit.>.[or, adopting the terminology of <cit.>, 2.5 post-Keplerian (PK) order, to highlight the fact that (strong) self-gravity effects are encompassed.] What was hence done in <cit.>-<cit.> is the computation of ST waveforms at 2PK relative order (although part of this computation requires information on the ST 3PK dynamics, which is, for now, unknown). In that context, the aim of <cit.> (henceforth paper 1) was to go beyond the (yet poorly known) PK dynamics of modified gravities by extending the EOB approach to scalar-tensor theories.More precisely, we started from the ST two-body 2PK Lagrangian obtained by Mirshekari and Will <cit.> (no spins, nor finite-size, “tidal" effects) and deduced from it the corresponding centre-of-mass frame 2PK Hamiltonian.That two-body 2PK Hamiltonian was then mapped to that of geodesic motion in an effective, “ST-deformed" metric, which has the important property to reduce to the 1998 Buonanno-Damour EOB metric <cit.> in the general relativity limit. When extended to encompass the currently best available (5PN) GR-EOB results, the corresponding ST-EOB Hamiltonian of paper 1 is therefore well-suited to test scalar-tensor theories when considered as parametrised corrections to GR. However, the scope of this GR-centered EOB Hamiltonian is, by construction, restricted to a regime where the scalar field effects are perturbative with respect to general relativity. In their 1998 paper, Buonanno and Damour successfully reduced the general relativistic two-body problem to an effective geodesic motion in a static, spherically symmetric (SSS) metric. In their approach, they ensured that the effective-one-body dynamics is centered on a particular one-body problem in general relativity, namely, the geodesic motion of the reduced mass of the system μ=m_A m_B/M in the Schwarzschild metric produced by a central body, M=m_A+m_B, to which it indeed reduces to in the test-mass limit (i.e., ν=0 with ν=μ/M). Consequently, the associated predictions were smoothly connected to those of the motion of a test mass in the Schwarzschild metric (which is known exactly),ensuring an accurate resummation of the two-body dynamics that could be pushed up to the strong field regime of the last few orbits before plunge.With the same motivation this paper proposes a mapping where the ST-EOB Hamiltonian reduces, in contrast with what was done in paper 1, to the scalar-tensor one-body Hamiltonian in the test mass limit, which describes the motion of a test particle in the metric and scalar field generated by a central SSS body. Although the conservative dynamics derived from this Hamiltonian and that proposed in paper 1 (and from the Mirshekari-Will Lagrangian) are the same at 2PK order, when taken as being exact, they define different resummations and hence, a priori different dynamics in the strong field regime which is reached near the last stable orbit. In particular, we shall highlight the fact that our new, ST-centered, EOB Hamiltonian is well-suited to investigate ST regimes that depart strongly from general relativity.The paper is organised as follows :In section <ref> we present the Hamiltonian describing the motion of a test particle orbiting in the metric and scalar field generated by a central body (when written in Just coordinates) in scalar-tensor theories, henceforth refered as the real one-body Hamiltonian. In order for the paper to be self-contained, in section <ref> we recall the expression of the two-body Hamiltonian in the centre-of-mass frame obtained in paper 1 at 2PK order. In section <ref>we then reduce the two-body problem to an EOB “ν-deformed” version of the ST one-body problem, by means of a canonical transformation and imposing the EOB mapping relation between their Hamiltonians. We finally study the resummed dynamics it defines ; in particular, we compute the innermost stable circular orbit (ISCO) location and associated orbital frequency in the case of Jordan-Fierz-Brans-Dicke theory. Corrections to general relativity ISCO predictions are compared to the results obtained in paper 1.§ THE SCALAR-TENSOR REAL ONE-BODY PROBLEM §.§ The metric and scalar field outside a SSS bodyIn this paper we limit ourselves to the single, massless scalar field case. Adopting the conventions of Damour and Esposito-Farèse (DEF, see e.g. <cit.> or <cit.>), the Einstein-frame action reads in vacuum, that is, outside the sources (setting G_*=c= 1) :S_ EF^ vac[g_μν,φ]=1/16π∫ d^4x√(-g) (R-2g^μν∂_μφ∂_νφ) ,where R is the Ricci scalar and g=g_μν. The vacuum field equations follow : R_μν=2∂_μφ∂_νφ ,□ φ=0 ,where R_μν is the Ricci tensor and □ φ=∂_μ(√(-g)g^μν∂_νφ).The vacuum, static and spherically symmetric (SSS) solutions to the Einstein-frame field equations (<ref>), henceforth, real one-body metric g^*_μν and scalar field φ_*, have a simple analytical expression in Just coordinates (see, e.g., <cit.>) :[In the following, a star (*) shall stand for quantities that refer to the real one-body problem. ] ds_*^2 =-D_*dt^2+dρ^2/D_*+C_*ρ^2(dθ^2+sin^2θ dϕ^2) , with D_*(ρ)=(1-𝔞_*/ρ)^𝔟_*/𝔞_*, C_*(ρ)=(1-𝔞_*/ρ)^1-𝔟_*/𝔞_*,andφ_*(ρ)=φ_0+𝔮_*/𝔞_*ln(1-𝔞_*/ρ) ,where φ_0 is a constant scalar background that must not be considered as an arbitrary integration constant, but rather as imposed, say, by the cosmological environment <cit.><cit.>, while the other integration constants 𝔞_*, 𝔟_* and 𝔮_* have the dimension of a mass and satisfy the constraint :𝔞_*^2=𝔟_*^2+4𝔮_*^2 . We note that when 𝔮_*=0, i.e. 𝔞_*=𝔟_*, the scalar field is a constant, the metric (<ref>) reduces to Schwarzschild's, and Droste and Just coordinates coincide. Note also that pure vacuum (black hole) solutions exhibit singular scalar field and curvature invariants at ρ=𝔞_*. For that reason, SSS black holes cannot carry massless scalar “hair" (thus 𝔮_*=0) and hence do not differ from Schwarzschild's, see e.g. <cit.> and <cit.>. One easily checks that expanding (<ref>)-(<ref>) at infinity and in isotropic coordinates (ρ=ρ̅+𝔞^*/2+⋯), the metric and scalar field behave as g̅^*_μν=η_μν+δ_μν(𝔟_*/ρ̅)+𝒪(1/ρ̅^2) ,φ_*=φ_0-(𝔞_*/ρ̅)+𝒪(1/ρ̅^2) ,where δ_μν is the Kronecker symbol. In order to relate the constants of the vacuum solution to the structure of the body generating the fields, we need the Einstein-frame action inside the source,S_ EF[g_μν,φ,Ψ]=1/16π∫ d^4x√(-g) (R-2g^μν∂_μφ∂_νφ)+S_m[Ψ,𝒜^2 (φ) g_μν] ,where 𝒜(φ) characterizes the ST theory and Ψ generically stands for matter fields, that are minimally coupled to the Jordan metric, g̃_μν≡𝒜^2 (φ) g_μν. The field equations read R_μν=2∂_μφ∂_νφ+8π(T_μν-1/2g_μνT) ,□ φ=-4πα(φ)T ,where T_μν≡ -2/√(-g)δ S_m/δ g^μν is the Einstein-frame energy-momentum tensor of the source, T≡ T^μ_μμ and whereα(φ)≡dln𝒜(φ)/dφmeasures the universal coupling strength between the scalar field and matter. The constants 𝔟_* and 𝔮_* can then be matched to the internal structure of the central body through integration of (<ref>) and (<ref>) as𝔟_*=2∫_0^ρ_0 d^3x√(-g) (-T^0_0+T^i_i) ,𝔮_*=-∫_0^ρ_0 d^3x√(-g) α(φ) T ,where ρ_0 denotes the radius of the central body.[For example, one rewrites (<ref>) as (√(-g)g^ρρφ')'=-4π√(-g)α(φ)T (where a'≡ da/dρ) and integrates both sides between the center of the star, where the fields are supposed to be regular, and its radius ρ_0. The left-hand-side hence reads ∫_0^ρ_0(√(-g)g^ρρφ')'dρ=√(-g)g^ρρφ'|_ρ=ρ_0=𝔮_* sinθ, using the vacuum expressions (<ref>-<ref>), by continuityat ρ=ρ_0. Hence, one has 𝔮_*=-(4π/sinθ)∫_0^ρ_0dρ√(-g)α T=-∫_0^ρ_0dρdθdϕ√(-g)α T, i.e. (<ref>). One similarly obtains 𝔟_* through integration of the t-t component of Einstein's equation (<ref>), see <cit.> for the details.] The numerical values of these integrals generically depend on the asymptotic value of the scalar field at infinity φ_0. Indeed, one can for example model a star as a perfect fluid, together with its equation of state. Given some central density and value for the scalar field φ_c≡φ(ρ=0), one integrates (<ref>) and the matter equations of motion from the regular center of the body up to ρ_0 where the pressure vanishes. The metric and scalar field are then matched to the exterior solution (<ref>)-(<ref>), fixing uniquely 𝔟_*, 𝔮_* and φ_0 in terms of the central density and φ_c. When the equation of state and the baryonic number of the star are held fixed, the exterior fields (i.e. 𝔟_* and 𝔮_*) are completely known as functions of φ_c only, or, equivalently, of the scalar field value at infinity, φ_0, see, e.g. <cit.> for an explicit computation.§.§ Skeletonizing the source of the gravity field In order to clarify the analysis to come in the forthcoming sections, we now “skeletonize" the body creating the gravity field ; that is, we phenomenologically replace S_m in (<ref>) by a point particle action, as was suggested by Eardley in <cit.> :S_m^skel[X^μ, g_μν, φ]=-∫ M_*(φ) dS ,where dS=√(-g_μνdX^μ dX^ν) and where X^μ(S) denotes the location of the skeletonized body. The Einstein-frame mass M_*(φ) depends on the value of the scalar field at X^μ(S) (substracting divergent self contributions), on the specific theory and on the body itself (contrarily to (<ref>) where the coupling to the scalar field was universal), hence encompassing the effects of the background scalar field on its equilibrium configuration.[Note that Eardley-type terms do not depend on the local gradients of g_μν and φ and hence cannot account for finite-size, “tidal" effects ; see e.g. <cit.>. In this paper, all tidal effects will be neglected. ] For a discussion on the validity of the skeletonization procedure, see <cit.> and <cit.>. The question adressed now is to relate the function M_*(φ) to the parameters describing the exterior solutions, that is 𝔟_* and 𝔮_*, given a scalar field value at infinity φ_0. The field equations are given byR_μν=2∂_μφ∂_νφ+8π(T_μν-1/2g_μνT) ,with T^μν=∫ dSM_*(φ)δ^(4)(x-X)/√(-g)dX^μ/dSdX^ν/dS , and □ φ=4π∫ dSM_*(φ)A_*(φ)δ^(4)(x-X)/√(-g) , where we introduced the body-dependent function (“capital alpha")A_*(φ)≡dln M_*(φ)/d φ ,which measures the coupling between the skeletonized body and the scalar field.Note that because of the body-dependent function M_*( φ), the effective scalar field equation is different from (<ref>) with T^μν given in (<ref>), because (<ref>) was derived from the universally coupled action (<ref>). Note also that since black holes cannot carry scalar hair, A_* must vanish in that case, i.e. M_* must then reduce to a constant and one recovers general relativity.We now solve these equations in the rest-frame of the skeletonized body, setting X⃗=0⃗. Outside it, the metric and scalar field are of the form (<ref>) and (<ref>). Moreover, solving the field equations (<ref>) perturbatively around the metric and scalar field backgrounds, i.e. g̅^*_μν=η_μν+h_μν, φ_*=φ_0+δφ, in harmonic coordinates ∂_μ(√(-g̅)g̅^μν)=0, easily yields, at linear order g̅^*_μν=η_μν+δ_μν(2M_*(φ_0)/ρ̅)+𝒪(1/ρ̅^2) ,φ_*=φ_0-M_*(φ_0)A_*(φ_0)/ρ̅+𝒪(1/ρ̅^2) ,where the φ_0-dependence of the fields recalls the fact that the skeletonized body is “sensitive" to the background value of the scalar field in which it is immersed, that is, φ_0, as already discussed below (<ref>).[while, as in GR, the asymptotic (constant) metric at infinity can always be “gauged away" to Minkowski by means of an appropriate coordinate change. ] Moreover, by comparing (<ref>) to (<ref>), one obtains the following relations (knowing that the harmonic and isotropic coordinates identify at linear order) :𝔟_*=2M_*^0 ,𝔮_*=M_*^0A_*^0 ,𝔞_*=2M_*^0√(1+(A^0_*)^2) ,see (<ref>), where and from now on, a zero index denotes a quantity evaluated for φ=φ_0. Hence, by means of the matching conditions (<ref>), we have traded the integration constants of the vacuum solution 𝔟_* and 𝔮_*, which are related to the source stress-energy tensor by (<ref>), for their “skeleton" counterparts, M_*^0 and A_*^0, which are the values of the function M_*(φ) and its logarithmic derivative evaluated at the background φ_0. §.§ The real one-body problem : the motion of a test particlein the fields of a skeletonized body in ST theoriesWe now turn to the motion of a self-gravitating test particle m_*(φ), coupled to the fields obtained above, i.e., generated by the central body only. The dynamics is described again by an Eardley-type action,S_*[x^μ]=-∫ m_*(φ_*) ds_* ,where ds_*=√(- g^*_μνdx^μ dx^ν) and where φ_* and g^*_μν are the real one-body metric and scalar field, given explicitly in Just coordinates in (<ref>), (<ref>) together with (<ref>). Note that the function m_*(φ_*) characterizing the particle can be related too to the properties of an extended test body following the steps presented above, but where the scalar environment is not φ_0 anymore, and is replaced by the value of the scalar field generated by the central body φ_*, at the location of the test particle, φ_*(x^μ(s_*)). To simplify notations it is convenient to replace m_*(φ_*) by the rescaled function V_*(φ_*)≡(m_*(φ_*)/m_*^0)^2 ,such that S_*[x^μ]=-m_*^0∫ √(V_*) ds_* , where we recall that m_*^0=m_*(φ_0) is the value of m_*(φ_*) when the test particle is infinitely far away from the central body. Therefore, the scalar-tensor Lagrangian for our test particle, defined as S_*≡∫ dtL_*, reads (restricting the motion to the equatorial plane, θ=π/2) :L_*=-m_*^0√(-(V_*g^*_μν)dx^μ/dtdx^ν/dt)=-m_*^0√(V_*(D_*-ρ̇^2/D_*-C_* ρ^2ϕ̇^2)) ,ρ̇≡dρ/dt ,ϕ̇≡dϕ/dt , withD_*(ρ)=(1-a_*/ρ̂)^b_*/a_*, C_*(ρ)=(1-a_*/ρ̂)^1-b_*/a_*,where we have introduced the dimensionless radial coordinateρ̂≡ρ/M^0_* ,and where the rescaled constants b_* and a_* follow from (<ref>),b_*=2 , a_*=2√(1+(A^0_*)^2) . In contrast, the expression of V_*(φ_*(ρ)) (or, equivalently, m_*) as an explicit function of ρ depends on the specific ST theory and on the internal structure of the test particle. At 2PK order to which we restrict ourselves in this paper, it will prove sufficient to replace it by its Taylor expansion around φ_0. To do so, let us introduce the three quantitiesα_*(φ)≡dln m_*/dφ ,β_*(φ)≡dα_*/dφ ,β'_*(φ)≡dβ_*/dφ ,such that, expanding m_*(φ) around φ_0 (where we recall that φ_0 is the value at infinity of the scalar field imposed by cosmology) yieldsm_*(φ)=m^0_*[1+α^0_*(φ-φ_0)+1/2(α^0_*^2+β^0_*)(φ-φ_0)^2+1/6(3β^0_*α^0_*+α^0_*^3+β'^0_*)(φ-φ_0)^3+⋯] .Now, the scalar field generated by the central body is given in (<ref>) together with (<ref>). Hence V_* reads, at 2PK order,V_*(ρ̂)=(m_*(φ_*(ρ̂))/m_*^0)^2=1+v^*_1/ρ̂+v^*_2/ρ̂^2+v^*_3/ρ̂^3+𝒪(1/ρ̂^4),where the dimensionless constants v^*_1, v^*_2 and v^*_3 depend on the functions M_*(φ) and m_*(φ) characterizing the central body and the test particle and are given by v^*_1=-2α^0_*A^0_* ,v^*_2=(2(α^0_*)^2+β^0_*)(A^0_*)^2-2α^0_*A^0_*√(1+(A^0_*)^2) ,v^*_3=-(4/3(α^0_*)^3+1/3β'^0_*+2α^0_*β^0_*)(A^0_*)^3+(4(α^0_*)^2+2β^0_*)(A^0_*)^2√(1+(A^0_*)^2) -8/3α^0_*A^0_*(1+(A^0_*)^2) . To summarize, we have obtained in this section the Lagrangian that describes the dynamics of a test particle orbiting around a central (skeletonized) body in scalar-tensor theories of gravity. At 2PK order, it is entirely described by five coefficients, a_*, b_*, v_1^*, v_2^*, v_3^*, which are in turn expressed in terms of the five fundamental parameters : M_*^0, A_*^0 describing the central body, and α_*^0, β_*^0, β'_*^0 describing the orbiting particle.[Note that b_*=𝔟_*/M_*^0 (with 𝔟_*=2M_*^0) is a parameter since M_*^0 has been factorized out in the definition of ρ̂=ρ/M_*^0. ]§ THE REAL TWO-BODY DYNAMICS AT 2PK ORDER, A REMINDERIn this section, we recall the results from paper 1 <cit.> that will be needed in the forthcoming sections.§.§ The two-body 2PK Hamiltonians in scalar-tensor theoriesThe two-body dynamics is conveniently described in the Einstein-frame (following DEF), by means of an Eardley-type actionS_ EF[x_A^μ, g_μν, φ]=1/16π∫ d^4x√(-g) (R-2g^μν∂_μφ∂_νφ)-∑_A∫ ds_A m_A(φ) ,where ds_A=√(- g_μνdx_A^μ dx_A^ν), and where x^μ_A(s_A) denotes the position of body A. The masses m_A(φ) depend on the (regularized) local value of the scalar field and are related to their Jordan-frame counterparts through m_A(φ)≡𝒜(φ) m̃_A(φ).In the negligible self-gravity limit, the “Jordan masses" reduce to constants, m̃_A(φ)=cst, so that the motion is a geodesic of the Jordan metric g̃_μν=𝒜^2 g_μν. In contrast, general relativity is recovered when the “Einstein masses" are constants, m_A(φ)=cst. We now define a set of body-dependent quantities, consistently with (<ref>) and (<ref>), α_A(φ)≡dln m_A/dφ(=dln𝒜/dφ+dlnm̃_A/dφ) ,β_A(φ)≡dα_A/dφ ,β'_A(φ)≡dβ_A/dφ ,that appear in the 2PK two-body Lagrangian. In the negligible self-gravity limit, m̃_A=cst, and henceα_A→α≡dln𝒜/dφ ,β_A→β≡dα/dφ ,β'_A→β'≡dβ/dφ ,become universal, while in the general relativity limit, m_A=cst, implying α_A=β_A=β'_A=0. The conservative part of the scalar-tensor two-body problem has been studied at 1PK order by Damour and Esposito-Farèse (DEF) in <cit.> and at 2PK order by DEF in <cit.> and Mirshekari and Will (MW) in <cit.>, performing a small orbital velocities, weak field expansion (V^2∼ m/R) around η_μν and a constant cosmological background φ_0. Because of the harmonic coordinates in which it has been computed, the two-body Lagrangian depends linearly on the accelerations of the bodies at 2PK level.In paper 1, we started from this MW Lagrangian, L(Z⃗_A/B,Ż⃗̇_A/B,Z̈⃗̈_A/B). Once translated in terms of the DEF conventions presented above (see also paper 1, appendix A), we eliminated the dependence in the accelerations Z̈⃗̈_A/B by means of suitable contact transformations of the formZ⃗'_A(t)=Z⃗_A(t)+δZ⃗_A(Z⃗_A/B,Ż⃗̇_A/B) ,that is, four-dimensional 2PK coordinate changes. We found a whole class of coordinate systems, labeled by fourteen parameters f_i, in which the Lagrangian is ordinary (see paper 1 appendix B and below). By means of a further Legendre transformation, we obtained the associated Hamitonians H(Q,P) in the center-of-mass frame, the conjugate variables being Z⃗=Z⃗_A-Z⃗_B and P⃗=P⃗_A=-P⃗_B, and in polar coordinates : (Q,P)≡(R,Φ, P_R, P_Φ) where P_R=N⃗·P⃗ and P_Φ=R(N⃗×P⃗)_z. The resulting isotropic, translation-invariant, ordinary Hamiltonians are given at 2PK order in paper 1, section III C,Ĥ≡H/μ=M/μ+(P̂^2/2-G_AB/R̂)+Ĥ^ 1PK+Ĥ^ 2PK+⋯where we have introduced the rescaled quantitiesP̂^2≡P̂_R^2+P̂_Φ^2R̂^2withP̂_R≡P_R/μ , P̂_Φ≡P_Φμ M , R̂≡R M ,and the reduced mass, total mass and symmetric mass ratio :μ≡m_A^0m_B^0 M , M≡ m_A^0+m_B^0 , ν≡μ M ,where m_A^0 and m_B^0 are the values of the functions m_A(φ) and m_B(φ) at φ=φ_0. At 2PK order, the two-body Hamiltonians depend on seventeen coefficients (h_i^n PK) (which are very lenghty and are given explicitely in appendix C of paper 1), which in turn depend on the fourteen f_i parameters and on the eleven following combinations of the eight fundamental mass parameters (<ref>) [m_A^0, α_A^0, β_A^0 and β_A^'0 and B counterparts, characterizing at 2PK order the functions m_A/B(φ)] : m_A^0 ,G_AB≡ 1+α_A^0α_B^0 ,γ̅_AB≡ -2α_A^0α_B^0/1+α_A^0α_B^0 ,β̅_A≡1/2β_A^0(α_B^0)^2/(1+α_A^0α_B^0)^2 , δ_A≡(α_A^0)^2/(1+α_A^0α_B^0)^2 ,ϵ_A≡(β'_Aα_B^3)^0/(1+α_A^0α_B^0)^3 ,ζ≡β_A^0α_A^0β_B^0α_B^0/(1+α_A^0α_B^0)^3 ,and (A↔ B) counterparts, where we recall that a zero index indicates a quantity evaluated at infinity, φ=φ_0. In the general relativity limit, m_A=cst, the Hamiltonian considerably simplifies since these combinations reduce toG_AB=1 ,andγ̅_AB=β̅_A=δ_A=ϵ_A=ζ=0 .§.§ The canonical transformationThe EOB mapping consists in imposing a functional relation between the two-body Hamiltonian H(Q,P), and an effective Hamiltonian H_e (that we shall build in the next section), by means of a canonical transformation,(Q,P)→(q,p) ,where (q,p)≡(ρ,ϕ,p_ρ,p_ϕ). The canonical transformation is generated by the (time-independent and isotropic) generic function G(Q,p) introduced in <cit.>, section III D, which depends on nine parameters at 2PK order,G(Q,p)μ M=R̂ p̂_ρ[(α_1 P^2+β_1p̂_ρ^2+γ_1R̂)+(α_2 P^4+β_2 P^2p̂_ρ^2+γ_2p̂_ρ^4+δ_2 P^2R̂+ϵ_2p̂_ρ^2R̂+η_2R̂^2)+⋯] ,where we introduced the reduced quantitiesP^2≡p̂_ρ^2+p̂_ϕ^2R̂^2 ,R̂≡R M ,p̂_ρ≡p_ρ/μ ,p̂_ϕ≡p_ϕ/μ M .The associated canonical transformation readsρ(Q,p)=R+∂ G∂ p_ρ ,ϕ(Q,p)=Φ+∂ G∂ p_ϕ , P_R(Q,p)=p_ρ+∂ G∂ R , P_Φ(Q,p)=p_ϕ+∂ G∂Φ ,and leads to 1PK and higher order coordinate changes. Note that the Φ-independence of G(Q,p) yields P_Φ=p_ϕ. Moreover, for circular orbits, p_ρ=0⇔ P_R=0, we note that ϕ=Φ and hence only the radial coordinates differ ρ≠ R. The two-body Hamiltonian (<ref>) is thus rewritten in the intermediate coordinate system H'(Q,p)=H(Q,P(Q,p)) using the last two equations in (<ref>) which yield (dropping the prime)Ĥ=M/μ+( P^2/2-G_AB/R̂)+Ĥ^ 1PK+Ĥ^ 2PK+⋯ ,where the explicit expressions for Ĥ^ 1PK and Ĥ^ 2PK are given in appendix D of paper 1. It depends on the eight fundamental parameters (<ref>), on the fourteen parameters f_i characterizing the coordinate system in which the two-body Hamiltonian H(Q,P) was written, and on the nine parameters of the canonical transformation (<ref>).§ THE SCALAR-TENSOR EOB HAMILTONIANIn this section we relate the canonically transformed, two-body Hamiltonians H(Q,p) to the Hamiltonian H_e of an effective test-particle in the fields of an effective central body. To this aim, we shall propose a ST-centered Hamiltonian H_e that contrasts with what was done in paper 1, where H_e was centered on the GR limit. §.§ The effective Hamiltonian In view of reducing the two-body dynamics to that of an effective test particle coupled to the generic SSS fields of an effective single body, and taking inspiration from (<ref>), let us consider the action (setting again θ=π/2) :S_e[x^μ]=-∫ m_e(φ_e) ds_ewhere ds_e=√(-g^e_μνdx^μ dx^ν) and where x^μ[s_e] is the world-line of the effective particle characterized by the function m_e(φ_e). As in (<ref>), we write the effective metric in Just coordinatesds_e^2=-D_e dt^2+dρ^2/D_e+C_e ρ^2dϕ^2 ,where D_e and C_e are effective functions to be determined later. We now replace, for notational convenience, m_e(φ_e) by the functionV_e≡(m_e(φ_e)/μ)^2 ,such that S_e[x^μ]=-μ∫√(V_e) ds_e ,which is the third effective function to be determined, and where μ is identified to the real two-body reduced mass, defined in (<ref>). The associated Lagrangian, defined as S_e≡∫ dt L_e, reads therefore L_e=-μ√(-(V_e g^e_μν)dx^μ/dtdx^ν/dt)=-μ√(V_e(D_e-ρ̇^2/D_e-C_eρ^2ϕ̇^2)) ,whereρ̇≡ dρ/dt ,ϕ̇≡ dϕ/dt . Note that L_e identifies to the Lagrangian of a geodesic in the body-dependent conformal metric, (V_eg^e_μν).One easily deduces the effective momenta and Hamiltonian, p_ρ≡∂ L_e∂ρ̇ , p_ϕ≡∂ L_e∂ϕ̇ , H_e≡ p_ρρ̇+p_ϕϕ̇-L_e ,that isĤ_e≡H_e/μ=√(V_eD_e+D_e^2p̂_ρ^2+D_e/C_ep̂_ϕ^2/ρ̂^2) ,where we used the reduced (dimensionless) variablesρ̂≡ρ M ,p̂_ρ≡p_ρμ ,p̂_ϕ≡p_ϕμ M ,p̂^2≡p̂_ρ^2+p̂_ϕ^2ρ̂^2 ,M being identified to the real total mass, see (<ref>). In order to relate the effective Hamiltonian H_e to the two-body (perturbative) Hamiltonian H, we now restrict H_e to 2PK order also. To this end, one could in principle expand V_e, D_e and C_e in the form of 1/ρ̂ series. However, our aim being to build an effective dynamics as close as possible to the scalar-tensor test-body problem, we shall rather introduce the non perturbative, “resummed" ansatz for the metric functions D_e and C_e :D_e(ρ)≡ (1-a/ρ̂)^b/a, C_e(ρ)≡(1-a/ρ̂)^1-b/a,as suggested by (<ref>), and where a and b are two effective parameters that we shall determine in the following. As already remarked below equation (<ref>), the effective dynamics is equivalent to the geodesic motion in the conformal metric (V_eg^e_μν). The ansatz (<ref>) that we shall use rather than a simple 1/ρ̂ expansion of D_e and C_e is hence crucial, since the latter would be equivalent, to within a mere coordinate change (r^2=C_eV_eρ^2), to the GR-centered approach of paper 1. In contrast, a specific ansatz for the function V_e can be proposed in the framework of a specific ST theory and when the internal structure of the two real bodies is known, see discussion below (<ref>). (For an example, see subsection <ref>.) For the moment, we hence expand V_e a 2PK order, similarly to what was done in (<ref>) :V_e(ρ)=1+v_1/ρ̂+v_2/ρ̂^2+v_3/ρ̂^3+⋯ ,where v_1, v_2 and v_3 are three further effective parameters to determine later. Expanding the effective Hamiltonian (<ref>) and (<ref>-<ref>) hence readsĤ_e=1+Ĥ^ K_e+Ĥ^ 1PK_e+Ĥ^ 2PK_e+⋯with, at 1PK,Ĥ^ K_e=p̂^2/2+v_1-b/2ρ̂ , H^ 1PK_e=-p̂^4/8+1/4 ρ̂[p̂^2 (2 a-3 b-v_1)-2 a p̂_ρ^2]+1/8 ρ̂^2[-2 a b+b^2-2 b v_1-v_1^2+4 v_2] ,and, at 2PK,H^ 2PK_e=p̂^6/16+1/16 ρ̂[p̂^4 (5 b+3 v_1-4a)+4 a p̂^2 p̂_ρ^2]+1/16 ρ̂^2[4 a p̂_ρ^2 (-2 a + 3 b + v_1) + (8 a^2 + 9 b^2 + 6 b v_1 + 3 v_1^2 -2 a (9 b + 2 v_1) - 4 v_2) p̂^2]+1/48ρ̂^3[-8 a^2 b - b^3 + 6 a b (b - v_1) + 3 b^2 v_1 + 3 b (v_1^2 - 4 v_2) + 3 (v_1^3 - 4 v_1 v_2 + 8 v_3)] . In order to relate the two-body Hamiltonians of the previous section <ref> and the present effective Hamiltonian H_e(q,p), we finally express the latter in the same coordinate system H'_e(Q,p)=H_e(q(Q,p),p) using the first two relations in (<ref>). The resulting effective Hamiltonian reads (dropping again the prime)Ĥ_e=1+(𝒫^2/2+v_1-b/2R̂)+Ĥ_e^ 1PK+Ĥ_e^2 PK+⋯where we recall that P^2≡p̂_ρ^2+p̂_ϕ^2/R̂^2 and where H_e^ 1PK and H_e^ 2PK are explicitly given in appendix <ref> of this paper. §.§ The EOB mappingBy means of the generic canonical transformation (<ref>-<ref>), the real and (a priori independent) effective Hamiltonians H(Q,p) and H_e(Q,p) have been written in a common coordinate system, (Q,p) ; see (<ref>) and (<ref>).Now, as discussed in e.g. <cit.>, <cit.> and<cit.>, and as proven to be indeed necessary at all orders in GR as well as in ST theories in<cit.>, both Hamiltonians shall be related by means of the quadratic functional relation (we recall that ν=μ/M) :H_e(Q,p)/μ-1=(H(Q,p)-M/μ) [1+ν/2(H(Q,p)-M/μ)] .The identification (<ref>) proceeds order by order and term by term to yield a unique solution for H_e, that is for the funtions introduced in the previous subsectionD_e(ρ)≡ (1-a/ρ̂)^b/a, C_e(ρ)≡(1-a/ρ̂)^1-b/a, V_e(ρ)=1+v_1/ρ̂+v_2/ρ̂^2+v_3/ρ̂^3+⋯ ,whose effective parameters now depend on the combinations (<ref>) and are the main technical result of this paper : b=2 , v_1=-2α_A^0α_B^0 , a=2 ℛ ,v_2= 2 - 4 G_AB + 2 (1 + ⟨β̅⟩)G_AB^2 -2α_A^0α_B^0 ℛ , v_3/4=1 - 5/3 G_AB + (1 +⟨β̅⟩ +2/3⟨δ⟩) G_AB^2- 1/3(1 +3 ⟨β̅⟩ +1/4⟨ϵ⟩ +2 ⟨δ⟩) G_AB^3+(1 - 2G_AB + (1 + ⟨β̅⟩) G_AB^2)ℛ + ν[ 17/3 G_AB -1/3(19 + 4 ⟨β̅⟩+6ζ) G_AB^2 + (2/3 - 3/4 (β̅_A+β̅_B) + 1/12 (ϵ_A +ϵ_B).. ..+ 1/6 (δ_A + δ_B) + 3/2⟨β̅⟩) G_AB^3 ] ,where we have introducedℛ≡√(1+⟨δ⟩ G_AB^2+ν[8G_AB-2(1+⟨β̅⟩)G_AB^2]) ,and the “mean" quantities⟨β̅⟩≡m_A^0β̅_B+m_B^0β̅_A/M ,⟨δ⟩≡m_A^0 δ_A + m_B^0 δ_B/M ,⟨ϵ⟩≡m_A^0 ϵ_B + m_B^0 ϵ_A/M . We note that as they should, these parameters can alternatively be deduced from the effective metric found in paper 1, using the 2PK-expanded coordinate change r^2=C_eV_eρ^2, where r is the Schwarschild-Droste coordinate used there.[Note also that the present results (<ref>-<ref>) have been simplified using the relation γ̅_AB=-2+2/G_AB, relating γ̅_AB to the dimensionless combination G_AB, see (<ref>). The reader willing to establish G_* (i.e. Newton's constant) again should note that it only appears through ρ̂≡ρ/(G_*M).] As a first consistency check, we note that the effective coefficients (<ref>) do not depend on the f_i parameters introduced in section <ref>, i.e., on the coordinate system (R,Φ) in which the two-body Hamiltonian has been initially written, as expected by covariance of the theory. Indeed, the f_i parameters are absorbed in the 2PK part of the canonical transformation (<ref>), whose parameters readα_1=-ν/2 ,β_1=0 ,γ_1=G_AB[1/2ν+(1+1/2γ̅_AB)ℛ] ,α_2=1/8(1-ν)ν ,β_2=0 ,γ_2=ν^2/2 ,δ_2=G_AB[ f_6 m_A^0/M + f_1 m_B^0/M - ν(f_1 + f_6 +(-f_3+f_5 + f_6) m_A^0/M+ (f_1 + f_2 - f_4) m_B^0/M -3/2-γ̅_AB+ν/8)] ,ϵ_2=G_AB[-ν^2/8 + f_10m_A^0/M + f_7 m_B^0/M - ν(f_7 + f_10 + (f_9 + f_10) m_A^0/M + (f_7 + f_8) m_B^0/M)] ,η_2= G_AB^2[f_13m_A^0/M+f_12m_B^0/M+ ν(f_11- f_12- f_13+ f_14) + ν(-7/4-γ̅_AB - ⟨β̅⟩ + β̅_A + β̅_B/2 + ν/4)] . The real two-body Hamiltonian (<ref>), whose full expression is relegated to section III C and appendix C of paper 1, has hence been reduced to a compact effective Hamiltonian, where most of the two-body Hamiltonian complexityis hidden in the canonical transformation (<ref>), (<ref>) (e.g., information regarding the initial coordinate system) and in the mapping relation (<ref>).§.§.§ The ν=0 limit Setting formally ν=0 in (<ref>-<ref>), the parameters reduce to, when written in terms of the fundamental quantities (<ref>) : b=2 ,v_1=-2α_A^0α_B^0 , a=2 ℛ ,v_2= 2(α_A^0α_B^0)^2+(m_Aα_A^2)^0β_B^0+(m_Bα_B^2)^0β_A^0/M-2α_A^0α_B^0ℛ , v_3=-4/3(α_A^0α_B^0)^3-1/3(m_Aα_A^3)^0β'_B^0+(m_Bα_B^3)^0β'_A^0/M-2α_A^0α_B^0(m_Aα_A^2)^0β_B^0+(m_Bα_B^2)^0β_A^0/M -8/3(1+(m_Aα_A^2)^0+(m_Bα_B^2)^0/M)α_A^0α_B^0+(4(α_A^2α_B^2)^0+2(m_Aα_A^2)^0β_B^0+(m_Bα_B^2)^0β'_A^0/M)ℛ ,withℛ=√(1+(m_Aα_A^2)^0+(m_Bα_B^2)^0/M) .Identifying now (<ref>) to the parameters (<ref>) and (<ref>) of the real one-body problem presented in section <ref> does yield a unique solution : (A^0_*)^2=m_A^0(α_A^0)^2+m_B^0(α_B^0)^2/m^0_A+m_B^0 ,α^0_*=α_A^0α_B^0/A^0_* ,β^0_*=(m_Aα_A^2)^0β_B^0+(m_Bα_B^2)^0β_A^0/(m_Aα_A^2)^0+(m_Bα_B^2)^0 ,β'^0_*=(m_Aα_A^3)^0β'^0_B+(m_Bα_B^3)^0β'_A^0/(m^0_A+m_B^0)(A^0_*)^3 ,together with m^0_*=μ, M^0_*=M. We hence conclude that the dynamics described by H_e is a ν-deformation of a scalar-tensor test-body problem, describing an effective test particle characterized byln m_*(φ)=ln m_*^0+α_*^0(φ-φ_0)+β_*^0(φ-φ_0)^2+β'_*^0(φ-φ_0)^3+⋯ ,orbiting around an effective central body characterized byln M_*(φ)=ln M_*^0+A_*^0(φ-φ_0)+⋯ , whose fundamental parameters[M^0_*, A^0_*, m^0_*, α^0_*, β^0_* and β'^0_*] are related to the real, two-body ones through (<ref>). Since ν→ 0 means, say, m_B^0>>m_A^0, one retrieves consistentlyM_*^0→ m_B^0 , A_*^0→α_B^0 ,m_*^0→ m_A^0 ,α_*^0→α_A^0 ,β_*^0→β_A^0 ,β'_*^0→β'_B^0 , that is, A becomes a test body orbiting around the central body B.We note also that ν-deformations do not enter the coefficients b and v_1 in the generic ν≠ 0 case, see (<ref>), which are hence particularly simple ; we hence recover a feature of the linearized effective dynamics which is common with that of the general relativity case (see Buonanno and Damour in <cit.>), and which is related to the very specific form of the quadratic functional relation (<ref>).[We also recall that the gravitational coupling G_AB=1+α_A^0α_B^0, appearing in the two-body Hamiltonian (see (<ref>), subsection <ref>, and paper 1 section III C), encompasses the linear addition of the metric and scalar interations at linear level <cit.>. The present mapping has consistently split it again, between the effective metric and scalar sectors, i.e. b and v_1, see (<ref>), contrarily to the GR-centered, fully metric mapping of paper 1, where G_AB appeared at each post-Keplerian orders in the form (G_ABM)/r, r being the Schwarzschild-Droste coordinate used there. ] §.§.§ General relativityFinally, in the general relativity limit (<ref>), (<ref>) and (<ref>) become the well-known reduced and total masses m_*(φ)=μ and M_*(φ)=M, and the effective coefficients (<ref>) reduce to a=2 √(1+6ν) , b=2 ,v_1=v_2=v_3=0 .In other words, V_e=1, i.e. the effective scalar field effects disappear. The (non-perturbative) metric sector is now written in Just coordinates and differs from the results of Buonanno and Damour <cit.> who worked out their analysis in Schwarzschild-Droste coordinates. In the present paper, we hence have on hands a resummation of the 2PN general relativity dynamics that differs from the one explored in <cit.>. The comparison and consistency of the two shall be commented upon subsection <ref>. When moreover ν=0, a=b and the metric consistently reduces to Schwarzschild's, see comment below (<ref>). §.§ ST-EOB dynamicsInverting the EOB mapping relation (<ref>) yields the “EOB Hamiltonian",H_ EOB=M√(1+2ν(H_e/μ-1)) ,whereH_e/μ=√(D_eV_e+D_e^2p̂_ρ^2+D_e/C_e(p̂_ϕ/ρ̂)^2) ,[where D_e, C_e and V_e are given in (<ref>) and (<ref>)] which defines a resummation of the two-body 2PK Hamiltonian, H.[We recall that by construction, when restricted to 2PK level, H_ EOB yields a dynamics which is canonically equivalent to that derived from H. ] In the following we focus on some features of the resultant resummed dynamics, in the strong field regime. Hence and from now on, the 2PK-truncated function V_e is to be considered as exact, along with D_e and C_e. §.§.§ Effective dynamicsAs we shall see, the ST-EOB dynamics will follow straightforwardly from that derived from the effective Hamiltonian H_e. This can be obtained from Hamilton's equations (q̇=∂ H_e/∂ p, ṗ=-∂ H_e/∂ q), or, as already remarked below (<ref>), can be equivalently interpreted as a geodesic of the (body-dependent) conformal metric g̃_μν=V_eg^e_μν :ds̃_e^ 2≡ -D_eV_e dt^2+V_e/D_edρ^2+C_eV_e ρ^2dϕ^2 . The staticity and spherical symmetry of this metric imply the conservation of the energy and angular momentum of the orbit (per unit mass μ),u_t=-D_eV_edt/dλ≡-E , u_ϕ=C_eV_eρ^2dϕ/dλ≡ L ,λ being an affine parameter along the trajectory. When moreover the 4-velocity is normalized as u^μ u_μ=-ϵ (where ϵ=1 for μ≠ 0, ϵ=0 for null geodesics), the radial motion is driven by an effective potential F_ϵ ,(dρ/dλ)^2=1/V_e^2F_ϵ(u) ,whereF_ϵ(u)≡ E^2-D_eV_e(ϵ+j^2u^2/C_eV_e) , j≡L/M , u≡1/ρ̂=M/ρ , and D_e(u)= (1-au)^b/a , C_e(u)=(1-au)^1-b/a , V_e(u)=1+v_1u+v_2u^2+v_3u^3 .§.§.§ ISCO locationWe now focus on circular orbits when ϵ=1, i.e., F_ϵ=1(u)=F'_ϵ=1(u)=0 ; j^2 and E are therefore related to u throughj^2(u)=-(D_eV_e)'/(u^2D_e/C_e)' , E(u)=√(D_eV_e(1+j^2(u)u^2/C_eV_e)) .A characteristic feature of the strong-field regime is the innermost stable circular orbit (ISCO), which is reached when the third (inflection point) condition is satisfied F”_ϵ=1(u)=0, i.e. when u_ ISCO is the root, if any, of the equation :F'_ϵ=1(u_ ISCO)=F”_ϵ=1(u_ ISCO)=0⇒(D_eV_e)”/(D_eV_e)'=(u^2D_e/C_e)”/(u^2D_e/C_e)' .§.§.§ Light-ring locationWhen ϵ=0, F_ϵ=0(u)=E^2-j^2u^2D_e/C_e and one can define a light-ring (LR), i.e. the radius of null circular orbits, through F'_ϵ=0(u_ LR)=0 :u_LR=1/b+a/2⇔ρ_LR=M(2+ℛ) ,where ℛ is given in (<ref>). In particular one retrieves ℛ=1, i.e. ρ_LR=3M (Schwarzschild's LR location) in the test-mass (ν→0), general relativity limit (<ref>). §.§.§ ST-EOB orbital frequencyWe now turn to the resummed two-body dynamics defined by the EOB Hamiltonian (<ref>). Since H_ EOB and H_e are conservative, we have :(∂ H_ EOB/∂ H_e)=1/√(1+2ν(E-1))since H_e=μ E is a constant on-shell. Therefore, the resummed equations of motiondρ/dt=∂ H_ EOB/∂ p_ρ ,dϕ/dt=∂ H_ EOB/∂ p_ϕ ,dp_ρ/dt=-∂ H_ EOB/∂ρ ,dp_ϕ/dt=-∂ H_ EOB/∂ϕ=0 ,are identical to the effective ones, i.e. derived from the effective Hamiltonian, H_e(q,p), to within the (constant) time rescaling t→ t√(1+2ν(E-1)). In particular, for circular orbits, the orbital frequency readsΩ(u)≡dϕ/dt=∂ H_ EOB/∂ H_e∂ H_e/∂ p_ϕ=D_e/C_eju^2/ME√(1+2ν(E-1)) ,where E(u) and j(u) are given for circular orbits in (<ref>). Its ISCO value is reached when u=u_ ISCO, as defined in (<ref>). Note that the orbital frequency has been derived in the Just coordinate system, (q,p), which is related to the real one, (Q,P), through the canonical transformation presented in subsection <ref>. Moreover, for circular orbits (p_ρ=P_R=0), Φ=ϕ,and hence (<ref>) is the observed orbital frequency. See also subsections <ref> and <ref>.§.§ An example : the Jordan-Fierz-Brans-Dicke theory§.§.§ A simple one-parameter modelWe now illustrate the previous results through the example of the Jordan-Fierz-Brans-Dicke theory <cit.>, <cit.>, which depends on a unique parameter α, such that[For a comparison with the Jordan-frame parameter ω, such that 3+2ω=α^-2, see <cit.> appendix A.]S_ JFBD[ g_μν, φ,Ψ]=1/16π∫ d^4x√(-g) (R-2g^μν∂_μφ∂_νφ)+S_m[Ψ,𝒜^2 (φ) g_μν] ,where𝒜(φ)=e^αφ ,α=dln𝒜/dφ=cst ,while general relativity is retrieved when α=0. The two-body dynamics is then described by replacing S_m by its “skeleton" version,S_m^skel[x_A^μ,g_μν,φ]=-∑_A∫ m_A(φ) ds_A ,where, for the sake of simplicity, we shall further neglect self-gravity effects, i.e. m_A(φ)=𝒜(φ) m̃_A, where m̃_A are constants, see discussion above (<ref>). In that case, since 𝒜(φ) is known and the Jordan masses m̃_A are constants, there is no need to expand m_A(φ) as in (<ref>) since it is entirely determined asm_A(φ)=m_A^0e^α(φ-φ_0) ,m_A^0=cst . Therefore, the fundamental parameters (<ref>) become universal (<ref>) and reduce toα_A=dln m_A/dφ=α ,β_A=0 ,β'_A=0 ,and the post-Keplerian (two-body) parameters (<ref>) greatly simplify as well toG_AB=1+ α^2 ,γ̅_AB=-2α^2/1+α^2 ,δ_A=δ_B=α^2/(1+α^2)^2 ,β̅_A=β̅_B=0 ,ϵ_A=ϵ_B=0 ,ζ=0 .Hence, the coefficients (<ref>) of the functionsD_e=(1-a/ρ̂)^b/a, C_e=(1-a/ρ̂)^1-b/a, V_e=1+v_1/ρ̂+v_2/ρ̂^2+v_3/ρ̂^3+⋯ ,depend only on α and ν=μ/M and reduce to b=2 , v_1=-2α^2 ,a=2ℛ ,v_2=2α^4-2α^2ℛ ,v_3=4/3α^2(3α^2ℛ-(2+2α^2+α^4)-ν(14+12α^2-2α^4)) , with ℛ(ν)=√((1+α^2)(1+2(3-α^2)ν)) .§.§.§ An improved V_e functionAs discussed in subsection <ref>, the effective dynamics is a ν-deformation of a ST test-body problem, which, in the present case, describes a test particle m_*(φ)=μe^α(φ-φ_0) orbiting around a central body M_*(φ)=M e^α(φ-φ_0), where μ=m_A^0m_B^0/M and M=m_A^0+m_B^0, see (<ref>) and below.Therefore, in keeping with our approach consisting in centering as much as possible the effective dynamics on the test-body problem, we can “improve" V_e by factorizing out its exact, ν=0 expression :V_e= V_exact^ν=0P(ν) , P(ν)=1+p_1/ρ̂+p_2/ρ̂^2+p_3/ρ̂^3+⋯ ,where, by definition, see (<ref>),V_exact^ν=0≡(m_*(φ_e)/m_*^0)^2=e^2αφ_e ,and where φ_e is the scalar field generated by the central body, see (<ref>) :φ_e=φ_0+α/2√(1+α^2)ln(1-2√(1+α^2)/ρ̂) ,ρ̂=ρ/M .The 2PK identification of (<ref>)-(<ref>) with (<ref>)-(<ref>) gives thenV_e=(1-2√(1+α^2)/ρ̂)^α^2/√(1+α^2)P(ν) , P(ν)=1+p_1/ρ̂+p_2/ρ̂^2+p_3/ρ̂^3 , withp_1=0 , p_2=2α^2[ℛ(0)-ℛ(ν)] , p_3=-8/3α^2(7+6α^2-α^4)ν ,where P(ν=0)=1. In doing so, in the test-mass limit, D_e, C_e as well as V_e reduce to their exact, non perturbative expressions, to which they are smoothly connected.§.§.§ The ST-EOB orbital frequency at the ISCOWe now have on hands all the necessary material to study the ISCO location, u_ISCO≡ M/ρ_ISCO, and associated orbital frequency, MΩ_ISCO, as defined in the previous subsection, using (<ref>), (<ref>) and (<ref>). The results are even in α, as expected from (<ref>) and (<ref>) and are gathered in figure <ref>, for 0<α^2<1. The limit α=0 reduces to general relativity. When moreover ν=0, one recovers the well-known Schwarzschild values u_ISCO=1/6, MΩ_ISCO=0.06804 (since then the Just and Droste-Schwarzschild coordinates coincide, see comment below <ref>). Note that when α=0 but ν≠ 0, u_ISCO is less than 1/6. This does not contradict the general relativity results of Buonanno and Damour <cit.>, who worked in Droste coordinates rather than Just's ; rather, this illustrates the fact that the effective radii are physically irrelevant, contrarily to the orbital frequency MΩ_ISCO which is an observable : for α=0 and for all ν≠ 0, the ISCO frequency turns out to be always larger than the Schwarzschild one (see right panel of figure <ref>), as in <cit.>. For instance, when ν=1/4, we find MΩ_ISCO=0.07919, i.e. slightly higher than the value 0.07340 quoted in <cit.>. The ∼ 7% difference in the numerical values is reasonable considering that the two resummations (see (<ref>)) are different and built on 2PK information only.Now, when α≠0, i.e. when the scalar field is switched on, the ISCO frequency increases roughly linearly in α^2, as can be seen from the right panel of figure <ref>, with a slope.d(MΩ_ISCO)/d(α^2)|_ν=1/4≃ 0.13and.d(MΩ_ISCO)/d(α^2)|_ν=0≃ 0.063 . Interestingly, when restricted to a perturbative regime α<<1, these results are qualitatively consistent with the ones obtained from the distinct, GR-centered resummation of <cit.>, where ST effects were considered as perturbations of general relativity. There, we started from the best available EOB-NR metric, known in GR at 5PN order, see <cit.>, <cit.>, and <cit.>. We then perturbed this effective metric by scalar-tensor 2PK corrections and studied their impact on the strong field dynamics. The ISCO frequency was also found there to increase linearly with the “PPN", Eddington parameterϵ_1PK≡⟨β̅⟩-γ̅_AB (which reduces to ϵ_1PK∼ 2α^2 in the present case, see (<ref>), (<ref>) and (<ref>)), the slope being numerically of the same order of magnitude,hence illustrating the robustness of the EOB description of the strong field regime.[In particular, we found d(G_ABMΩ)/dϵ_1PK≃ 0.13 in the equal-mass case. In the present paper we will not proceed to any detailed, quantitative comparison of the two resummations since the present ST-centered approach is limited in this section to the JFBD case and since paper 1 included some extra 5PN GR information. ] More importantly, we have developped, throughout this paper, a ST-centered EOB Hamiltonian that reduces to the exact test-body Hamiltonian in the test-mass limit. In consequence, the ISCO predictions are well-defined even when |α|∼ 1, that is, can be pushed to a regime that strongly departs from general relativity : there, the estimated ISCO location and frequency significantly deviate from the GR ones and remain smoothly connected to the test-mass (ν=0) limit (see figure <ref>), which we know exactly even in the strong field regime.[It must be noted that when α>α_ crit≃ 1.6, the exact test-body problem (which is reached when ν=0) does not feature any ISCO anymore, since then (<ref>) has no root. This phenomenon is encompassed by our mapping ; when ν is non zero and increases, the value of α_ crit smoothly decreases to reach α_ crit(ν=1/4)≃ 1.03. ] We hence have illustrated, in the simple case of the Jordan-Fierz-Brans-Dicke theory, the complementarity of two EOB resummations of the scalar-tensor dynamics :(i) The first one, introduced in paper 1, which is built on rich (5PN) general relativity information, is oriented towards regimes where ST effects are considered as perturbations of GR [while the dynamics is ill-defined in non-perturbative regimes ; this necessitates, e.g., the use of appropriate Padé resummations of the ST perturbations as soon as ϵ_1PK≳ 10^-1, see <cit.> for details].(ii) The second, ST-centered one, that we have developped throughout this paper, which has been shown to be well-suited to describe regimes that may depart strongly from general relativity ; the price to pay being that it is based on 2PK information only.§ CONCLUDING REMARKS The reduction to a simple, effective-one-body motion has been a key element in thetreatment of the two-body problem in general relativity. In the pionnering 1998 paper <cit.> of Buonanno and Damour, the 2PN effective dynamics was found to be a ν-deformation of the test-body problem in GR, namely, the geodesic motion of a test particle μ in the Schwarzschild metric generated by a central body M. Remarkably, the fruitfulness of the EOB approach spreads beyond the scope of general relativity : indeed, by means of a canonical transformation and the same EOB quadratic relation (<ref>), we reduced the 2PK two-body dynamics in scalar-tensor theories to a ν-deformed version of the ST test-body problem ; namely, the motion of atest particle [μ, α_*^0, β_*^0, β'_*^0] orbiting in the fields of a central body [M, A_*^0]. The present mapping has led, just like that of paper 1 <cit.>, to a much simpler and compact description of the two-body dynamics in the 2PK regime, “gauging away" the irrelevant information in a canonical transformation.The (conservative) dynamics derived from the two ST-EOB Hamiltonians presented in <cit.> and in the present paper are, by construction, canonically equivalent at 2PK order but, when taken as being exact, they define two distinct resummations of the dynamics in the strong field regime. The fact that both lead to consistent ISCO predictions (in their overlapping ST regimes) is a hint that they may have captured accurately some of the strong field features of binary coalescence in ST theories.To summarize, we have on hands two complementary EOB dynamics : (i) the geodesic motion in an effective metric in Schwarschild-Droste coordinates, encompassing the most accurate (5PN) GR information, which is particularly well-suited to test scalar-tensor theories when considered as parametrised corrections to general relativity <cit.>, and (ii) a ST effective test-body problem, in Just coordinates, that allows to investigate regimes that depart strongly from GR, as was illustrated by the JFBD example (see subsection <ref>). An exhaustive study of generic ST theories (that depend on five parameters (<ref>)) is left to future works. Note that one cannot perform the 2PK Droste-Just coordinate change r^2=C_eV_eρ^2 without spoiling either the resummation towards the ST test-body problem of (ii) or the 5PN accurate GR information of (i). Now, Solar System and binary pulsar experiments have already put stringent constraints on ST theories, namely,(α_A^0)^2<4× 10^-6 for any body A, and α^2<2× 10^-5 in (non self-gravitating) JFBD theory, see, e.g., <cit.> and <cit.>). Since the parameters (<ref>) contain terms that are all driven by at least (α_A/B^0)^i, i≥ 2, these constraints seem to imply that scalar-tensor effects are negligible. However, gravitational wave astronomy allows to observe new regimes of gravity that might escape these constraints. For example, stars that are subject to dynamical scalarization <cit.> can develop nonperturbative α_A^0 parameters during the few last orbits before plunge (they can numerically reach order unity <cit.>), that is, in the strong field regime which is precisely explored by our EOB approach. Also, from the cosmological point of view, GR is indeed an attractor of ST theories <cit.>, <cit.>, and hence, gravitational wave detectors, which are designed to observe sources at high redshifts can probe epochs when ST effects may have been stronger. Hence, the tools developped in the present paper, which goes beyond the scope of <cit.>, could turn out to become useful in practice.Finally, we recall that SSS black holes cannot carry scalar hair in the class of ST theories we are considering here (provided that the no hair theorems hold in the highly dynamical regime of a merger), see, e.g., the comments below equation (<ref>) and references quoted there. An interesting alternative would be to induce hair by means of a massless gauge vector field, as for, e.g., Einstein-Maxwell-Dilaton theories <cit.> <cit.>, which will be the subject of future works. § ACKNOWLEDGEMENTSI am very grateful to Nathalie Deruelle who guided me throughout this project, and who carefully read and commented the manuscript of the present paper.§ CANONICALLY-TRANSFORMED EFFECTIVE HAMILTONIANSPerforming the canonical transformation (<ref>-<ref>), the effective 2PK Hamiltonian (<ref>) is rewritten in the intermediate coordinate system (q,p)→(Q,p) :Ĥ_e=1+(𝒫^2/2+v_1-b/2R̂)+Ĥ_e^ 1PK+Ĥ_e^2 PK+⋯whereĤ_e^1PK=p̂_r^4 (2 α _1+3 β _1)-p̂_r^2P^2 (α_1+3 β _1)+ P^4 (-α _1-1/8)+1/4R̂[P^2 (2 a+2 α _1 b-3 b-4 γ _1-(2 α _1+1) v_1)-2 p̂_r^2 (a-2 α _1 (b-v_1)-3 β _1 (b-v_1)-2 γ _1)]+1/8R̂^2[b (-2 a+b+4 γ _1)-2 v_1 (b+2 γ _1)-v_1^2+4 v_2] , Ĥ_e^2PK=-1/2p̂_r^6 (36 α _1 β _1+12 α _1^2+27 β _1^2-4 β _2-10 γ _2)+1/2p̂_r^4P^2 (2 α _1 (9 β _1-1)+8 α _2+27 β _1^2-3 β _1+2 β _2-10 γ _2)+1/2p̂_r^2P^4 (α _1 (18 β _1+1)+9 α _1^2-6 α _2+3 β _1-6 β _2)+1/16(24 α _1^2+8 α _1-16 α _2+1)P^6+1/16R̂[8 p̂_r^4 (2 α _1 (3 (a-2 b β _1-b-2 γ _1)+(6 β _1-1) v_1)+3 β _1 (3 a-3 b-6 γ _1-v_1)+2 b β _2+5 b γ _2-4 α _1^2 (b-v_1)-9 β _1^2 (b-v_1)+4 δ _2-2 β _2 v_1-5 γ _2 v_1+6 ϵ _2)-4 p̂_r^2P^2 (-2 α _1 (-3 a-6 b β _1+6 b+6 γ _1+(6 β _1+2) v_1)+18 a β _1-a-27 b β _1-6 b β _2-36 β _1 γ _1+8 α _1^2 (b-v_1)-8 α _2 (b-v_1)+2 γ _1+4 δ _2-9 β _1 v_1+6 β _2 v_1+12 ϵ _2)+ P^4 (α _1 (-24 a+36 b+48 γ _1)-4 a-8 α _1^2 b+8 α _2 b+5 b+8 γ _1-16 δ _2+(8 α _1^2+12 α _1-8 α _2+3) v_1)]+1/16R̂^2[ P^2 (8 a^2+8 α _1 a b+v_1 (-4 a+8 α _1 (b+2 γ _1)+6 b+12 γ _1-8 δ _2)-18 a b-24 a γ _1-4 α _1 b^2+9 b^2-16 α _1 b γ _1+36 b γ _1+8 b δ _2+24 γ _1^2-16 η _2-4 (4 α _1+1) v_2+(4 α _1+3) v_1^2).-4 p̂_r^2 (2 a^2-6 a b β _1-v_1 (a+4 α _1 (b+2 γ _1)+6 β _1 (b+2 γ _1)-2 γ _1-4 δ _2-6 ϵ _2)+α _1 (2 b (-2 a+b+4 γ _1)+8 v_2).-3 a b-6 a γ _1+3 b^2 β _1+12 b β _1 γ _1+6 b γ _1-4 b δ _2-6 b ϵ _2+6 γ _1^2-4 η _2-v_1^2 (2 α _1+3 β _1)+12 β _1 v_2)]+1/48R̂^3[-3 v_1 (2 a b-b^2-8 γ _1 (b+γ _1)+8 η _2+4 v_2)-12 (b γ _1 (-2 a+b+2 γ _1)-2 b η _2+v_2 (b+4 γ _1))-b (b-4 a) (b-2 a)+3 v_1^2 (b+4 γ _1)+3 v_1^3+24 v_3] . unsrt 10Damour:2016bks Thibault Damour and Alessandro Nagar. 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"authors": [
"Félix-Louis Julié"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170927214348",
"title": "Reducing the two-body problem in scalar-tensor theories to the motion of a test particle : a scalar-tensor effective-one-body approach"
} |
Exact path-integral evaluation of locally interacting systems: The subtlety of operator ordering Nobuhiko Taniguchi December 30, 2023 ================================================================================================ Density estimation is a classical problem in statistics and has received considerable attention when both the data has been fully observed and in the case of partially observed (censored) samples. In survival analysis or clinical trials, a typical problem encountered in the data collection stage is that the samples may be censored from the right. The variable of interest could be observed partially due to the presence of a set of events that occur at random and potentially censor the data. Consequently, developing a methodology that enables robust estimation of the lifetimes in such setting is of high interest for researchers.In this paper, we propose a non-parametric linear density estimator using empirical wavelet coefficients that are fully data driven. We derive an asymptotically unbiased estimator constructed from the complete sample based on an inductive bias correction procedure. Also, we provide upper bounds for the bias and analyze the large sample behavior of the expected 𝕃_2 estimation error based on the approach used by Stute (1995), showing that the estimates are asymptotically normal and possess global mean square consistency.In addition, we evaluate the proposed approach via a theoretical simulation study using different exemplary baseline distributions with different sample sizes. In this study, we choose a censoring scheme that produces a censoring proportion of 40% on average. Finally, we apply the proposed estimator to real data-sets previously published, showing that the proposed wavelet estimator provides a robust and useful tool for the non-parametric estimation of the survival time density function.§ INTRODUCTION Density estimation is a classical problem in statistics and has received considerable attention when both the data has been fully observed and also in the case of partially observed (censored) samples. See <cit.> for thorough discussions about this topic. In areas such as survival analysis, the estimate of the lifetime density function is of a major importance. In fact, the knowledge of how the lifetimes behave in medical follow-up research or reliability analysis is paramount to get insights, draw conclusions, derive results, make comparisons and/or characterize the underlying death/failure process.In general, the density estimation problem can be approached from either a parametric or non-parametric perspective. In the first case, an assumption is made about the particular distribution or family of distributions to which the density of interest belongs. As can immediately be observed, that approach causes the estimated function to be completely dependant on the such assumption which may prove of high benefit in the case when it is correct or close-to correct. However, if the elicited family for the target density is not correct, the parametric approach may lead to unsatisfactory results.Because of the uncertainty about parametric family, the non-parametric approach for density estimation has become a popular topic of research in statistics. In particular, popular methods for density estimation include kernel and nearest neighbors methods <cit.>. Another approach for the aforementioned problem consists of the use of orthogonal series (see <cit.>). In this approach. wavelets can be utilized since they can generate orthonormal bases for functions belonging to 𝕃_2(ℝ).One of the first uses of wavelets in density estimation could be traced back to papers by Doukhan and Leon (1990), Antoniadis and Carmona (1991) Kerkyacharian and Picard (1992) and Walter (1992). Moreover, due to their locality in both time and frequency and their exceptional approximation properties, wavelets provide a good choice for density estimation. See e.g. Meyer (1992), Daubechies (1992)<cit.>, Donoho and Johnstone (1994, 1995, 1998) for detailed discussions about the properties of wavelets in this context. Also, in Vidakovic (1999)<cit.> an extensive and thorough discussion of wavelets and their application in statistical modeling can be found.Even though wavelets offer major advantages for curve estimation, there is a potential problem associated with their use in density estimation: there is no guarantee that the estimates are positive or integrate to 1 when using general scaling functions ϕ. As described in <cit.>, the negative values may appear often in the tails of the target distribution. Nonetheless, that can be addressed; a possible remedial approach is the estimation of the square root of the density which allows then to square back to get a non-negative estimate integrating to 1 (as can be see in Pinheiro and Vidakovic (1997) <cit.>).In survival analysis or clinical trials, a typical problem encountered in the data collection stage is that the samples may be censored from the right. The variable of interest may be prevented to be fully observed due to the presence of random events (typically assumed to be independent of the variable of interest) and potentially censor the data. A common example of right censoring in clinical trials is the situation in which a patient leaves the study before its termination or was still alive by the end of the observation period. In these cases, only a subset of the observations are fully observed lifetimes; the others are partially observed and it is only known that the actual lifetime was greater than equal to the time at which the subject ceased to be observed (i.e. the censored time).Let X_1,...,X_N be i.i.d. survival times with a common unknown density function f. Also, let T_1,...,T_N be i.i.d. censoring times with a common unknown density g. Typically (and in the sequel) it is assumed that for i=1,...,N X_i⊥ T_i (here, ⊥ stands for statistical independence). In the context of partially observed data, instead of fully observing X_1,...,X_N, we observed an i.i.d. sequence {Y_i, δ_i}_i=1^N, where Y_i=min(X_i, T_i) and δ_i=1_(X_i≤ T_i). The function 1_(·) stands for the indicator function.In this paper, we propose a linear estimator based on an orthogonal projection onto a defined multiresolution space V_J using empirical wavelet coefficients that are fully data driven. We derive an asymptotically unbiased estimator constructed from the complete sample based on a an inductive bias correction. Also, we provide estimates for the bias and large sample behavior of the expected 𝕃_2 error based on the approach used by Stute (1995). In addition, we evaluate the performance of the proposed estimator via a simulation study using different exemplary unimodal and multimodal baseline distributions under different sample sizes. For this purpose, we chose an exponential censoring scheme that produces a censoring proportion of 40% on average. Finally, we apply the proposed estimator to real data-sets previously used in other published results in the field of non-parametric density estimation.Our results are based on wavelets periodic on the interval [0,1] and are derived under the assumption that both densities f and g are continuous and the survival function of the censoring random variable T is bounded from below by an exponentially decaying function. Also, we assume that the scaling function ϕ is absolutely integrable and the multiresolution space index J used for the projection is chosen as a function of the sample size N as J=⌊log_2(N)-log_2(log(N)) ⌋. The only assumption that we impose on the target density f is that it belongs to the s-sobolev space H^s.§.§ Overview of previous and current work in the area In the context of wavelets applied to density estimation with complete data, Donoho, et al. (1992) <cit.> proposed a wavelet estimator based on thresholded empirical wavelet coefficients and investigate the minimax rates of convergence over a wide range of Besov function classes B_σ p q. They choose the resolution of projection spaces such that the estimator achieves the proper convergence rates. As it can be seen in recent literature, their work is fundamental for subsequent research in the field.A work by Vanucci (1998) <cit.> provides overview of different wavelet-based density estimators, emphasizing their properties and comparison with classical estimators. In her paper, the author provides a general description of an orthonormal wavelet basis, focusing on the properties that are essential for the construction of wavelet density estimators. Also, a description of linear and thresholded density estimators is provided.This works constitutes a comprehensive reference for density estimation in the context of complete data.Following the available results in the context of complete-data density estimation (i.e. no censoring), Pinheiro and Vidakovic (1997) <cit.> propose estimators of the square root of a density based on compactly supported wavelets. Their estimator is a bona-fide density with 𝕃_1 norm equal to 1, taking care of possible negative values resulting from the usual estimation of the density f.Now in the context of density estimation with censored data, Antoniadis et al. (1999) <cit.> proposed a wavelet method based on dividing the time axis into a dyadic number of intervals and counting the number of occurrences within each one. Then, they use wavelets smoothers based on wavelets on the interval (see <cit.>) to get the survival function of the observations. Also, they obtain the best possible asymptotic mean integrated square error (MISE) convergence rate under the assumption that the target density f is r-times continuously differentiable and the censoring density g is continuous.Later on, Li (2003)<cit.> provides a non-linear wavelet-based density estimator under random censorship that uses a thresholded series expansion of the sub-density f_1(x)=f(x)1_{x≤ T} where T<τ_H and τ_H=inf{x : F_Y(x)=1 }. This approach is based on compactly supported ϕ and ψ (father and mother wavelet, respectively) and detail coefficients d_jk are thresholded according to d̃_jk=d̂_jk1_{|d̂_jk |>δ} for a suitable defined threshold δ and parameter j=q for the wavelet expansion. In his work, Li provides and asymptotic expansion for the MISE and calculate the convergence rates under smoothness and regularity assumptions on the target density f. This work is then further extended in Li (2007) <cit.>, where the minimax optimality of the thresholded wavelet-based estimator is investigated over a large range of Besov function classes.One of the most recent works in the context of censored data was developed by Zou and Liang (2017) <cit.>. They define a non-linear wavelet estimator for the right censoring model in the case when the censoring indicator δ is missing at random. They develop an asymptotic expression for the MISE which is robust under the presence of discontinuities in f. Their estimator reduces to the one proposed by Li (2003) when the censoring indicator missing at random does not happen and a bandwidth in non-parametric estimation is close to zero. §.§ About Periodic Wavelets For the implementation of the functional estimator, we choose periodic wavelets as an orthonormal basis. Even though this kind of wavelets exhibit poor behaviour near the boundaries (when the analyzed function is not periodic, high amplitude wavelet coefficients are generated in the neighborhood of the boundaries) they are typically used due to the relatively simple numerical implementation and compact support. Also, as was suggested by Johnstone (1994), this simplification affects only a small number of wavelet coefficients at each resolution level.Periodic wavelets in [0,1] are defined by a modification of the standard scaling and wavelet functions:ϕ^per_j,k(x)=∑_l ∈ℤϕ_j,k(x-l),ψ^per_j,k(x)=∑_l ∈ℤψ_j,k(x-l) .It is possible to show, as in <cit.>, that {ϕ^per_j,k(x), 0≤ k ≤ 2^j-1 , j≥ 0} constitutes an orthonormal basis for 𝕃_2[0,1]. Consequently, ∪_j=0^∞ V_j^per=𝕃_2[0,1], where V_j^per is the space spanned by {ϕ^per_j,k(x), 0≤ k ≤ 2^j-1 }. This allows to represent a function f with support in [0,1] as: f(x)=⟨ f(x),ϕ^per_0,0(x) ⟩ϕ^per_0,0(x) + ∑_j≥ 0∑_k=0^2^j-1⟨ f(x),ψ^per_j,k(x) ⟩ψ^per_j,k(x) .Also, for a fixed j=J, we can obtain an orthogonal projection of f(x) onto V_J denoted as P_J(f(x)) given by: P_J(f(x))=∑_k=0^2^J-1⟨ f(x),ϕ^per_J,k(x) ⟩ϕ^per_J,k(x) Since periodized wavelets provide a basis for 𝕃^2([0,1]), we have that ∥ f(x) - P_J(f(x)) ∥_2→ 0 as J →∞. Also, it can be shown that ∥ f(x) - P_J(f(x)) ∥_∞→ 0 as J →∞. Therefore, we can see that P_J(f(x)) uniformly converges to f as J →∞. Similarly, as discussed in <cit.> it is possible to assess the approximation error for a certain density of interest f using a truncated projection (i.e. for a certain chosen detail space J). For example, using the s-th Sobolev norm of a function defined as: ∥ f(x) ∥_H^s=√(∫(1+|x|^2)^s|f(x)|^2dx) ,one defines the H^s sobolev space, as the space that consists of all functions f whose s-Sobolev norm exists and is finite. As it is shown in <cit.>: ∥ f(x) - P_J(f(x)) ∥_2≤ 2^-J· s·∥ f ∥ _H^s[0,1] .From (<ref>), for a pre-specified ϵ>0 one can choose J such that ∥ f(x) - P_J(f(x)) ∥_2≤ϵ. In fact, a possible choice of J could be: J ≥ -⌈1/slog_2(ϵ/∥ f ∥ _H^s[0,1]) ⌉ .Therefore, it is possible to approximate a desired function to arbitrary precision using the MRA generated by a wavelet basis. In this context, extensive work has been done regarding the proper choice of the scale level J for the estimator in the MRA. In fact, <cit.> suggests that the choice J=⌊log_2(N)-log_2(log(N)) ⌋ can guarantee consistency of the estimator, under the proper regularity conditions on the scaling functions and underlying density f.§ SURVIVAL DENSITY ESTIMATION FOR RIGHT-CENSORED DATA USING PERIODIZED WAVELETS§.§ Problem statement, assumptions and derivation of the estimator for a density f(x). Consider a sample of iid lifetimes (non-negative) of the form X̃_1,...,X̃_N drawn from a random variable X̃∼f̃(·), with unknown density f̃∈𝕃_2(ℝ).Furthermore, let τ_X̃=inf{x̃:F̃_X̃(x̃)=1 }, where F̃_X̃(x̃) corresponds to the cumulative density function (cdf) of the random variable X̃.Define the target density (i.e. the density to be estimated) as f̃_c(x̃)=f̃(x̃)1_{x̃≤τ_X̃}, which corresponds to f̃(·) constrained to the interval [0,τ_X̃]. This definition implies that f̃_c(x̃)=f̃(x̃), for x̃≤τ_X̃.From the observed sample X̃_1,...,X̃_N, and a pre-specified τ>0, define the normalized random variable X=1/τX̃. Then, it follows:f_X(x)=τ f_X̃(τ x)1_{x≤τ_X̃/τ} , for the domain-restricted density f̃_c(x̃). *Remarks * If τ = τ_X̃ the normalized random variable X has support in [0,1] with density given by f(x)=f_X(x).* In practice, since f̃ is not known, it is possible to select τ=max{X̃_1,...,X̃_N}; this, since in general X̃_(N)→^ℙτ_X̃ where the operator →^ℙ denotes convergence in probability.* Note that the definition f̃_c(x̃)=f̃(x̃)1_{x̃≤τ_X̃} corresponds exactly to the conditional density f̃_X̃|X̃≤τ_X̃(x̃). In the sequel, it will be assumed that the random variable X was obtained presented above, with a probability density of the form (<ref>).§.§.§ Representing f(x) using WaveletsUsing a multiresolution analysis (MRA) based on periodized wavelets in [0,1], the density f(·) can be expressed as:f(x)=∑_j∈ℤ∑_k≥ 0d_jk·ψ_jk^per(x) . Using the hierarchical structure of the MRA, for a pre-specified multiresolution scale J=J_0, (<ref>) can be expressed as:f(x)=∑_k∈ℤc_J_0,k·ϕ_J_0,k^per(x)+∑_j≥ J_0∑_k∈ℤd_jk·ψ_jk^per(x) , for ϕ_jk^per(x)=2^j/2ϕ^per(2^jx-k), and ψ_jk^per(x)=2^j/2ψ^per(2^jx-k) for j,k∈ℤ.Because periodic extensions of wavelets in [0,1] are used, the support of the scaling function ϕ_jk^per(x) and the wavelet function ψ_jk^per(x) is[k· 2^-j,(k+1)· 2^-j] where k=0,...,2^j-1, and by the Strang-fix condition j≥ 0.From (<ref>), the summation over the MRA scale index j goes from J_0 to ∞. This implies that it is possible to approximate f(·) by truncating the summation up to scale index J^*. Therefore, it follows:f̂_J^*(x)=∑_k∈𝐊(J_0)c_J_0,k·ϕ_J_0,k^per(x)+∑_j≥ J_0^J^*∑_k∈𝐊(j)d_jk·ψ_jk^per(x) , where 𝐊(J_0)={k∈ℕ | 0≤ k≤2^J_0-1} and 𝐊(j)={k∈ℕ | 0≤ k≤2^j-1}. In the sequel, the value of J^* will be assumed to be selected as a function of the sample size N.In the wavelet series approximation of f(·) defined by (<ref>), the coefficients c_J_0,k and d_jk are given by the orthogonal projection of f(·) onto each subspace V_J_0^per and W_j^per in the MRA[In fact, from the MRA approach we have that V_J^*^per=V_J_0^per⊕∪_j=J_0^J^*W_j^per.]. Here, V_J_0^per and W_j^per correspond to the functional spaces spanned by {ϕ_J_0,k^per ,0≤ k ≤ 2^J_0-1 }, and {ψ_j,k^per ,0≤ k ≤ 2^j-1 , J_0≤ j ≤ J^*} respectively. Using this definitions, it follows:c_J_0,k = ∫_0^1f(x)·ϕ^per_J_0,k(x)dx=⟨ f(x),ϕ^per_J_0,k(x) ⟩ ,d_jk = ∫_0^1f(x)·ψ^per_j,k(x)dx=⟨ f(x),ψ^per_j,k(x) ⟩ . Clearly, since f is a probability density, (<ref>) and (<ref>) can be represented as:c_J_0,k = 𝔼_f[ϕ^per_J_0,k(X)] ,d_jk = 𝔼_f[ψ^per_j,k(X)]. Substituting (<ref>) and (<ref>) in (<ref>), f̂_J^*(x) takes the form:f̂_J^*(x)=∑_k∈𝐊(J_0)𝔼_f[ϕ^per_J_0,k(X)]·ϕ^per_J_0,k(x)+∑_j≥ J_0^J^*∑_k∈𝐊(j)𝔼_f[ψ^per_j,k(X)]·ψ^per_jk(x) . Using (<ref>) and assuming X_1,...,X_N∼ f(·) are iid, for f(·) unknown, it is possible to estimate the coefficients c_J_0,k and d_jk from the sample as follows:c̃_J_0,k = 1/N∑_i=1^Nϕ^per_J_0,k(X_i) ,d̃_j,k = 1/N∑_i=1^Nψ^per_j,k(X_i) . Therefore, the data-driven estimated density f̂_J^*(x) can be expressed as:f̂_J^*(x)=∑_k∈𝐊(J_0)( 1/N∑_i=1^Nϕ^per_J_0,k(X_i) ) ·ϕ^per_J_0,k(x)+∑_j≥ J_0^J^*∑_k∈𝐊(j)( 1/N∑_i=1^Nψ^per_j,k(X_i) ) ·ψ^per_jk(x) . From (<ref>), it follows that f̂_J^*(x) was constructed based on fully observed realizations of the lifetime random variable X. Therefore, a natural extension is the modification of (<ref>) to allow the introduction of partially observed (censored) samples; in particular, we will focus on the case of right-censored data.§.§ Estimating f̂_J^*(x)in the case of partially observed data. Consider a random variable X that is distributed with an unknown density f(x). Furthermore, suppose an observed sample {Y_i,δ_i}_i=1^N that is composed on both fully, and partially observed realizations of X. In the sample, Y_i is defined as:Y_i=min(X_i,T_i)i=1,...,N , for T_1,...,T_N being iid random variables from an unknown distribution T∼ g(t), which is the right-censoring sequence that causes some realizations from X to be partially observed, and is assumed to be independent of X. Also δ_i, representing the censoring indicator, is defined as:δ_i=1_(X_i≤ T_i) i=1,...,N , where 1_(X_i≤ T_i)=1 if and only if (X_i≤ T_i) and 0 otherwise. Therefore, δ_i=0 represents a life-time X_i that was observed only up to time T_i, for which we can only conclude that X_i>T_i.Since the observed data is {Y_i,δ_i}_i=1^N, from (<ref>) and (<ref>), the joint distribution of the pair ( Y,δ) can be obtained as follows:ℙ(Y≤ y,δ=1) = ℙ( min(X,T)≤ y, X≤ T ) = ∫_-∞^yℙ( T≥ x )f(x)dx = ∫_-∞^y( 1-G(x) )f(x)dx, where G(x)=ℙ( T≤ x ). Similarly, for ℙ(Y≤ y,δ=0) and a fixed y, it follows: ℙ(Y≤ y,δ=0) = ℙ( min(X,T)≤ y, X>T ) = ∫_-∞^+∞ℙ( T≤min(x,y) )f(x)dx = ∫_-∞^yℙ( T≤ x )f(x)dx + ∫_y^+∞ℙ( T≤ y )f(x)dx = ∫_-∞^yG(x)f(x)dx+G(y)∫_y^+∞f(x)dx = ∫_-∞^yG(x)f(x)dx+G(y)(1-F(y)). From (<ref>) and (<ref>) it follows:f_Y,δ(y,δ)=f(y)^δ(1-G(y))^δg(y)^1-δ(1-F(y))^1-δ . Similarly, from (<ref>), the marginal density of the complete-data sample Y can be expressed as:f_Y(y)=f_X(y)(1-G_T(y))+g_T(y)(1-F_X(y)) , where the subscripts X and T are placed to emphasize the relation between each density function and its corresponding random variable. Assuming 0<G_T(y)<1, f(x), from (<ref>) it follows that f(x) can be expressed as:f_X(y)=f_Y(y)/1-G_T(y)-(1-F_X(y))g_T(y)/1-G_T(y) . As was mentioned in <ref>, the next sections assume that the observed data has been normalized according to τ = max{Y_1,...,Y_N}, to restrict the support of the random variable X to the interval [0,1].§.§.§ Complete Data EstimatorFrom (<ref>) and (<ref>), (<ref>) and (<ref>), the wavelet coefficients c_J_0,k in the orthogonal wavelet expansion can be expressed as:c_J_0,k = ∫_0^1f(x)·ϕ^per_J_0,k(x)dx = ∫_0^1(f_Y(y)/1-G_T(y)-(1-F_X(y))g_T(y)/1-G_T(y)) ·ϕ^per_J_0,k(x)dx . Therefore:c_J_0,k=𝔼_Y[ ϕ^per_J_0,k(Y)/(1-G(Y))]-𝔼_T[ (1-F(Y))ϕ^per_J_0,k(Y)/(1-G(Y))] . Similarly, for the coefficients d_j,k, it follows:d_j,k=𝔼_Y[ ψ^per_j,k(Y)/(1-G(Y))]-𝔼_T[ (1-F(Y))ψ^per_j,k(Y)/(1-G(Y))] .*Remarks: * Expressions (<ref>) and (<ref>) are valid assuming 0<G(y)<1 for y∈ [0,1]. * In the case of non-censored data, G=δ_∞ (i.e. Dirac at ∞) and, for i=1,...,Nδ_i=1. Therefore, f_Y,δ=f(x). Thus, (<ref>) and (<ref>) collapse into 1/N∑_i=1^Nϕ_Jk^per(Y_i) and 1/N∑_i=1^Nψ_Jk^per(Y_i) respectively, which is the usual orthogonal-series density estimator scheme.Using an empirical approach as in (<ref>) and (<ref>), it follows: c̃_J_0,k=1/N∑_i=1^Nϕ^per_J_0,k(Y_i)/1-G(Y_i)-1/N∑_i=1^N1_(δ_i=0)(1-F(Y_i))ϕ^per_J_0,k(Y_i)/(1-G(Y_i)) , provided 0<G(Y_i)<1, for i=1,...,N.Finally, the data-driven estimated density f̂_J^*(x) can be expressed as:f̂_J^*(x)=∑_k∈𝐊(J_0)(1/N∑_i=1^Nα^ϕ_i·ϕ^per_J_0,k(Y_i) ) ·ϕ^per_J_0,k(x)+∑_j≥ J_0^J^*∑_k∈𝐊(j)( 1/N∑_i=1^Nα^ψ_i·ψ^per_j,k(Y_i) ) ·ψ^per_jk(x) , where:α^ϕ_i=α^ψ_i=1/1-G(Y_i)-1_(δ_i=0)(1-F(Y_i))/1-G(Y_i) , for i=1,...,N.As can be seen from (<ref>) and (<ref>), the computation of (<ref>) implies addressing the following issues: * Estimation of G(Y_i) and F(Y_i) for i=1,...,N.* Computation of α^ϕ_i, for i=1,...,N.* Computation of ϕ^per_J_0,k(Y_i) and ψ^per_j,k(Y_i) for i=1,...,N, j=J_0,...,J^* and 0 ≤ k ≤ 2^j-1. Naturally, G(Y_i) and F(Y_i)can be obtained using the Kaplan-Meier estimator, which is well known for its robustness in the presence of censored data. Similarly, ϕ^per_J_0,k(Y_i) and ψ^per_j,k(Y_i) we can computed using Daubechies-Lagarias algorithm. Denote { (Y_(i),δ̃_(i)) }_i=1^N as the ranked sample {(Y_i,δ_i)}_i=1^N with respect to Y_i, where δ̃_(i)=1-δ_(i). Using Kaplan-Meier, it follows:Ĝ_N(Y_(i)) = Ĝ(Y_(i))=∑_k=1^i( δ̃_(k)/N-k+1∏_j=1^k-1( 1-δ̃_(j)/N-j+1 )) , F̂_N(Y_(i)) = F̂(Y_(i))=∑_k=1^i( δ_(k)/N-k+1∏_j=1^k-1( 1-δ_(j)/N-j+1 )) , for i=1,...,N. Thus, the estimated density f̂_J^*(x) can be expressed as:f̂_J^*(x) = ∑_k∈𝐊(J_0)(1/N∑_i=1^Nα^ϕ_(i)·ϕ^per_J_0,k(Y_(i)) ) ·ϕ^per_J_0,k(x) +∑_j≥ J_0^J^*∑_k∈𝐊(j)( 1/N∑_i=1^Nα^ψ_(i)·ψ^per_j,k(Y_(i)) ) ·ψ^per_jk(x) , where:α^ϕ_(i)=α^ψ_(i)=1/1-Ĝ(Y_(i)) - 1_(δ_i=0)(1-F̂(Y_(i)))/1-Ĝ(Y_(i)) , for 0<Ĝ(Y_(i))<1, i∈⊂{ 1,...,N }, 𝐊(J_0)={0,1,...,2^J_0-1 }, and 𝐊(j)={0,1,...,2^j-1; j≥ J_0}. From section <ref>, for a properly chosen multiresolution index J, the estimated density f̂_J(x) can be approximated by a truncated projection P_J(f(x)) onto a multiresolution space V_J spanned by the functions {ϕ_Jk^per, 0≤ k ≤ 2^J-1}. Under this setting, f̂_J^*(x) takes the form:f̂_J(x)=∑_k=0^2^J-1c̃_̃J̃k̃·ϕ^per_J,k(x) , where:c̃_̃J̃k̃=1/N∑_i=1^Nα^ϕ_(i)·ϕ^per_J,k(Y_(i)) .§.§.§ Partial-Data Estimator assuming G(y) is known.From definition (<ref>), using an iterative bias-correction procedure it is possible to obtain an unbiased estimator for (<ref>), which is given by:f̂^PD(x)=∑_k=0^2^J-1c̃_Jk·ϕ^per_J,k(x) , where:c̃_Jk = 1/N∑_i=1^N1_(δ_i=1)/1-Ĝ(Y_i)ϕ^per_Jk(Y_i) ,and 𝔼[c̃_Jk]=c_Jk . The corresponding derivation can be found in section <ref> of the appendix.*Remark From (<ref>), it is possible to observe that the "partial data" definition comes from the fact that the estimator uses only the samples corresponding to actual observations of the survival time X, as opposed to (<ref>) which uses the complete sample Y_1,...,Y_N. A similar estimator is proposed by Efromovich in <cit.> using a fourier basis instead of wavelets.§.§ Statistical properties of the Estimator assuming G(y) is known . §.§.§ Mean Square Consistency. Now we investigate the mean-square convergence of the estimator f̂^PD(x). §.§.§ Proposition 1 Define:μ_J(x) = 𝔼[f̂^PD(x) ]=f_J(x),σ^2_J(x) =Var[ f̂^PD(x) ] . Assume the following conditions are satisfied: * The scaling function ϕ that generates the orthonormal set {ϕ_Jk^per, 0≤ k ≤ 2^J} has compact support and satisfies ||θ_ϕ(x)||_∞=C<∞, for θ_ϕ(x):=∑_r∈ℤ|ϕ(x-r)|.* ∃ F∈𝕃_2(ℝ) such that |K(x,y)|≤ F(x-y), for all x,y ∈ℝ, where K(x,y)=∑_k∈ℤϕ(x-k)ϕ(y-k).* For s=m+1, m≥1, integer, ∫ |x|^sF(x)dx<∞.* ∫ (y-x)^lK(x,y)dy=δ_0,l for l=0,...,s.* The density f belongs to the s-sobolev space W_2^s([0,1]), defined as:W_2^s([0,1])={f | f∈𝕃_2([0,1]), ∃f^(1),...,f^(s) s.t. f^(l)∈𝕃_2([0,1]), l=1,...,s }. Then, it follows:sup_f∈ W_2^s([0,1])𝔼[||f̂^PD(x)-f(x) ||_2^2]≤ C_12^J/N+C_22^-2sJ ,and for J=⌊log_2(N)-log_2(log(N)) ⌋:σ^2_J(x)=𝒪(log(N)^-1) ,𝔼[∥ f(x)-f̂^PD(x) ∥_2^2]≤ 𝒪(N^-slog(N)^s) for C_1>0 , C_2>0 independent of J and N, provided ∃ α_1 | 0<α_1<∞, C_T∈ (0,1) such that (1-G(y)) ≥ C_Te^-α_1y for y ∈ [0,1), and 0 ≤ F(y) ≤ 1 ∀ y ∈ [0,1].The proof can be found in section <ref> of the appendix.Based on (<ref>), it is possible to observe that σ^2_J(x) → 0 as N →∞,which implies that f̂^PD(x) is consistent for f(x), for all x∈[0,1] and f∈ W_2^s([0,1]).§.§.§ RemarksNote that from (<ref>), it is possible to choose the multiresolution level J such that the upper bound for the 𝕃_2 risk is minimized. In this context, it is possible to show that J^*(N)=1/2s+1log_2(2s C_2/C_1)+1/2s+1log_2(N) achieves that result. Moreover, under this choice of J, it follows:sup_f∈ W_2^s([0,1])𝔼[||f̂^PD(x)-f(x) ||_2^2]≤C̃N^-2s/2s+1 .§.§ Statistical properties for Partial Data Estimator assuming G(y) unknown. In the previous section, we showed that f^PD(x) is unbiased for f_J(x) and mean square consistent for f(x)∈ W_2^s([0,1]), assuming G known and the multiresolution index J for the orthogonal projection onto the space V_J was chosen as J=⌊log_2(N)-log_2(log(N)) ⌋. Naturally, assuming G is known may be questionable because of both the nature of the non-parametric density estimation approach, and its practical application. In most of real life cases neither the target density f, nor the censoring density g are known, so making assumptions about them could undermine the robustness and quality of the estimated functions.In this section we approach the problem of deriving the partial-data estimator using the data driven wavelet coefficients proposed in (<ref>). In particular, we investigate the statistical properties of the partial data estimator through the application the methodology proposed by Stute (1995) <cit.> that approximates Kaplan-Meier integrals by the average of i.i.d. random variables plus a remainder that decays to zero at a certain rate. §.§.§ Asymptotic unbiasedness. As was proposed in (<ref>), c̃_Jk =1/N∑_i=1^N1_(δ_(i)=1)/1-Ĝ(Y_i)ϕ^per_Jk(Y_i). Using the methodology and results proposed by Stute in <cit.>, and assumptions defined in <ref>, it follows: ∑_i=1^NW_(i)ϕ_Jk^per(Y_(i)) = 1/N∑_i=1^Nδ_iϕ_Jk^per(Y_i)γ_0(Y_i)+1/N∑_i=1^NU_i+R_N , where W_(i)=dF̂_N(x) is the Kaplan-Meier probability mass function of the random variable X based on the sample, γ_0(Y_i)=1/1-G_T(Y_i) and U_i=(1-δ_i)γ_1(Y_i)-γ_2(Y_i) for i=1,...,N.Similarly, γ_1(x)=γ_1,Jk(x) and γ_2(x)=γ_2,Jk(x) are given by the following expressions: γ_1,Jk(x)=1/1-F_Y(x)∫_x^τ_Hϕ_Jk^per(u)f_X(u)du ,γ_2,Jk(x)=∫_-∞^τ_HC(min{x,u})ϕ_Jk^per(u)f_X(u)du,where C(x)=∫_-∞^x^-g_T(u)du/(1-F_Y(u))(1-G_T(u)) . In addition, assume the following conditions are satisfied (from Stute <cit.>): ∫ϕ^2(x)γ_0^2(x)f_Y,δ=1(x)dx< ∞ ,∫ |ϕ(x)|√(C(x))f_X(x)dx< ∞ . Condition (<ref>) corresponds to the requirement of finite second moment (modified) on the scaling function ϕ(x), while condition (<ref>) incorporates a modification on the first moment of ϕ(x) with respect to f_X that allows to control de bias in ∫ϕ_Jk^per(u)f̂_N(u)du. For further details, see <cit.> and <cit.>.From the definitions above, it follows:𝔼[ ϕ_Jk^per(Y)δγ_0(Y)]=c_Jk , assuming x<τ_H for τ_H=inf{x:F_Y(x)=1}.Also, from (<ref>) and (<ref>), it follows that dF̂_N(x)=f̂_N(x); indeed:dF̂_N(x)=0 x ∉{Y_(1),...,Y_(N)} δ_(i)/N-i+1∏_j=1^i-1(1-δ_(j)/n-j+1) x = Y_(i) , i=1,...,N After some algebra, it follows:dF̂_N(x)=δ_(i)/N-i+1∏_j=1^i-1(n-j/n-j+1)^δ_(j) . Moreover, 1/1-Ĝ_N(Y_(i)) can be expressed as:1/1-Ĝ_N(Y_(i))=N/N-i+1∏_j=1^i-1(n-j/n-j+1)^δ_(j) . Therefore, putting together (<ref>) and (<ref>), it follows:δ_(i)/N(1-Ĝ_N(Y_(i)))=δ_(i)/N-i+1∏_j=1^i-1(n-j/n-j+1)^δ_(j)=dF̂_N(x) . These results altogether imply:∫ϕ_Jk^per(u)f̂_N(u)du =c̃_Jk . From Stute (1995), results (<ref>)-(<ref>) imply that (<ref>) can be expressed as:∫ϕ_Jk^per(u)f̂_N(u)du =1/N∑_i=1^Nδ_iϕ_Jk^per(Y_i)γ_0(Y_i)+1/N∑_i=1^NU_i+R_N , where U_i i.i.d. for i=1,...,N with 𝔼[U_1]=0 , 𝔼[U_1^2]=σ^2<∞ and |R_N|=𝒪(N^-1log(N)).Therefore:𝔼[ ∫ϕ_Jk^per(u)f̂_N(u)du ]= 𝔼[ 1/N∑_i=1^Nδ_iϕ_Jk^per(Y_i)γ_0(Y_i) ]+𝔼[ 1/N∑_i=1^NU_i]+𝒪(N^-1log(N)) , =c_Jk+𝒪(N^-1log(N)). Thus, bias(c̃_Jk)= 𝒪(N^-1log(N)), which implies that the partial data approach is asymptotically unbiased. The exact bias can be obtained by following the details presented in <cit.>.§.§.§ 𝕃_2 Risk Analysis. Following the same methodology and assumptions used in the previous section, we investigate the estimation error for the partial data approach, in the case where G is unknown.§.§.§ Proposition 2 Under the assumptions and definitions stated in <ref> and <ref>, by choosing J=⌊log_2(N)-log_2(log(N)) ⌋, it follows: sup_f∈ W_2^s([0,1])𝔼[∥ f(x)-f̂^PD(x) ∥_2^2]= 𝒪(N^-slog(N)^s) . The corresponding proofs can be found in section <ref> of the appendix.§.§.§ Remarks* Observe that by following the same methodology as in <ref>, it is possible to obtain:sup_f∈ W_2^s([0,1])𝔼[||f̂^PD(x)-f(x) ||_2^2]≤ C_12^J/N+C_22^-2sJ , for C_1=||F||_2^2e^2γ/C^2 and C_2>0, independent of N and J.* The last result implies that by choosing J^*(N)=1/2s+1log_2(2s C_2/C_1)+1/2s+1log_2(N), the 𝕃_2 risk of the estimator f̂^PD(x) (when G is unknown) is also mean square consistent, and achieves a convergence rate of the order ∼ N^-2s/2s+1. This implies that as long as the empirical survival function of the censoring random variable obtained from the Kaplan-Meier estimator is bounded from below by an exponentially decaying function, the knowledge of the its cdf does not affects the statistical properties of the estimator. §.§.§ Limiting Distribution. In this section, we investigate the limiting distribution of the partial data estimator f̂^PD(x). Similarly as in sections <ref> and <ref>, we will use results proposed in <cit.> as framework for our analysis.As seen in (<ref>), (<ref>), Theorem 1.1 of <cit.> and the SLLN (Strong Law of Large Numbers), the following results hold:1/N∑_i=1^Nδ_iϕ_Jk^per(Y_i)/1-G(Y_i) →^ℙc_Jk ,R_N →^ℙ0, where (<ref>) follows from the SLLN (assuming the expectation is finite), and (<ref>) from the fact that |R_N|=𝒪_ℙ(1/√(N)), as shown in <cit.>. Using Slutzky's theorem (see <cit.>), it follows:c̃_Jk-1/N∑_i=1^Nδ_iϕ_Jk^per(Y_i)/1-G(Y_i)-R_N=^𝔻1/N∑_i=1^NU_i , where U_i=(1-δ_i)γ_1(Y_i)-γ_2(Y_i), i=1,...,N are i.i.d. zero-mean and finite variance random variables with 𝔼[ U_1^2]=σ^2. Also, from the definitions of γ_1(x) and γ_2(x), it follows that σ^2=σ^2_Jk since it depends on the scaling function ϕ_Jk^per(x).Now, by the CLT (Central Limit Theorem) it follows: 1/√(N)∑_i=1^NU_i→^𝔻 N(0,σ_Jk^2) . Combining results (<ref>), (<ref>), Slutzky's theorem implies:√(N)(c̃_Jk-c_Jk) →^𝔻 N(0,σ_Jk^2) . Similarly, it follows:√(N)(f̂^PD(x)-f(x) ) =∑_k=0^2^J-1√(N)(c̃_Jk-c_Jk)ϕ_Jk^per(x) .§.§.§ Proposition 3For c>0, β >1 and x in a neighborhood of 1, assume the following conditions hold:* (1-F_X)∼c(1-G_T)^β * C(x)≤1/(1-F_X(x))(1-G_T(x))Then, it follows:√(N)(f̂^PD(x)-f(x) ) →^𝔻 N(0 , ∑_k=0^2^J-1σ_Jk^2(ϕ_Jk^per(x))^2+2 ∑_k<lσ_J,klϕ_Jk^per(x)ϕ_Jl^per(x) ) , for k,l=0,...,2^J-1, σ_Jk^2=𝔼[((1-δ)γ_1,Jk(Y)-γ_2,Jk(Y) )^2] and σ_J,kl=𝔼[ δ^2ϕ_Jk^per(Y)ϕ_Jl^per(Y)/(1-G(Y))^2-c_Jkc_Jl], provided assumptions detailed in <ref>, (<ref>), (<ref>) are satisfied and J=⌊log_2(N)-log_2(log(N)) ⌋. The corresponding proof can be found in section <ref> of the appendix.§.§.§ Remarks * Note that condition (<ref>) indicates that there is enough information about the tails of the target density f; also, the larger the values of β, the heavier the tails of the censoring distribution, compared to the tails of the survival time distribution.* As described in <cit.> and <cit.>, the condition of β>1 is required so that the bias of c̃_Jk-c_Jk achieves a convergence rate better that a N^-1/2 for some non-vanishing a which may cause that (<ref>) is no longer valid.* As it can be seen in (<ref>), the fact that f̂^PD(x) presents asymptotic normality brings to discussion the possibility that the estimates may be negative, as was previously mentioned in <ref> and discussed in <cit.>. § SIMULATION STUDY In this section, we investigate the estimation performance of f̂^PD(x) and evaluate it with respect to the AMSE (Average Mean Squared Error) via a simulation study. For this purpose, we choose a set of exemplary baseline functions that resemble important features that continuous survival times that can be encountered in practice could posses. To simplify the simulations, we chose functions that are supported in an interval close to [0,1]. A brief description of each chosen function follows:* Delta. This corresponds to a R.V. X ∼ N(0.5 ,0.02^2). The idea is to have an extreme spatially heterogeneous curve that has support over a small region. The goal is to represent situations when a short but abrupt deviation from a process may happen.* Normal. This corresponds to the usual Normal distribution with parameters μ=0.5 and σ=0.15.* Bimodal. This corresponds to a mixture of 2 Normal distributions and has the form f(x)=0.5X_1 +0.5X_2 where X_1∼ N(0.4 ,0.12^2) and X_2∼ N(0.7 ,0.08^2).* Strata. This corresponds to a mixture of 2 Normal distributions and has the form f(x)=0.5X_1 +0.5X_2 where X_1∼ N(0.2 ,0.06^2) and X_2∼ N(0.7 ,0.08^2). The idea is to represent a function that is supported over 2 separate subintervals.* Multimodal. This functions corresponds to a mixture of 3 Normal distributions and has the form f(x)=1/3 X_1 +1/3 X_2 +1/3 X_3 where X_1∼ N(0.2 ,0.06^2),X_2∼ N(0.5 ,0.05^2) andX_3∼ N(0.7 ,0.05^2). The idea of this function is to represent multimodal survival times which are expected to occur in heterogeneous populations. An advantage of using simulated data in the case of censored data is that the values for both X and T are known for all samples; also, the controlled-environment approach allows the investigation of the estimator's performance for different sample sizes and censoring schemes. For testing purposes, we choose a censoring random variable T ∼ Exp(λ) with λ=0.8, which produces approximately 45% censored samples at each generated datasets. Also, we use samples sizes N=100,200,500,1000 and measure the global error given by: M̂ŜÊ=1/B∑_b=1^B1/N∑_i=1^N(f(x_i)-f̂_N,b(x_i) )^2 , where B is the number of replications of the experiment and N is the number of samples. For all experiments we choose B=1000 and the wavelet filter Symmlet5. To implement simulations, we generate 2 independent random samples {X_i}_i=1^N and {T_i}_i=1^N. X_i random variables were drawn from each one of the aforementioned distributions, while T_i∼^i.i.d.Exp(λ). Also, we included in the simulation study the complete data estimator as we found of interest to observe its performance and compare it to the partial data approach. §.§ Simulation Results. In this section, we summarize the results obtained for each baseline distribution. In particular, the following results are provided:* Tables <ref> to <ref> present details for AMSE results obtained for each baseline distribution used in the study.* In figures <ref> - <ref>, dashed lines (red and blue) correspond to the average estimates for f̂^PD(x), computed at each data point x from all B=1000 replications. The black line indicates the actual density function and the light blue and blue continuous lines represents the best estimates among all replications (i.e. the one with the smallest AMSE).* In figures <ref> - <ref>, dashed lines (red and green) correspond to the empirical 95% quantiles computed at each data point x from all B=1000 replications, for f̂^b(x) and f̂^PD(x) respectively. The blue and magenta lines show the average density estimates for the complete and partial data approach, respectively. The black line indicates the actual density function.* Figure <ref> shows the AMSE vs. sample size plot.* Figure <ref> exemplifies the asymptotic normality behavior of the density estimates, as proposed in <ref>. §.§ Remarks and comments. * From the resulting figures, it is possible to observe that the proposed estimator is able to recover the underlying density in the presence of right-censored observations. Also, estimates (Best and Mean) with respect to the sample size, suggests a bias effect in the vicinity of the underlying distribution modes.* In terms of the sensibility of the estimator's performance to the scaling functions, we observed during our experiments that results obtained using Symmlets, Coiflets and Daubechies wavelets are similar.* From the quantiles plots, the empirical quantiles of the estimated densities contain the actual values of the target density in most of its support. Moreover, for all baseline distributions except for the Multimodal, this is the case. On the contrary, the regions where the 95% empirical quantiles do not contain the true density value are observed to occur in the vicinities of the distribution modes. This could be caused by the choice of the multiresolution index J, the post-processing smoothing procedure and/or by the censoring effect.* As the sample size increases, it was observed that the interval |f̂_̂N̂_0.975(x)-f̂_̂N̂_0.025| monotonically decreases in coherence with the theoretical convergence results shown in section <ref>.* From the AMSE plot (<ref>), it is possible to observe that all baseline distributions present a similar error decay behavior. Moreover, results contained in tables <ref> to <ref>, imply that as N grows, the standard deviation and range of AMSE decays in accordance with the convergence rates proposed for both estimators.* Figure <ref>, suggest normality of the estimated density values, which is coherent with results presented in section <ref>. This property of the estimators allows the construction of confidence intervals and the application of standard statistical inference tools that could be useful in practical situations. However, to make this applicable, the Variance of f̂^PD(x) in accordance with (<ref>) needs to be estimated.* In most of presented figures it is possible to observe that at the extremes of the support sometimes the estimated density values are slightly negative. This effect is consistent with the boundary effect noted in <cit.> by Antoniadis. As was mentioned in the introduction, a possible remedial measure could be application the approach proposed by <cit.>. Another possibility is usingf̂_+(x)=max{0,f̂^PD(x) }, as proposed in <cit.>.§ REAL DATA APPLICATION AND COMPARISON WITH OTHER ESTIMATORS. In this section we consider the implementation of the proposed estimator on the datasets utilized by Antoniadis et al. in <cit.>. To compare our approach with other popular estimators, we will also use the non-parametric Kernel density estimator with optimal bandwidth and the smoothed histogram using local polynomials based on the actual samples.The first application considers the data studied by Haupt and Mansmann (1995)[The data set is available at CART for Survival Data. Statlib Archive <http://lib.stat.cmu.edu/S/survcart>.]. In their research, they analized the survival times for patients with liver metastases from a colorectal tumour without other distant metastases. In their data, they have a total of 622 patients from which 43.64% of the samples are censored. The obtained results are given in Fig.<ref> (a).Our next practical application, considers the study of marriage dissolution based on a longitudinal survey conducted in the U.S.[Data set available at <http://data.princeton.edu/wws509/datasets> and was adapted from an example in the software aML (SeeLillard and Panis (2000), aML Multilevel Multiprocess Statistical Software, Release 1.0, EconWare, LA, California.)] The unit of observation is the couple and the event of interest is the time from marriage to divorce. Interviewed and widowhood are considered as censoring events. Couples with different educational levels and ethnicity were considered. The original data considered 3371 couples with 30.61% of samples being censored. The obtained results are given in Fig.<ref> (b).From figure <ref> (a), it can be observed that the complete data estimator (in red) shows boundary effects, since after 45 months, according to the data there are almost no patients alive. However, both complete data and partial data estimators are able to catch the individual modes shown by the histogram without over smoothing as compared to the smoothed histogram (in green). Also, the estimators are able to keep the proportions between the histogram modes as compared to the Kernel density estimator with universal bandwidth (in black).From figure <ref> (b), it is possible to observe the fairly exponential behavior of the density estimates. Both the complete data and the partial data are able to follow the rate of decay of the Histogram envelope and do not overestimate the density values in the right tails, which is consistent with the data (from data, it is highly unlikely that a certain couple would last married longer than 45 years); both local polynomial and kernel density estimator fail to account for that fact, while assigning significant density to times above 40 years. § CONCLUSIONS AND DISCUSSION.This paper introduced an empirical wavelet-based method to estimate the density in the case of randomly censored data. We proposed estimators based on the partial and complete sample, showing statistical properties of bias, consistency and limiting distribution. Also, we derived convergence rates for the expected 𝕃_2 error using J=⌊(log_2(N)-log_2(log(N)))⌋ for the multiresolution index.Both estimators were implemented and tested using different baseline distributions via a theoretical simulation study, showing good performance in the presence of significantly censored data. This simulation study shows that in theory, the estimator attains the large sample behavior that was proposed: it is asymptotically unbiased and mean-square consistent. Regarding the effect of censoring in the estimates, we observed that our method is robust enough to handle censoring proportions of nearly 50% while achieving acceptable estimation results. Moreover, in the case of no censoring, the method converges to the usual orthogonal wavelet-series estimator (See remarks in section <ref>).From a real data application viewpoint, the proposed method was capable to uncover modes that were be hard to detect by other methods in the used datasets, avoiding the problem of modes over-smoothing that methods such as non-parametric kernels exhibited. Also, the estimator was capable of capturing exponential rates of decay of the underlying density, preventing the overestimation of density values in regions of the support with near-zero empirical mass.Based on the results seen in the simulation study and the real data testing, we can argue that our estimator yields interesting interpretations and results; it has good asymptotic properties and is relatively easy to implement. Also, it offers a useful and competitive alternative for the problem of density estimation with censored data, with respect to multimodal identification and exponential decay adjustment.Finally, some of the drawbacks that were observed throughout this paper were the possibility of obtaining negative values for the density estimates (highly likely at the tails) and also boundary problems resulting from the periodic wavelet extension approach. Also, another important remark worth noting is the fact that it is possible that the estimated density does no integrate to 1. Nonetheless, for most of these problems there are possible solutions such as the ones proposed in <cit.> and <cit.>. unsrt§ DERIVATION OF THE UNBIASED PARTIAL-DATA ESTIMATOR.In this section we provide the derivation for the partial-data estimator proposed in <ref>. From (<ref>) and (<ref>), it follows:𝔼(f̂_J(x))=∑_k=0^2^J-1𝔼[ c̃_̃J̃k̃]·ϕ^per_J,k(x) . Using (<ref>), the expectation in the left hand side (lhs) of (<ref>) is given by: 𝔼[ c̃_̃J̃k̃]=𝔼[ 1/N∑_i=1^Nϕ^per_J,k(Y_(i))/1-Ĝ(Y_(i))] -𝔼[ 1/N∑_i=1^N1_(δ_i=0)(1-F̂(Y_(i)))/1-Ĝ(Y_(i))ϕ^per_J,k(Y_(i)) ] . Assuming iid samplesand G(y) known, the first expectation on the rhs of (<ref>) can be obtained as:𝔼[ 1/N∑_i=1^Nϕ^per_J,k(Y_(i))/1-Ĝ(Y_(i))]=𝔼_Y[ϕ^per_J,k(Y)/1-G(Y)] . Similarly, provided iid samples, and both F(y) and G(y) known, the expectation of the second term in the rhs of (<ref>) can be obtained as: 𝔼[ 1/N∑_i=1^N1_(δ_i=0)(1-F̂(Y_(i)))/1-Ĝ(Y_(i))ϕ^per_J,k(Y_(i))]= 𝔼_Y,δ=0[(1-F(Y))ϕ^per_J,k(Y)/1-G(Y)] . Since f_Y,δ(y,δ=0)=g(y)(1-F(y)), it follows: .9 !𝔼_Y,δ=0[(1-F(Y))ϕ^per_J,k(Y)/1-G(Y)]=𝔼_T[(1-F(T))ϕ^per_J,k(T)/1-G(T)]- 𝔼_T[F(T)(1-F(T))ϕ^per_J,k(T)/1-G(T)] . Finally, combining (<ref>) and (<ref>), it follows:.9 !𝔼[ c̃_̃J̃k̃]=𝔼_Y[ϕ^per_J,k(Y)/1-G(Y)]-𝔼_T[(1-F(T))ϕ^per_J,k(T)/1-G(T)]+ 𝔼_T[F(T)(1-F(T))ϕ^per_J,k(T)/1-G(T)] . Using (<ref>) and (<ref>), (<ref>) takes the form:𝔼[ c̃_̃J̃k̃]=c_Jk+𝔼_T[F(T)(1-F(T))ϕ^per_J,k(T)/1-G(T)] , which further implies that for (<ref>), it follows:𝔼(f̂_J(x))=f_J(x)+∑_k=0^2^J-1𝔼_T[F(T)(1-F(T))ϕ^per_J,k(T)/1-G(T)]ϕ^per_J,k(x) . To facilitate notation, define b_J,k=𝔼_T[F(T)(1-F(T))ϕ^per_J,k(T)/1-G(T)]. Thus, (<ref>) can be represented as:𝔼(f̂_J(x))=f_J(x)+∑_k=0^2^J-1b_J,k·ϕ^per_J,k(x) . Using the same approach as in (<ref>), b_J,k (i.e. the wavelet coefficient that define the bias of f̂_J(x) can be estimated from the sample as follows:b̃_J,k=1/N∑_i=1^N1_(δ_i=0)F̂(Y_i)(1-F̂(Y_i))ϕ^per_J,k(Y_i)/1-Ĝ(Y_i) . Therefore, the biased-corrected version of the estimator can be represented as:f̂_J^*(x)=f̂_J(x)-∑_k=0^2^J-1b̃_J,k·ϕ^per_J,k(x) , f̂_J^*(x)=∑_k=0^2^J-1c̃^*_J,k·ϕ^per_J,k(x) , where:.9 !c̃^*_J,k=c̃_J,k-b̃_J,k=1/N∑_i=1^N(1/1-Ĝ(Y_(i)) - 1_(δ_(i)=0)(1-F̂(Y_(i)))/1-Ĝ(Y_(i))-1_(δ_(i)=0)F̂(Y_(i))(1-F̂(Y_(i)))/1-Ĝ(Y_(i)))·ϕ^per_J,k(Y_(i)) . Note that (<ref>) can be further simplified into:c̃^*_J,k=1/N∑_i=1^N(1-1_(δ_(i)=0)(1-F̂(Y_(i)))(1+F̂(Y_(i)))/1-Ĝ(Y_(i)))ϕ^per_J,k(Y_(i)) . Computing the expectation of the bias-correction coefficient b̃_Jk, it follows:𝔼_Y[b̃_Jk]=b_Jk-𝔼_T[F(T)^2(1-F(T))/1-G(T)ϕ^per_Jk(T) ] . Therefore, the bias of b̃_Jk can be corrected by defining b̃^*_Jk=b̃_Jk+𝔼_T[F(T)^2(1-F(T))/1-G(T)ϕ^per_Jk(T) ]. Using the empirical argument as in (<ref>), b̃^*_Jk can be estimated by:b̃^*_Jk=b̃_Jk+1/N∑_i=1^N1_(δ_(i)=0)F(Y_i)^2(1-F(Y_i))/1-G(Y_i)ϕ^per_Jk(Y_i) . This implies that the updated bias-corrected estimator of b_Jk can be represented as:b̃^*_Jk=1/N∑_i=1^N1_(δ_(i)=0)F(Y_i)(1-F(Y_i))(1+F(Y_i))/1-G(Y_i)ϕ^per_Jk(Y_i) . Taking the expectation of b̃^*_Jk, it follows:𝔼_Y[b̃^*_Jk]=b_Jk-𝔼_T[F(T)^3(1-F(T))/1-G(T)ϕ^per_Jk(T) ] . Following the same methodology used to derive (<ref>), an updated bias-corrected estimate of b̃^*_Jk, denoted by b̃^**_Jk can be represented as:b̃^**_Jk= 1/N∑_i=1^N1_(δ_(i)=0)F(Y_i)(1-F(Y_i))(1+F(Y_i)+F(Y_i)^2))/1-G(Y_i)ϕ^per_Jk(Y_i) . Taking the expectation of b̃^**_Jk, it follows:𝔼_Y[b̃^**_Jk]=b_Jk-𝔼_T[F(T)^4(1-F(T))/1-G(T)ϕ^per_Jk(T) ] . This implies that the bias-corrected estimate of b_Jk represented as b̃^***_Jk=b̃^**_Jk+𝔼_T[F(T)^4(1-F(T))/1-G(T)ϕ^per_Jk(T) ] can be iteratively updated. Thus, following the same process as before, it follows:b̃^***_Jk=1/N∑_i=1^N1_(δ_(i)=0)F(Y_i)(1-F(Y_i))(1+F(Y_i)+F(Y_i)^2+F(Y_i)^3))/1-G(Y_i)ϕ^per_Jk(Y_i) . From the last set of equations, it follows that this process can be repeated sequentially, infinitely many times. This implies that:b̃̃̃_Jk=1/N∑_i=1^N1_(δ_(i)=0)F(Y_i)(1-F(Y_i))∑_l=0^∞F(Y_i)^l/1-G(Y_i)ϕ^per_Jk(Y_i) , provided 0<F(Y)<1. Therefore, it follows that ∑_l=0^∞F(Y_i)^l is a convergent series. In fact, it is a geometric power series that satisfies:∑_l=0^∞F(Y_i)^l=1/1-F(Y_i) . Therefore, this implies that (<ref>) takes the form:b̃̃̃_Jk=1/N∑_i=1^N1_(δ_(i)=0)F(Y_i)/1-G(Y_i)ϕ^per_Jk(Y_i) . Clearly, b̃̃̃_Jk is an unbiased estimate of b_Jk. Therefore, we conclude that the unbiased estimate of the c_Jk coefficient, denoted by c̃̃̃_Jk is given by:c̃̃̃_Jk=c̃_Jk-b̃̃̃_Jk=1/N∑_i=1^N1_(δ_(i)=1)/1-G(Y_i)ϕ^per_Jk(Y_i) , thus, it is possible to define the partial-data density estimator f̂^PD(x) as:f̂^PD(x)=∑_k=0^2^J-1c̃_Jk·ϕ^per_J,k(x) , where: c̃_Jk=1/N∑_i=1^N1_(δ_(i)=1)/1-Ĝ(Y_i)ϕ^per_Jk(Y_i) , which is unbiased for f_J(x), provided G(y) is known and 0<F(Y)<1.§ PROOF OF PROPOSITION 1Assume the following conditions are satisfied: * The scaling function ϕ that generates the orthonormal set {ϕ_Jk^per, 0≤ k ≤ 2^J} has compact support and satisfies ||θ_ϕ(x)||_∞=C<∞, for θ_ϕ(x):=∑_r∈ℤ|ϕ(x-r)|.* ∃ F∈𝕃_2(ℝ) such that |K(x,y)|≤ F(x-y), for all x,y ∈ℝ, where K(x,y)=∑_k∈ℤϕ(x-k)ϕ(y-k).* For s=m+1, m≥1, integer, ∫ |x|^sF(x)dx<∞.* ∫ (y-x)^lK(x,y)dy=δ_0,l for l=0,...,s.* The density f belongs to the s-sobolev space W_2^s([0,1]), s>1 defined as:W_2^s([0,1])={f | f∈𝕃_2([0,1]), ∃f^(1),...,f^(s) s.t. f^(l)∈𝕃_2([0,1]), l=1,...,s }. Then, it follows:sup_f∈ W_2^s([0,1])𝔼[||f̂^PD(x)-f(x) ||_2^2]≤ C_12^J/N+C_22^-2sJ ,and for J=⌊log_2(N)-log_2(log(N)) ⌋:σ^2_J(x)=𝒪(log(N)^-1) ,𝔼[∥ f(x)-f̂^PD(x) ∥_2^2]≤ 𝒪(N^-slog(N)^s) for C_1>0 , C_2>0 independent of J and N, provided ∃ α_1 | 0<α_1<∞, C_T∈ (0,1) such that (1-G(y)) ≥ C_Te^-α_1y for y ∈ [0,1), and 0 ≤ F(y) ≤ 1 ∀ y ∈ [0,1]. §.§ Proof Note that f̂^PD(x) can be expressed as follows:f̂^PD(x)=1/N∑_i=1^Nw_iK_J(Y_i,x) , where w_i=δ_i/1-G(Y_i), and K_J(x,Y_i)=2^J∑_k∈ℤϕ(2^Jx-k)ϕ(2^Jy-k), for i=1,...,N. Since it is assumed that ∃ α_1 | 0<α_1<∞, C_T∈ (0,1) such that (1-G(y)) ≥ C_Te^-α_1y for y ∈ [0,1), this implies that 0≤ w_i≤e^α_1/C_T, for i=1,...,N.Also, it is possible to bound the 𝕃_2 risk of the estimator f̂^PD(x) as follows: 𝔼[||f̂^PD(x)-f(x) ||_2^2]≤ 2{𝔼[||f̂^PD(x)-𝔼[f̂^PD(x) ] ||_2^2]+||𝔼[f̂^PD(x) ]-f(x) ||_2^2} , where the first term in the rhs of (<ref>) corresponds to Var(f̂^PD(x)) and the second, to bias(f̂^PD(x)).§.§.§ Bound for 𝔼[||f̂^PD(x)-𝔼[f̂^PD(x) ] ||_2^2]From (<ref>), it follows:f̂^PD(x)-𝔼[f̂^PD(x)]=1/N∑_i=1^N(w_iK_J(x,Y_i)-𝔼[w_iK_J(x,Y_i)]) . Define Z_i(x)=w_iK_J(x,Y_i)-𝔼[w_iK_J(x,Y_i)] and Z̃_i(x)=K_J(x,Y_i)-𝔼[K_J(x,Y_i)]. Clearly, 𝔼[Z_i(x)]=𝔼[Z̃_i(x)]=0. This implies:|f̂^PD(x)-𝔼[f̂^PD(x) | ≤e^α_1/C_T1/N|∑_i=1^NZ̃_i(x) | , since 0≤ w_i≤e^α_1/C_T, for i=1,...,N. Therefore, it follows: |f̂^PD(x)-𝔼[f̂^PD(x) |^2 ≤ e^2α_1/C_T^21/N^2|∑_i=1^NZ̃_i(x) |^2 𝔼[∫_0^1|f̂^PD(x)-𝔼[f̂^PD(x) |^2dx ]≤ e^2α_1/C_T^21/N^2𝔼[∫_0^1|∑_i=1^NZ̃_i(x) |^2dx ] . From conditions (i) and (ii), Fubini's thorem implies:𝔼[∫_0^1|f̂^PD(x)-𝔼[f̂^PD(x) |^2dx ]≤ e^2α_1/C_T^21/N^2∫_0^1𝔼[|∑_i=1^NZ̃_i(x) |^2]dx ≤ e^2α_1/C_T^21/N∫_0^1𝔼[Z̃_1(x)^2 ]dx , where (<ref>) follows from the fact that Z̃_i(x) are iid, with 𝔼[ Z̃_i(x)]=0, and 𝔼[Z̃_i(x)^2]<∞. This, together with the application of Rosenthal's inequality implies 𝔼[|∑_i=1^NZ̃_i(x) |^2] ≤∑_i=1^N𝔼[Z̃_i(x)^2 ]=N 𝔼[Z̃_1(x)^2 ].Since 𝔼[Z̃_1(x)^2]=𝔼[K_J(x,Y_1)^2 ]-(K_Jf_Y(x) )^2≤𝔼[K_J(x,Y_1)^2 ], where K_Jf_Y(x)=∫_0^1K_J(x,u)f_Y(u)du, and the fact that |K_J(x,y) |=2^J|K(2^Jx,2^Jy)|, it follows from (<ref>) and condition (ii):𝔼[||f̂^PD(x)-𝔼[f̂^PD(x)] ||_2^2] ≤e^2α_1/C_T^21/N∫_0^1𝔼[K_J(x,Y_1)^2 ]dx∫_0^1𝔼[K_J(x,Y_1)^2 ]dx≤2^J∫_0^1[∫_-2^Jy^2^J(1-y)F^2(v)dv ]f_Y(y)dy ≤2^J||F||_2^2 . Therefore, substituting (<ref>) into (<ref>), it follows:𝔼[||f̂^PD(x)-𝔼[f̂^PD(x)] ||_2^2] ≤||F||_2^2e^2α_1/C_T^22^J/N .§.§.§ Bound for ||𝔼[f̂^PD(x) ]-f(x) ||_2^2According to corollary 8.2 <cit.>, if f∈ W_2^s([0,1]) then ||K_Jf-f||_2^2=𝒪(2^-2Js). Furthermore, assume conditions (i)-(iv) are satisfied. Since 𝔼[f̂^PD(x)]=K_Jf(x), it follows: ||𝔼[f̂^PD(x) ]-f(x) ||_2^2≤ C_2 2^-2Js . Finally, putting together (<ref>) and (<ref>), it follows:sup_f∈ W_2^s([0,1])𝔼[||f̂^PD(x)-f(x) ||_2^2]≤ C_12^J/N+C_22^-2sJ , as desired, for C_1=||F||_2^2e^2α_1/C_T^2 and C_2>0, independent of N and J.From (<ref>), by choosing J=⌊log_2(N)-log_2(log(N)) ⌋, it follows that σ^2_J(x)=𝒪(log(N)^-1). Furthermore, this also implies that sup_f∈ W_2^s([0,1])𝔼[||f̂^PD(x)-f(x) ||_2^2] = 𝒪(N^-slog(N)^2, which completes the proof.§.§.§ RemarksNote that from (<ref>), it is possible to choose the multiresolution level J such that the upper bound for the 𝕃_2 risk is minimized. In this context, it is possible to show that J^*(N)=1/2s+1log_2(2s C_2/C_1)+1/2s+1log_2(N) achieves that result. Moreover, under this choice of J, it follows: sup_f∈ W_2^s([0,1])𝔼[||f̂^PD(x)-f(x) ||_2^2]≤C̃N^-2s/2s+1 . § PROOF OF PROPOSITION 2 Under the assumptions and definitions stated in <ref> and <ref>, and choosing J=⌊log_2(N)-log_2(log(N)) ⌋, it follows:sup_f∈ W_2^s([0,1])𝔼[∥ f(x)-f̂^PD(x) ∥_2^2]= 𝒪(N^-slog(N)^s) .§.§ ProofAssume conditions (i)-(iv) established in <ref> are satisfied. Furthermore, assume ∃ γ >0 and a constant C∈(0,1) such that 1-Ĝ(y)≥ Ce^-γ y, for y∈[0,1). Note that f̂^PD(x) can be expressed as follows:f̂^PD(x)=1/N∑_i=1^Nw_iK_J(Y_i,x) , where w_i=δ_i/1-Ĝ(Y_i), and K_J(x,Y_i)=2^J∑_k∈ℤϕ(2^Jx-k)ϕ(2^Jy-k), for i=1,...,N. Since it is assumed that ∃ γ >0 and a constant C∈(0,1) such that 1-Ĝ(y)≥ Ce^-γ y, for y∈[0,1), this implies that 0≤ w_i≤e^γ/C, for i=1,...,N. Thus, following the same methodology as in <ref>, it follows that by choosing J=⌊log_2(N)-log_2(log(N)) ⌋:sup_f∈ W_2^s([0,1])𝔼[∥ f(x)-f̂^PD(x) ∥_2^2]= 𝒪(N^-slog(N)^s) . §.§.§ Remarks* Observe that by following the same methodology as in <ref>, it is possible to obtain:sup_f∈ W_2^s([0,1])𝔼[||f̂^PD(x)-f(x) ||_2^2]≤ C_12^J/N+C_22^-2sJ , for C_1=||F||_2^2e^2γ/C^2 and C_2>0, independent of N and J.* The last result implies that by choosing J^*(N)=1/2s+1log_2(2s C_2/C_1)+1/2s+1log_2(N), the 𝕃_2 risk of the estimator f̂^PD(x) when G is unknown is also mean square consistent, and achieves a convergence rate of the order ∼ N^-2s/2s+1. § PROOF OF PROPOSITION 3From (<ref>), and for N large it follows that the rhs of (<ref>) corresponds to the sum of normally distributed random variables ∼ N(0,σ_Jk^2) which is indeed a normally distributed random variable. To obtain its variance, it can be used the fact that Cov(√(N)(c̃_Jk-c_Jk) ,√(N)(c̃_Jl+c_Jl) ) = N 𝔼[(c̃_Jk-c_Jk)(c̃_Jl-c_Jl) ]. Thus, (<ref>) implies:𝔼[N(c̃_Jk-c_Jk)(c̃_Jl-c_Jl) ]=N (𝔼[c̃_Jkc̃_Jl]-c_Jkc_Jl)-(c_Jk-c_Jl)𝒪(log(N)) . Using (<ref>), it follows:c̃̃̃_Jkc̃̃̃_Jl =A_1+A_2+A_3+A_4+A_5+A_6+A_7+A_8+A_9 , where:A_1 = 1/N^2∑_i=1^N∑_j=1^Nδ_iδ_jϕ_Jk^per(Y_i)ϕ_Jl^per(Y_j)/(1-G_T(Y_i) )(1-G_T(Y_j) )A_2 = 1/N^2∑_i=1^N∑_j=1^Nδ_iϕ_Jk^per(Y_i)U_jl/1-G_T(Y_i)A_3 = 1/NR_Nl∑_i=1^Nδ_iϕ_Jk^per(Y_i)/1-G_T(Y_i)A_4 = 1/N^2∑_i=1^N∑_j=1^Nδ_jϕ_Jl^per(Y_j)U_ik/1-G_T(Y_j)A_5 = 1/N^2∑_i=1^N∑_j=1^NU_ikU_jlA_6 = 1/NR_Nl∑_i=1^NU_ikA_7 = 1/NR_Nk∑_i=1^Nδ_jϕ_Jl^per(Y_j)/1-G_T(Y_j)A_8 = 1/NR_Nk∑_i=1^NU_ilA_9 =R_NkR_Nl . From the last set of equations, it is possible to observe that the following pairs have the same structure (i.e. they are symmetric counter parts of each other) (A_2,A_4), (A_3,A_7) and (A_6,A_8). Now, assuming that 𝔼[ δ^2ϕ_Jk^per(Y)ϕ_Jl^per(Y)/(1-G(Y))^2] is finite (provided (<ref>), (<ref>), and the assumptions stated above) for A_1, it follows:𝔼[A_1]= 1/N^2𝔼[ ∑_i=1^N∑_j=1^Nδ_iϕ_Jk^per(Y_i)U_jl/1-G_T(Y_i)] = 1/N𝔼[ δ^2ϕ_Jk^per(Y)ϕ_Jl^per(Y)/(1-G(Y))^2]+N-1/Nc_Jkc_Jl . Consider possible upper bounds for γ_1,Jk(x) and γ_2,Jk(x). Using the corresponding definitions stated in <ref>, it follows:γ_1,Jk(x) =1/(1-F_X(x) ) (1-G_T(x) )∫_x^1ϕ_Jk^per(u)f_X(u)du ≤ ∥ f_X∥_∞ M 2^-J/2/c(1-G_T(x))^β+1≤ e^α_1(β+1)/2∥ f_X∥_∞ M 2^-J/2/c C_T^β+1/2 . Similarly, for γ_2,Jk(x), it follows:γ_2,Jk(x)≤ ∫_0^1|ϕ_Jk^per(u)|f_X(u)du/(1-F_X(u))(1-G_T(u))≤ ∫_0^1|ϕ_Jk^per(u)|f_X(u)du/c(1-G_T(u))^β+1≤ e^α_1(β+1)/2∥ f_X∥_∞ M 2^-J/2/c C_T^β+1/2 . Therefore, the last result implies that for k,l = 0,...,2^J-1 and ĩ∈{0,1}:γ_ĩ,Jk(x)γ_ĩ,Jl(x)≤ ∥ f_X∥_∞^2 M^2 2^-J/c^2(1-G_T(x))^2(β+1)≤ e^α_1(β+1)∥ f_X∥_∞^2 M^2 2^-J/c^2C_T^β+1≤ 𝒪(N^-1log(N)) . Using the last result,it follows:𝔼[(1-δ)γ_1,Jk(Y)γ_2,Jl(Y) ]≤ e^α_1(β+1)∥ f_X∥_∞^2 M^2 2^-J/c^2C_T^β+1∫_0^1(1-G(u))f_X(u)du≤ e^α_1(β+1)∥ f_X∥_∞^2 M^2 2^-J/c^2C_T^β+1 . Clearly, from the last result the same upper bound holds for 𝔼[(1-δ)^2γ_1,Jk(Y)γ_1,Jl(Y) ] and𝔼[γ_2,Jk(Y)γ_2,Jl(Y) ].Now, for the pair (A_2,A_4), it follows:𝔼[A_2]= 1/N^2𝔼[ ∑_i=1^N∑_j=1^Nδ_iϕ_Jk^per(Y_i)U_jl/1-G_T(Y_i)]=-1/N𝔼[δϕ_Jk^per(Y)γ_2,Jk(Y)/1-G_T(Y)] ≤ 1/Ne^α_1(β+1)/2∥ f_X∥_∞ M 2^-J/2/c C_T^β+1/2∫_0^1| ϕ_Jk^per(u)|c(1-G_T(u))^β-1g_T(u)du ≤ 1/Ne^α_1(β+1)/2∥ f_X∥_∞∥ g_T∥_∞ M^2 2^-J/C_T^β+1/2≤ 𝒪(N^-2log(N)) , In the case of the pair (A_3,A_7) we have:𝔼[A_3]= 1/N𝔼[R_Nl∑_i=1^Nδ_iϕ_Jk^per(Y_i)/1-G_T(Y_i)] ≤ 𝒪(N^-1log(N))c_Jk For the term A_5 we have the following: 𝔼[A_5]= 1/N^2𝔼[∑_i=1^N∑_j=1^NU_ikU_jl] = 1/N𝔼[U_kU_l] Therefore, using the definition of U_k: .9 !𝔼[A_5] = 1/N𝔼[ (1-δ)^2γ_1,Jk(Y)γ_1,Jl(Y) - (1-δ)γ_1,Jk(Y)γ_2,Jl(Y)-(1-δ)γ_1,Jl(Y)γ_2,Jk(Y)+γ_2,Jk(Y)γ_2,Jl(Y)] From the last result and (<ref>), it is clear that: 𝔼[A_5] ≤𝒪(N^-2log(N)) Now, for the pair (A_6,A_8) it is clear from the zero mean condition of U_k and the fact that R_N=𝒪(N^-1log(N)) that: 𝔼[A_6]≤ 𝒪(N^-2log(N))𝔼[A_9]≤ 𝒪(N^-2log(N)^2) Putting together (<ref>)-(<ref>) in (<ref>) we get: .9 !𝔼[c̃̃̃_Jkc̃̃̃_Jl] ≤1/N𝔼[ δ^2ϕ_Jk^per(Y)ϕ_Jl^per(Y)/(1-G(Y))^2]+N-1/Nc_Jkc_Jl + 𝒪(N^-2log(N))+𝒪(N^-2log(N)^2)+𝒪(N^-1log(N))(c_Jk+c_Jl) Therefore, (<ref>) becomes: 𝔼[N(c̃̃̃_Jk-c_Jk)(c̃̃̃_Jl-c_Jl) ] ≤𝔼[ δ^2ϕ_Jk^per(Y)ϕ_Jl^per(Y)/(1-G(Y))^2]-c_Jkc_Jl+𝒪(N^-1log(N)^2) Therefore, for N large the last result suggests that: Cov(√(N)(c̃̃̃_Jk-c_Jk) ,√(N)(c̃̃̃_Jl+c_Jl) ) ≈𝔼[ δ^2ϕ_Jk^per(Y)ϕ_Jl^per(Y)/(1-G(Y))^2-c_Jkc_Jl] Finally, in light of the last result and the properties of the Normal Distribution, result (<ref>) follows. Therefore, .93 !f̂^PD(x) ∼^app.N(f(x) ,1/N∑_k=0^2^J-1σ_Jk^2(ϕ_Jk^per(x))^2+2/N∑_k<l𝔼[ δ^2ϕ_Jk^per(Y)ϕ_Jl^per(Y)/(1-G(Y))^2-c_Jkc_Jl]ϕ_Jk^per(x)ϕ_Jl^per(x)) | http://arxiv.org/abs/1709.09298v2 | {
"authors": [
"German A. Schnaidt Grez",
"Brani Vidakovic"
],
"categories": [
"stat.AP"
],
"primary_category": "stat.AP",
"published": "20170927011741",
"title": "An Empirical approach to Survival Density Estimation for randomly-censored data using Wavelets"
} |
Frustrated spin order and stripe fluctuations in FeSe Aleksejs Zajakins Draft Oct 2016 ===================================================== In the next few years the classification of radio sources observed by the large surveys will be a challenging problem, and spectral index is a powerful tool for addressing it. Here we present an algorithm to estimate the spectral index of sources from multiwavelength radio images. We have applied our algorithm to SCORPIO <cit.>, a Galactic Plane survey centred around 2.1 GHz carried out with ATCA, and found we can measure reliable spectral indices only for sources stronger than 40 times the rms noise. Above a threshold of 1 mJy, the source density in SCORPIO is 20 percent greater than in a typical extra-galactic field, like ATLAS <cit.>, because of the presence of Galactic sources. Among this excess population, 16 sources per square degree have a spectral index of about zero, suggesting optically thin thermal emission such as Hii regions and planetary nebulae, while 12 per square degree present a rising spectrum, suggesting optically thick thermal emission such as stars and UCHii regions. Galaxy: stellar content – Radio continuum: stars – radio continuum: ISM – techniques: interferometric – radio continuum: galaxies.§ INTRODUCTIONWe are entering a golden age for radio astronomy. There are several new interferometers upcoming, including the Square Kilometre Array (SKA; ) and its precursors, such as the Australian SKA Pathfinder (ASKAP, ) and MeerKAT <cit.>, that will deliver an extraordinary number of new discoveries. In particular, we expect a lot of data from the forthcoming radio continuum surveys, both in the Galactic Plane (GP) and at high Galactic latitude, such as the Evolutionary Map of the Universe (EMU; ), to be carried out with ASKAP, and the MeerKAT International GigaHertz Tiered extra-galactic Exploration Survey (MIGHTEE; ) and the MeerKAT High Frequency Galactic Plane Survey (MeerGAL), both to be carried out with MeerKAT. We will need to classify a large number of sources, especially in the GP, where the Galactic population adds to the extragalactic population. There are essentially six ways to classify a generic radio source, depending on the data available and whether it is resolved or not: * studying its morphology, which requires that the source is resolved; * studying the radio emission mechanism, which requires that the spectral energy distribution (SED) in the radio wavelengths is known; * studying the complete SED, which requires the identification of the source counterparts at different wavelengths; * studying the polarization, which requires a very high signal-to-noise ratio (S/N); * studying the time domain, which requires a sufficient time resolution; In this paper we will focus on the second method, since it is solely based on radio multiwavelength or wide-band observations and no other information is required. The radio SED depends on the continuum emission mechanism, which could mainly be thermal (bremsstrahlung) or non-thermal (synchrotron).A parameter that characterises the SED at the radio wavelengths is the spectral index α, defined as S=S_0(ν/ν_0)^α, where S is the flux density of the source at a given frequency ν and S_0 is the flux density at the frequency ν_0. The spectral index can change with the frequency, but both synchrotron and bremsstrahlung lose this dependence in the two optical depth limits τ≫1 and τ≪1. In particular, in the case of a homogeneous source of bremsstrahlung we have α=2 for τ≫1 while, for τ≪1, α=-0.1 <cit.>. In the case of n∝ r^-2 (like a stellar wind), where n is the electronic density and r is the distance from the centre, α=0.6 (; ). In the case of a homogeneous source of synchrotron α=2.5 for τ≫1 while, for τ≪1, α depends on the spectral index δ of the energy distribution of the relativistic electrons, defined as α=-δ-1/2 so, for typical values of δ, it ranges between -1 and -0.5 <cit.>. The SED can also be modified by synchrotron self-absorption and free-free absorption, causing a low frequency turnover, and by electron cooling, causing a high frequency break. Complex models are invoked to reconcile these models with the data, which are beyond the scope of this paper, but they are discussed extensively by Collier et al., in preparation.The frequency at which the flux density of a source reaches a maximum is called the turnover frequency. Measuring the turnover frequency can allow us to constrain additional physical parameters, such as magnetic field intensity or electronic density. To derive the spectral index and all the connected informations for a large number of sources, considering that the future surveys will detect millions of sources <cit.>, we developed an algorithm to automatically compute the spectral index. Our algorithm has been tested by using the images of Stellar Continuum Originating from Radio Physics In Ourgalaxy (SCORPIO, , hereafter Paper I) survey. SCORPIO is a 2×2 deg^2 survey in the GP centred at Galactic coordinates l=344°.25, b=0°.66, carried out with the Australian Telescope Compact Array (ATCA) between 1.1 and 3.1 GHz that acts as a pathfinder for EMU. A pilot experiment took place in 2011 and in 2012, observing about a quarter of the selected field. The rms of the map was ≈25-30 μ Jy/beam, with a resolution of 14.0 by 6.5 arcsec (Paper I). Notice that SCORPIO is actually larger than 4 deg^2 because, to have a better coverage of the region, we preferred to observe a larger area. This can also be applied to the pilot, in fact its area is actually 2.333 deg^2, compared to the declared 1 deg^2.In Paper I we produced a catalogue of 614 point sources in the pilot field, listing their flux densities, position and possible infrared counterpart. The data release of the full SCORPIO field, with a catalogue of about 2000 sources, is in progress (Trigilio et al., in preparation). In this paper we present a statistical analysis of the radio SED of the 614 point sources extracted and catalogued in Paper I, to statistically discriminate Galactic from extra-galactic point sources. In Section 2 we give a brief overview of the observations and data reduction. In Section 3 we describe our algorithm to automatically reconstruct the source SED and derive the spectral index. In Section 4 we present our results, thoroughly discussing their relation with ATLAS (Australia Telescope Large Area Survey, ) ones, highlighting and explaining the differences. Conclusions are reported in Section 5.§ DATAThe observations were performed in the periods 21-24 April 2011 and 3-11 June 2012, using ATCA in 6A and 6B configurations at 16-cm. The field was observed using a mosaic of 38 pointings. A detailed description of the observations and data reduction is reported in Paper I. Here we briefly describe the main procedures we followed to produce the final images. §.§ Data cube generationFlagging was performed in<cit.>, using the task . Radio Frequency Interference (RFI) was removed by flagging 30 to 40 per cent of the visibilities. The usable range of frequencies covers 1.350 to 3.100 GHz, corresponding to Δν/ν≈0.8. Calibration was performed using the standardtasks and the standard bandpass calibrator, the radio galaxy 1934-638, which had an assumed flux density of 12.31 Jy at 2.1 GHz <cit.>. We divided our data in 7 sub-bands (see Table <ref>).The imaging was performed in . The deconvolution was executed with thetask using the Högbom algorithm <cit.> on every pointing. The cell size was set to 1.5 arcsec and the restored Gaussian beam was set to 14.0 by 6.5 arcsec. The pointings were then joined into one mosaic using thetask. Finally we obtained 7 maps at 7 different contiguous frequency bands. This division is very convenient to calculate the SED of the sources, considering that the 16-cm band is often the turn-over region for galactic sources. An all-band map has been obtained using all the uv data to accomplish the best S/N. The source extraction was conducted on a region of the all-band map as in <cit.>. We chose to restrict our studies to the largest region with a uniform and low rms, by taking into account the size of the primary beam at the highest frequency sub-band, given that at this frequency the primary beam is the smallest (see paper I). The maps of the local noise were used to identify sources on the basis of their S/N. Local maxima above 5σ were identified as sources. A peak position and flux density value were measured by interpolating between pixels. A centroid position, integrated flux density and source area were also calculated by integrating contiguous pixels down to 2.5σ (as in ), and sources were identified as overlapping if the integration area contained more than one source. Our source extraction process yielded a sample of 614 point sources. While the majority of the sources extracted by the <cit.> algorithm are point-like, a few of them are slightly resolved. The most extended one is SCORPIO_169a, with an area of 8.3 times the beam area (Paper I). This source has a linear angular scale of ∼35”, while our Largest Angular Scale is ∼2.3', so we are sensitive to scales significantly larger than our largest compact source.The study of SCORPIO extended sources is beyond the scope of this paper, but we are using the extraction algorithm presented in <cit.> and analysing them in Ingallinera et al., in preparation. §.§ Primary beam correctionThe mosaicing process requires the correction of the telescope primary beam in order to get accurate brightness in the field. A non accurate knowledge of the primary beam prevents a good flux density measurement, in particular for the sources far from the center of the individual pointings. As a consequence, the spectral indices of the sources will be affected non-uniformly in the field.In the last few years measurements of the primary beam shape for the new 16-cm CABB receivers were made across the entire usable frequency range (1.1 to 3.1 GHz). Although an accurate primary beam model is now available, it had not been implemented in MIRIAD when SCORPIO imaging was performed. Instead, in Paper I, we used the Gaussian primary beam model made by <cit.> (see Table <ref>) (Paper I). The product of the FWHM and the frequency d^', is, according to them, constant for the first three sub-bands, then it changes and remains constant for the last 4 ones because they used different receivers, where the primary beam is assumed to vary as ν^-1 in a single receiver. New primary beam measurements show that this is not true[<http://www.narrabri.atnf.csiro.au/people/ste616/beamshapes/beamshape_16cm.html>] (in Table <ref>, the new ones are reported as d values).The primary beam FWHM tends to decrease less rapidly then ν^-1 below 2.1 GHz. It can be caused by a deterioration in the focus above 1.4 GHz. For a single pointing the correction factor would be:P=exp{4ln2[(x/d)^2-(x/d^')^2]}where x is the distance from the pointing centre multiplied by the frequency. In general, for n pointings we have:P=∑_i=1^nexp[-4ln2(x_i/d^')^2]·exp[-4ln2(x_i/d)^2]/∑_i=1^n{exp[-4ln2(x_i/d^')^2]}^2 In this paper, we use the new version of MIRIAD, in whichhas the right primary beam correction, so we decided to redo the mosaicking, applying the newtask to the pointings to obtain the corrected maps. The slight differences in the fluxes between the data in this paper and the data in Paper I is due to this correction. § IMAGE ANALYSISEach of our 614 sources is characterised by a SED that provides us information on the mechanism of the radio emission (thermal or non thermal), and, if we find the turn-over frequency, some physical parameter such as magnetic field or electron density. This information can help us to discriminate between galactic and extra-galactic sources and to characterise the source, giving us hints on its nature. §.§ Flux Density Extraction Algorithm In order to automatically estimate the source flux densities at different frequencies, we wrote a Python script that runs in CASA (Common Astronomy Software Applications, ) and uses thetask (see flowchart in Fig. <ref>). The script reads the position of each source from the Paper I catalogue and creates a box around it and 4 other boxes, that are 15 pixels along the x axis and 10 pixels along the y axis away from the vertices. The box dimensions are chosen to be 3 times the beam major and minor axis respectively (the major axis is oriented along the north-south direction). The script then callsin the central box andin the other four ones.performs a bidimensional Gaussian fit in the chosen region and returns a Python dictionary dic containing all the fit parameters (such as integrated flux density, Gaussian peak position, major and minor axis dimension, etc.). Thetask implements an algorithm that models and subtracts the background from the fit thereby removing the effect of diffuse emission. The script usesto measure the rms and median noise in each of the adjacent boxes, and then calculates the median background noise. At the end the script returns two matrices with N lines and M columns, where N is the number of sources andM the number of sub-bands. Every element a_ij of the first matrix is the fit parameter dic of the i^th source for the j^th sub-band, in a similar way every element b_ij of the second one is the local rms of the i^th source for the j^th sub-band. If the Gaussian fit does not converge for the i^th source and the j^th sub-band, a_ij is simply a 0 (see Fig. <ref>).To check the reliability of the algorithm, we applied it to the old all-band map and compared the results with the source flux densities extracted using the <cit.> algorithm. The value of the average ratio r=S_f/S_c, where S_f is the flux density measured by the <cit.> algorithm and S_c is the one measured using our algorithm, is r=0.97±0.13. This value shows that the two methods are consistent.We also ran our algorithm on the new all-band map, to check for differences with the old map flux densities. We found a ratio of r=0.94±0.44 between the flux densities measured with our algorithm in the new all-band map and S_c. This higher difference is consistent with the primary beam correction.§.§ Spectra construction and control algorithmFor the purposes of this work, we initially assume that all sources have a power-law spectrum. Our algorithm performs a weighted linear fit of logS as a function of logν, where S is the flux density and ν the frequency and the single point error isΔ S=√(e_fit^2+e_rms^2+e_cal^2)where e_fit is the fit uncertainty, computed by , e_rms is the noise and e_cal is the calibration error. To check that the SED has a linear behaviour it fits a second degree polynomial and compares the chi squared (χ^2) of the linear and parabolic fits. In the 30 sources in which the χ^2 of the parabolic fit has a lower value, we ignored the low-frequency turnover by excluding all data at a frequency below that of the peak and redo the fit to get α. The algorithm performs the following tests for each source in each of the 7 sub-bands to flag “bad points” (see Fig. <ref> for a simplified flow-chart of the procedure): * The flux density S is equal or less than 0. This is the case of non-converged Gaussian fit or Gaussian fit of negative values on the map, for e.g. caused by artifacts: S≤0,∼36 percent of the 4298 (614 sources multiplied by the 7 sub-bands) flux density measurements were flagged by this criterion;* the difference between the position of the fitting Gaussian peak and the position given by the source extraction algorithm is greater than half the width of the beam w. This is the case of a Gaussian fit made on a near brighter source with high sidelobes:P_peak-P_list≥w/2,∼6 percent of the points were flagged by this criterion* The difference between the area of the fitting Gaussian at a given frequency and the average of the area of the other frequencies of the same source is greater than 2 times their standard deviation. This is the case of fitting a faint source that can be affected by noise: |A-A̅|≥ 2σ,where A and A̅ are the area of the fitting Gaussian for a sub-band and the mean area of the other sub-bands respectively and σ is its standard deviation. About 1 percent of the points were flagged by this criterion.* The flux density is significantly greater than the average S̅. S>nS̅.We tested several values of n between 1 and 2 over ∼50 sources. The results have been visually checked and it turns out that n=1.8 was the minimum value above which the fit was unreliable.About 9 percent of the points were flagged by this criterion.If a flux density measurement shows one of those irregularities, it is discarded and does not contribute to the linear fit.§.§ Grouped Components A sub-sample of 65 of the 614 sources are classified by the source finding algorithm as “double sources”<cit.>. They can be components of larger extended sources or simply be near to each other by projection. To fit these multiple componentsneeds to read a file that contains a list of estimates of the position of the Gaussians peaks, their dimensions, orientation and brightness. We built a script that automatically writes this file for each group of components, keeping the Gaussian dimensions and orientation fixed. Then the flux extractor algorithm works in the same way it works for the single sources. The algorithm treats each component of a double source as a real component of a larger source, adding the integrated flux densities and using the result as the flux density of the complex source.§.§ MGPS-2 matches To check our spectral indices we considered the 43 SCORPIO sources that match with the second epoch Molonglo Galactic Plane Survey (MGPS-2, ) sources (Paper I). MGPS-2 is a survey of the Galactic Plane, carried out with the Molonglo Observatory Synthesis Telescope (MOST) at a frequency of 843 MHz and with a restoring beam of 45”× 45”δ, where δ is the declination so it is suited to expand the SCORPIO sources' SED and check for spectral index errors. If we exclude the double sources and those embedded in diffuse emission (with confusion, because of the large MGPS-2 beam), there are only 16 sources left. We calculated their spectral index, including the 843-MHz MGPS-2 point. In Fig. <ref> we show the spectra of the 16 sources. Except for SCORPIO1_178 and SCORPIO1_288 the SEDs are consistent with the 843 MHz flux density. In the case of SCORPIO1_178 the discrepancy is due to the fact that the MGPS-2 centroid is not centred on the SCORPIO source and its beam encompasses another brighter source, leading to a higher flux density, while in the case of SCORPIO1_288 it can be due to a turn-over effect. In Fig. <ref> we show the effect of the large beam size of MGPS-2 in the case of an isolated source (left panel) and SCORPIO1_178 (right panel).§ SPECTRAL INDICES ANALYSIS Since the sub-band maps have a lower S/N than the all-band map, some of the faint sources are detected in less than 2 sub-bands and hence no spectral index can be derived. In the end we extracted the SED for 510 sources of the 614 detected, of which 74 have no rejected frequency points. However not all of the 510 spectral indices are accurate due to the errors on the flux density extraction in the sub-bands. To define the criteria that only select sources with an accurate spectral index, we ran a simulation. We used thetask in<cit.> to simulate 10 ATCA fields with a total of 301 point sources with α=0, and a flux density almost uniformly distributed between 500 μJy and 200 mJy at a the same frequency and with the same bandwidth of SCORPIO. We mapped the whole field and derived the spectral index by following the same method used for the SCORPIO data. As shown in Fig. <ref>, simulated sources with a S/N smaller than ∼40 can have a spectral index error greater than 0.5. Therefore in the following analysis we will only refer to the sources brighter than 40σ. We also tried to divide the band in 3 sub-bands instead of 7. We found out that the standard deviation of the spectral indices of sources with a S/N greater than 40 and smaller than 100 is 0.2 in the case of 3 sub-bands, 0.03 in the case of 7 sub-bands. This is due to the smaller number of points, leading to a worse linear fit. In the 3 sub-bands case we should choose a 100σ threshold. Therefore we did not use the 3 sub-bands.This limiting S/N is also illustrated by Fig. <ref>. The median of the spectral index errors depends on the S/N as an hyperbola and beyond ∼40σ it decreases rapidly. At 40σ the median on the spectral index errors is ∼ 0.2, thus it is assumed as the minimal value necessary for a good separation. Our threshold of 40σ agrees with results by <cit.> that suggest a value between 16σ and 100σ as the limit of reliability for spectral indices. Finally, considering that in the pilot field the rms is ∼ 25-30 μJy/beam (Paper I), we adopted a lower limit of 1 mJy for the brightness of the sources. Imposing this limit fixes the number of our sources to 306. In the Paper I catalogue we list only 260 sources with flux densities greater than 1 mJy. This discrepancy has two causes. The first is the difference in the flux densities due to the updated primary beam model. The second is the different method used, in the catalogue and in this paper, to extract the flux density. To have a consistent way to extract it in the SCORPIO-ATLAS comparison, we decided to use, as our measured flux densities, the value of the linear fit of the SEDs at 2.1 GHz. As noticed in Sec. <ref>, the differences in methods and primary beam correction can lead to a slightly higher flux density in our measures compared to the <cit.> ones.§.§ Results In Fig. <ref> we show the source density distribution of all the 306 sources brighter than 1 mJy as a function of their spectral indices. The distribution is asymmetric, with a main peak at α∼-0.9 and a long tail at α≳0.We modelled the spectral index distribution as the sum of several populations, both Galactic and extra-galactic, and fit it with a variable number of Gaussians. We used the Bayesian Information Criterion (BIC) <cit.> to select the best-fit model, which will have the smallest BIC. We used the BIC formula as defined in <cit.>:BIC=-2·lnL̂+k·lnnwhere L̂ is the maximised value of the likelihood function, n is the number of data points and k is the number of free parameters of the model. We used the criteria described in <cit.> to exclude models (see Table <ref>). The BIC parameter penalises fits with a larger number of parameters so we can be confident in choosing models with a high number of parameters. Table <ref> shows that a model consisting of 3 Gaussians (see Fig. <ref>) has a better BIC than the other considered models, such as combinations of any number of Gaussians from 1 to 5 and a skewed Gaussian. The latter is a function similar to an asymmetric Gaussian due to a “skew” parameter, defined as f(t)=Aϕ(t)Φ(kt), where A is the amplitude parameter, k is the skew parameter, ϕ(t) is the normal distribution and Φ(t)=∫_-∞^tϕ(x)dx is the cumulative distribution function. As reported in Sec. <ref> we detected 65 group of components that we treated as components of 31 sources. With the same method used for single sources, we checked if they have a different spectral index distribution with respect to the former (Fig. <ref>). We have a total of 11.5 group of components per square degree with a flux density greater than 1 mJy. Among them, ∼9.5 deg^-2 have a negative spectral index, 1.6 deg^-2 have a spectral index around 1 and 0.4 deg^-2 around 2. These numbers are compatible with the all-source results but our sample is too small to make a strong statement. In the following we compare the statistical results of the spectral index analysis on the SCORPIO data with those of ATLAS, a survey performed at high Galactic latitude. ATLAS can be used as a template for the extra-galactic population of SCORPIO, as the Galaxy is optically thin at radio wavelengths and the cosmological distribution of galaxies is isotropic at these scales. §.§ Comparison with THORTHOR is a survey of a ∼100 deg^2 area in the GP, carried out with the Very Large Array (VLA) in spectral line and continuum from 1 to 2 GHz with an angular resolution of 10-25”. The noise level is ∼0.3-1 mJy beam^-1. They detected around 4400 sources above 5σ. For ∼1800 of them the authors measured the spectral index. The SCORPIO spectral indices histogram is similar to the THOR one with one main difference: the peak centred between -0.1 and 0 is much higher compared to the synchrotron peak in the THOR survey. This is probably due to the difference in the source flux densities. While in the SCORPIO survey we selected all the sources above 1 mJy, in the THOR survey, which has a higher rms noise, they selected all sources that have a reliable intensity for all the six spectral windows in which they divided their data. This, due to the higher rms, makes their flux limit much higher than ours resulting in 18 sources deg^2 with a spectral index, against our 120. The fact that their flux limit is not directly chosen makes a direct comparison between THOR and SCORPIO not feasible. If we consider that the extra-galactic sources increase fast in number with the sensitivity and that the Hii region are usually bright, we can reasonably state that it is possible to detect almost all of them even with THOR sensitivity while a lot of galaxies are lost. This makes their 0 peak higher than ours when we compare them with the synchrotron ones. §.§ Comparison with ATLASATLAS <cit.> is a survey conducted on two fields, the Chandra Deep Field-South (CDF-S) and the European Large Area ISO Survey-South 1 (ELAIS-S1) at 1.4 GHz with ATCA. The third ATLAS data release <cit.> includes observations taken using the CABB receiver with a 500 MHz bandwidth covering 1.3-1.8 GHz. Most importantly, the fields are far away from the GP, thus we expect to find only an extra-galactic population.In <cit.>, the ATLAS DR3 CABB data were divided into two sub-bands and two separate mosaics of each field were created, one using the lower sub-band centred at 1.40 GHz and the other one using the higher sub-band centred at 1.71 GHz. For consistency with SCORPIO, we extracted the spectral indices of the ATLAS sources running the algorithm described in Section <ref> on the two mosaic maps. Then we normalised the number of sources per square degree for both the surveys and only considered the sources brighter than 1 mJy at 2.1 GHz (extrapolated using the spectral index). As shown in Fig. <ref>, the synchrotron peak is the predominant component in ATLAS. There is a minor peak around 0, which can also be explained by extra-galactic sources, the Gigahertz Peaked Spectrum (GPS) sources (GPS; ), that can have the turnover frequency between 1.4 and 1.71 GHz. We tried to fit this distribution with different models, as in Section <ref>, and we found that the Gaussian model has a BIC value of 85.4 , higher than the skewed gaussian one (81.7). Consequently the latter is the one we selected (see Fig. <ref> for the model). Assuming that virtually all the ATLAS sources are extra-galactic, we used the ATLAS model as a template for the extra-galactic population of SCORPIO. We found 20 percent more sources per square degree in SCORPIO than in ATLAS (120 versus 100) and we found that a model using the ATLAS skewed Gaussian plus another skewed Gaussian gives a BIC of 76.8, better than almost all the models shown in Table <ref> (with the exception of the one that uses three Gaussians). Nevertheless the model including the ATLAS extra-galactic template has, as anticipated before, a physical motivation making it the more reliable one. Fig. <ref> shows that the ATLAS Gaussian fits quite well the peak at α=-0.9 and, most importantly, another skewed Gaussian fits the other peaks. Maybe larger statistics could help us in concluding that they are part of more than one galactic population, as expected, but, with our data, we can only say that they are part of a generic galactic population.In Fig. <ref> the difference between the SCORPIO and the ATLAS population per spectral index bin is shown. The error e_i associated with the i^th bin is:e_i=√(N_Si)+√(N_Ai)where N_Si and N_Ai are respectively the number of SCORPIO sources and ATLAS sources in the i^th bin. We can recognize 3 regions of the spectral index α: * -2.5<α<-0.5: the number of excess sources is not significantly different from zero, implying that all SCORPIO sources in this spectral index range may be extra-galactic. In this spectral index range there can also be unresolved or almost unresolved Young Supernovae Remnants (YSNRs). To know what size we should expect from different age SNRs, we consider the size of the SN1987A remnant. The SN1987A remnant is located in the Large Magellanic Cloud, at ∼51.4 kpc from the Sun. It expanded from 0.21 to 0.39 pc between 1995 and 2010 with an almost linear expansion rate of about 0.01 pc/year <cit.>. Therefore a 100 years old similar supernovae would have a linear dimension of ∼1.1 pc and an angular dimension of ∼11” at a 20 kpc distance from the Sun. Given the 14.0”×6.5” resolution of SCORPIO, we may not be able to resolve it. We now estimate how many YSNRs are expected. We assume that in the last 1000 years there were 7 supernovae in our Galaxy within 5 kpc from the Earth in the GP. Given the position of the Sun in the GP, from the point of view of the Galactic Centre this corresponds to a ∼50 deg circular sector. Assuming that the distribution is uniform in the GP, we estimate about one supernova every ∼20 years in our Galaxy. Considering that ∼66 percent of the OB stars lies in the direction of the Galactic Center <cit.> in a 120 deg^2 area, we can assume a supernova every ∼40 square degree younger than 100 years in the direction of the SCORPIO field.Moreover we have to consider that SN discovery is tipically performed at optical wavelength. Thus, even if the remnants of these supernovae are bright in radio (at peak we expect almost 1000 Jy in L-band at 20 kpc ), they could be very faint in the optical due to the GP absorption: considering that m = A + M + μ, assuming a SN with a peak absolute magnitude of M = −17 mag, we would have an apparent magnitude as high as m∼26 mag, having in the direction of SCORPIO an extinction A = 26.7 <cit.> and a distance module, corresponding to 20 kpc, of μ∼16.5. This means that it is possible for a supernova in our Galaxy to happen undiscovered but to leave a remnant bright enough to be detected at radio wavelength. Even with this premise, considering a density of 0.025 supernova younger than 100 years per square degree in the direction of SCORPIO, we would not be able to detect more than one YSNR in the relatively small SCORPIO field of view (∼5 square degree for the whole field), confirming the extragalactic origin of the sources in this α range; * -0.5<α<0.5: the excess here is ∼16±10 per square degree. These are typical spectral indices of Galactic source that show thermal emission in an optically thin environment, e.g. Hii regions, Planetary Nebulae (PNe) and Luminous Blue Variable (LBV) (e.g. , , , 2014, ). * 0.5<α<2.5: there is a significant excess of about 12±8 sources per square degree with a spectral index in this range, suggesting stellar winds, interacting stellar winds and thermal emission in an optically thick environment, e.g. Wolf Rayet, LBV, OB stars and compact and ultracompact Hii regions (e.g. , , ). Obviously we expect some overlap between spectral index regions. Furthermore, the different frequency range of the ATLAS survey makes it likely to have a surplus of extra-galactic flat sources due to possible turnover positions. Note that the difference between the total sources surplus in SCORPIO (20 percent) and the one in the interval -2.5<α<2.5 is due to the smaller bandwidth in ATLAS, resulting in a larger number of spurious spectral indices.We cross-matched our point sources with the <cit.> H ii regions catalogue, extracted from infrared colours, and we found 7 matching sources, reported in Table <ref>: 4 of them do not have a counterpart in the Southern Galactic Plane Survey (SGPS, ), suggesting that many of the “radio quiet” H ii regions emit in radio being actually “radio weak”; the remaining 3 are H ii region candidates. The spectral indices of SCORPIO1_123b, SCORPIO1_169, SCORPIO1_218 and SCORPIO1_483 are consistent with thermal free-free emission in an optically thick medium in the first case, and in an optically thin medium in the latter ones. The other 3 sources present a much less reliable spectral index because they are resolved and embedded in diffuse emission.§ SUMMARY AND CONCLUSIONSWe have described: * An algorithm, based on the CASA task , to automatically extract the flux density of a large number of point-like sources by fitting Gaussians and to calculate their spectral indices; * The application of this algorithm to the SCORPIO data to derive the SCORPIO sources spectral index distribution, along with the discussion about the different source populations that contribute to the distribution; * The comparison between SCORPIO and an extra-galactic survey, ATLAS. We have found that, with a bandpass like the SCORPIO one, the S/N for the object in the field has to be at least 40 in order to rely on the spectral index α with an uncertainty ≲0.2. We have found that, in the Galactic Plane, the source count with flux densities greater than 1 mJy is about 20 percent higher than the source count at high galactic latitude. We found 16 Galactic sources per square degree with a spectral index of about α=0, suggesting optically thin thermal emission such as Hii regions and planetary nebulae, while the remaining 12 sources per square degree present a spectral index 0.5<α<2.5, pointing to an optically thick thermal emission such as stars and compact Hii regions.These results are very important for planning forthcoming radio surveys. This work will be eventually extended to the whole SCORPIO field. 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H., Kesteven M.J, 1992, Measurements of the ATCA Primary Beam, AT Memo 39.3/024 [Wit et al.2012]BIC Wit E., Van Den Heuvel E., Jan-Willem Romeijn, 2012, Statistica Neerlandica 66 (3), 217, 236 [Wright & Barlow1975]Wright Wright A. E., Barlow M. J., 1975, MNRAS, 170, 41 § CATALOGUEWe created a new catalogue of the spectral indices of the 510 sources listed in paper I for which we could calculate it (see Table <ref> for the first 10 entries). The columns are: * ID of the source in the SCORPIO catalogue;* galactic longitude;* galactic latitude;* flux density at 2.1 GHz;* error on the flux density (see details on how we calculated that on Paper I)* spectral index;* error on the spectral index. | http://arxiv.org/abs/1710.04702v1 | {
"authors": [
"F. Cavallaro",
"C. Trigilio",
"G. Umana",
"T. M. O. Franzen",
"R. P. Norris",
"P. Leto",
"A. Ingallinera",
"C. S. Buemi",
"J. Marvil",
"C. Agliozzo",
"F. Bufano",
"L. Cerrigone",
"S. Riggi"
],
"categories": [
"astro-ph.GA",
"astro-ph.HE",
"astro-ph.SR"
],
"primary_category": "astro-ph.GA",
"published": "20170926124319",
"title": "SCORPIO-II: Spectral indices of weak Galactic radio sources"
} |
=1 mybluebox[#1] ifnextchar[@mybluebox[#1]@mybluebox[#1][0pt]@mybluebox[#1][#2]#3 #3 #1#2<̧çi̧ţ.̧>̧ 𝔰𝔬 𝔰𝔲 𝔲̆ [] MRℙP∫∫ ψ̅ ϕ̅ O Φ Φ Δ_Φ δG a_n,ℓ γ_n,ℓ ø ζ δ̣ ↔********(0) (1) (2) (3) (4) (n)ℒ γ̂M Γ Q D minminλ maxmax τ Z Ẑ ðd̂ mod J q vac h light heavy su(2)_k gap_R hs[λ](<ref>) §§.§ §.§.§ asym O łλ .1 in hw α→ |ρ⟩ ⟨ ⟩ ØO [ØØ] V W(t)VW(t)_VV_W(t)W(t)_ W G Regge H̋ α̅ 20' W_∞[λ] F_vac,∞^(1) q_v^(s) q_w^(s) F A V W§.§ §.§.§ §ı∞Σ F ØO ⟩ ⟨ ∂̅ D̅ τ vaceig asym ⟨ ⟩ hw Tr_fhs[λ] σ Ê ∙ O .1 inΔ _2F_1 F(c,h_i,h_p,x) h_p ŁΛ γ_mn γ α | http://arxiv.org/abs/1709.09159v2 | {
"authors": [
"Simone Giombi",
"Eric Perlmutter"
],
"categories": [
"hep-th"
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"primary_category": "hep-th",
"published": "20170926175058",
"title": "Double-Trace Flows and the Swampland"
} |
=1EPJ Web of Conferences Lattice2017 #1Eq. (<ref>)a_HArg φT_chop θ_a ρ_dm (m_a/m)^2_max θ_A,ext θ^ str_A t_start k_start t_cut r_min r_max κ_min k_max n_ax z̅_2 α_WS a_H0 1/2 𝒪 ℒ 𝒟Tr Institut für Kernphysik, Technische Universität Darmstadt, Schlossgartenstraße 2, D-64289 Darmstadt, Germany First I will review the QCD theta problem and the Peccei-Quinn solution, with its new particle, the axion.I will review the possibility of the axion as dark matter.If PQ symmetry was restored at some point in the hot early Universe, it should be possible to make a definite prediction for the axion mass if it constitutes the Dark Matter.I will describe progress on one issue needed to make this prediction – the dynamics of axionic string-wall networks and how they produce axions.Then I will discuss the sensitivity of the calculation to the high temperature QCD topological susceptibility. My emphasis is on what temperature range is important, and what level of precision is needed. Axion dark matter and the Lattice GuyMoore1Acknowledges financial support by TU Darmstadt December 30, 2023 ==============================================================§ OVERVIEW The axion <cit.> is a proposed particle, the angular excitation of a new “Peccei-Quinn” (PQ) field φ that would solve the strong CP problem <cit.> and which is also a very interesting dark matter candidate <cit.>, thereby solving two puzzles with one mechanism.That's why I think it well motivated to study the axion as a dark matter candidate.The axion model has one undetermined parameter, the vacuum value of φ, f_a; the axion mass m_a scales as f_a^-1.The value of f_a also plays a role in determining the amount of axion dark matter produced in the early Universe.So do some nontrivial dynamics which we will explain in detail below.If we can understand the nontrivial dynamics of the axion field during cosmology, that lets us find a fixed relation between f_a and the (measured) dark matter abundance.It therefore allows a clean determination of the axion mass, under some simple assumptions.This is valuable in the experimental search for the axion and it motivates us to solve the cosmological axion dynamics.Supposing we make the following assumptions:*The axion exists.*The axion field starts out “random” (in a sense we will define precisely below) either during or shortly after inflation (or whatever physics featured in the Universe at very high energy density).*Gravity and the Universe's energy budget followed the “standard” picture (General Relativity and the known Standard Model species dominating the energy density) at temperatures T < 2 GeV.*Axions make up all of the observed dark matter.Then, as I will explain, we have enough information to determine the axion mass.But to do so we will need to solve two problems:*We need to know the temperature dependence of the QCD topological susceptibility at temperatures between 540 and 1150 MeV.*We need to control the axion field dynamics during this temperature range, to solve for the efficiency of axion production.The session following this talk addresses item <ref> and the remainder of this talk will lay out the groundwork more completely and will then address item <ref>.§ STRONG CP PROBLEM AND THE AXION Let me refresh your memories on the strong CP problem. There are two gauge-invariant dimension-4 scalars which can enter the gauge-field part of the QCD Lagrangian:= 1/2g^2 G_μν G_μν + Θ/32π^2ϵ_μναβ G^μν G^αβ .The latter term is P and T odd because it contains the antisymmetric tensor ϵ_μναβ.The operator which Θ multiplies is the topological density and it integrates to the instanton number.Its zero-momentum two-point function defines the topological susceptibility,χ(T) ≡⟨∫ d^4 x 1/32π^2ϵ_μναβ G_μν G_αβ(x)1/32π^2ϵ_σρκζ G_σρ G_κζ(0) ⟩_T , ⟨…⟩_T means the expectation value in the thermal ensemble at temperature T.Such a term is strongly constrained by the absence of a measured neutron electric dipole moment.The experimental limit of <cit.> |d_n,meas| < 2.9× 10^-26e cmcontradicts the lattice results for the dipole moment from Θ<cit.> d_n = -3.8 × 10^-16e cm×Θunless |Θ|<10^-10.At this conference we saw new results for the lattice Θ-dependent dipole moment which show that the above result may be too high and its error bar was certainly underestimated (see <cit.> and these proceedings). However it is clear that the absence of a neutron electric dipole moment places an extremely tight constraint on Θ.This is hard to understand because we know P and T are not fundamental symmetries; and any physics at a high scale which violates them generically gives rise to a Θ which does not decrease as we move to lower scales.For instance, consider a very heavy Dirac quark species Q, with Lagrangian_Q = Q̅D Q + ( m Q̅ P_ L Q + m^* Q̅ P_ R Q ).Note that m can be complex, since (Q̅ P_ L Q)^† = Q̅ P_ R Q.But an imaginary part is T and P odd, since the role of left and right projector, P_ L and P_ R, switch under parity and because T is antiunitary.We can remove this mass through a chiral rotation of Q,Q → (e^-i argm/2 P_ L + e^-i argm/2 P_ R) Q,at the cost of reintroducing it, via the Fujikawa mechanism <cit.>, as a shift in the value of Θ,Θ→Θ - argm.Therefore even a very heavy quark can influence the P and T symmetry properties of low energy QCD.But what if there is a symmetry forbidding the mass term for this quark?For instance, suppose P_ L Q is charge 1 and P_ R Q is charge-0 under some global U(1) symmetry?Then the mass term breaks this symmetry, but a complex scalar φ with charge 1 under the symmetry could induce a mass via a Yukawa interaction and a vacuum value.The possible Lagrangian terms for such a scalar are_φ = ∂_μφ^* ∂_μφ + m^2/8f_a^2( f_a^2 - 2φ^* φ)^2 + ( y φQ̅ P_ R Q + y^* φ^* Q̅ P_ L Q ).The combination y φ plays the role of m^* in the previous case.But nowis a dynamical quantity.We will be interested in temperatures around 1 GeV and φ varying on scales of the Hubble scale at that time – tens of meters!Therefore from the point of view of QCD we can take φ to be space-independent, and perform andependent rotation on Q, making the theta termΘ→Θ +.Here we have absorbed the phase in y into a phase redefinition of φ.We can also absorb Θ in the same way, so thatalone plays the role of Θ-angle. For notational compactness we will henceforth write φ = v e^i, with =.This specific way of coupling QCD topology to a complex scalar is called the KSVZ axion <cit.>.There are other mechanisms but the low-energy phenomenology is essentially identical and this mechanism is particularly clear to understand.From the point of view of QCD, the Θ-angle is replaced by a possibly spacetime-varying dynamical field .What about from the point of view of the field φ?Since we want physics on the meter length scale, we can integrate out QCD, leading to an effective potential:V_eff() = -T/Ωln∫ (A_μψ̅ψ) Det (D+m) e^-∫ d^4 x 1/2g^2 G_μν G_μν× e^i∫ d^4 x 1/32π^2ϵ_μναβ G_μν G_αβ≃ χ(T) ( 1 - cos),with Ω the volume of space included in the path integration. In the second line we have made a dilute instanton approximation, which is that the integration exponentiates over the two-point function of the topological density, controlled by the topological susceptibility χ(T) introduced already in chidef. This is not a good approximation for largeand low temperatures <cit.>, but it works well when instantons are dilute, which is true for T > 500 MeV, and for small values of , which will be all we encounter below this temperature.So we can actually use this approximation all the time. Independent of this approximation, it is easy to see that the effective potential is smallest (thechoice is most energetically favored) for =0 and therefore when P and T symmetry are restored.Note thatonly enters as the coefficient in a complex phase, in an otherwise real and positive integral.The integral is maximized, and the free energy minimized, if the phase is always unity.Any nonzero value ofgives rise to phase cancellations and therefore suppresses the partition function, raising the free energy.Although we derived it in Euclidean space, we can also use this effective potential in Minkowski space to study the spacetime evolution of the φ field. In summary, the Minkowski effective Lagrangian for the φ field is-_φ = ∂_μφ^* ∂^μφ + m^2/8f_a^2( f_a^2 - 2φ^* φ)^2 + χ(T) ( 1 - cos ).We will use this to determine the dynamics of the field in the next sections.§ AXION IN COSMOLOGY Let us see what happens to the axion field during cosmological evolution. §.§ Value of susceptibilityThe form of Laxion2 makes it clear that, in order to study the axion's role in cosmology, we are going to need to know the temperature dependence of the topological susceptibility χ(T). It does not yet tell us what temperature range will be interesting. Figure <ref> shows our knowledge at the cartoon level. At low temperature or vacuum, chiral perturbation theory works and <cit.> χ(T → 0) ≃m_u m_d/(m_u+m_d)^2 m_π^2 f_π^2 ≃ (76MeV)^4.At high temperatures we have conventional perturbation theory, which forecasts <cit.> that χ(T) ∝ T^-7-N_f/3.However the exact coefficient is sensitive to the physics of electric screening and is not known accurately. This is why we need lattice results for this quantity! Recently there have been several <cit.>, which give generally compatible results but generally at temperatures below 600 MeV (or in the quenched approximation).It takes new techniques to reach higher temperatures, and only one recent paper <cit.> achieves this, reaching temperatures of 1500 MeV.Also, one group <cit.> finds results which are discrepant with the others, indicating that the matter is not yet settled.Here we will assume that the results of <cit.> are correct.This may well be the case, but we leave the discussion of the relative merits of these approaches and results to the panel, who have more expertise. Needless to say it would be valuable to know definitively that χ(T) is well determined. §.§ Space-uniform axion field So let's assume for now that we know χ(T).For simplicity let us also assume that the axion takes the same value everywhere in space, (x,t) = (t).It is simplest to work in terms of conformal time, so the metric is g_μν = a^2(t) η_μν with a the scale factor. (Later we will use a to represent the lattice spacing.This is actually the same thing, since we will work in comoving coordinates; the lattice spacing is proportional to the scale factor and we may as well use a proportionality of 1.) In the radiation era a(t) ∝ t and T ∝ t^-1.The radial component of φ = v e^i is inactive, v=f_a, and the angular part obeys= f_a^2 t^2 ( 1/2 (∂_t )^2 + t^2 χ̃(t^-1) (1 - cos) ),∂_t^2+ 2/t∂_t = - t^2 χ̃(t^-1) sin ,where χ̃(t^-1) is a rescaled form of the susceptibility. This leads to damped, anharmonic oscillations.The oscillations start roughly at the time t_* when t_*^2 χ̃(t_*^-1) = t_*^-2, or in physical units, when m_a ≡√(χ(T)/f_a^2) obeys m_a t_* = 1 or equivalently m_a / H=1 with H the Hubble scale.After this time the oscillations accelerate as t^2 χ̃ increases, and they damp away.The damping arises both from the 2∂_t /t term (Hubble drag) and from the time variation of the susceptibility.After several oscillations the axion particle number becomes an approximate adiabatic invariant, with number density parametrically of form (t_*/t)^2 (f_a^2/t_*). We see that the number density is quadratic in f_a, while the axion mass is m_a ∝ f_a^-1.Because χ(T) is a very strong function of temperature, t_* depends only weakly on f_a, and so the generated axion energy density is almost linear in f_a.Therefore, the larger the value of f_a, the larger the produced axion abundance.However the axion abundance also depends on the unknown initial angle (t=0).Therefore the dark matter density depends on two variables and it is impossible to make a clean prediction for the value of f_a. We can make a baseline prediction, however, by averaging over the value of the starting angle (t=0).Doing so, one finds the axion mass should be 32 μeV, and t_* corresponds to a temperature of T_* = 1.6 GeV. §.§ Space-random axion field It is far more likely that the Universe started out with a spatially random value for , with no correlations on scales longer than the Hubble scale.Arguments for this picture are presented in <cit.> and are summarized as follows:*It is likely that inflation occurs with a high scale, H^2 > f_a^2 / 60.In this case, over 60 efoldings of inflation, quantum fluctuations stretched (squeezed) by inflation into classical fluctuations would randomize the value of the axion field over the course of inflation.The observation of cosmological tensor modes would more-or-less settle this issue.*After inflation, the Universe reheats to a temperature which can be as high as T_rh∼ 0.1 √(Hm_pl). Even if H ≪ f_a, if the reheat temperature is T_rh > f_a, there would be thermal symmetry restoration for φ.Then when the temperature falls below this scale, φ would independently take on a vacuum value at different points in space, which would be uncorrelated.*The case where inflation and reheating are both low-scale is actually tightly constrained by the absence of observed isocurvature fluctuations (different fluctuations inthan in the radiation temperature), which require roughly H < 10^-5 f_a.Most inflation model-builders would consider this rather unlikely.I emphasize that we do not know thatwas randomized in the early Universe (assuming the axion exists).But it appears likely, and it motivates studying the consequences. I will also assume that the axion makes up all of the dark matter in the universe, so we may equate the final axion matter density with the dark matter density, which is known to obey /s = 0.39eV with s the entropy density <cit.>.The space-inhomogeneous case is much more complicated than the space-homogeneous case.Nevertheless, in the remainder of the presentation I will show how to solve it.§ AXION STRING/WALL NETWORK The φ field varies with amplitude of order f_a ∼ 10^11 GeV over a length scale controlled by H ∼ T^2/m_pl∼ 10^-18 GeV.This huge hierarchy makes the dynamics those of a classical field to extremely high accuracy.The Lagrangian Laxion2 (times t^2 to account for Hubble expansion) and resulting classical equations of motion are easy to put on the lattice and solve as a function of time, from random initial conditions.In broad brushstrokes, our approach is to do just this, evolving the system until only small fluctuations inremain and their evolution has become adiabatic.Then we integrate the associated axion number,= ∫d^3 p/(2π)^3 f(p) = ∫d^3 p/(2π)^3(p^2+m_a^2) φ^* φ(p) + φ̇^* φ̇/√(p^2+m_a^2)and compare it to the result of the angle-averaged misalignment baseline.In fact such a simulation is not sufficient, because of the large hierarchy in Laxion2 between the mass scale m ∼ 10^11 GeV of radial excitations and the mass scale m_a = √(χ(T))/f_a ∼ H ∼ 10^-18 GeV of angular fluctuations.The simulations have to take place at the m_a scale, which means that the radial-mass scale cannot be resolved.Naively this should not matter, as radial excitations should decouple.But it does matter, because the theory contains topological string defects which play a role in the dynamics, and the string tension depends logarithmically on the ratio m/m_a. Let's explain this in a little more detail. §.§ String defects First note thatis only defined modulo 2π.Therefore in traversing a circle,might return to its starting value, but it might only return modulo 2π, that is, ∮∂_idx^i = 2π N.The integer N is a winding number which counts a “flux” of string defects through the circle. If we deform a loop, N can only change when the loop passes through a singularity in thefield.The locus of these singularities defines the axionic cosmic string.We illustrate the idea with Figure <ref>, which shows a 2D slice out of a simulation, representing the complex field as a field of arrows with length and direction.The field direction has singularities where the arrows have zero length; going around the singularity, the direction of φ revolves by ± 2π.The singular point extends in 3D into a line where the field has zero value; any loop circling this line will have the direction of φ revolve by ± 2π as the one circles around the string.Such a defect – essentially a vortex in the φ field – is called an axionic cosmic string, and it is topologically stable; no local changes to the value of φ can cause it to disappear.If PQ symmetry is restored in the early Universe, thenstarts out uncorrelated at widely separated points and will generically begin with a dense network of these strings (the Kibble mechanism for string production <cit.>).The strings evolve, straightening out, chopping off loops, and otherwise reducing their density, arriving at a scaling solution <cit.> where the length of string per unitvolume scales with time t as t^-2.They may play a dominant role in establishing axion production in the scenario under discussion <cit.>.Let us analyze the structure of a string in a little more detail. Consider a straight string along the z axis; in polar (z,r,ϕ) coordinates the string equations of motion are solved by √(2)φ = v(r) f_a e^iϕ, with v(r) ≃ 1 for all r ≫ 1/m; so = ϕ (up to a constant which we can remove by our choice of x-axis).The string's energy is dominated by the gradient energy due to the space variation of :T_str = / = ∫ r dr dϕ( V(φ^* φ) + ∇φ^* ∇φ)≃ π∫ rdr ( ∂_ϕφ^*/r ∂_ϕφ/r) ≃π∫^H^-1_1/m r drf_a^2/r^2 = π f_a^2 ln(m/H) ≡π f_a^2 κ ,where the integral over r is cut off at small r by the scale where v(r) ≠ 1 (the string core), and at large distances by the scale where the string is not alone in the Universe but its field is modified by other strings or effects; this should be the larger of H and m_a.We define κ = ln(m/H) as the log of this scale ratio.Now m is at most f_a ∼ 10^11 GeV, and to ensure that the radial particles decay by the scale of 1 GeV we need m > 10^3 GeV.Therefore κ∈ [48,67].This logarithm, κ, controls several aspects of the strings' dynamics.It controls the string tension, as we just saw.More relevant, while the string tension is πκ f_a^2, the string's interactions with the long-range φ field scale as f_a^2without the κ factor.Therefore the string's long-range interactions become less important, relative to the string evolution under tension, as κ gets larger.The long-range interactions are responsible for energy radiation from the strings, as well as for long-range, often attractive, interactions between strings.Since these effects tend to deplete and straighten out the string network, the large-κ theory will have denser, kinkier strings.Indeed, in the large κ limit the string behavior should go over to that of local (Nambu-Goto) strings <cit.>. Unfortunately, a numerical implementation must resolve the length scale m, ma ≤ 1, and cannot exceed m/H ∼ 1000; numerical studies of the scalar field system have κ < 7, nearly an order of magnitude too small. §.§ Wall defects Besides the strings, there are also wall defects.These occur late in the simulation when m_a ≫ H.The potential term χ(T) (1-cos) then forces ≃ 0 nearly everywhere – modulo 2π.But suppose some region has ≃ 0 and another has ≃ 2π.There must be some 2D surface between them with ≃π.This is a wall defect.The region near the defect wherediffers significantly away from its minimum has thickness ℓ∼ 1/m_a, which is easily resolved on the lattice.The surface tension of such a surface turns out to be 8 m_a f_a^2.A 2D slice of a configuration, illustrating such a domain wall attached to a string, is shown in Figure <ref>.These wall defects are not a problem to simulate.But they play an important role in the dynamics.Every string hastake every value [0,2π] as one goes around the string.That includes =π.Therefore every string is attached to a domain wall.When m_a becomes m_a ≫κ H, the force from the domain wall tension becomes large enough to pull around the strings, leading to the collapse of the string network and the annihilation of all strings.It is only after this network collapse that one can speak about axion number.Because of the factor κ in the needed tension, a large-κ simulation will feature a more persistent string network.One final problem for scalar-only simulations, pointed out in <cit.>, is that the domain walls actually lose even their metastability as soon as m_a^2/m^2 > 1/39.This drives up the required size of simulations so that large m can be achieved.§ SIMULATING HIGH-TENSION STRINGS We see from the previous section that simulations of the φ field alone are not reliable.Although one can make the scale m very heavy compared to H,m_a, the string tension depends logarithmically on this scale, and is nearly a factor of 10 too small.This profoundly affects the dynamics of the string network, and therefore renders the results unreliable.We need a method to simulate high-tension strings coupled to . We found such a method in <cit.> and present it here. §.§ Effective theory We are interested in the large-scale structure of string networks and the infrared behavior of any (pseudo)Goldstone modes they radiate.For these purposes it is not necessary to keep track of all physics down to the scale of the string core.Rather, it is sufficient to describe the desired IR behavior with an effective theory of the strings and the Goldstone modes around them.This consists of replacing the physics very close to the string core with an equivalent set of physics.It has long been known how to do this <cit.>.The string cores are described by the Nambu-Goto action <cit.>, which describes the physics generated by the string tension arising close to the string core.The physics of the Goldstone mode is described by a Lagrangian containing the scalar field's phase.And they are coupled by the Kalb-Ramond action <cit.>:= _NG + _GS + _KR ,_NG=κ̅π f_a^2 ∫ dσ√(y'^2(σ)(1-ẏ^2(σ))) ,_GS= f_a^2 ∫ d^3 x∂_μ∂^μ ,_KR= ∫ d^3 x A_μν j^ μν , H_μνα= f_a ϵ_μναβ∂^β = ∂_μ A_να +, j^ μν= -2π f_a ∫ dσ( v^μy'^ν - v^νy'^μ) δ^3 (x-y(σ)).Here σ is an affine parameter describing the string's location y^μ(σ,t), v^μ=(1,ẏ)=dy^μ/dt is the string velocity, and H_μνα and A_μν are the Kalb-Ramond field strength and tensor potential, which are a dual representation of . Effectively _NG tracks the effects of the string tension, which we name κ̅π f_a^2, stored locally along its length.Next, _GS says that the axion angle propagates under a free wave equation, as expected for a Goldstone boson, and its decay constant is f_a.And _KR incorporates the interaction between strings and axions, also controlled by f_a.The interaction can be summarized by saying that the string forcesto wind by 2π in going around the string (in the same sense that the eJ_μ A^μ interaction in electrodynamics can be summarized by saying that it enforces that the electric flux emerging from a charge is e).It should be emphasized that in writing these equations, we are implicitly assuming a separation scale ; at larger distances from a string r> we consider ∇φ energy to be associated with ; for r < the gradient energy is considered as part of the string tension <cit.>, meaning that κ̅ incorporates all tension contributions from scales shorter than .Any other set of UV physics which reduces to the effective description of Ltot would present an equally valid way to study this string network.Our plan is to find a model without a large scale hierarchy, such that the IR behavior is also described by Ltot with a large value for the string tension. Optimally, we want a model which is easy to simulate on the lattice with a spacing not much smaller than .Reading LNG through jmn in order, the model must have Goldstone bosons with a decay constant f_a and strings with a large and tunable tension T_str = κ̅π f_a^2, with κ̅≫ 1.There can be otherdegrees of freedom, but only if they are very heavy (with mass m ∼^-1), and we will be interested in the limit that their mass goes to infinity.Finally, the string must have the correct Kalb-Ramond charge.Provided everything is derived from an action, this will be true if the Goldstone boson mode always winds by 2π around a loop which circles a string. §.§ The model We do this by writing down a model of two scalar fields φ_1,φ_2, each with a U(1) phase symmetry.A linear combination of the phases is gauged; specifically, the fields are given electrical charges q_1 ∈𝒵 and q_2 = q_1-1 under a single U(1) gauge field. The orthogonal phase combination represents a global U(1) symmetry which will give rise to our Goldstone bosons.The role of the gauge symmetry will be to attach an abelian-Higgs string onto every global string, which will enhance the string tension.The added degrees of freedom are all massive off the string, achieving our intended effective description.The model falls under the general rubric of “frustrated cosmic strings”<cit.>, but our motivation and some specifics (particularly our initial conditions) are different.Specifically, the Lagrangian is- (φ_1,φ_2,A_μ) = 1/4e^2 F_μν F^μν + | (∂_μ -i q_1 A_μ) φ_1 |^2 + | (∂_μ -i q_2 A_μ) φ_2 |^2 = + m_1^2/8 v_1^2( 2φ_1^* φ_1 - v_1^2 )^2 + m_2^2/8 v_2^2( 2φ_2^* φ_2 - v_2^2 )^2+ λ_12/2(2φ_1^* φ_1 - v_1^2 ) ( 2φ_2^* φ_2 - v_2^2 ).For simplicity we will specialize to the caseλ_12=0,m_1=m_2=√(e^2(q_1^2 v_1^2 + q_2^2 v_2^2))≡ m_e.The model has 6 degrees of freedom; two from each scalar and two from the gauge boson.Symmetry breaking, φ_1 = e^iθ_1 v_1√(2) and φ_2 = e^iθ_2 v_2 √(2), spontaneously breaks both U(1) symmetries and leaves five massive and one massless degrees of freedom. Specifically, expanding about a vacuum configuration, the fluctuations and their masses arev_1→ v_1 + h_1,m = m_1 v_2→ v_2 + h_2,m = m_2 A_i ≠ 0,m = √(e^2(q_1^2 v_1^2 + q_2^2 v_2^2))≡ m_e (θ_1,θ_2)→ (θ_1,θ_2) + ω (q_1,q_2), (θ_1,θ_2)→ (θ_1,θ_2) + ( q_2/q_1^2+q_2^2 , -q_1/q_1^2 + q_2^2)m = 0.We see that the choices in masses_equal have made all heavy masses equal. [ We set λ_12=0 so that the fluctuations in |φ_1| and |φ_2| are unmixed; our other choices ensure that all heavy fields have the same mass.We could consider other cases but we see no advantage in doing so if the goal is to implement the model on the lattice.The lattice spacing is limited by the inverse of the heaviest particle mass, while the size of the string core and the mass of extra degrees of freedom off the string will be set by the inverse of the lightest particle mass.So we get a good continuum limit with the thinnest strings, and therefore the best resolution of the network, by having all heavy masses equal.] To clarify, note that a gauge transformation A_μ→ A_μ + ∂_μω changes θ_1 →θ_1 + q_1 ω and θ_2 →θ_2 + q_2 ω.Therefore the linear combination of θ_1,θ_2 fluctuations with δθ_1 ∝ q_1 and δθ_2 ∝ q_2 isprecisely the combination which can be shifted into A^μ by a gauge change, and is therefore the combination which is “eaten” by the A-field to become the third massive degree of freedom.The remaining phase difference q_2 θ_1 - q_1 θ_2 is gauge invariant,q_2 θ_1 - q_1 θ_2 →_ω q_2(θ_1 + q_1 ω) - q_1(θ_2 + q_2 ω) = q_2 θ_1 - q_1 θ_2 + 0 ωand represents a global, Goldstone-boson mode. §.§ The strings We initialize φ_1 and φ_2 with the same space-random initial phase, which ensures that all strings will have each scalar wind by 2π, and the strings will have global charge q_1-q_2=1.To find the tension of such a string, we write the Ansatz √(2)φ_1(r,ϕ) = e^i ϕ f_1(r) v_1,√(2)φ_2(r,ϕ) = e^i ϕ f_2(r) v_2, A_ϕ(r) = g(r)/r ,and derive the equations of motion from L-2field,g” - g'/r= e^2 v_1^2 f_12 q_1(q_1 g-1) + e^2 v_2^2 f_2^2 q_2(q_2 g-1), f_1” + f_1'/r= f_1/r^2 (1-q_1 g)^2 + m^2/2 f_1(f_1^2-1), f_2” + f_2'/r= f_2/r^2 (1-q_2 g)^2 + m^2/2 f_2(f_2^2-1).Here f_1,f_2 represent the progress of the two scalar fields towards their large-radius asymptotic vacuum values, while 2π g(r) is the magnetic flux enclosed by a loop at radius r, which trends at large r towards the total enclosed magnetic flux. The large-r behavior is well behaved only if f_1→ 1, f_2 → 1, andlim_r→∞g(r) = q_1 v_1^2 + q_2 v_2^2/q_1^2 v_1^2 + q_2^2 v_2^2 = 1/2π .The magnetic flux is therefore a compromise between the value 1/q_1, which cancels large-distance gradient energies for the first field, and 1/q_2, which cancels large-distance gradient energies for the second field.The gradient energy at large distance is given byT_str ≃ 2π∫ r dr ( |D_ϕφ_1|^2 + |D_ϕφ_2|^2 ) ≃π∫ r dr ( v_1^2/r^2( 1 - q_1 g )^2 + v_2^2/r^2( 1 - q_2 g )^2 ) ≃π∫dr/r v_1^2 v_2^2/q_1^2 v_1^2 + q_2^2 v_2^2 .Comparing Tension, kappa with tension-fa, we identify the Goldstone-mode decay constant asf_a^2 = v_1^2 v_2^2/q_1^2 v_1^2 + q_2^2 v_2^2 . For a more intuitive explanation, consider Figure <ref>. It shows the set of possible phases (θ_1,θ_2) for the two scalar fields, in the case (q_1,q_2) = (4,3).The figure includes a dotted line to indicate which phase choices are gauge-equivalent.Moving along the dotted line corresponds to changing the gauge, or moving through space along a gauge field; a vector potential of the right size can cancel a gradient energy along this field direction.The orthogonal direction, which is unaffected by a gauge field, is the global (axion) field direction.A change in this direction from one blue dotted line to the next represents a full 2π rotation in the (axial) Goldstone direction, which explains the value of f_a found in fa-is.Figure <ref> then shows how each field varies around a string.As we consider loops farther and farther from the string's center, more and more flux is enclosed, so more and more of the gradients along the blue-dotted direction are canceled by the A_ϕ field.For the innermost loop there is no enclosed flux, and the gradient energy is given by the distance between the point (θ_1,θ_2)=(0,0) to the point (2π,2π).For a loop enclosing the entire flux, all gradient energy arising from the gauge-direction is canceled, almost but not fully removing the gradient energy.Only the gradient energy arising from the shortest path from one blue-dotted line to the next cannot be compensated.This represents the residual global charge of the string.This path length is 2π f_a.Now let us estimate the effective value of κ̅, the added contribution to the string tension in units of the long-distance Goldstone-mode contribution.The energy of the string's core is the energy of an abelian Higgs string with m_h = m_e and with f^2 = v_1^2 + v_2^2, which isT_str,abelian≃π ( v_1^2 + v_2^2 ).The value of κ̅ is thereforeκ̅= T_str/π f_a^2≃v_1^2 + v_2^2/v_1^2 v_2^2/q_1^2 v_1^2 + q_2^2v_2^2 = (v_1^2+v_2^2)(q_1^2 v_1^2+q_2^2 v_2^2)/v_1^2 v_2^2⟶_v_1=v_2 2(q_1^2 + q_2^2).Detailed calculations show that this is indeed the added tension. The full value of κ is κ = κ̅+ ln(m/H) where m,H are the values actually used in the numerical simulation; typically m/H ∼ 1000.So choosing q_1=4 and v_1=v_2 gives κ̅= 50 and κ = 57, in the middle of the physically interesting range. Figure <ref> shows how the density of the string network depends on the string tension.The network density is expressed in terms of the dimensionless scaling variable ξ,L_sep^-2≡ V^-1∫_all stringγ dl, ξ≡t^2/4 L^2 .We see that it increases by over a factor of 3 as one goes from scalar-only (κ≃ 7) to high-tension (κ≃ 57) simulations.It is not clear that our lattices are large enough to see the onset of scaling behavior for the highest-tension networks we studied.We certainly don't see scaling behavior for the abelian Higgs simulations, but this is another story.§.§ Numerical implementation I will not insult this audience by explaining how to implement a bosonic U(1) theory on the lattice.I use the noncompact formulation of U(1) and a next-nearest neighbor (a^2) improved action.There are no issues of renormalization of parameters because we are studying classical field theory.A subtlety in implementing electric fields with improvement is handled as in <cit.>.We pause only to mention one subtlety in how we implement χ (1-cos).The function cos, with = q_2 θ_1 - q_1 θ_2, is highly singular near the string core.Such singular behavior creates problems under space discretization, so we have to “round off” the behavior inside the string core, with the substitutionχ(t) ( 1 - cos)⇒χ(t) F(2φ_1^*φ_1) F(2φ_2^* φ_2) ( 1 - cos( q_2 _1 - q_1 _2 ) ) , F(r)≡{[ 25/16 r ( 8/5 - r ) ,r< 4/5 ,; 1 , r > 4/5 . ].The smoothing function F(r) is chosen such that F(1)=1, F'(1)=0, F(0)=0, and F'(r) is continuous. This modification only affects the dynamics inside string cores, but the χ(t) (1-cos) term is much smaller than the other potential terms there.Indeed, this term is everywhere small and it is only important because it operates over much of the lattice volume, while the radial potential has an effect only in the tiny fraction of the lattice corresponding to string cores.§ RESULTS We studied this model <cit.> on 2048^3 lattices using a single compute node containing two Xeon Phi (KNL) processors.After systematic studies of lattice spacing and continuum limits, we also studied how the network evolution and the axion production depend on the string tension.The most notable result is that the axion production is actually smaller for the case with strings than the angle-average of the “misalignment” mechanism of EOMax.This contradicts the “conventional wisdom”<cit.> that axion production should be the sum of a misalignment contribution, a string contribution, and a wall contribution.We claim that this view double counts; the energy in domain walls is the energy of field misalignment, from values ∼π.This energy represents most of the potential axion production from misalignment.But the energy in the wall network is mostly absorbed by the strings when the walls pull on the strings, giving their energy to string velocity.Then the strings chop up into loops, which appear to produce axions quite inefficiently.Combining our numbers with expressions for the energy budget g_* and topological susceptibility from <cit.>, and the observed dark matter density, we find m_a = 26.2 ± 3.4μeV.§ TOPOLOGY: TEMPERATURE RANGE We have emphasized that an input value for χ(T) from the lattice is essential to deriving these results.But we don't need χ(T) at all temperatures; some are more important than others.We will extend the results of <cit.> by investigating over what temperature range the topological susceptibility is really needed.At sufficiently high temperature χ(T) is small.Therefore χ(T)(1-cos) plays little role in the field dynamics.Its importance is controlled by the combination m_a t which rises with time as approximately t^5.8.Therefore at high temperature, large errors in χ(T), or no value at all, is not a problem.To study this, we replace χ(T) with the following “chopped” form:χ(T) →{[ χ(T)T <;χ() T >.;].Technically it is t^2 χ(T) we chop in this way, because of the t^2 factor in, eg, EOMax.We then study the axion production as a function of the temperature .Whenceases to depend on , we know that χ() is not relevant.But so long as the result withis different than the unchopped limit, we need χ(T) at that temperature.We see from Fig.<ref> that above about 1150 MeV, it makes little difference if we change χ(T).But below this value, we are quite sensitive.So it is necessary to determine χ(T) all the way up to 1150 MeV.On the IR side, perhaps surprisingly, there is also a range of temperatures where the susceptibility is not important.That is because the strings and walls are gone and thevalue is small and oscillating rapidly, with m_a ≫ H.Then the axion number is an approximate adiabatic invariant, which reacts smoothly to Hubble expansion and mass shifts and does not feel the anharmonicity of the potential, 1-cos≃^2/2.All that matters is the final value of χ(T→ 0), which is well known.To find out where this temperature range starts, we make a similar modification:χ(T) →{[ χ(T)T >;χ() T <.;].That is, we freeze χ(T) (really, t^2 χ(T)) from rising after some point, and see if that changes the axion number produced.The results are shown in Fig. <ref>. They indicate that the susceptibility is irrelevant below about 550 MeV; the behavior above that temperature is important.Therefore the axion dynamics is sensitive to χ(T) in the range 550 to 1150 MeV; outside of that range it is not.§ CONCLUSIONS The axion is a well motivated hypothetical particle, because it might explain two mysteries – the P and T symmetry of QCD, and the nature of the Dark Matter – with a single mechanism and particle. The physics of the axion in cosmology is rich, governed by a network of string defects which are swept together when domain walls form. Its explication requires two new pieces of physics.We need to understand this network evolution and axion production better.And we need to know the QCD topological susceptibility as a function of temperature, χ(T), because it sets the tension of the domain walls and controls the physics which destroys the string network.I have presented the latest details on the network evolution, introducing a new technique which allows simulation of high-tension strings without excessive numerical resources.The results indicate rather inefficient axion production and therefore a rather light axion compared to previous studies, with m_a = 26.2 ± 3.4μeV.It remains to form a consensus in the lattice community that the topological susceptibility is well measured.My work indicates that the susceptibility is needed in a temperature range from 550 to 1150 MeV.Below this range the evolution is adiabatic.Above this range the susceptibility does not yet play a role in axion dynamics. Clearly it is challenging to determine the susceptibility at such high temperatures.But I am confident this is a challenge which the lattice community will accept with gusto.§ ACKNOWLEDGMENTS I am indebted to close collaboration with Vincent Klaer and Leesa Fleury, and to useful conversations with Mark Hindmarsh.This work has been supported both by the TU Darmstadt's Institut für Kernphysik and the GSI Helmholtzzentrum, and by the Department of Physics at McGill University and the Canadian National Science and Engineering Research Council (NSERC). | http://arxiv.org/abs/1709.09466v1 | {
"authors": [
"Guy D. Moore"
],
"categories": [
"hep-ph",
"hep-lat"
],
"primary_category": "hep-ph",
"published": "20170927121408",
"title": "Axion dark matter and the Lattice"
} |
[email protected] Department of Physics, The University of Texas at Austin, Austin, TX 78712, [email protected] Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan title author#1#2#3#4#5@arrow 0099@#1#2#3#4#5 There is a closeconnection between various new phenomena in Weyl semimetals and the existence oflinear band crossings in the single particle description. We show, by a full self-consistent mean-field calculation, how this picture is modified in the presence of long-range Coulomb interactions. The chiral symmetry breaking occurs at strong enough interactions andthe internode interband excitonic pairing channel is found to be significant, which determines the gap-opened band profile varying with interaction strength. Remarkably, in the resultant interacting phase, finite band Chern number jumps in the three-dimensionalmomentum space are retained, indicating the robustness of the topologically nontrivial features.Instability and topological robustness of Weyl semimetals against Coulomb interaction Xiao-Xiao Zhang December 30, 2023 =====================================================================================§ INTRODUCTIONThe physics related to the Weyl point or Weyl node, as the three-dimensional (3D) analog of the two-dimensional (2D) Dirac physics<cit.>, is sparking keen interests both theoretically and experimentally<cit.>. This started from the revival of the old concept of Weyl fermion<cit.> in the context of various condensed matter systems without time-reversal symmetry and/or inversion symmetry<cit.>. Besides the recent solid-state realizations in a family of nonmagnetic and noncentrosymmetric transition metal monoarsenides/monophosphides<cit.>, it is found or predicted as well in photonic crystals<cit.>, magnon bands<cit.>, and even photo-driven systems<cit.>. In contrast to the real-space emergent monopole structures<cit.>, a Weyl node has a momentum-space monopole nature<cit.>. Based on this and its special Landau level formation under a magnetic field<cit.>, many new phenomena are discussed and experimentally investigated in this gapless quantum phase of matter, i.e., the Weyl semimetal, including but not limited to the anomalous Hall effect<cit.>, the chiral magnetic effect<cit.>, and the observation of the negative magnetoresistance<cit.>. The Weyl nodes as degeneracies of codimension three are generally stable against local perturbations unless opposite-chirality nodes separated in the momentum space arecoupled to break the chiral symmetry. The disorder might not open a gap either, due to the randomly vanishing pinned Fourier component and the inadequate strength in practice. The other indispensable aspect is the interaction effect. There are some theoretical studies with different focuses and approximations to facilitate analytic analyses, including mean-field or renormalization group calculations within or beyond local interactions<cit.> and the formation of spin or charge density waves<cit.> or Luttinger liquids<cit.> under an external magnetic field. On the other hand, it is well known that an excitonic semimetal-insulator phase transition could occur under the influence of strong enough long-range Coulomb interaction<cit.>. The Coulomb interaction can bind electrons and holes to excitons, quasibosonic bound states, which can condense at low temperatures. Previous exciton condensate studies focus on semiconductor bilayer systems <cit.> and a 2D quantum well placed in an optical cavity<cit.> because a Bose-Einstein condensate (BEC) type low density exciton (polariton) limit exists in these 2D systems. Can we apply some similar analysis to a Weyl semimetal and address the natural question of whether itis stable against the Coulomb interaction and in what sense?To answer this, we study a simple type-I Weyl semimetal with vanishing density of states, i.e., the chemical potential is tuned at the Weyl points, under the long-range Coulomb interaction upon which we apply a standard Hartree-Fock approximation. The theory would become less valid when the chemical potential is away from the Weyl points due to complex interaction effects beyond the mean-field level, such as the strong screening in 3D at a finite density of states which renormalizes the interaction down to a short-range form<cit.>.Within the mean-field level taking all possible electron-hole pairing channels into account, we numerically carry out the self-consistent calculation without any other a priori approximation in order to draw unambiguous conclusions.Our main findings are twofold. First, a strong enough long-range Coulomb interaction connects the left and right nodes and breaks the chiral symmetry and the translational symmetry as well, leading to a finite gap opening. We explain how the quasiparticle band profile evolves using twoorder parameters, the dressed single particle band energy and the internode interband coupling. Second, using the self-consistent Hamiltonian obtained to calculate Chern numbers of many 2D slices in themomentum space, we find that the nontrivial topology of the system is robust and retained despite losing the Weyl nodes. This supports the topologically nontrivial and axionic nature of the density wave phase<cit.>. The paper is organized as follows. In Sec. <ref> we introduce ourmodel Hamiltonianand explain how we apply mean-field calculations . In Sec. <ref> we present our band structure results and Chern number calculations where the phase transition occurs under strong enough interactions. We also discuss the quasiparticle band evolution and demonstrate that the transition is continuous. In Sec. <ref> we summarizeand comment on related issues. § MODELWe consider a general continuum model of a time-reversal symmetry breaking Weyl semimetal with two Weyl nodes located symmetrically at ±K⃗=± K ẑ (ẑ is defined in momentum space rather than real space), labeled as s=R/L node or interchangeably s=±, respectively.The Hamiltonian can be written in the basis spanned by the state |β_σ^s⟩ of the Weyl electron (pseudo)spin σ=↑/↓ and the node pseudospin s=R/L Ĥ_0=∑_k⃗ψ_k⃗^†ħ v_F k⃗·σ⃗ s^z [ħ v k⃗·σ⃗0;0 -ħ v_F k⃗·σ⃗;]ψ_k⃗ +Ĥ_I, where k⃗ is the reduced momentum relative to the nodes at ± K ẑ and v_F is the Fermi velocity which varies in different materials, σ⃗ is the electron (pseudo)spin Pauli matrix, s⃗ is the node pseudospin Pauli matrix, ψ_k⃗=(c_↑^R,c_↓^R,c_↑^L,c_↓^L)^T, and the electron annihilation/creation operator c_σ^s(†)is for the state |β_σ^s⟩ with |β_↑^s⟩=(1,0)^T ,|β_↓^s⟩=(0,1)^T. Taking account of the presence of Weyl nodes (Appendix <ref>), the Coulomb interaction for this system has the formĤ_I=1/2Ω∑_σσ',ss'∑_k⃗,k⃗',q⃗ [V(q⃗) c_σk⃗+q⃗^s† c_σ' k⃗'-q⃗^s'† c_σ' k⃗'^s' c_σk⃗^s+ V(q⃗+2sK⃗) s_ss'^x c_σk⃗+q⃗^s† c_σ' k⃗'-q⃗^s'† c_σ' k⃗'^s̅'̅ c_σk⃗^s̅] where Ω is the 3D system volume,V(p⃗)=e^2/(ε_0ε_r |p⃗|^2) with the vacuum/relative permittivity ε_0 / ε_r, and s̅≠ s.Equation (<ref>) includes all possible intranode/internode scattering processes allowed by momentum conservation. This reduced momentum representation has the merit of expressing all the mean-field interactions in a momentum-diagonal manner. Applying the Hartree-Fock approximation<cit.> to the interaction Hamiltonian in Eq. (<ref>) as detailed in Appendix <ref>, we finally getĤ_MF=∑_k⃗ψ_k⃗^† (H_0+H_Hartree+H_Fock) ψ_k⃗where[H_Hartree]_σσ'^ss' = V(2K⃗)∑_k⃗'(ρ^ss'_σσ'+ρ^ss'_σ̅σ̅')_k⃗'δ_σσ' s^x_ss',and [H_Fock]_σσ'^ss' = -∑_k⃗'[V(k⃗-k⃗')ρ^ss'_σσ'k⃗'+ δ_ss' V(k⃗-k⃗' + 2sK⃗)ρ^s̅s̅'_σσ'k⃗'].Here, σ̅≠σ, and the density matrix is defined relative to a reference value determined by the filling of the noninteracting ground state ρ^ss'_σσ'k⃗ = ⟨c_σ' k⃗^s'† c_σk⃗^s|-⟩ρ̅^ss'_σσ'k⃗,where ρ̅^ss'_σσ'k⃗ = δ_ss'⟨β_v^s|β_σ'^s|⟨%s|%s⟩⟩β_σ^s|β_v^s (see the next paragraph).We note that the long-range Coulomb interaction allows for both the internode and intranode couplings. Even though the Hartree contribution to the internode coupling might be small due to the decay of the Coulomb interaction at large momentum transfer, the Fock contribution still accommodates possible strong internode coupling. In our self-consistent calculation, given all nonzero initial terms, the density matrix ρ and hence the mean-field Hamiltonian Ĥ_MF are iteratively updated at each iteration using the lowest two eigenvectors until the convergence. Note that the trace of ρ at each momentum is always zero.For each node, the noninteracting system can be diagonalized in the band basis representation |β_n^s⟩ of band n=c,v (conduction/valence band) and node s. We have |β_c^R⟩_k⃗=|β_v^L⟩_k⃗=(cosθ/2,sinθ/2^ϕ)^T and |β_v^R⟩_k⃗=|β_c^L⟩_k⃗=(-sinθ/2,cosθ/2^ϕ)^T, where the momentum k⃗ has polar and azimuth angle θ,ϕ.In order to avoid the tedious band states' overlap functions, as shown above, we formulate the interaction Hamiltonian in the spin basis |β_σ^s⟩ where we take care of the noninteracting ground state using the relative density matrix ρ. Since we include all the possible interaction channels, it is fundamentally equivalent to work with either the spin basis or the band basis, which are related by a unitary transformation U^ss'_nσ=s^0_ss'⟨β_n^s|β_σ^s'|$⟩. For instance, we actually determine the aforementioned reference matrixρ̅and the form ofH_HartreeusingUand the overall charge neutrality constraint∑_n ⟨c_n k⃗^s† c_n k⃗^s|=⟩ 1. Especially, we will later use the band basis density matrix and Hamiltonian (henceforth denoted with a tilde) of the formρ̃ = U^†T ρU^TandH̃ = U H_MF U^†, whereupon more transparent physical understandings become available.We use a modified Rydberg unit in the calculation, settingħ=e^2/2=4πε_0=1whereħis the Planck constant,eis the elementary charge, andε_0is the vacuum permittivity. This leads to a characteristic velocityv_0=4πε_0 e^2/2ħ=1(10.9×10^5 m/sin SI unit). Combined with the material dependent relative permittivityε_rand the Fermi velocityv_F, we have the quantityv_0/ε_rv_Fcharacterizing the strength of the interaction. Therefore, smallerv_Fandε_reffectively mean stronger Coulomb interaction effects and only the combined value ofε_r v_F, referred to as the relative velocityv_rhenceforth, matters. The realistic typical ranges arev_F=0.5–3×10^5 m/sandε_r=10–20<cit.>, which means aboutv_r=0.5–6in our unit. Because of the massless linearly dispersed band structure and the long range Coulomb interaction, an important feature of this system is the lack of an intrinsic lengthscale<cit.>. Even if one sums up to a certain momentum magnitude (a sharp ultraviolet bandwidth cutoffv_rKin our case), the obtained band energies will be just proportional to that cutoff. Therefore, in our theory, the concrete predictions are the band profiles, phase transitions, topological features rather than the exact gap or band energies. Indeed, we observe this feature in our numerical calculations and an inspection of its self-consistency is given in Appendix <ref>. § RESULTSOur numerical results are based on a34×34×34cubic momentum grid withk_x,k_y,k_z ∈[-K/2,K/2].The momenta and energies (sincev_0=1) are thus indicated with a unitk_c=K/2in all the figures. The minimal momentum magnitude is thereforek_Γ=K/66and up to this accuracy, we refer to momenta along an axis or at theΓpoint in the following. Here we deliberately detour theΓpoint to avoid the gauge choice ambiguity at the node. In this setup, left and right nodes located at±K ẑare well separated and expressed in the diagonal blocks in the4×4Hamiltonian. Only the long-range Coulomb interaction can induce off-diagonal terms that lead to chiral symmetry breaking and gap opening. Tuning the relative velocityv_ras aforementioned and keeping Coulomb interaction in its vacuum form, we can explore different phases and physical properties by looking at the renormalized quasiparticle band structure and the Chern number profile. §.§ Mean-field band profileIn Fig. <ref>, we plot renormalized four eigenvalues alongthek_zaxis. The plots alongk_xandk_yaxes are the same for the unbroken rotational symmetry with respect to thek_zaxis, and their difference with Fig. <ref> is insignificant. Unlike in the 2D case where arbitrarily weak attractive interaction will create bound states, a strong enough interaction is required to create internode or intranode electron-hole bound states, if any, in 3D Weyl semimetals. At a large relative velocityv_r=1, the interaction strength is not strong enough to bind electron-hole pairs and the band profile is unchanged, i.e.,E=±|k⃗|with double degeneracy. Note that the tiny gap in Fig. <ref>(a) simply comes from the momentum resolutionk_Γand all the energies have perfect linearity. To clarify whether we have electron-hole excitonic pairs, we also plot the dominant interaction induced internode electron-hole pairing term magnitude|H̃_cv^RL|alongk_zshown as red dots in Fig. <ref>. Note that we haveH̃_cv^RL=H̃_cv^LRdue to the inversion symmetry.Atv_r=1,H̃_cv^RLin Fig. <ref>(a) converges to zero everywhere, illustrating the absence of any bound states.Other than this case, we observe gap openings in Fig. <ref> which are accompanied by the strong internode s-wave pairings, which are nonzero at zero momentum, shown in Fig. <ref>.Meanwhile, a finite number of electron-hole pairs are created by the interaction andthe exciton density readsn_ex=trρ̃_cc/Ω=∑_k⃗(ρ̃_cck⃗^RR+ρ̃_cck⃗^LL)/Ω, which equals the similar valence band trace ofρ̃_vvby virtue of the particle-hole symmetry. Because we place the Weyl nodes on thek_zaxis, interaction terms containing2K ẑself-consistently persist. It not only breaks the rotational symmetry with respect to thek_xork_yaxes, but also, more importantly, completely lifts the double degeneracy, although the splitting is small due to the decay of the Coulomb potential at large momenta. For instance, we haveρ̃_cc^RR ≠ρ̃_cc^LLin consequence.To clearly inspect this phase transition, in Fig. <ref>(a), we plot two order parameters, the internode interband pairing magnitude|H̃_cv^RL(k⃗=0)|and the exciton densityn_exin a small range ofv_rnear the critical valuev_r^c=0.96, which is similar to a recent study<cit.>.Both|H̃_cv^RL(k⃗=0)|andn_exshow a typical second-order continuousphase transition that is expected from normal exciton condensates<cit.>. Using the wavefunctions, we also plot bands with node weight in Figs. <ref>(b) and <ref>(c) to illustrate the lifting of the R/L node degeneracy and the no longer conserved node pseudospin under strong interaction. One particle-hole symmetric pair of bands (second and third highest bands) chooses one major node while the other (first and fourth highest bands) chooses the opposite. At small momenta, R/L nodes are mixed in agreement with later discussion that the interaction induced internode pairing mainly contributes at small momenta when the phase transition just occurs. When interaction is not strong enough, these pairs of bands have entirely mixed colors in calculation due to the R/L degeneracy. When we further decreasev_r, the quasiparticle gap becomes larger and the band extrema move from zero to finite momenta relative to the cutoff momentum. This hump-like profile is similar to the BEC-BCS crossover<cit.> where exciton condensate pair-excitation energy extrema move to finite momenta at larger exciton density. To illustrate band evolution after the phase transition, we write down a simplified Hamiltonian in the band basis based on our mean-field result and the symmetry,H̃'= [v_rk-ξ_k⃗00 Δ_k⃗;0 -v_rk+ξ_k⃗ Δ^*_k⃗0;0 Δ_k⃗v_rk-ξ_k⃗0; Δ^*_k⃗00 -v_rk+ξ_k⃗ ],where we only consider the internode interband couplingΔ_k⃗and the dressed single particle energyξ_k⃗, i.e.,H̃_cv^RLandH̃_cc^RRpreviously discussed, and neglect the degeneracy-lifting effect. Both the intranode interband and internode intraband terms are negligibly small and dropped for simplicity.The corresponding eigenvalues are doubly degenerateE_k⃗=±√((v_rk-ξ_k⃗)^2+|Δ_k⃗|^2). The pair-excitation gap at each momentum is2|E_k⃗|. Atv_r=0.9in Fig. <ref>(b) where the phase transition just occurs, a small amount of internode electron-hole pairs are bound and exciton condensates form. The condensates lead to the small but finite internode interband pairing and the dressing of single-particle energy.As shown in Fig. <ref>(b),Δ_k⃗is maximized atk=0and decays with increasingk. Since the single particle partv_r know outweighsξ_k⃗,v_r k - ξ_k⃗remains almost linearly increasing with slopev_r.The gap minimum is thus located at zero momentum as a result of the dominant single-particle contribution. At large momenta, since the interaction effect is diminished, the band structure resembles the noninteracting linear one.Asv_rbecomes even smaller shown in Fig. <ref>(c),Δ_k⃗outweighs the diagonal partv_r k-ξ_k⃗at small momenta due to the cancellation betweenv_r kandξ_k⃗.The gap minimum then moves to finite momenta becauseΔ_k⃗decreases slowly as the momentum increases. At large momenta, the noninteracting term again becomes more pronounced, leading to hump like bands in Fig. <ref>(c).For completeness, we also plot the case of very smallv_rin Fig. <ref>(d) in which the top (bottom) of the upper (lower) band is at small momenta.Unlike the previous ones, it seemingly inherits little linear band remnants. This originates from the fact that the single particle linear band is not strong enough to diminish the Coulomb interaction to small values at large momenta, which is seen from the dominance and slow decay of the interaction inducedH̃_cv^RLin Fig. <ref>(d).ThenΔ_k⃗and hence the gap decrease as the momentum increases.If one were to avoid this, nonlinearity could be introduced to the Weyl bands at large momenta, which is, however, not within the scope of the current study.Last but not least, we emphasize the importance of includingξ_k⃗, which is from the intraband exchange interaction, in a full mean-field calculation in order to explain how renormalized quasiparticle bands evolve. In some previous studies, it was ignored to analyze the pairing gap alone<cit.>, which is unjustified in a complete theory and insufficient to capture all the physics such as hump like quasiparticle bands. §.§ Topological propertyMany of the topologically nontrivial features of the Weyl semimetal stem from the Weyl node as a source or sink of the flux of the momentum-space Berry phase. The simple and thorough way to see is to scan the Brillouin zone and calculate the Chern numbers slice by slice along several directions, say,k_x,k_y,k_z. A Chern number jump along any direction from±1/2to∓1/2clearly indicates the presence of a Weyl node of charge∓1. Here we will answer the question over the fate of such topological properties in the presence of the interactions. For any continuum model, the exact quantization will never be attainable unless one pushes the range of momentum toward the infinity. Taking thek_z-slice (k_x-k_y-plane) Chern numberC(k_z)of the noninteracting Weyl semimetalas an example, practically despite the imperfect quantization and the decay for larger and larger|k_z|as shown in Fig. <ref>(a), one can still identify a sharp jump of about±1at the node position.However, since one typically has to sum up the Berry curvature over the momentum space using the TKNN-type Kubo formula<cit.>, there lies another severe problem, viz., the density of the sampling mesh, which is in general very limited in 3D numerical calculations. One way out is the Wilson loop method that counts the winding of the Wannier center in a cylinder geometry and applies to various distinct topological systems<cit.>. Here, to make the most direct use of our calculated data, we adopt another strategy of remarkably fast convergence even with a 2D momentum mesh of tens or hundreds of points<cit.>. Based on lattice gauge theory, it sums up the gauge invariant plaquette Berry flux, e.g., for thek_x-k_yplane Chern number, C = 1/2π∑_k⃗ln[U_x(k⃗) U_y(k⃗+x̂) U_x(k⃗+ŷ)^-1 U_y(k⃗)^-1], C = 1/2π∑_k⃗[A_x(k⃗) + A_y(k⃗+x̂) -A_x(k⃗+ŷ) - A_y(k⃗)],where the summation is over a discrete momentum mesh and the lattice Berry connectionA_i(k⃗)=ln[⟨Ψ(k⃗)|Ψ(k⃗+î)|/⟩|⟨Ψ(k⃗)|Ψ(k⃗+î)||⟩]with the normalized Bloch state|Ψ(k⃗)⟩solved from our self-consistently converged Hamiltonian. Surprisingly, as shown in Figs. <ref>(b) and <ref>(c), up to a3%deviation from±1, we find that each of the four bands retain the Chern number jump along every direction at the original node position, i.e, theΓpoint of the reduced momentum. This appears identically in cases after the phase transition whereas we show only thev_r=0.8one for simplicity.C(k_x,y), unlikeC(k_z), has the same shape as the noninteracting case, which is again due to the asymmetry betweenk_zandk_x,y. Also, adjacent bands possess opposite slice Chern number at any momentum and hence opposite jumps. This means that, despite the fact that the R/L node is no longer a good quantum number due to the interaction induced mixing, the bands still partially inherit the topological features.As suggested by some previous studies, the axionic character and hence theθterm due to the chiral anomaly can survive from the dynamical mass generation due to the chiral symmetry breaking<cit.>. Here we demonstrate directly from a topological number calculation that the band topology is indeed more robust than the gapless Weyl nodes themselves. Thus, the Coulomb interaction does not necessarily deteriorate the topological electromagnetic responses such as the anomalous Hall effect and the chiral magnetic effect in Weyl semimetals, for instance. The nonzero Chern number jump in the absence of band touchings exceeds the conventional picture of bulk-boundary correspondence, where the violation of adiabaticity is required to nullify the topological index. Only rarely does this happen in the noninteracting picture by reducing or enhancing the symmetry and the accompanied topological class<cit.>. More relevantly, this is caused by interaction effects<cit.>. Our case can be understood as topological numbers of interacting Green's functions that surpass the single particle picture. The zeros rather than the poles of the Green's function play the role of generating topological numbers<cit.>. In our case, this information is encapsulated in the complicated momentum dependence of the pairings in the self-consistent Hamiltonian, in contrast to constant gap-opening terms that cannot lead to the above topological feature. A complementary aspect of the robust Chern number jump is that exciton condensates prevent the gap closing<cit.> which is necessary for a continuous phase transition between topologically trivial and nontrivial states.§ CONCLUDING REMARKSWe study how the long-range Coulomb interaction affects the properties of a generic Weyl semimetal with the chemical potential at the Weyl nodes.There are recently some studies on the instability of Weyl semimetals with interaction<cit.>. In this paper, we provide a yet missing complete mean-field study considering all possible Coulomb interaction induced phases and let the self-consistent procedure manifest the major channel.The gap-opened phase has a charge density wave character from the viewpoint of translational symmetry breaking due to the internode interband coupling. The coherence of this coupling is s-wave like since the Coulomb interaction favors isotropic interband pairing. Our main findings are that the Weyl nodes are not stable against strong enough Coulomb interactions while nontrivial topological Chern number jumps can survive after the gap opening. Our model itself cannot predict directly that anomalous Hall effect survives under strong Coulomb interaction, but it supports from a topological number viewpoint that topological responses are more robust than Weyl nodes themselves. A lattice model study of strong Coulomb interaction induced commensurate charge density wave order could explore whether a 3D magnetic insulator with nonzero Hall effect exists, which is beyond our current scope.Some interesting questions for future studies might further include relating this topologically nontrivial state to the axionic predictions in a more direct manner and calculating the electromagnetic responses. We also expect similar mean-field calculation could be done for the Dirac semimetal of much interest, using a doubled Hilbert space to account for the Kramers degeneracy. Besides the internode interband coupling dominant in this study, an intranode interband coupling is also possible to invalidate the symmetry protection and open the gap. The intranode coupling may not break translation symmetry but could break ann-fold rotational symmetry which leads to an interaction induced nematic state<cit.>.§ ACKNOWLEDGMENTSThe authors appreciate helpful discussions with A.H. MacDonald and M. Ezawa. X.-X.Z is grateful to Q. Niu for the hospitality during his stay at Austin (supported by the ALPS program). This work was supported by ARO (No. 26-3508-81), the Welch Foundation (No. F1473), JSPS Grant-in-Aids for Scientific Research (No. 26103006 and No. 16J07545) and CREST (No. JPMJCR16F1). § MEAN-FIELD APPROXIAMTION OF THE INTERACTIONStarting from the most general form of the Coulomb interactionĤ_I=1/2Ω∑_p⃗,p⃗',q⃗ V(q⃗) c_p⃗+q⃗^† c_p⃗'-q⃗^† c_p⃗' c_p⃗,we expand the electron operator and make use of the reduced momentumk⃗=p⃗-sK⃗to getc_p⃗ = c_p⃗+K⃗^L + c_p⃗-K⃗^Rfor the two nodes and find six terms allowed by the momentum conservation. In the spin basis, we further havec_k⃗^s = ∑_σ=↑↓ |β_σ^s⟩ c_σk⃗^s, then the resulting expression is given in a compact form by Eq. (<ref>). Performing the summation over node indexs'in Eq. (<ref>), we in fact obtain three termsĤ_I1= V(q⃗) c_σk⃗+q⃗^s† c_σ' k⃗'-q⃗^s† c_σ' k⃗'^s c_σk⃗^sĤ_I2= V(q⃗) c_σk⃗+q⃗^s† c_σ' k⃗'-q⃗^s̅† c_σ' k⃗'^s̅ c_σk⃗^sĤ_I3= V(q⃗+2sK⃗) c_σk⃗+q⃗^s† c_σ' k⃗'-q⃗^s̅† c_σ' k⃗'^s c_σk⃗^s̅, wherein we neglect the common prefactor and summations1/2Ω∑_k⃗,k⃗',q⃗∑_σσ', sfor simplicity. Firstly, by the Hartree approximation that contracts the direct operators, these three parts become Ĥ_Hatree1= V(0) ρ_σ'σ' k⃗'^ss c_σk⃗^s† c_σk⃗^sĤ_Hatree2= V(0) ρ_σ'σ' k⃗'^s̅s̅ c_σk⃗^s† c_σk⃗^sĤ_Hatree3= 2V(2K⃗) ρ_σ'σ' k⃗'^ss̅ c_σk⃗^s† c_σk⃗^s, respectively. Note that the first two Hartree terms cancel out because of the charge neutrality constraint which is reflected in the definition of the density matrices in Sec. <ref>. Secondly, by the Fock approximation that contracts the exchange operators, the three parts become Ĥ_Fock1= -2V(k⃗-k⃗') ρ_σ'σk⃗'^ss c_σ' k⃗^s† c_σk⃗^sĤ_Fock2= -2V(k⃗-k⃗') ρ_σ'σk⃗'^s̅s c_σ' k⃗^s̅† c_σk⃗^sĤ_Fock3= -2V(k⃗-k⃗'+2sK⃗) ρ_σσ' k⃗'^s̅s̅ c_σk⃗^s† c_σ' k⃗^s, respectively. Combining the above, we arrive at the mean-field Hamiltonian Eq. (<ref>).§ SELF-CONSISTENCY EQUATIONSHere we write down the mean-field self-consistency equations of a general two-band model, which is accessible to analytic analysis. We assign a linearly dispersing noninteracting part to it.H=∑_k⃗(a_ck⃗^†, a_vk⃗^†) (ξ_k⃗σ_z-Δ_k⃗σ_x) [ a_ck⃗; a_vk⃗ ],whereξ_k⃗=v_Fk-1/2Ω∑_k⃗'V(k⃗-k⃗')(1-ξ_k⃗'/E_k⃗'),Δ_k⃗=1/2Ω∑_k⃗'V(k⃗-k⃗')Δ_k⃗'/E_k⃗',E_k⃗=√(ξ_k⃗^2+Δ_k⃗^2).Here we assume the chemical potential at the band touching point. We can rewrite Eq. (<ref>) in integral form and recognize that all terms depend on the magnitude of the momentum only:ξ_k=v_Fk-e^2/4πϵ∫ (1-ξ_|k⃗-q⃗|/E_|k⃗-q⃗|)sinθ dqdθ dϕ,Δ_k=e^2/4πϵ∫Δ_|k⃗-q⃗|/E_|k⃗-q⃗|sinθ dqdθ dϕ ,E_k=√(ξ_k^2+Δ_k^2).The1/q^2in Coulomb interaction cancels with Jacobian factorq^2. Atk⃗=0, Eq. (<ref>) can be simplified ξ_0=-e^2/ϵ∫_0^k_max(1-ξ_q/E_q)dqΔ_0=e^2/ϵ∫_0^k_maxΔ_q/E_qdqE_0=√(ξ_0^2+Δ_0^2).We immediately notice that ifξ_q,Δ_q,E_q ∝q(a possible self-consistent solution) at finite momentum, then zero-momentum Hamiltonian terms are all linearly proportional to the cutoff we choose. This reassures our discussion of the lack of intrinsic length scale in the main text. | http://arxiv.org/abs/1709.09316v2 | {
"authors": [
"Fei Xue",
"Xiao-Xiao Zhang"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.mtrl-sci",
"cond-mat.str-el"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170927030702",
"title": "Instability and topological robustness of Weyl semimetals against Coulomb interaction"
} |
Ground-based detection of a cloud of methanol from Enceladus: When is a biomarker not a biomarker? [================================================================================================== We define a class of reflected backward stochastic differential equation (RBSDE) driven by a marked point process (MPP) and a Brownian motion, where the solution is constrained to stay above a given càdlàg process. The MPP is only required to be non-explosive and to have totally inaccessible jumps. Under suitable assumptions on the coefficients we obtain existence and uniqueness of the solution, using the Snell envelope theory. We use the equation to represent the value function of an optimal stopping problem, and we characterize the optimal strategy.Keywords: reflected backward stochastic differential equations, optimal stopping, marked point processes.§ INTRODUCTION Nonlinear backward stochastic differential equations (BSDE) driven by a Brownian motion were first introduced by Pardoux and Peng in the seminal paper <cit.>. Later, BSDE have found applications in several fields of mathematics, such as stochastic control, mathematical finance, nonlinear PDEs (see for instance <cit.>). As the driving noise, the Brownian motion has been replaced by more general classes of martingales; the first example is perhaps <cit.>, see <cit.> for a very general situation.In particular, occurrence of marked point processes in the equation has been considered since long. In <cit.>, related to optimal control and PDEs respectively, an independent Poisson random measure is added to the driving Wiener noise. Motivated by several applications to stochastic optimal control and financial modelling, more general marked point processes were considered in the BSDE. Examples can be found in <cit.> for L^2 solutions, <cit.> for the L^1 case and <cit.> for the L^p case. In connection with optimal stopping and obstacle problems, in <cit.> a reflected BSDE is introduced, where the solution is forced to stay above a certain continuous barrier process. This class of BSDE finds applications in various problems in finance and stochastic games theory. A number of generalizations has followed, both with variations on the nature of the barrier process and the type of noise. In the Brownian case, in <cit.> the author solves the problem when the obstacle is just càdlàgin <cit.>, the authors allow the obstacle to be only L^2. On the other hand, in <cit.> the authors solve the problem when a Poisson noise is added, and the barrier is càdlàg with inaccessible jump times. This is later generalized in <cit.> where the barrier can have partially accessible jumps too. Other specific results are <cit.> where a BSDE with two generators is solved in a Wiener framework and <cit.> in a Lévy framework; the papers <cit.> and <cit.> where the noise is a Teugels Martingale associated to a one-dimensional Lévy process. The paper <cit.> that considers a marked point process with compensator admitting a bounded desnity with respect to the Lebesgue measure.Finally, very general barriers beyond the càdlàg case were recently considered in <cit.>.It is the aim of the present work to address the case when the obstacle to be a càdlàg process and, in addition to the Wiener process, a very general marked point process occurs in the equation. The only assumptions we make is that it is non-explosive and has totally inaccessible jumps. This is equivalent to the requirement that the compensator of the counting process of the jumps has continuous trajectories. However, we do not require absolute continuiuty with respect to the Lebegue measure. To our knowledge, only in <cit.>, in <cit.>, and in <cit.>even more general cases have been addressed, butwithout reflection. The equation has the formY_t=ξ+∫_t^Tf_s(Y_s,U_s)dA_s +∫_t^Tg_s(Y_s,Z_s)ds-∫_t^T∫_EU_s(e)q(dtde)-∫_t^TZ_sdW_s+K_T-K_tY_t≥ h_t.Here W is a Brownian motion and q, independent from W, is a compensated integer random measure corresponding to some marked point process (T_n,ξ_n)_n≥ 1: see <cit.> as general references on the subject.The data are the final condition ξ and the generators f and g. A is a continuous stochastic increasing process related to the point process. The Y part of the solution is constrained to stay above a given barrier process h, and the K term is there to assure this condition holds This equation is then used to solve a non-markovian optimal stopping problem, where the running gain, stopping reward and final reward are the data used in the BSDE. Under additional assumptions on the barrier process, an optimal stopping time is characterized.This work generalizes the results previously obtained by allowing a more general structure in the jump component. This introduces some technical difficulties and some assumptions. For instance, we work in “weighted L^2 spaces", with a weight of the form e^β A_t, and the data must satisfy this integrability conditions. Direct use of standard tools, like the Gronwall lemma, becomes difficult in our case, so we have to resort to direct estimates. Since there is no general comparison theorem for BSDE with so general marked point process, we do not use a penalization method, but rather a combination of the Snell envelope theory and contraction theorem.The paper is organized as follows: in section <ref> we first recall some results on marked point processes and describe the setting and the problem we want to solve. In section <ref> we prove the existence and uniqueness of a Reflected BSDE driven by a marked point process and a Wiener process when the generators do not depend on the solution of the BSDE.This is solved in some L^2 space, appropriate for the Brownian motion. When the (given) generator and the other data are adapted only to the filtration generated by the point process, the solution can be found in a larger space. We then linkthese equations to an optimal stopping problem. Lastly in section <ref> we solve the BSDE in the general case with the help of a contraction argument. Here we use the L^2 frameworkfor both the case with only marked point process or with both driving processes.§ PRELIMINARIES, ASSUMPTIONS, FORMULATION OF THE PROBLEMS§.§ Some reminders on point processesWe start by recalling some notions about marked point processes and then defining the objectives of this paper. For a comprehensive treatment of marked point processes, we refer the reader to <cit.>, <cit.> or <cit.>. Let (Ω,ℱ,ℙ) be a complete probability space and let E be a Borel space, i.e. a topological space homeomorphic to a Borel subset of a compact metric space (sometimes called Lusin space; we recall that every separable complete metric space is Borel). We call E the mark space and we denote by ℰ its Borel σ-algebra. A marked point process (MPP) is a sequence of random variables (T_n,ξ_n)_n≥ 0 with values in [0,+∞]× E such that ℙ-a.s. * T_0=0.* T_n≤ T_n+1∀ n≥ 0.* T_n<∞⇒ T_n<T_n+1∀ n≥ 0.We will always assume the marked point process in the paper to be non-explosive, that is T_n→+∞ ℙ-a.s. To each marked point process we associate a random discrete measure p on ((0,+∞)× E,ℬ((0,+∞)⊗ℰ):p(ω,D)=∑_n≥ 1_(T_n(ω),ξ_n(ω))∈ D. We refer to p also as marked point process. For each C∈ℰ, define the counting process N_t(C)=p((0,t]× C) that counts how many jumps have occurred to C up to time t. Denote N_t=N_t(E). They are right continuous increasing process starting from zero. Each point process generates a filtration 𝔾=(𝒢_t)_t≥ 0 as follows: define for t≥ 0𝒢_t^0=σ(N_s(C):s∈[0,t], C∈ℰ)and set 𝒢_t=σ(𝒢_t^0,𝒩), where 𝒩 is the family of ℙ-null sets of ℱ. 𝔾 is a right-continuous filtration that satisfies the usual hypotheses. Denote by 𝒫^𝒢 the σ-algebra of 𝒢-predictable processes.For each marked point process there exists a unique predictable random measure ν, called compensator, such that for all non-negative 𝒫^𝒢⊗ℰ-measurableprocess C it holds that ∫_0^+∞∫_EC_t(e)p(dtde)=∫_0^+∞∫_EC_t(e)ν(dtde).Similarly,there exists a unique right continuous increasing process with A_0=0, the dual predictable projection of N, such that for all non-negative predictable processes D∫_0^+∞D_tdN_t=∫_0^+∞D_tdA_t.It is known that there exists a function ϕ on Ω×[0,+∞)×ℰ such that we have the disintegrationν(ω,dtde)=ϕ_t(ω,de)dA_t(ω). Moreover the following properties hold: * for every ω∈Ω, t∈[0,+∞),C↦ϕ_t(ω,C) is a probability on (E,ℰ).* for every C∈ℰ, the process ϕ_t(C) is predictable.We will assume in the following that all marked point processes in this paper have a compensator of this form.From now on, fix a terminal time T>0. Next we need to define integrals with respect to point processes.Let C be a 𝒫^𝒢⊗ℰ-measurable process such that∫_0^T∫_E |C_t(e)|ϕ_t(de)dA_t<∞.Then we can define the integral∫_0^T∫_E C_t(e)q(dtde)=∫_0^T∫_EC_t(e)p(dtde)-∫_0^T∫_EC_t(e)ϕ_t(de)dA_tas difference of ordinary integrals with respect to p and ϕ dA. In the paper we adopt the convention that ∫_a^b denotes an integral on (a,b] if b<∞, or on (a,b) if b=∞. Since p is a discrete random measure, the integral with respect to p is a sum:∫_0^t∫_E C_s(e)p(dsde)=∑_T_n≤ tC_T_n(ξ_n) Given a process C as above, the integral defines a process ∫_0^t∫_E C_s(e)q(dsde) that, by the definition of compensator, is a martingale.§.§ Probabilistic settingIn this paper we will assume that (Ω,ℱ,) is a complete probability space and p(dtdx) a marked point process on a Borel space (E,ℰ) as before,whose compensator is ϕ_t(dx)dA_t. In addition we assume we are givenan independent Wiener process W in ℝ^d. Let 𝔾=(𝒢_t)_t≥ 0 (resp. 𝔽=(ℱ_t)_t≥ 0) be the completed filtration generated by p (resp. p and W), which satisfies the usual conditions. Let 𝒯_t be the set of 𝔽-stopping times greater than t. Denote by 𝒫 (resp. Prog) be the predictable (resp. progressive)σ-algebra relative to 𝔽. For β>0, we introduce the following spaces of equivalence classes we will be using in the following* L^r,β(A) (resp. L^r,β(A,𝔾)) is the space of all 𝔽-progressive (resp. 𝔾-progressive) processes X such that||X||_L^r,β(A)^r=[∫_0^Te^β A_s|X_s|^rdA_s]<∞. * L^r,β(p) (resp. L^r,β(p,𝔾)) is the space of all 𝔽-predictable (resp. 𝔾-predictable) processes U such that||U||_L^r,β(p)^r =[∫_0^T∫_E e^β A_s|U_s(e)|^rϕ_s(de)dA_s]<∞. * L^r,β(W,ℝ^d) (resp. L^r,β(W,ℝ^d,𝔾)) is the space of 𝔽-progressive (resp. 𝔾-progressive) processes Z in ℝ^d such that||Z||_L^r,β(W)^r=[∫_0^Te^β A_s|Z_s|^rds]<∞ * ℐ^2 (resp. ℐ^2(𝔾)) is the space of all càdlàg increasing 𝔽-predictable (resp. 𝔾-predictable) processes K such that [K^2_T]<∞.One last tool we will need in the following is the martingale representation theorem: if M is acàdlàgsquare integrable 𝔽-martingale on [0,T], then there exist two processes U and Z such that∫_0^T∫_E|U_t(e)|ϕ_t(de)dA_t+∫_0^T|Z_t|^2dt<∞M_t=M_0+∫_0^t∫_EU_s(e)ϕ_s(de)dA_s+∫_0^tZ_sdW_s. §.§ Assumptions and formulation of the problem Let (Ω,ℱ,), (E,ℰ),p(dtdx), W, 𝔽 be as before. We will consider the following reflected BSDE. Y_t=ξ+∫_t^Tf(s,Y_s,U_s)dA_s+∫_t^Tg(s,Y_s,Z_s)ds-∫_t^T∫_EU_s(y)q(dsdy) -∫_t^TZ_s(y)dW_s+K_T-K_t, ∀ t∈[0,T]a.s.Y_t≥ h_t, ∀ t∈[0,T]a.s. ∫_0^T(Y_s-h_s)dK^c_s=0and Δ K_t≤(h_t^--Y_t)^+_{ Y_t^-=h_t^-}∀ t∈[0,T]a.s.,A solution is a quadruple (Y,U,Z,K) that lies in (L^2,β(A)∩ L^2,β(W))× L^2,β(p)× L^2β(W)×ℐ^2, with Y càdlàg, that satisfies (<ref>). The condition on the last line in (<ref>) is called the Skorohod condition, orthe minimal push condition. It can be expressed in an alternative way: see Remark <ref> below. Let us now state the general assumptions that will be used throughout the paper. Additional specific assumptions will be presentedin section <ref>. The first one is an assumption on the compensator A of the counting process N relative to p. (A)The process A is continuous. (B) * The final condition ξ:Ω→ℝ is ℱ_T-measurable and[e^β A_Tξ^2]<∞. * For every ω∈Ω, t∈[0,T], r∈ℝ a mappingf(ω,t,r,·):L^2(E,ℰ,ϕ_t(ω,dy))→ℝis given and satisfies the following: * for every U∈L^2,β(p) the mapping(ω,t,r)↦ f(ω,t,r,U_t(ω,·))is 𝑃𝑟𝑜𝑔⊗ℬ(ℝ)-measurable, where 𝑃𝑟𝑜𝑔 denotes the progressive σ-algebra.* There exist L_f≥ 0, L_U≥ 0 such that for every ω∈Ω, t∈[0,T], y,y'∈ℝ, u,u'∈L^2(E,ℰ,ϕ_t(ω,dy)) we have|f(ω,t,y,u(·))-f(ω,t,y',u'(·))|≤L_f|y-y'| +L_U(∫_E|u(e)-u'(e)|^2ϕ_t(ω,de))^1/2 * we have[∫_0^Te^β A_s|f(s,0,0)|^2dA_s]<∞.* The mapping g:Ω×[0,T]×ℝ×ℝ^d→ℝ is given * g is Prog×ℬ(ℝ)×ℬ(ℝ^d) measurable.* There exist L_g≥ 0, L_Z≥ 0 such that for every ω∈Ω, t∈[0,T], y,y'∈ℝ, z,z'∈ℝ^d|g(ω,t,y,z)-g(ω,t,y',z')|≤ L_g|y-y'|+L_Z|z-z'| * we have[∫_0^Te^β A_s|g(s,0,0)|^2ds]<∞. * h is a càdlàg 𝔽-adapted process such that h_T≤ξ. There exists a δ>0 such that[sup_t∈[0,t]e^(β+δ)A_th_t^2] We recall that Assumption <ref>is equivalent to the fact that the jumps of the point process are totally inaccessible (relative to 𝔽): see <cit.> Corollary 5.28.We will often use the following consequence:since K is required to be predictable, its jumps (that are all non-negative)are disjoint from the jumps of p;so at any jump time of K we also have a jump of Y with the same size,but of opposite sign, in symbols we have a.s.Δ K_t Δ K_t>0=(-Δ Y_t)^+Δ K_t>0,t>0.The Skorohod condition on the last line in (<ref>)tells us that the process K grows only when the solution is about to touch thebarrier.We claim that it is in fact equivalent to∫_0^T(Y_s^--h_s^-)dK_s=0,a.s.To check the equivalence, note first that∫_0^T(Y_s^--h_s^-)dK_s=∫_0^T(Y_s-h_s)dK^c_s+∑_0<s≤ T^(Y_s^--h_s^-)Δ K_s ,a.s.If the Skorohod condition in (<ref>) holds then both terms in the right-hand side are zero, since jumps of K can only happen when Y_t^-=h_t^-. Conversely, assume that (<ref>) holds. Then clearly ∫_0^T(Y_s^--h_s^-)dK^c_s=0 and so ∫_0^T(Y_s-h_s)dK^c_s=0. Also, ∑_0<s≤ T^(Y_s^--h_s^-)Δ K_s=0, so {t:Δ K_t>0}⊂{t:Y_t^-=h_t^-} and, recalling (<ref>), we have a.s.Δ K_t=Δ K_tΔ K_t>0 =(-Δ Y_t)^+ Δ K_t>0≤(-Δ Y_t)^+ Y_t^-=h_t^-=(Y_t^--Y_t)^+Y_t^-=h_t^-=(h_t^--Y_t)^+Y_t^-=h_t^-. In the simpler case when there is no Brownian component the reflected BSDE (<ref>) becomesY_t= ξ+∫_t^T f(s,Y_s,U_s)dA_s-∫_t^T∫_EU_s(y)q(dsdy)+K_T-K_t,∀ t∈[0,T]a.s. Ycàdlàg and Y∈ L^2,β(A,𝔾), U∈ L^2,β(p,𝔾), K ∈ℐ^2(𝔾) Y_t≥h_t ∀ t∈[0,T]a.s. ∫_0^T(Y_s- h_s)dK^c_s=0and Δ K_t≤( h_t^--Y_t)^+_{ Y_t^-= h_t^-}∀ t∈[0,T]a.s.Here we only assume we are given the space (Ω,ℱ,) and themarked point process p. The assumptions we need are the same asin <ref> and <ref>, provided we set g=0 and 𝔾=𝔽. § REFLECTED BSDE WITH GIVEN GENERATORS AND OPTIMAL STOPPING PROBLEM In this section wefirst study the reflected BSDE in the case when the generators g and f do not depend on (Y,Z,U) but are a given processes that satisfy (B^')f and g are𝔽-progressive processes such that[∫_0^Te^β A_s|f_s|^2dA_s+∫_0^Te^β A_s|g_s|^2 ds]<∞. Equation (<ref>) reduces toY_t=ξ+∫_t^Tf_sdA_s+∫_t^Tg_sds-∫_t^T∫_EU_s(y)q(dsdy)-∫_t^TZ_sdW_s+K_T-K_tY∈ L^2,β(A)∩ L^2,β(W), U∈ L^2,β(p), Z∈ L^2,β(W), K ∈ℐ^2 Y_t≥ h_t ∀ t∈[0,T]a.s. ∫_0^T(Y_s-h_s)dK^c_s=0and Δ K_t≤(h_t^--Y_t)^+_{ Y_t^-=h_t^-}∀ t∈[0,T]a.s.In this case, the solution Y to the equation is also the value function of an optimal stopping problem, as we will see later. First we define the càdlàg process η_t as η_t=∫_0^t∧ T f_sdA_s+∫_0^t∧ T g_sds+h_t t<T+ξt≥ T In the following we will often use this kind of inequalities:(∫_0^tf_sdA_s)^2=(∫_0^te^-β A_s/2e^β A_s/2|f_s|dA_s)^2 ≤∫_0^te^-β A_sdA_s∫_0^te^β A_sf_s^2dA_s =1-e^β A_t/β∫_0^te^β A_sf_s^2dA_s≤1/β∫_0^te^β A_sf_s^2dA_sUnder assumptions <ref>-(i)(iv) and <ref>, η is of class [D] and[sup_0≤ t ≤ T|η_t|^2]<∞Fix a stopping time τ. Clearly|η_τ|^2 ≤ 4(∫_0^T |f_s|dA_s)^2+4(∫_0^T |g_s|ds)^2 +4|h_τ|^2τ<T+4|ξ|^2≤4/β∫_0^Te^β A_sf_s^2dA_s +4T∫_0^Te^β A_s|g_s|^2ds+4sup_t∈[0,T]e^β A_t|h_t|^2+4 e^β A_Tξ^2,and since the right-hand side has finite expectationwe obtain the class [D] property. Likewise, by taking the supremum over all t∈[0,T], and expectation after that, we obtain the second property. Now, using the Snell envelope theory, we show that there exists a solution to the equation above. Appendix <ref> lists the properties that we will need in the following.Let assumptions <ref>, <ref>-(i)(iv) and <ref> hold for some β>0, then there exists a uniquesolution to (<ref>).The uniqueness property is stated and proved separately in Proposition <ref> below. Existence is proved in several steps.Step 1. We start by defining Y_t, for all t≥ 0, as the optimal value of the stopping problem:Y_t=_τ∈𝒯_t∫_t^τ∧ T f_sdA_s+∫_t^τ∧ T g_sds +h_ττ<T+ξτ≥ T.From (<ref>) it followsthat Y_t is integrable for all t and Y_t=ξ for t≥ T. We have the following a priori estimate on Y, that we will prove later.Assume <ref>-(i)(iv) and <ref> above on ξ,f,h,ξ. Then[sup_t∈[0,T]e^β A_tY_t^2]<∞.It followsthatY_t+∫_0^t∧ Tf_sdA_s+∫_0^t∧ Tg_sds =_τ∈𝒯_tη_τso Y_t+∫_0^t∧ Tf_sdA_s +∫_0^t∧ Tg_sds is the Snell envelope of η, that is the smallest supermartingale such that Y_t+∫_0^t∧ Tf_sdA_s+∫_0^t∧ Tg_sds≥η_t. Since η is càdlàg, its Snell envelope R(η), and hence Y, have a càdlàg modification. We refer to the appendix for a review of the properties of the Snell envelope that we will use. Also, from now on all supermartingalesthat we consider in this proof are assumed to be càdlàg. Also, since η satisfies (<ref>) by Lemma <ref>,Y+∫_0^·∧ T f_sdA_s + ∫_0^·∧ Tg_sds is of class [D] and thus it admits a unique Doob-Meyer decompositionY_t+∫_0^t∧ Tf_sdA_s∫_0^t∧ Tg_sds=M_t-K_t,where M is a martingale and K is a predictableincreasing process starting from zero. From Lemma <ref> andit follows that K_T^2<∞, so that M is a square integrable martingale.Furthermore, K can be decomposed into K^c+K^d,the continuous and discontinuous part, and we have that Δ K_t=Δ K_t_{R(η)_t^-=η_t^-}(see <ref>). However it is immediate to seethat R(η)_t^-=η_t^- if and only if Y_t^-=h_t^-_{t≤ T} + ξ_{t> T} and it follows that Δ K_t=Δ K_t_{Y_t^-=h_t^-},t∈ [0,T] .By the martingale representation theorem, there exists some U and Z such that∫_0^T∫_E|U_t(e)|ϕ_t(de)dA_t+∫_0^T|Z_t|^2dt<∞M_t=M_0+∫_0^t∫_EU_s(e)q(dsde)+∫_0^tZ_sdW_s.Choosing τ=t in (<ref>) we see that a.s. Y_t≥ h_t for all t<Tand Y_T=ξ, so Y_t≥ h_t for all t≤ T a.s. Plugging (<ref>) in (<ref>) we conclude that the first equality in (<ref>) is verified. Step 2. In this step we prove that the Skorohod conditions in (<ref>) hold. From (<ref>) it follows that Δ K_t ≤ (-Δ Y_t)^+ and, taking into account (<ref>), we obtainΔ K_t≤ (-Δ Y_t)^+ _{ Y_t^-=h_t^-} =(Y_t^--Y_t)^+_{ Y_t^-=h_t^-},that gives us the second condition. Consider now Ỹ_t=Y_t+∫_0^tf_sdA_s+∫_0^tg_sds+K^d_t=M_t-K_t^c and η̃_t=η_t+K^d_t. We claim that Ỹ_t is the Snell envelope of η̃_t. Indeed, it is a supermartingale that dominates η̃_t. Let Q_t be another supermartingale that dominates η̃_t. Then Q_t-K_t^d is still a supermartingale, and dominates η_t. Then, since Y_t+∫_0^tf_sdA_s+∫_0^tg_sds=R(η)_t, Q_t≥Ỹ_t. Then Ỹ_t is the smallest supermartingale that dominates η̃_t, and thus its Snell envelope. Next, Y_t+∫_0^tf_sdA_s+K_t^d=M_t-K_t^c is regular (we recall that a process X is regular if X_t^-=^pX_t, where ^pX_t denotes the predictable projection, see also <ref>; all uniformly integrable càdlàgmartingales are regular). Then, the stopping time defined asD_t^*=inf{ s≥ t : M_s≠ R(η̃)_s}=inf{ s≥ t : K^c_s> K^c_t}is the largest optimal stopping time, and it satisfies:Ỹ_D_t^*=η̃_D_t^*Ỹ_s∧ D_t^* is a 𝔽-martingaleSee (<ref>). Define thenD_t=inf{s≥ t : Ỹ_s≤η̃_s}Since Ỹ_D_t^*=η̃_D_t^* we haveD_t≤ D_t^*, and it follows that0=∫_t^D_t(Ỹ_s-η̃_s)dK^c_s=∫_t^D_t(Y_s-h_s)dK^c_s,which implies K^c_D_t=K^c_t for arbitrary t, and hence ∫_0^T(Y_s-h_s)dK^c_s=0, that together with (<ref>) gives us the Skorohod conditions. Step 3. We conclude the proof showing that the processes are in the right spaces. We have already noticed that [K_T^2]<∞. Next we define the sequence of stopping times: S_n=inf{ t∈[ 0,T] :∫_0^te^β A_s|Y_s|^2dA_s+∫_0^t∫_Ee^β A_s|U_s(e)|^2ϕ_s(de)dA_s. .+∫_0^te^β A_s|Z_s|^2ds>n},and consider the “Ito Formula" applied to e^β(A_t+t)Y_t^2 between 0 and S_n. We havee^β(A_S_n+S_n)Y_S_n^2 =Y_0^2+β∫_0^S_ne^β(A_s+s)Y_s^2dA_s+β∫_0^S_ne^β(A_s+s)Y_s^2ds+2∫_0^S_n∫_Ee^β(A_s+s)Y_s^-U_s(e)q(dsde)+2∫_0^S_ne^β(A_s+s)Y_sZ_sdW_s-2∫_0^S_ne^β(A_s+s)Y_sf_sdA_s-2∫_0^S_ne^β(A_s+s)Y_sg_sds-2∫_0^S_ne^β(A_s+s)Y_s^-dK_s+∫_0^S_ne^β(A_s+s)Z_s^2ds+∑_0<s≤ S_ne^β(A_s+s)Δ K^2_s+∫_0^S_n∫_E e^β(A_s+s)U_s^2(e)p(dsde)Now we use the fact that∫_0^t∫_E U_s(e)p(dsde)=∫_0^t∫_E U_s(e)ϕ_s(de)dA_s+∫_0^t∫_E U_s(e)q(dsde),and, by Remark <ref>,∫_0^te^β(A_s+s) Y_s^-dK_s=∫_0^te^β(A_s+s)(Y_s^--h_s^-)dK_s_=0+∫_0^te^β(A_s+s) h_s^-dK_s.Neglecting the positive terms Y_0^2 and ∑_0<s≤ S_ne^β(A_s+s)Δ K^2_s the previous equation becomese^β(A_S_n+S_n)Y_S_n^2 ≥β∫_0^S_ne^β(A_s+s)Y_s^2dA_s+β∫_0^S_ne^β(A_s+s)Y_s^2ds+2∫_0^S_ne^β(A_s+s)Y_s^-U_s(e)q(dsde)+2∫_0^S_ne^β(A_s+s)Y_sZ_sdW_s-2∫_0^S_ne^β(A_s+s)Y_sf_sdA_s-2∫_0^S_ne^β(A_s+s)Y_sg_sds -2∫_0^S_ne^β(A_s+s)h_s^-dK_s+∫_0^S_n∫_E e^β(A_s+s)U_s^2(e)ϕ_s(de)dA_s+∫_0^S_n∫_E e^β(A_s+s)U_s^2(e)q(dsde)+∫_0^S_ne^β(A_s+s)Z_s^2ds, By the definition of S_n and remembering that Y satisfies (<ref>), and using Burkholder-Davis-Gundy inequality we have that∫_0^t∧ S_ne^β(A_s+s)Y_sZ_sdW_sis a martingale. Indeed we have[sup_t∈ [0,T]|∫_0^t∧ S_ne^β(A_s+s)Y_sZ_sdW_s|] ≤[(∫_0^S_ne^2β(A_s+s)Y_s^2Z_s^2ds)^1/2]≤ e^β T[sup_te^β A_t/2|Y_t|(∫_0^S_ne^β A_sZ_s^2ds)^1/2]≤ n^1/2e^β T[sup_te^β A_tY_t^2]<∞.Similarly, since[∫_0^t∫_Ee^β(A_s+s) |Y_s^-U_s(e)|ϕ_s(de)dA_s]≤[∫_0^te^β(A_s+s) Y_s^2dA_s] +[∫_0^t∫_E e^β(A_s+s) U_s^2(e)ϕ_s(de)dA_s]≤ 2n<∞,we obtain that ∫_0^t∧ S_n∫_Ee^β(A_s+s) Y_s^-U_s(e)q(dsde) is a martingale. Reordering terms and taking expectation we obtainβ[∫_0^S_ne^β(A_s+s)Y_s^2dA_s]+∫_0^S_n∫_E e^β(A_s+s)U_s^2(e)ϕ_s(de)dA_s+β[∫_0^S_ne^β(A_s+s)Y_s^2ds]+∫_0^S_ne^β(A_s+s)Z_s^2ds≤e^β(A_S_n+S_n)Y_S_n^2 + 2∫_0^S_ne^β(A_s+s)Y_sf_sdA_s+2∫_0^S_ne^β(A_s+s)Y_sg_sds+ 2∫_0^S_ne^β(A_s+s)h_s^-dK_s≤sup_t e^β(A_t+t)Y_t^2 + β/2∫_0^S_ne^β(A_s+s)Y_s^2dA_s +β/2∫_0^S_ne^β(A_s+s)Y_s^2ds+ 1/β∫_0^Te^β(A_s+s)f_s^2dA_s +2/β∫_0^Te^β(A_s+s)g_s^2ds+γsup_t e^(β+δ)(A_t+t)h_t^-^2 +1/γ(∫_0^S_ne^(β-δ)A_s+s/2dK_s)^2,where γ>0 is a constant whose value will be chosen sufficiently large afterwards. We only need to estimate the last term with the integral in dK. In order to do that we apply Ito's formula to e^(β-δ)A_t+t/2Y_t between 0 and a stopping time τ, obtaining the following relation(∫_0^τe^(β-δ)A_s+s/2dK_s)^2 =(Y_0-e^(β-δ)A_τ+τ/2Y_τ+β-δ/2∫_0^τ e^(β-δ)A_s+s/2Y_sdA_s.+β-δ/2∫_0^τ e^(β-δ)A_s+s/2Y_s ds-∫_0^τ e^(β-δ)A_s+s/2f_s dA_s-∫_0^τ e^(β-δ)A_s+s/2g_s ds+∫_0^τ∫_E e^(β-δ)A_s+s/2U_s(e) q(dsde).+∫_0^τ e^(β-δ)A_s+s/2Z_s dW_s )^2Notice that the following holds:(∫_0^τ e^(β-δ)A_s+s/2Y_sdA_s)^2≤∫_0^τ e^-δ (A_s+s)dA_s∫_0^τ e^β (A_s+s)Y_s^2dA_s≤1/δ∫_0^τ e^β (A_s+s)Y_s^2dA_s (∫_0^τ e^(β-δ)A_s+s/2Y_s ds)^2≤∫_0^τ e^-δ A_se^-δ sds∫_0^τ e^β(A_s+s)Y_s^2 ds≤1/δ∫_0^τ e^β(A_s+s)Y_s^2dsand similarly(∫_0^τ e^(β-δ)A_s+s/2f_sdA_s)^2 ≤1/δ∫_0^τ e^β (A_s+s)f_s^2dA_s (∫_0^τ e^(β-δ)A_s+s/2g_sds)^2 ≤1/δ∫_0^τ e^β (A_s+s)g_s^2dsWe note that for a 𝒫⊗ℰ measurable process H we have[(∫_0^t∫_E H_s(e)q(dsde))^2]≤[∫_0^t∫_E H_s^2(e)ϕ_s(de)dA_s].This can be checked for instance by applying the Ito formula to compute N^2_t where N_t=∫_0^t∫_E H_s(y)q(dsdy) and taking expectation after appropriate localization. Now by taking expectation and using Ito Isometry we obtain the following bound for (∫_0^τe^(β-δ)A_s+s/2dK_s)^2:[(∫_0^τe^(β-δ)A_s+s/2dK_s)^2]≤ 16sup_t e^β(A_t+t)Y_t^2 +8/δ∫_0^τ e^β(A_s+s)g_s^2s +2(β-δ)^2/δ∫_0^τ e^β(A_s+s)Y_s^2ds+2(β-δ)^2/δ∫_0^τ e^β(A_s+s)Y_s^2dA_s+8/δ∫_0^τ e^β(A_s+s)f_s^2dA_s+8∫_0^τ e^β (A_s+s)Z_s^2ds+8∫_0^τ∫_E e^β (A_s+s)U_s^2(e)ϕ_s(de)dA_s.By plugging this last estimate into (<ref>), by choosing α, γ such thatγ>max(8,4(β-δ)^2/βγ)we obtain[∫_0^S_ne^β(A_s+s)Y_s^2dA_s]+[∫_0^S_ne^β(A_s+s)Y_s^2ds]+∫_0^S_n∫_E e^β(A_s)U_s^2(e)ϕ_s(de)dA_s+∫_0^S_ne^β(A_s)Z_s^2ds≤ C( sup_t e^β A_tY_t^2+2(1/β +1/δγ)∫_0^Te^β A_sf_s^2dA_s..+∫_0^Te^β A_sg_s^2ds+γsup_t e^(β+δ)A_th_t^-^2),for some constant C independent of n. Now let S=lim_n S_n and by the last estimate, considering how S_n are defined, we have S=T. This implies that Y∈ L^2,β(A)∩ L^2,β(W), Z∈ L^2,β(W) and U∈ L^2,β(p).By the definition of Y we havee^β A_t/2|Y_t|≤[ e^β A_T/2|ξ|+e^β A_t/2∫_t^T|f_s|dA_s.+e^β A_t/2∫_t^T|g_s|ds+sup_0≤ s≤ Te^β A_s/2|h_s||ℱ_t]Proceeding as in Remark <ref> we have∫_t^T|f_s|dA_s≤e^-β A_t/2/β^1/2(∫_t^Te^β A_s|f_s|^2dA_s)^1/2and it follows thate^β A_t/2|Y_t|≤[e^β A_T/2|ξ|+1/β^1/2(∫_0^Te^β A_s|f_s|^2dA_s)^1/2. +∫_0^Te^β A_s/2|g_s|ds +sup_0≤ s≤ Te^β A_s/2|h_s||ℱ_t] S_tUnder assumption <ref>-(i)(iv) and <ref>, S is a square integrable martingale. Then by Doob's martingale inequality[sup_0≤ t ≤ Te^β A_t|Y_t|^2] ≤ C[S_T^2]<∞.Contrary to the diffusive (or diffusive and Poisson) case, the fact that [sup_t∈[0,T]e^β A_tY_t^2]<∞ does not imply that Y∈ L^2,β(A). For this to happen we would need additional conditions on A, for example [A_T^2]<∞. Next we prove uniqueness. Let assumptions <ref>, <ref>-(i)(iv) and <ref> hold for some β>0, then thesolution to (<ref>) isunique.Let (Y',U',Z',K') and (Y”,U”,Z”,K”) be two solutions. DefineY̅=Y'-Y”U̅=U'-U”Z̅=Z'-Z”K̅=K'-K”,then (Y̅,U̅,Z̅,K̅) satisfiesY̅_t=-∫_t^T∫_E U̅(e)q(dsde)-∫_t^TZ_sdW_s+K̅_T-K̅_t. We compute d(e^β (A_t+t)Y̅^2_t) by the Ito formula and we obtain-Y̅^2_0=β∫_0^T e^β (A_s+s)Y̅^2_sdA_s+β∫_0^T e^β (A_s+s)Y̅^2_sds-2∫_0^TY̅_s^-dK̅_s +2∫_0^T∫_E e^β (A_s+s)Y̅_s^-U_s(y)q(dsdy)+∫_0^Te^β(A_s+s)Y_sZ_sdW_s+∫_0^Te^β(A_s+s)Z_s^2ds+∑_0<s≤ Te^β(A_s+s)(ΔY̅_s)^2The last term can be divided in totally inaccessible jumps (from the martingale in q(dsde)) and predictable jumps, from the K process, thus:∑_0<s≤ Te^β(A_s+s)(ΔY̅_s)^2≥∑_0<T_n≤ Te^β(A_s+s)U_T_n^2(ξ_n)=∫_0^T∫_EU_s^2(e)p(dsde)=∫_0^T∫_E U_s^2(e)q(dsde)+∫_0^T∫_E U_s^2(e)ϕ_s(de)dA_sProceeding as in (<ref>) and (<ref>) we prove that the stochastic integrals with respect to W and q are martingales. By neglecting Y_0^2 and taking expectation in (<ref>), we obtainβ[∫_0^Te^β(A_s+s)Y̅^2_sdA_s]+β[∫_0^Te^β(A_s+s)Y̅^2_sds] +[∫_0^T∫_Ee^β(A_s+s)U̅^2_s(y)ϕ_s(dy)dA_s]+∫_0^Te^β(A_s+s)Z_s^2ds ≤ 2[∫_0^Te^β(A_s+s)Y̅_s^-dK̅_s].Now, taking into account Remark <ref> we have∫_0^TY̅_s^-dK̅_s =∫_0^T(Y'_s^--h_s^-)dK'_s_=0-∫_0^T(Y'_s^--h_s^-)dK”_s_≥ 0+-∫_0^T( Y”_s^--h_s^-)dK'_s_≥ 0+∫_0^T( Y”_s^--h_s^-)dK”_s_=0≤ 0,and thusβ||Y̅||^2_L^2,β(A)+β||Y̅||^2_L^2,β(W) +||U̅||^2_L^2,β(p)+||Z̅||^2_L^2,β(W)≤ 0,which gives the uniqueness of Y, U and Z. From (<ref>) we obtainK̅_T=K̅_t ∀ t ∈[0,T].Then K̅_T=0 since K̅_0=0 and consequently K̅_t=0 for all t. Consider now the optimal stopping problem with running gains f,g, early stopping reward h and non stopping reward ξ. This means we are interested in the quantityv(t) =_τ∈𝒯_t∫_t^τ f_sdA_s+∫_0^τ g_sds+h_τ_{τ<T}+ξ_{τ≥ T}.Notice that we have two running gains, f integrated with respect to the process A, and g integrated with respect to Lebesgue measure in time. This could be used for example if we want to describe two different time dynamics, one depending on the speed of the point process.It is possible to show thatthe solution to the RBSDE solves the optimal stopping problem and it is possible to identify an ϵ-optimal stopping time. Under additional assumptions, it is possible to find an optimal stopping time. For this we need a definition, given in <cit.> for admissible families over stopping times, that we adapt to our simpler case: We say that a process ϕ is left(resp. right) upper semi-continuous over stopping times in expectation (USCE) if for all θ∈𝒯_0, [ϕ_θ]<∞ and for all sequences of stopping times (θ_n) such that θ_n↑θ (resp. θ_n↓θ) it holds that[ϕ_θ]≥lim sup_n→∞[ϕ_θ_n]. If ϕ is a left upper semi continuous progressive process, then ϕ is left upper semi continuous along stopping times. If also [sup_t|ϕ_t|] holds, then it is left USCE. Indeed we havelim sup_n→∞[ϕ_θ_n]≤[lim sup_n→∞ϕ_θ_n]≤[ϕ_θ].by using Reverse Fatou's lemma with sup_t|ϕ_t| as dominant. Let assumptions <ref>, <ref>-(i)(iv) and <ref> hold. Then we have: * The solution to the RBSDE (<ref>) is a solution to the optimal stopping problem Y_t=_τ∈𝒯_t∫_t^τ f_sdA_s+∫_0^τ g_sds+h_τ_{τ<T}+ξ_{τ≥ T}. * For all ϵ>0, define D_t^ϵ asD_t^ϵ=inf{ s≥ t : Y_s≤ h_s+ϵ}∧ T.Then D_t^ϵ is an ϵ-optimal stopping time in the sense thatY_t≤_τ∈𝒯_t∫_t^D_t^ϵ f_sdA_s+∫_0^D_t^ϵ g_sds+h_D_t^ϵ_{D_t^ϵ<T}+ξ_{D_t^ϵ≥ T}+ϵ. *If in additionh_t_{ t<T}+ξ_{ t≥ T} is left USCE, thenτ_t^*=inf{ s≥ t : Y_s≤ h_s }∧ T.is optimal and is the smallest of all optimal stopping times. The condition on the third point may seem unusual, but it is satisfied for example if h_t is left upper semi continuous on [0,T] and h_T<ξ. Let τ∈𝒯_t and consider the first equation (<ref>) between t and τ:Y_t=Y_τ+∫_t^τ f_sdA_s+∫_t^τ g_sds-∫_t^τ∫_EZ_s(y)q(dsdy)+K_τ-K_t.By taking conditioning at ℱ_t we haveY_t =Y_τ+∫_t^τ f_sdA_s+∫_t^τ g_sds+K_τ-K_t≥h_τ_{τ<T}+ξ_{τ≥ T}+∫_t^τ f_sdA_s+∫_t^τ g_sds,since the integral on q is a martingale, K is increasing and Y_t≥ h_τ_{ t<T}+ξ_{ t=T}. To prove the reverse inequality, consider ϵ>0 and the corresponding D_t^ϵ. It holds that Y_D_t^ϵ≤ h_D_t^ϵ+ϵ on {D_t^ϵ<T}. And on {D_t^ϵ=T} we have Y_u>h_u+ϵ for all t≤ u < T. Then , between t and D_t^ϵ, Y_s^->h_s^- and thus ∫_t^D_t^ϵ(Y_s^--h_s^-)dK_s=0 ⇒ K_D_t^ϵ=K_t.Considering all this in (<ref>) we have Y_t =Y_D_t^ϵ+∫_t^D_t^ϵ f_sdA_s+∫_t^D_t^ϵg_sds≤h_D_t^ϵ_{D_t^ϵ<T}+ξ_{D_t^ϵ=T}+∫_t^D_t^ϵ f_sdA_s+∫_t^D_t^ϵg_sds+ϵ.This together with (<ref>) proves points one and two. For the third point, notice that D_t^ϵ are non increasing in ϵ and that D_t^ϵ≤τ^*. Thus D_t^ϵ→ D_t^0≤τ^* when ϵ→ 0. Now since h_t_{ t<T}+ξ_{ t=T} is left USCE and the integral part is too, we have from (<ref>)[Y_t]≤lim sup_ϵ→ 0[h_D_t^ϵ_{D_t^ϵ<T}+ξ_{D_t^ϵ=T}+∫_t^D_t^ϵ f_sdA_s+∫_t^D_t^ϵ g_sds] ≤[h_D_t^0_{D_t^0<T}+ξ_{D_t^0=T}+∫_t^D_t^0 f_sdA_s+∫_t^D_t^0 g_sds].Thus we have[Y_t]= [h_D_t^0_{D_t^0<T}+ξ_{D_t^0=T}+∫_t^D_t^0 f_sdA_s+∫_t^D_t^0 g_sds],soD_t^0 is optimal (see <ref>). We only need to prove that D_t^0=τ^*. We already know that D_t^0≤τ^*. On the other hand, since D_t^0 is optimal it holds that Y_D_t^0=η_D_t^0, and thus by the definition of τ^*, τ^*≤ D_t^0. This also proves that τ^* is the smallest optimal stopping time. A further interesting property holds when the reward is left USCE:Under assumptions<ref>-(i)(iv) and <ref>, if h_τ_{τ<T}+ξ_{τ≥ T} is also left USCE, then K in the solution of (<ref>) is continuous.The proof is given in <cit.> in the case were the reward is a positive progressive process ϕ of class [D]. We can adapt to our case by using the transformationI=inf_tη_tN_t=Iη̃_t=η_t-N_t.We have that η̃_t is USCE, as [η̃_t]=[η_t]-[I]. Indeed let θ_n↑θ, thenlim sup_n→∞[η̃_θ_n]≤[η̃_θ].Then if R(η) denotes the Snell envelope of η, it holds that R(η̃)=R(η)-N_t. The Doob-Meyer decomposition for the càdlàg supermartingale R(η̃) holds:R(η̃)_t=M̃_t-K̅_tWith K̅ continuous thanks to Proposition B.10 in <cit.>. Then Y_t+∫_0^tf_sdA_s=R(η)=R(η̃)+N_t=M̃_t+N_t-K̅_t, but since the decomposition is unique, ∫_0^t∫_E Z_s(y)q(dsdy)=M_t=M̃+N_t and K_t=K̅_t. Thus the term K is continuous. If we are interested only in (<ref>), and we have a filtration generated only by a MPP and g≡0, the proofs above are still applicable. In this case, there is no particular reason to use a L^2 space, since the martingale representation theorem for marked point processes works in L^1 (see <cit.>). We thus obtain the following: Let assumption <ref> hold. Let ξ be a 𝒢_T-measurable random variable. Let f,h be𝔾-progressive processes. Assume that|ξ|+∫_0^T|f_s|dA_s+sup_t∈[0,T]|h_t|<∞.Then there exists a unique solution to the systemY_t=ξ+∫_t^Tf_sdA_s-∫_t^T∫_EU_s(y)q(dsdy)+K_T-K_tY_t≥h_t ∀ t∈[0,T]a.s. ∫_0^T(Y_s-h_s)dK^c_s=0and ΔK_s≤(h_s^--Y_s)^+_{Y_s^-=h_s^-}.where Y is a càdlàg 𝔾-adapted process such that [|Y_t|]<∞ for all t, K is a 𝔾-predictable càdlàg increasing process with K_0=0 and [K_T]<∞ and U is a 𝒫(𝔾)⊗ℰ-measurable process such that [∫_0^T∫_E|U_s(e)|ϕ_s(de)dA_s]<∞. Existence of a solution is obtained as in <ref>. The process η_t satisfies then the weaker condition sup_t|η_t|<∞, but this is enough to apply the Snell's envelope results (see appendix <ref>, in particular (<ref>)). Integrability is straightforward. Now let (Y',U',K') and (Y”,U”,K”) be two solutions, their difference satisfiesY'_t-Y”_t=Y'_0-Y”_0+∫_0^t∫_E(U'_s(e)-U”_s(e))q(dsde)-(K'_t-K”_t).Uniqueness of the component Y comes from the fact that if (Y,U,K) satisfies the equation, the càdlàg process Y satisfiesY_t=_τ∈𝒯_t[.∫_t^τ∧ Tf_sdA_s+h_ττ<T+ξτ≥ T|𝒢_t],which can be shown as in proposition <ref>, adapted to the this case with less integrability. Relation (<ref>) becomes∫_0^t∫_EU'_s(e)q(dsde)-K'_t=∫_0^t∫_EU”_s(e)q(dsde)-K”_t.Since the predictable jumps of K and the totally inaccessible jumps of the integrals with respect to q are disjoint, we have that U'_T_n(ξ_n)=U”_T_n(ξ_n) for all n. Then∫_0^T∫_E|U'_s(e)-U”_s(e)|ϕ_s(de)dA_s =∫_0^T∫_E|U'_s(e)-U”_s(e)|p(dsde)=∑_n≥ 1^|U'_T_n(ξ_n)-U”_T_n(ξ_n)|=0,and thus U'_s(e)=U”_s(e) ϕ_s(de)dA_sdℙ-a.e. Then K'_t=K”_t a.s. and uniqueness is proven. We have then a result for optimal stopping analogous to proposition <ref>:Assume that the conditions of proposition <ref> hold. Then * The solution to the RBSDE (<ref>) is a solution to the optimal stopping problem Y_t=_τ∈𝒯_t∫_t^τ f_sdA_s+∫_0^τ g_sds+h_τ_{τ<T}+ξ_{τ≥ T}. * For all ϵ>0, define D_t^ϵ as D_t^ϵ=inf{ s≥ t : Y_s≤ h_s+ϵ}∧ T. Then D_t^ϵ is an ϵ-optimal stopping time in the sense that Y_t≤_τ∈𝒯_t∫_t^D_t^ϵ f_sdA_s+∫_0^D_t^ϵ g_sds+h_D_t^ϵ_{D_t^ϵ<T}+ξ_{D_t^ϵ≥ T}+ϵ. * If in additionh_t_{ t<T}+ξ_{ t≥ T} is left USCE, then τ_t^*=inf{ s≥ t : Y_s≤ h_s }∧ T. is optimal and is the smallest of all optimal stopping times. Moreover, the process K is continuous.§ REFLECTED BSDE We now turn to the case where the generators depend on the solution, that is equation (<ref>).Denote by λ the Lebesgue measure on [0,T], and introduce now L^2,β(Ω×[0,T],ℱ⊗ℬ([0,T]),(A(ω,dt)+λ(dt)), the space of all 𝔽-progressive processes such thatY^2_L^2,β(A+λ)=∫_0^Te^β A_sY_s^2(dA_s+ds)<∞.For brevity we denote is as L^2,β(A+λ) in the following. It is a Hilbert space equipped with the norm above. It is clear that a process is in L^2,β(A+λ) if and only if lies in Y∈ L^2,β(A)∩ L^2,β(W). Let assumption <ref> and <ref> hold for some β>L_p^2+2L_f. Then there exists a unique solution to (<ref>). We con We will use a contraction theorem on 𝕃^β=L^2,β(Ω×[0,T],ℱ⊗ℬ([0,T]),(A(ω,dt)+λ(dt))ℙ(dω))× L^2,β(p)× L^2,β(W). We construct a mapping Γ that to each (P,Q,R)∈ L^2,β(A+λ)× L^2,β(p)× L^2,β(W) associates (Y,U,Z) solution to equation (<ref>) when the generators are given by f_t(P_t,Q_t) and g_t(P_t,R_t). Such map is well defined: indeed if we fix (P,Q,R)∈ L^2,β(A+λ)× L^2,β(p)× L^2,β(W), thanks to assumption <ref>, the generators are known process that satisfy assumption <ref> and proposition <ref> and <ref> give us the existence and uniqueness of (Y,U,Z)∈ L^2,β(A+λ)× L^2,β(p)× L^2,β(W). Notice that thanks to the Lipschitz conditions on g and f, if we take two triplets (P',Q',R')≡(P”,Q”,R”) in L^2,β(A+λ)× L^2,β(p)× L^2,β(W), then f_s(Y',U')≡ f_s(Y”,U”) in L^2,β(A) and g_s(Y'.Z')≡ g_s(Y”,Z”) in L^2,β(W). Consider now (P',Q',R') and (P”,Q”,R”) in 𝕃^β, and consider their images through Γ, (Y',U',Z')=Γ(P',Q',R') and (Y”,U”,Z”)=Γ(P”,Q”,R”). Denote Y̅=Y'-Y”, P̅=P'-P” and so on. Denote also f̅_t=f_t(P'_t,Q'_t)-f_t(P”_t,Q”_t) and similarly denote g̅. (Y̅,U̅,Z̅,K̅) satisfiesY̅=∫_t^Tf̅_sdA_s+∫_t^Tg̅_sds-∫_t^T∫_EU̅_s(e)q(dsde)-∫_t^TZ̅_sdW_s+K̅_T-K̅_t. We now apply Ito's Lemma to e^β A_se^γ sY̅^2_s obtaining, after taking expectation,β∫_0^Te^β A_se^γ sY̅^2_sdA_s +γ∫_0^Te^β A_se^γ sY̅^2_sds +∫_0^Te^β A_se^γ sZ̅^2_sdW_s+∫_0^T∫_Ee^β A_se^γ sU̅^2_sϕ_s(de)dA_s≤ 2∫_0^Te^β A_se^γ sf̅_s^2dA_s+2∫_0^Te^β A_se^γ sg̅_s^2ds +2∫_0^TY̅_s^-dK̅_s.As in the proof of proposition <ref>, we have that∫_0^TY̅_s^-dK̅_s≤ 0.Denote by ||·||_β,γ,A the norm (equivalent to ||·||_L^2,β(A))(∫_0^Te^β A_se^γ sY̅^2_sdA_s)^1/2,and similarly denote the norms ||·||_β,γ,p and ||·||_β,γ,W. Using the Lipschitz properties of f and g this gives β||Y̅||^2_β,γ,A+γ||Y̅||^2_β,γ,W+||U̅||^2_β,γ,p+||Z̅||^2_β,γ,W≤ ≤ 2L_f∫_0^T e^β A_se^γ s|Y̅_s||P̅_s|dA_s+2L_p∫_0^T e^β A_se^γ s|Y̅_s|(∫_E|Q̅^2_s|)^1/2dA_s +2L_g∫_0^T e^β A_se^γ s|Y̅_s||P̅_s|ds+2L_W∫_0^T e^β A_se^γ s|Z̅_s||R̅_s|ds.Using the inequality 2ab≤α a^2+b^2/α for a,b≥ 0 we obtain: β||Y̅||^2_β,γ,A+γ||Y̅||^2_β,γ,W+||U̅||^2_β,γ,p+||Z̅||^2_β,γ,W ≤L_f/√(α_p)||Y̅||^2_β,γ,A+L_f√(α)||P̅||^2_β,γ,A+L_p^2/α||Y̅||^2_β,γ,A+α||Q̅||^2_β,γ,p +L_g/√(α)||Y̅||^2_β,γ,W+L_g√(α)||P̅||^2_β,γ,W+L_W^2/α||Y̅||^2_β,γ,A+α||R̅||^2_β,γ,A.Rewriting we obtain the following relation:||U̅||^2_β,γ,p+||Z̅||^2_β,γ,W+(β-L_p^2/α-L_f/√(α))||Y̅||^2_β,γ,A+(γ-L_W^2/α-L_g/√(α))||Y̅||^2_β,γ,W ≤L_f√(α)||P̅||^2_β,γ,A+ α||Q̅||^2_β,γ,p+ L_g√(α)||P̅||^2_β,γ,W+ α||R̅||^2_β,γ,A.Since β>L_p^2+2L_f, it is possible to choose α∈(0,1) such thatβ>L^2_p/α+2L_f/√(α),and for that α, choose γ such that γ>L^2_W/α+2L_g/√(α). The relation (<ref>) rewrites asL_f/√(α)||Y̅||^2_β,γ,A+L_g/√(α)||Y̅||^2_β,γ,W+||U̅||^2_β,γ,p+||Z̅||^2_β,γ,W ≤L_f√(α)||P̅||^2_β,γ,A+ α||Q̅||^2_β,γ,p+ L_g√(α)||P̅||^2_β,γ,W+ α||R̅||^2_β,γ,A=α(L_f/√(α)||P̅||^2_β,γ,A+ ||Q̅||^2_β,γ,p+ L_g/√(α)||P̅||^2_β,γ,W+ ||R̅||^2_β,γ,A). NowL_f/√(α)||P̅||^2_β,γ,A+L_g/√(α)||P̅||^2_β,γ,W=∫_0^Te^β A_se^γ sP̅_s^2(L_f/√(α)dA_s+L_g/√(α)ds)is a norm equivalent to P̅_L^2,β(A+λ). We have thus that Γ is a contraction on 𝕃 for the equivalent norm(Y,U,Z)^2_𝕃^β,γ=L_f/√(α)||Y||^2_β,γ,A+L_g/√(α)||Y||^2_β,γ,W+||U̅||^2_β,γ,p+||Z̅||^2_β,γ,W.Since the space is complete, the contraction theorem assures us the existence of a unique triplet (Y,U,Z) in 𝕃^β such that (Y,U,Z)=Γ(Y,U,Z), and (Y,U,Z,K) is the solution to (<ref>), where K is the one associated to (Y,Z,U) by the map Γ. Since we knowThis last result generalizes the case of Brownian and Poisson noise, allowing for a more general structure in the jump part. If we are interested only on a BSDE driven by a marked point process, the proof above still applies when the filtration 𝔾 is generated only by p and the data are adapted to it. Then we have the counterpart of theorem <ref> Let assumptions <ref> and <ref>(i,ii,iv) hold for some β>L_p^2+2L_f, but with the data adapted to the filtration 𝔾. Then the system (<ref>) admits a unique solution in L^2,β(A)× L^2,β(p)×ℐ^2. This is proven exactly as the case with also a Brownian motion. First, we show as in <ref>, the solution lies in L^2,β(A)× L^2,β(p)×ℐ^2 and, using Ito's formula, that it is unique. Next we build a contraction on this space, and obtain existence and uniqueness when the generator depends on (Y,U). A similar result does not hold in general in L^1. Counter examples are given in <cit.>, whereadditional hypotheses are then added to obtain an existence and uniqueness result. We also refer to <cit.> where the case L^p is analysed. § SOME REMARKS ON THE SNELL ENVELOPE THEORY The Snell envelope theory has been treated in various works. <cit.> considers the case for a positive process without any restrictions on the filtration, obtaining general results. For a bit less general results, but still enough for our work, <cit.> develops the theory for non-negative càdlàg processes, while <cit.> treats the case where the process is càdlàg and left continuous over stopping times, and satisfies the condition[sup_t|η_t|]<∞.The recent work <cit.> treats the subject in the framework of family of random variables indexed by stopping times, using quite general assumptions. In the following, let (Ω,ℱ,ℙ) be a probability space and let 𝔽=(ℱ_t)_t≥ 0 be a filtration satisfying the usual conditions. Let η be a cadlag process. Several properties that hold for positive processes can be shown under the condition (<ref>), as we will see in proposition <ref>.Werecall the following definition:An optional process R of class [D] is said to be regular if R_t^-=^pR_t for any t<T, where ^pX indicates the predictable projection. Let η be a càdlàg process satisfying (<ref>). DefineR_t=_τ∈𝒯_tη_τ It holds that *R_t is the Snell envelope of η_t. This means it is the smallest càdlàg supermartingale that dominates η_t, i.e. R_t≥η_t for all t ℙ-a.s.*A stopping time τ^* is optimal in (<ref>) (i.e. R_t=η_τ^*) if and only if one of the following conditions hold * R_τ^*=η_τ^* andR_s∧τ^*is a 𝔽-martingale* [R_t]=[η_τ^*] *R_t is of class [D], hence it admits decompositionR_t=M_t-K_t,where M is a martingale, K a predictable increasing process with K_0=0. K can be decomposed as K=K_t^c+K_t^d, where K^c indicates the continuous part and K^d the discontinuous part. Moreover we have, a.s. { t:Δ K_t>0}⊂{ t: R_t^-=η_t^-} or equivalently, Δ K_t=Δ K_t_{R(η)_t^-=η_t^-},t≥ 0.*If the process R_t is regular in the sense that R_t^-=^pR_t, where ^pR indicates the predictable projection, defining the stopping timeD_t^*=inf{ s≥ t : R_s≠ M_s},then D_t^* is an optimal stopping time and it is in fact the largest optimal stopping time.DefineI=inf_t∈[0,T]η_tandN_t=I, and since η_t-I≥ 0 for all t, we have η_t-N_t≥ 0 for all t.N_t is a uniformly integrable martingale thanks to (<ref>). Consider η̃_t=η_t-N_t≥ 0 and R̃_t=R_t-N_t. Notice that thenR̃_t=R_t-N_t=_τ∈𝒯_tη_τ-N_τ=_τ∈𝒯_tη̃_τ,i.e. R̃ is the Snell envelope of the positive process η̃. R inherits all the properties from R̃. Let us see why the fourth property holds, as the rest are obtained similarly. If R_t is regular, so is R̃_t because we are adding a uniformly integrable martingale, which is regular (all uniformly quasi-left-continuous integrable càdlàg martingales are regular, see <cit.> Def 5.49). The result then holds by <cit.> pag 140. | http://arxiv.org/abs/1709.09635v1 | {
"authors": [
"Nahuel Foresta"
],
"categories": [
"math.PR",
"60H10 (Primary), 60G55, 60G40 (Secondary)"
],
"primary_category": "math.PR",
"published": "20170927171859",
"title": "Optimal stopping of marked point processes and reflected backward stochastic differential equations"
} |
Linking the troposphere and the thermosphere of a hot gas giantDipartimento di Fisica e Astronomia `Galileo Galilei', Univ. di Padova, Vicolo dell’Osservatorio 3, Padova I-35122, Italy Observatoire astronomique de l’Université de Genève, Université de Genève, 51 chemin des Maillettes, CH-1290 Versoix, Switzerland University of Bern, Center for Space and Habitability, Sidlerstrasse 5, CH-3012, Bern, Switzerland Space-borne low- to medium-resolution (ℛ∼10^2-10^3) and ground-based high-resolution spectrographs (ℛ∼10^5) are commonly used to obtain optical and near infrared transmission spectra of exoplanetary atmospheres. In this wavelength range, space-borne observations detect the broadest spectral features (alkali doublets, molecular bands, scattering, etc.), while high-resolution, ground-based observations probe the sharpest features (cores of the alkali lines, molecular lines). The two techniques differ by several aspects. (1) The Line Spread Function of ground-based observations is ∼10^3 times narrower than for space-borne observations; (2) Space-borne transmission spectra probe up to the base of thermosphere (P≳10^-6 bar), while ground-based observations can reach lower pressures (down to ∼10^-11 bar) thanks to their high resolution; (3) Space-borne observations directly yield the transit depth of the planet, while ground-based observations can only measure differences in the apparent size of the planet at different wavelengths. These differences make it challenging to combine both techniques. Here, we develop a robust method to compare theoretical models with observations at different resolutions. We introduce ^πη, a line-by-line 1D radiative transfer code to compute theoretical transmission spectra over a broad wavelength range at very high resolution (ℛ∼10^6, or Δλ∼0.01 Å). An hybrid forward modeling/retrieval optimization scheme is devised to deal with the large computational resources required by modeling a broad wavelength range (∼0.3-2 μ m) at high resolution. We apply our technique to HD189733b. In this planet, HST observations reveal a flattened spectrum due to scattering by aerosols, while high-resolution ground-based HARPS observations reveal sharp features corresponding to the cores of sodium lines. We reconcile these apparent contrasting results by building models that reproduce simultaneously both data sets, from the troposphere to the thermosphere.We confirm: (1) the presence of scattering by tropospheric aerosols; (2) that the sodium core feature is of thermospheric origin. When we take into account the presence of aerosols, the large contrast of the core of the sodium lines measured by HARPS indicates a temperature of up to ∼ 10 000 K in the thermosphere, higher than what reported in the literature. We also show that the precise value of the thermospheric temperature is degenerate with the relative optical depth of sodium, controlled by its abundance, and of the aerosol deck. Combining low- to high-resolution transit spectroscopy of HD189733b Lorenzo Pino 1,2 David Ehrenreich 2 Aurélien Wyttenbach 2 Vincent Bourrier 2 Valerio Nascimbeni 1 Kevin Heng 3 Simon Grimm3 Christophe Lovis 2 Matej Malik 3 Francesco Pepe 2 Giampaolo Piotto 1 December 30, 2023 ====================================================================================================================================================================================================§ INTRODUCTIONBy studying the variation in the apparent radius of a planet as a function of wavelength, i.e. by building its transmission spectrum, it is possible to retrieve the physical conditions and composition of its atmosphere. Despite efforts to use low-resolution spectroscopy and accurate photometry to obtain transmission spectra from the ground (e.g. ), space-borne low- to medium-resolution spectroscopy (ℛ∼10^2-10^3) and ground-based high-resolution spectroscopy (ℛ∼10^5) remain the most solid techniques to date.In the near-infrared (NIR), HST/WFC3 transmission spectra are available for tens of planets, comprising several hot Jupiters, some hot and warm Neptunes, and even some super-Earths (e.g. ). Observations of this kind are sensitive to water absorption bands at 1.1 μm and 1.4 μm, whose strength is often lower than predicted by theoretical models. Both aerosols <cit.> and subsolar abundance of water <cit.> could be responsible for weakening or even concealing these spectral features. Combining optical and infrared HST observations, <cit.> were able to break this degeneracy; their analysis favoured the aerosol explanation (see also ).On the other side of the transmission spectrum, optical observations with HST/ACS and HST/STIS led to the detection of an enhanced optical slope in the transmission spectra of several exoplanets <cit.>, which can be explained by scattering from small aerosols (see section <ref> and e.g. ). The strongest spectral features, such as the sodium and potassium doublets, are generated above the aerosol deck and are thus visible in transmission spectra (e.g. , ). The broad spectral coverage and the accuracy of the optical and NIR HST observations allowed us to firmly detect many species and spectral features in exoplanet atmospheres (Na, K, H_2O, aerosols, etc.), but it is limited by relatively low resolving powers (ℛ∼10^2-10^3, see Eq. <ref> and Table <ref>).Ground-based observatories are able to achieve higher resolving powers ofℛ∼10^5. At such high spectral resolution, single absorption lines are spectrally resolved and molecular fingerprints are uniquely identified <cit.>. Even the single lines of the sodium and potassium doublets are resolved, and it is possible to probe lower pressures at high altitude in the atmosphere through their sharp cores <cit.>.Space-born low-resolution and ground-based high-resolution transit spectroscopy are complementary. However, they are difficult to combine because the data-reduction process for the two techniques is different. Both techniques are time-differential, meaning that they compare stellar spectra taken at different times to record the transit radius. Ground-based, high-resolution transmission spectroscopy is `double-differential', i.e. it also requires a wavelength differentiation. The wavelength differentiation is necessary to eliminate variations in the measured flux only due to instrumental effects and the atmosphere of the Earth <cit.>. During wavelength differentiation the absolute level of the absorption of the planet is lost. Thus, assembling transmission spectra obtained by the two techniques requires a common normalization framework.Another challenge is the interpretation of the transit radius in terms of models. In particular, several authors have noted that the reference radius/pressurelevel and the abundance of atmospheric constituents are degenerate <cit.>. However, in this paper we do not infer absolute abundances for the atmospheric constituents of the atmosphere, and the presence of this degeneracy does not affect our conclusions.<cit.> presented a first attempt to combine high- and low-resolution transmission spectra. They combined high-resolution, ground-based, infrared (λ∼2.287-2.345 μ m) CRIRES eclipse observations with HST WFC3 observations (λ∼1.125-1.655 μ m) of lower resolution but more extended spectral coverage, which were used by <cit.> to retrieve the atmospheric conditions in HD209458b. The results show that high-resolution spectra on a narrow band are sufficient to increase the precision on the retrieved abundances of the molecular constituents of the atmosphere by at least one order of magnitude (several for CO).Here, we combine for the first time optical high-resolution, ground-based transmission spectroscopy and optical and NIR space-borne transmission spectroscopy of one of the most well-studied hot Jupiters, HD189733b <cit.>. Low- to medium-resolution data probe atmospheric layers from the troposphere (P∼10 bar-10^-4 bar), where clouds, molecular bands and the wings of the alkali doublets are generated, to the base of the thermosphere (down to P≳ 10^-6 bar). On the other hand, high-resolution optical data are sensitive to the fine cores of the alkali atoms <cit.> generated at higher altitudes in the lower thermosphere (P∼10^-4 bar-10^-11 bar). These observations are complementary to near-UV transit spectroscopy, which probes the transition between the upper thermosphere and the exosphere <cit.>, and to far-UV transit spectroscopy, which probes the exosphere. This is the outermost region of the atmosphere of the planet. Thermospheric heating by stellar X/EUV photons into the lower thermosphere is the source of the expansion of the HD189733b upper atmosphere <cit.>, which leads to the escape of hydrogen and heavier particles into the exosphere <cit.>.For HD189733b, the low-resolution transmission spectrum is flattened by the scattering of stellar light by aerosols suspended in the troposphere of the planet <cit.>; instead, the ground-based, high-resolution transmission spectrum reveals the presence of sharp absorption features in the core of the sodium doublet lines <cit.>. These results are apparently contrasting. By modelling the two data sets simultaneously we aim to reconcile them. Furthermore, <cit.> and <cit.> have measured a positive temperature gradient through the sodium doublet, indicative of thermospheric heating (theresult was confirmed by , via an independent analysis of the same data set). With our analysis we aim to investigate the biases and degeneracies introduced by the presence of tropospheric aerosols (neglected in previous works) in the retrieval based on high-resolution data.Using the code, which we present in section <ref>, we produce high-resolution (R∼10^6) transmission spectrum models in order to simulate HST and HARPS data sets. These simulations are described in sections <ref> and <ref>. High-resolution models with a broad spectral coverage are computationally very demanding. Instead of performing a full retrieval approach which requires the computation of up to millions of models <cit.>, we adopt a hybrid forward modelling/retrieval approach. In section <ref>, we describe this method. In section <ref>, we discuss the models that best reproduce the data and focus on their consistency with both low- and high-resolution spectra. Furthermore, we explore how the different data sets are sensitive to different atmospheric parameters and discuss their complementarity.§ THE CODE We introduce (a Python code for extra-solar transiting atmospheres), an improved version of the η code presented in <cit.> and expanded in <cit.> to compute transmission spectra of exoplanetary atmospheres. There are three main characteristics of : * High-resolution (ℛ∼10^6). This is necessary to compare models with ground-based, high-resolution data;* Broad wavelength coverage (200 nm-2 μm). This is necessary to compare models with space-borne, optical to NIR data;* Flexibility of the input composition and T-p profile. is used to model simultaneously several orders of magnitudes in pressure where common assumptions such as equilibrium chemistry may break down.In the following we summarize the basic concept of η (seeandfor a more complete description of the basic equations of the problem). We then describe the improvements of with respect to its former version.§.§ The η codeThe η code is a plane parallel, line-by-line radiative transfer code used to compute the transmission spectra δ(λ) of exoplanetary atmospheres. More precisely, δ(λ) is the transit depth, i.e. the flux absorbed by the planetary atmosphere in units of stellar flux. The total opacity arising from photoabsorption by atoms and molecules and scattering by molecules and aerosols in the atmosphere determines the spectral shape of the absorption. Practically, this code computes τ_b(λ), the total optical depth encountered by a light ray traversing the atmosphere at an impact parameter b at wavelength λ, and integrates over all impact parameters (see Eq. 1-3 inand figure <ref> in this paper). By assuming hydrostatic equilibrium and spherical symmetry, the slant optical depth for the species i along a chord at impact parameter b isτ_b, i(λ) =X_i n_0··∫_-∞^∞σ_b, i(λ, x)T_0T(√(b^2+x^2))exp[ -∫_r_0^√(b^2+x^2)drH(r)] dx ,where n is the total particle numerical density in the atmosphere, X_i = n_i/n is the volume mixing ratio of species i, T is the temperature and σ_b, i is the cross section profile along the considered chord. Quantities subscripted with 0 are taken at a reference level r_0 well inside the opaque part of the atmosphere of a gaseous giant or at the surface of a rocky planet. All of these quantities are functions of the position in the atmosphere. The typical local spatial vertical scale is given by the scale height defined byH(r) = k_BT(r)μ(r) g(r) ,where g(r) is the gravity acceleration and μ(r) is the molecular weight of the species present in the atmosphere. If the atmosphere is composed of more than one species, the total optical depth is the sumτ_b(λ) = ∑_iτ_b, i(λ) .Integration of Eq. (<ref>) over all the layers that contribute to a measurable atmospheric absorption gives a wavelength dependent atmospheric equivalent surface of absorption Σ(λ), which is the surface of a completely optically thick disk that absorbs as much light as the entire translucent atmosphere.A synthetic transmission spectrum is produced in three steps: computing the atmospheric structure (section <ref>), computing the cross section and thus the optical depth of the absorbing species at each altitude layer (Sects. <ref>, <ref>), and finally summing all the contributions from the layers.§.§ Atmospheric structure Hydrodynamical models of hot Jupiters show that the motion up to the lower part of the thermosphere is subsonic, which implies that hydrostatic balance is a good assumption (e.g. , ). In the temperature profile can be assigned as an arbitrary function of the height in the atmosphere to keep generality. We introduce the possibility of assigning this quantity as a function of either r or p (pressure). Keeping p as an independent variable allows us to account parametrically for temperature inversions, whose intensity is determined by the pressure in the atmosphere and not by the absolute height. The hydrostatic equilibrium equations are then integrated with the boundary condition that the pressure is p_0 at a reference layer r_0, both chosen by the user. If p is the independent variable:r (p ) = (1r_0 + ∫_p_0^pk_b T (p )m_Hμ (p ) G M_ppdp )^-1 . §.§ GeometryBecause of the spectral resolution required for our models (millions of wavelength points in each layer), the implementation of geometry must compromise between time consumption and memory usage. With reference to Fig. (<ref>), we divide the vertical axis z in layers of height h. The optical depth in each layer z_n is computed and stored. We verified with a step-doubling method that an adaptive grid with four layers per local scale height is enough to grant accuracy in our case. The axis crossing the atmosphere (x) is divided into bins of width Δ x_n which are the projection of h along the chord. From geometric considerations: Δ x_n = √((z_l, n + r_0)^2 - b^2) - √((z_l, n-1 + r_0)^2 - b^2) .The value of the optical depth in layer z_n can now be used as a proxy for the value of the optical depth in this bin. A typical run of the code then requires ∼ 3 GB of RAM and ∼ 10 minutes to run on an Intel^® Xeon^® CPU E5620 (2.40 GHz) using pre-computed opacity tables (see Sects. <ref>).§.§ Photoabsorption cross sections at high spectral resolution Difficulties in accurately modelling the cross section σ(λ) arise from poor theoretical knowledge of broadening, position and intensity of the lines and from the computational requirements set by the high number of lines to be considered in a line-by-line code.Atomic lines: For bound-bound processes the cross section can be written as <cit.>σ_at(ν) = π e^2m_ecfΦ(ν) ,where e and m_e are the elementary charge and mass of the electron, c the speed of light, f the oscillator strength of the transition and Φ(ν) the line profile. Many physical effects determine the atomic line shape. Here we model each line as a Voigt profile, accounting for four types of broadening: thermal Doppler broadening, turbulent broadening, intrinsic broadening, and collisional broadening. We compute the Voigt profile as the real part of the Faddeeva function, computed with standard Python libraries ().The ASD line list <cit.> provides the natural broadening widths, while for the collisional broadening of the sodium and potassium doublet, by far the dominant atomic transitions, we follow <cit.> and <cit.> and setγ_coll, [Na K] = [ 0.071;0.14 ] (T2000 )^-0.7cm^-1atm^-1· P· X_H_2 , where X_H_2 is the volume mixing ratio of H_2, the relevant perturbing species.Molecular transitions: We focus on water, which was detected in HD189733b using WFC3 data <cit.>. To model H_2O absorption, we adopt the HITEMP line list <cit.>[In appendix <ref> we discuss possible alternatives.]. Since the temperature in the atmosphere of HJs can reach well above 1000 K, it is necessary to use the HITEMP line list instead of the HITRAN line list <cit.>. The HITEMP line list includes transitions that may be of negligible intensity at laboratory temperature, but due to the temperature dependence of the line intensities might play a role at high temperatures <cit.>. This results in the inclusion of 2.7· 10^7 H_2O transitions in the wavelength range 3 300–20 000 Å that interests us. Each line has to be modelled precisely at very high resolution (ℛ∼10^6). The task requires state-of-the-art numerical techniques: we therefore rely on the HELIOS-K opacity calculator <cit.>. HELIOS-K is an ultrafast, open-source line-by-line opacity calculator for radiative transfer that extensively exploits parallel computing on GPUs to compute opacity tables generated from the HITRAN or HITEMP line list.The shape of each line is modelled as a Voigt profile with a characteristic intensity, broadening and central wavelength. These are in turn determined by the thermodynamical conditions of the molecular gas: * A change in temperature has the effect of changing the partition function of a given molecule, i.e. the number density of the molecules associated with an energy level with respect to the total number density of that molecule. This impacts the intensity of a given line. * A change in temperature or pressure changes the half width half maximum of a given line <cit.>.* A change in pressure induces a shift of the central wavelength of the line. All of these effects are considered in HELIOS-K, which is thus able to accurately model each line present in the HITEMP line list in the selected wavelength range.We precomputed a grid of opacity tables covering the temperature range 500–3000 K with a sampling of 50 K and the pressure range 10^-9-10 bar in 21 logarithmically spaced points. Each opacity table is calculated on a wavenumber grid with a resolution of 0.01 cm^-1 over the entire spectral range 3 300–20 000 Å, corresponding to a wavelength resolution between 0.0033 and 0.02 Å or ℛ≳ 5·10^5, which is enough to grant no opacity loss (see also ). We extend the calculation for each line out to 25 cm^-1 from its centre.Our high-resolution approach is aimed at limiting molecular opacity loss. It is not optimal, however, since it requires a large amount of disk space to store the opacity tables and long computational times to model the entire spectral range. For the purposes of this paper, computational time is not an issue: it is sufficient to compute a limited number of models (∼10^2, see section <ref>). In future implementations we will explore several possibilities to speed up the code, such as k-distribution tables <cit.>, the opacity sampling method () and the recently proposed super-line approach <cit.>. Continuum absorptionThe absorption lines emerge from a `continuum' absorption, due to Rayleigh scattering by several species and to scattering by aerosols when present. We implement Rayleigh scattering due to H_2. In we adopt the prescription by <cit.> commonly adopted in both solar system and exoplanets communities <cit.>:σ_H_2(λ) = 8.14·10^-13λ^4 + 1.28·10^-6λ^6+1.61λ^8 ,which is accurate to O (λ^-10 ). <cit.> implemented a different prescription for H_2, based on the ab initio calculations given in <cit.>. A third alternative, based on <cit.>, is adopted by other authors <cit.>. In Fig. (<ref>), we show a comparison between the Rayleigh cross sections for molecular hydrogen computed according to the prescriptions by <cit.>, <cit.>, and <cit.>. They differ by <10% in the wavelength range 3 300–20 000 Å. We conclude that our analysis is consistent with all prescriptions. However, possible studies sensitive to Rayleigh scattering in the UV spectral range will have to consider which is the most appropriate law to follow.We also include a simple treatment of aerosols as an opaque opacity source. Following <cit.>, we distinguish between a grey absorber, to which we call `clouds', and a chromatic absorber that we generically call `hazes'[The naming convention we have adopted is commonly used by Earth scientists. The planetary science community generally distinguish clouds and hazes in terms of formation pathways rather than in terms of size. See also a blog entry by Sarah Hörst on this topic:http://www.planetary.org/blogs/guest-blogs/2016/0324-clouds-and-haze-and-dust-oh-my.html]. With this distinction, clouds mask the transmission spectrum below a pressure level p_c that coincides with the clouds top. Clouds are aerosols with particles that are larger than the wavelength of incident light. What we call hazes are, on the other hand, small particle aerosols (smaller than the wavelength of incident light) whose cross sections depend on wavelength and on the particle size. We model the global haze contribution as a scattering cross section:σ_h = σ_h(λ_0) (λλ_0 )^-s .Here, σ_h(λ_0) = Aσ_H_2(λ_0), where A is the amplitude of the hazes scattering cross section in units of the H_2 Rayleigh scattering cross section. s is a slope that is determined by the composition and the sizes of the particles that scatter light <cit.>. Observations are not able to distinguish all the different populations of scatterers (but see e.g. ); on the other hand leaving s as a parameter allows us to have full generality when dealing with aerosol scattering.In the limit for λ→∞, Eq. (<ref>) reduces to a pure power law with s=4 (Rayleigh scattering with a wavelength independent refraction index). In this case the cross sections of hazes and Rayleigh scattering by molecular hydrogen can be combined in an “effective Rayleigh scattering” cross section:σ_Ra, eff(λ) = σ_H_2(λ_0) (λλ_0 )^-4[ 1 + A (λλ_0 )^4-s] .From Eq. (<ref>), it is clear that hazes affect the transmission spectrum only if the scale parameter A is high enough. Depending on the index s the effect may be more severe at longer wavelengths (s < 4) or shorter wavelengths (s>4).Despite its simplicity, several studies based on similar prescriptions for cloudsand hazes showed that this treatment is sufficient to capture degeneracies between the presence of clouds and water abundance (e.g. ).In standard conditions, symmetric molecules such as H_2 do not possess any dipole moment, and thus are not strong emitters. However, they may acquire a transient dipole during collisions with other molecules if the density is high enough. In extremely hydrogen-rich dense environments, such as cool low-metallicity stars <cit.> and some solar system planets <cit.>, the absorption due to this phenomenon (collisional-induced absorption, CIA) should usually not be neglected. In our case of transmission geometry of a hot Jupiter, CIA contribution is negligible in the wavelength range considered (as molecular opacities are dominant at the pressures sounded). For consistency with the literature (e.g. ), we nonetheless include in the CIA by H_2–H_2 and H_2–He couples from the density-normalized HITRAN coefficients presented in <cit.>. Finally, we neglect the effects of refraction and multiple-scattering, working in approximation of pure absorption (where scattering opacity is treated as absorption opacity). <cit.> and <cit.> claimed that these effects are unimportant in hot Jupiters. More recently, <cit.> argued that there may be a few cases in which this simplification is inaccurate (mainly cold and small planets, which we do not investigate here). <cit.> demonstrated that aerosol forward scattering can be an important consideration for planets whose host star is relatively large in angular size, as is the case for hot Jupiters.However, for the models presented here, haze/cloud forward scattering would likely reduce our predicted transit depths by less than a scale height (; Robinson, private communication). § SIMULATING OBSERVED TRANSMISSION SPECTRA A comparison between theoretical models and observations is only possible after the following steps: * Convolution with the line spread function (LSF). Since the FWHM of the LSF varies by orders of magnitude between low- and high-resolution (∼0.048 Å for HARPS, up to ∼ 107.7Å for HST), its impact on the transmission spectrum in these different regimes must be taken into account;* Binning at the data sampling;* Only for ground-based data: wavelength normalization. Contrary to space-based observations, ground-based, high-resolution spectra are double differential (in time and wavelength, ). The wavelength differentiation is required to remove time-varying signals due to the Earth's atmosphere or to the instrument. However, it removes the absolute reference for the absorption, a limitation that we need to take into account when making a comparison to models. §.§ Convolution with the instrumental LSFWe model the LSF of all the instruments as Gaussians with full width half maximum (FWHM) equal to their nominal resolving power. The resolving power is defined asℜ = cΔ v = λΔλ ,where Δ v and Δλ are the FWHM of one resolution element of the instrument considered in the velocity space and in the wavelength space, respectively. In Table <ref> we list the adopted parameters for the LSF of the instruments used for observations.We compute the chromatic absorption due to the atmosphere of the planet, or transit depth, δ(λ). Then, we convolve this quantity with the LSF of the instrument,δ_conv(λ) = (δ∗LSF)(λ) = ∫_-∞^+∞δ(λ) LSF(λ - λ') dλ' .To compute the integral on the right-hand term, it is necessary to limit its calculation in the wavelength space. We made sure that integration over 3 times the FWHM of the LSF causes no loss of flux for all simulated instruments.§.§ BinningAfter the stellar light is dispersed, it hits the CCD pixels. In a calibrated spectrograph, each pixel is associated with a wavelength. The net effect is that the photons are binned by wavelength, each bin corresponding to a pixel. To compare the model with the observations it is necessary to bin the model in the same bins the observations are provided with.Binning to the instrumental sampling must conserve flux, thus byδ_bin( λ_i) = ∫_λ_i - dx_i^λ_i + dx_iδ_conv(λ) dλ2 dx_i ,where λ_i and dx_i represent the centre and the half width of a pixel in the wavelength space.§.§ Wavelength normalizationTo remove flux variations only due to the instrument or the atmosphere of Earth, ground-based observations require a wavelength differentiation <cit.>. Practically, the differential transmission spectrum ℜ̃(λ) presented in <cit.> is normalized to a reference band. To compare the models to this data set, it is necessary to normalize the models in the same fashion. The relation between the transit depth and the differential transmission spectrum isℜ̃_bin(λ_i) = 1-δ_bin(λ_i)1-δ_bin(λ_ref) ,where δ_bin(λ_ref) is the transit depth after convolution with the LSF and binning, averaged in a reference band where no features are expected. Equation <ref> is demonstrated in Appendix <ref>. § TRANSMISSION SPECTRUM AND PLANETARY ABSORPTION The procedure we have outlined in section <ref> is commonly adopted to compare models and observations. However, it does not account for the fact that transmission spectra are built by dividing out stellar fluxes during the transit (in-transit, F_in) and out-of-transit (F_out). This is different from what is simulated by following the procedure outlined in the previous section, since neither convolution nor integration (or sum) are distributive with respect to division. For example, for convolution,LSF∗( f/g ) ≠(LSF∗ f )/ (LSF∗ g ) .Thus, a more realistic simulation should compute asδ_bin( λ_i) = F_in, bin( λ_i)F_out, bin( λ_i) = ∫_λ_i - dx_i^λ_i + dx_i F_in, conv(λ) dλ∫_λ_i - dx_i^λ_i + dx_i F_out, conv(λ) dλ , F_in/out, conv(λ) = ∫_∼3·FWHM(LSF) F_in/out(λ) LSF(λ - λ') dλ' .However, the computational cost of these more realistic simulations is doubled with respect to computing δ_bin and ℜ̃_bin directly from the theoretical planetary absorption δ(λ) (Eq. <ref>).If the integrand can be approximated as a constant in the domain of integration it can pass through the sign of integral. If this applies to both Eq. (<ref>) and Eq. (<ref>) the order of integration and division does not matter, and we can proceed following section<ref>.We test numerically if this is the case for our spectra. To perform the test, the stellar (F_out) should be known. To represent it we use a PHOENIX model representative of the spectrum of HD189733 (T=4900 K, log g=4.5, [ Fe/H]=0, α=0). The code computes the absorption δ due to the planet, thus it is possible to compute F_in = F_out (1-δ). This is the stellar flux, planetary absorption included, before it is observed with an instrument. Then, the transmission spectra can be computed following Eq. (<ref>), Eq. (<ref>), and Eq. (<ref>), and compared.In Fig. (<ref>) we show the difference between the transmission spectrum computed using the stellar spectrum (Eq.<ref> and Fig. <ref>, blue line), which is more accurate, and that using the absorption directly (Eq. <ref>, red line). The statistical difference between the models is non-significant (using the BIC, defined in section <ref>, ΔBIC ∼ 3 for HARPS, less than 1 for WFC3); this is also confirmed by the small residuals between the two models compared to the error bar (Fig. <ref>). We note, however, that for observations at higher signal-to-noise such as ESPRESSO, the E-ELTs, or JWST might deliver or with different host stars, this effect might become significant. Indeed, <cit.> showed that neglecting this effect, which they call resolution-linked bias (RLB), yields a difference of 12% in the peak transmittance for TRAPPIST-1b in the region around 1.4 μm. The effect is severe because cold stars likes TRAPPIST-1 have spectra dominated by forests of molecular lines. HD189733b is a K dwarf, thus the effect is less important.§ HD189733B CASE: METHODS We compare models of transmission spectra of HD189733b with optical, high-resolution differential transmission spectroscopy HARPS data () and optical, low-resolution HST STIS G430L and ACS HRC G800 (), medium-resolution HST STIS G750M () and NIR, low-resolution WFC3 () data. The observations used are listed in Table <ref>. All the HST data are taken in the version presented by , who corrects for possible systematic offsets between the data sets that are mainly due to stellar activity. For G750M, <cit.> presents the data in wavelength bins of 60 Å. However, a single point in the core of potassium is presented at higher sampling (bin width of 10 Å). Its wavelength domain is already covered by the neighbouring lower sampling data points. We thus exclude this point from our analysis.We calculate the χ^2 between the HST transit depths or the normalized HARPS transmission spectrum in a wavelength bin λ_i, and the model processed as described in section <ref> in the same wavelength bin. We also compute the corresponding Bayesian information criterion (BIC, ). The lowest BIC corresponds to the favoured model; a model whose BIC differs by more than 5 is statistically different. Moreover, the BIC takes into account the number of model parameters, preferring simpler models.§.§ HST data setWe find a model that minimizes the BIC_HST for the full HST data set (best fit HST, BF-HST). Throughout this operation we fix the following parameters: * Star radius: R_⋆ = 0.756 R_⊙;* Planetary mass: M_P = 1.138 M_J;* Constant solar composition, i.e. we adopt the following fixed mixing ratios:* X_H_2 = 0.9289989;* X_He = 0.07;* X_H_2O = 10^-3;* X_Na = 10^-6;* X_K=10^-7. We adopt p as an independent variable, and fix the T-p profile. For the lower part of the atmosphere (p>1 mbar), we adopt the profile presented in <cit.>. For the thermosphere (p<1 mbar), we adopt the profile presented in <cit.>.The remaining parameters are adjusted to minimize BIC_HST. These parameters are: * The reference radius, which we take as the radius at 10 bar: r_10 atm = r(p=10 atm) (but seefor a discussion on why this causes a degeneracy in inferring the values of mixing ratios at the order-of-magnitude level);* The hazes parameters σ_h(λ_0) and s;* The top pressure of the cloud deck p_c.The HST data set is further divided into optical (both STIS gratings and ACS) and WFC3. The optical data set is mainly sensitive to the haze parameters and to the reference radius, while the WFC3 data set is mainly sensitive to p_c and to the reference radius. Instead of performing a full retrieval, we exploit this difference to design a step-by-step optimization algorithm: Step 1: Adjust r_10 atm using the average absorption in the WFC3 band;Step 2: Adjustσ_h(λ_0) and s using STIS + ACS data;Step 3: Adjust the cloud level p_c using the WFC3 data. For step 3, the full WFC3 spectral information is used. In each step, all parameters that are not being adjusted are fixed to values from the previous steps. If the parameters were perfectly uncorrelated, the procedure would converge to a minimum of BIC_HST after one iteration. This is not the case (e.g. optical and WFC3 data sets both depend on the reference radius), thus we iterate the three steps until convergence. Convergence is achieved when ΔBIC_HST<5 among the six models obtained in the last two iterations. The initialization for the first iteration assumes an aerosol-free atmosphere.The intermediate nature between forward modelling and retrieval of our approach, which we refer to as `retrieval by hand', allows us to limit the number of models to be computed, and thus treat the problem numerically. At the same time, it allows us to reproduce the HST data set satisfactorily (see section <ref>).§.§ HARPS data setThe BF-HST is then adjusted to reproduce the HARPS data set. For the wavelength normalization, we adopt the reference band 5870-5882.22 Å, 5903.24-5916 Å (the same adopted by , grey background in Fig. <ref>). The merit functions are computed in two bands centred in the cores of the sodium lines: 5889.22 Å-5890.26 Å and 5895.20 Å-5896.24 Å (, yellow background inFig. <ref>). For each model, we compute the difference in BIC of the fit to the combined data set ΔBIC_HR+HST with respect to the reference model BF-HST. We also compute the difference in BIC of the fit to the low- and high-resolution data sets separately (ΔBIC_HR and ΔBIC_HST), to highlight which data set is driving the fit.To adjust the HARPS data we modify a different set of parameters that mainly affects the thermosphere, leaving the troposphere mostly unchanged: * Thermospheric T-p profile (P<0.1 mbar);* Sodium abundance, X_Na.The strongest water lines may be affected by the T-p profile modification and the sodium wings are produced in the troposphere. It is thus necessary to verify a posteriori that the quality of the fit to the HST data set is comparable to the quality obtained with the BF-HST for each modification.§ HD189733B CASE: RESULTS AND DISCUSSION §.§ HST data setAfter six iterations we obtain a model that reproduces adequately the whole HST data set (BF-HST, see Fig. <ref>). We find a reduced chi square χ_HST, ν=43^2 = 1.1. When considering only the optical data sets (STIS G430L and G750M, and ACS), we obtain χ^2_ν=17 = 0.8. For the WFC3 data set we find χ^2_ν=26 = 1.3. The bluest points of the WFC3 observations would be reproduced better by the model if the scattering slope were weaker in this region, but a reduction of the haze content would cause a decrease in the quality of the reduced chi square obtained with the optical data set. The final model parameters are * σ_h∼8000·σ_H_2(589.46 nm) (λ589.46 nm )^-3.17 ,* p_c∼ 0.4 mbar ,* r_10 atm∼1.1082 R_J .Consistent with the literature, we find that the spectrum is dominated by aerosol scattering. An enhanced scattering due to a mix of different small particles (which produces an effective opacity with slope different than 4, see ) is required to explain the slope in the optical. The reduced intensity of the water feature probed by WFC3 is reproduced with a grey opacity cloud-deck. §.§ HARPS data setWe show the results of the comparison with the HARPS data in Fig. <ref>. The absorption in both sodium lines of the BF-HST model is low compared to the data (Fig. <ref>). Since we adopted a T-p profile consistent with <cit.>, which in turn is a result of a fit to the same data set considered here, this comes as a partial surprise. The most likely reason is that <cit.> assumed an aerosol-free atmosphere. Since the cores of the sodium lines are detected, aerosols are likely confined to the troposphere. The sodium core features probed by HARPS are generated higher up, in the lower thermosphere. However, it was already noticed from medium-resolution observations that scattering by aerosols may in some cases partially mask the sodium spectral feature (e.g. ). To test whether aerosols in HD189733b are at high enough altitudes to have an effect on the high-resolution transmission spectrum we run an aerosol-free model (AF) that otherwise has the same parameters as the BF-HST.The aerosol-free model reproduces the data better, as evinced by ΔBIC_HR∼45 and by smaller residuals. We can thus confirm that the discrepancy between our results and the results by <cit.> is due to the presence of aerosols and their effect on wavelength normalization. Indeed, aerosols are not high enough to affect the inner cores of the lines, but they are high enough to occult the sodium wings in the reference bands (see Fig. <ref>). The difference in absorption between the core of the lines and the reference band is thus reduced in the presence of aerosols. Thanks to the correct wavelength normalization of the models, we are sensitive to this difference, and can distinguish between the BF-HST and its aerosol-free version.However, the low-resolution data set firmly excludes the aerosol-free case; ΔBIC_HST>10 000 supports the aerosol-rich case. To reconcile the space-borne and ground-based data sets we need to keep the aerosols and increase the contrast of the sodium feature. We thus modify the thermospheric T-p profile to increase the scale height of the layers where the cores of the sodium lines are generated. This is obtained by increasing the thermospheric temperature with respect to the BF-HST. The high thermospheric temperature (HTT) model transmission spectrum is shown in Fig. <ref>. For this model we find ΔBIC_HR∼13, indicating that it reproduces the HARPS data significantly better than the BF-HST. In Fig. <ref> we show the T-p profile that best reproduces the HARPS data (red curve and shadowed area), and the comparison with <cit.> (black curve). Furthermore, the HTT model is also consistent with the HST data (ΔBIC_HST<1), indicating that the troposphere is indeed left unchanged by the enhanced thermospheric heating. Finally, we note that our T-p profile is in qualitative agreement with observation <cit.> and models (e.g. ) of the HD209458b thermosphere, with temperatures of up to 10 000 K. HD209458b is in similar irradiation and gravity conditions to those of HD189733b, and both have expanded upper thermospheres.The presence of aerosols also introduces a degeneracy between sodium abundance and thermospheric temperature. Intuitively, the more abundant the sodium is, the higher up in the atmosphere its features originate. For an abundance that is high enough, an aerosol-free atmosphere can be mimicked when focusing on the narrow wavelength band probed by high-resolution data. In Fig. <ref>, we show in green a model with the same parameters as the BF-HST, but with 10 times its sodium abundance (10Na). The sodium lines are now generated higher up in the atmosphere, and part of their wings are generated above the aerosol scattering-deck (see Fig. <ref>, green line). As a result, when normalized, the enhanced sodium abundance model is nearly indistinguishable from the aerosol-free model in the χ^2 band (see Fig. <ref>). It also reproduces the HARPS data significantly better than the BF-HST. Models with 50 to 100 times the solar sodium abundance lead to even better matches with HARPS data. The quality of the fit to the HST data set is slightly decreased for such high sodium abundances because the wings of the doublets start to be visible. Overall, the BIC value resulting by the combined analysis of low- and high-resolution data favours the 50Na model.The degeneracy between thermospheric heating and sodium abundance is further illustrated by constructing a second HTT model, with sodium abundance fixed at 50 times the solar value (HTT 50Na). Indeed, the difference in the global BIC between HTT 50Na and 50Na is not significant (ΔBIC_HR+HST∼1.1). In other words, a strongly (50x) enriched atmosphere would not require an enhanced thermospheric heating to exhibit the observed sodium signature (see also Fig. <ref>). We summarize our results in Table <ref>.We draw the following conclusions: * The low- to medium-resolution data set can be reconciled with the high-resolution data set;* The presence of aerosols at the 0.1 mbar level, with which we explain the low- to medium-resolution data set, requires an enhanced thermospheric heating or enhanced sodium abundance in order to reproduce the high-resolution data set;* Having assumed solar sodium abundance, <cit.> has underestimated the role of thermospheric heating because they assumed an aerosol-free atmosphere.Rather than the absolute abundance of sodium, it is its optical depth compared to scattering by aerosols that determines the contrast of its lines in the high-resolution transmission spectrum. This point is distinct but linked to the degeneracy between abundance and reference level first pointed out by <cit.> and generalized by <cit.>. A complete analysis aimed at measuring sodium abundance or the precise value of the thermospheric temperature would thus need to fully explore this degeneracy, and is beyond the scope of this paper.Finally, we note that the residuals between high-resolution data and models are asymmetric (see Fig. <ref>). This could be due to the uncorrected Rossiter-McLaughlin effect <cit.>. This effect has to be modelled before any quantitative conclusion can be obtained from the high-resolution data set. However, it does not affect our conclusion that high-resolution observations are sensitive to the set of parameters that complements lower resolution observations when the two techniques are combined.§ CONCLUSIONSWe developed a method to simultaneously compare theoretical models with high-resolution (ℛ∼10^5), double-differential transmission spectra and low- to medium-resolution (ℛ∼10^2-10^3) transmission spectra. It relies on a dedicated 1D, line-by-line, high resolution (ℛ∼10^6) radiative transfer code, called ^πη. Within this framework we have done the following: * Showed that in the case of HD189733b, and likely many other planets, the common assumption that convolution and binning can be applied after dividing in-transit by out-of-transit spectra is justified by the error bars of current instrumentation (but see );* Built model atmospheres of the hot Jupiter HD189733b that are consistent with HST STIS, HST ACS, HST WFC3, and HARPS transmission spectra. These models are consistent with observations on an unprecedented range of pressures (tens of scale heights) from the troposphere to the thermosphere of the exoplanet. They show that the apparent discrepancy between flattened spectra observed at low-resolution and peaked features observed at high-resolution can be solved. This is obtained by adjusting a model that reproduces the low-resolution data to the high-resolution data, adopting a best set of parameters (sodium abundance, thermospheric T-p profile) that affects the cores of the sodium lines while leaving the troposphere mainly unaffected;* Showed that ground-based, high-resolution observations are sensitive to the presence of aerosols. Neglecting aerosols has led to an underestimation of the thermospheric heating in this planet by <cit.>. For a solar abundance of sodium, the same assumed by <cit.>, we obtain thermospheric temperatures of up to ∼10 000 K, in qualitative agreement with models and observations of the similar hot Jupiter HD209458b;* Identified a degeneracy between sodium abundance and thermospheric heating that affects double-differential techniques in the presence of scattering by aerosols, linked to but distinct from the well-known degeneracy between abundance and reference level <cit.>. Retrievals of the absolute sodium abundance must take into account the presence of this degeneracy.In summary, high-resolution ground-based observations provide unique insight into the thermosphere of exoplanets and are sensitive to alkali abundances when properly interpreted and combined with lower resolution observations. Our analysis opens new perspectives for the optimal exploitation of future facilities such as JWST on the one hand, and ESPRESSO and the E-ELTs on the other. We thank R. Allart and the entire atmospheric group of the Geneva Observatory for insightful discussions and spot-on comments. We are also grateful to the anonymous referee, whose comments helped clarify several aspects of this paper, some of them crucial. This work has been carried out in the frame of the National Centre for Competence in Research ‘PlanetS’ supported by the Swiss National Science Foundation (SNSF). L.P., D.E., V.B., K.H., C.L., and F.P. acknowledge the financial support of the SNSF. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 724427). aa§ DERIVATION OF EQ. <REF>What is measured The procedure followed by <cit.> to obtain ℜ̃ is the following: * All out-of-transit and in-transit spectra are normalized to a reference band:f̃_in/out(λ) = f_in/out(λ)<f_in/out(λ)>_λ_ref , * A master out F̃_out is built as the sum of the out-of-transit spectra and normalized;* The spectrum ratio is obtained as the sum of all the self-normalized in-transit spectra divided by the master out after having shifted them in the planetary rest frame:ℜ̃(λ)=.∑_transitf̃_̃ĩñ(λ, t)F̃_out(λ)|_P .What is simulated The quantity δ(λ) is the transit depth, i.e. the absorption due to the planet in units of stellar flux. This is then related to f_in/out byδ(λ) = f_out - f_inf_out = 1 - f_inf_out ,and thusf_inf_out = 1 - δ(λ) .Furthermore, our tests in section <ref> allow us to write<f_in>_λ_ref<f_out>_λ_ref≈< f_inf_out>_λ_ref = 1 - < δ(λ)>_λ_ref .If we assume that the only time varying signal is due to the transit of the atmosphere of the planet, when we divide <ref> and <ref> we obtain Eq. (<ref>).§ CHOICE AND THE USE OF THE WATER LINE LIST The so-called `Million- to Billion-line radiative transfer challenge' <cit.> is a well known problem. The exact solution to the radiative transfer equation would require each single molecular absorption line to be modelled precisely (the `line-by-line' approach). For several molecules (e.g. water, methane) billions of lines contribute to the opacity at high temperatures (>1 000 K). To add complexity to the problem, multiple line lists are available. The impact of the choice of one line list over the others has never been fully quantified. We review here the knowledge for water to date.The ExoMol project <cit.> is the most up-to-date effort to provide line lists suitable for hot environments. In some cases this line list clearly outperforms previous efforts (e.g. for methane HITRAN vs ExoMol, ). For water, however, there is general agreement that the line lists by <cit.> and <cit.> are solid and complete, and indeed the BT2 line list by <cit.> has been adopted by ExoMol (but a new line list is being developed inside the ExoMol project, Polyansky et al., in prep). The BT2 line list also constitutes the basis for the HITEMP water line list adopted in this paper. The HITEMP line list contains 25% of the lines of the BT2 line list. The lines are selected to reduce the size of the line list while avoiding opacity losses at high-temperatures. This line list is adopted by several groups to model exoplanetary atmospheres (e.g. ). Furthermore, <cit.> provide a thorough analysis of the WFC3 spectrum of WASP-63b using forward modelling and retrieval methods based on both line lists. The authors conclude that there is `general agreement amongst all the results', suggesting that in this case differences due to the choice of the line list are minor. <cit.> pointed out that the HITEMP and the <cit.>[based on the BT2 line list] line lists may differ when dealing with high-resolution data; however, the culprit is not uniquely identified in their analysis (shift in line positions, line intensity, completeness, and different treatment of broadening are all possible candidates). It is thus not clear to what extent these differences would propagate to the comparison with low-resolution data. A quantitative assessment of the impact of different choices for the opacity tables is certainly warranted; because this assessment is lacking (to the best of our knowledge) we consider the HITEMP line list to be an appropriate choice to simulate WFC3 data. | http://arxiv.org/abs/1709.09678v2 | {
"authors": [
"Lorenzo Pino",
"David Ehrenreich",
"Aurelien Wyttenbach",
"Vincent Bourrier",
"Valerio Nascimbeni",
"Kevin Heng",
"Simon Grimm",
"Christophe Lovis",
"Matej Malik",
"Francesco Pepe",
"Giampaolo Piotto"
],
"categories": [
"astro-ph.EP"
],
"primary_category": "astro-ph.EP",
"published": "20170927180513",
"title": "Combining low- to high-resolution transit spectroscopy of HD 189733b. Linking the troposphere and the thermosphere of a hot gas giant"
} |
Effective Image Retrieval via Multilinear Multi-index Fusion Zhizhong Zhang, Yuan Xie, Member, IEEE, Wensheng Zhang, Qi Tian, Fellow, IEEE, Z. Zhang, Y. Xie, and W. Zhang are with the Research Center of Precision Sensing and Control, Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, China and the School of Computer and Control Engineering, University of Chinese Academy of Sciences, Beijing, 101408, China. E-mail: {zhangzhizhong2014, yuan.xie}@ia.ac.cn, [email protected]. Tian is with the Department of Computer Science, University of Texas at San Antonio, San Antonio, TX 78249 USA; E-mail: [email protected]================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================This technical report presents an environment representation for use in vision-based navigation. The representation has two useful properties: 1) it has constant size, which can enable strong run-time guarantees to be made for control algorithms using it, and 2) it is structurally similar to a camera image space, which effectively allows control to operate in the sensor space rather than employing difficult, and often inaccurate, projections into a structurally different control space (e.g. Euclidean). The presented representation is intended to form the basis of a vision-based subsumption control architecture. § INTRODUCTIONAgents often encounter large and varying numbers of entities within a scene while performing navigation, which can be problematic for conventional planning and control approaches that reason explicitly over all entities. Consider, for instance, busy roadways or crowded sidewalks or convention halls: conventional planning approaches in these scenarios can suffer significant performance degradation as entity count increases <cit.>. While more scalable approaches exist, they often have strict requirements on system dynamics <cit.> or observability of agent policies <cit.>.To help address the scalability problem, this report builds on <cit.> to present a constant size environment representation for vision-based navigation. The representation is modeled after a camera image space, which is chosen because cameras are a ubiquitous sensor modality, and image space is typically discretized and constant size. The proposed representation allows planning and control routines to reason almost directly in sensor space thereby avoiding often complex and noisy transformations to and from a more conventional Euclidean space representation. This new representation can help vision-based mobile robots navigate complex multi-agent systems efficiently, and can aid satisfying the strict resource requirements often present in real-time, safety critical, and embedded systems <cit.>. Further, the representation enables additional guidance information to be added in while making guarantees about the preservation of hard constraint (Definition <ref>) information. This property makes it ideal for use in vision-based subsumption control architectures.§ BACKGROUNDThe approach in this report is based in part on potential fields <cit.>. These fields represent attractive and repulsive forces as scalar fields over a robot's environment that, at any point, define a force acting on the robot that can be interpreted as a control command. As noted in literature, this type of approach is subject to local minima and the narrow corridor problem <cit.>, particularly in complex, higher-dimensional spaces <cit.>. Randomized approaches can partially overcome these difficulties <cit.>, while extensions to globally defined navigation functions <cit.> can theoretically solve them but are often difficult to use in practice. This report uses low-dimensional potential fields, limiting the possibility of narrow corridors, and designs the fields such that additional information can be added in to help break out of minima <cit.>. Because the potential fields are modeled after an image space, controlling on them can be accomplished effectively through visual servoing techniques <cit.>.In order to define values for the potential fields, this approach draws on a wealth of related works in optical flow and monocular collision avoidance, notably <cit.>. The intuition of these approaches is that a sequence of monocular images provides sufficient information to compute time-to-contact (Definition <ref>), which informs an agent about the rate of change of proximity.The fields in this representation are intended for use in a subsumption architecture <cit.> where additional information about the system can be layered in while hard constraint information is guaranteed to be preserved. This closure property is proven to hold under a restricted input space with specially constructed potential transform functions.§ DEFINITIONSA potential field (also artificial potential field) is field of artificial forces that attracts toward desired locations and repels from undesirable locations.An affinely extended potential field is a potential field with a potential function that ranges over the affinely extended reals ℝ=ℝ∪{-∞,+∞}. A positive (or negative) affinely extended potential field is defined over ℝ but contains only positive (or only negative) infinite values.[For technical reasons, the potential fields are intentionally not defined over the projectively extended reals ℝ=ℝ∪{∞}.]An image space potential function is a mapping of an image pixel value I(x,y) to a tuple in ℝ^2 that consists of the potential value and its time derivative:I(x,y)↦ℝ^2 An asymptotic region R_A is a closed set of points on ℝ^2 such that the potential function takes a value ±∞ for any element of R_A. (This is related to the notion of a natural boundary in complex analysis.)Encroachment is the reduction in minimum proximity between two or more objects in a workspace 𝒲 as measured by a metric μ(·,·).Guided collision avoidance describes the strategy of choosing goal-directed motions from the space of collision avoiding controls in order to navigate while satisfying collision constraints.Navigation or multi-agent navigation describes the general process of navigating a space possibly shared with multiple other agents.Time-to-contact (τ), is the predicted duration of time remaining before an object observed by a camera will come into contact with the image plane of the camera. The time derivative of τ is written τ̇.A hard constraint is a system constraint that an agent is never allowed to violate.A soft constraint is a system constraint that an agent only prefers not to violate. § THE IMAGE SPACE POTENTIAL FIELDImage Space Potential (ISP) fields are affinely extended potential fields that are modeled after image planes. As with image planes, these potential fields can be discretized, and regions of interest (ROIs) can be defined for them. In this report it is assumed that the fields are origin and axis aligned with the agent's camera image plane, and that they have the same ROIs (Figure <ref>). §.§ Representing Hard & Soft Constraints In potential field representations the distinction between hard and soft constraints can be made in terms of the limiting value of the field as the robot approaches some state that would cause constraint violation: the limiting value of the field over states where hard constraint violation would occur can be infinite, such that no reward can overwhelm the cost, and the limiting value of the field over states where soft constraint violation would occur can be finite, such that a reward must be at least some value before the robot chooses to violate it. In order for the ISP field representation to be useful it must incorporate this notion of constraints, and it must maintain that notion through summation operations. It is shown below that sums of ISP fields do maintain this information through the use of asymptotic regions, and that they behave as expected so long as the following requirements are met:* All ISP fields involved in summation must have like affine extensions, i.e., all fields must either be positively or negatively affinely extended* ISP fields may only be multiplied by scalars in (0, +∞)* ISP fields may only be elementwise multiplied by scalar fields where all values are in (0, +∞) These properties ensure that operations on ISP fields are closed and that asymptotic regions are preserved. The following lemmas prove this: Let F_1 and F_2 be ISP fields, and let R_A be an asymptotic region in F_1. Define element a∈ F_2 as any arbitrary point and define scalar value s∈(0,+∞). In any field F_3=F_1+F_2 or F_4=s· F_1, R_A will persist as an asymptotic region. It suffices to show that the property holds for a single element of the field. Let ±∞ be the value of any element of F_1 from R_A. Then, by definition of addition and multiplication on the affinely extended real number line: a±∞ =±∞, a∓∞s·(±∞) =±∞, s∈(0,+∞) The restrictions that ISP fields have like affine extensions and that scalars belong to (0,+∞) guarantee the conditions in the right column. Thus, any point in R_A with infinite potential in F_1 will have infinite potential in F_3 or F_4.Let F_1 and F_2 be ISP fields. Define elements a_1∈ F_1 and a_2∈ F_2 as any arbitrary points in F_1 and F_2, and define scalar value s∈(0,+∞). For all a_3∈ F_3=F_1+F_2 and a_4∈ F_4=s· F_1, it will be that a_3,a_4∈ℝ It suffices to show that the property holds for a single element of the field. When a_1,a_2∈ℝ addition is closed. As noted in Lemma <ref> when either, or both, is infinite addition is also closed, assuming F_1 are both either positive or negative affinely extended fields. Similar arguments apply to scalar multiplication. However, if s were allowed to range to infinite values, the following would result in an indeterminate form for field value a, and closure would be broken: s· a ∉ℝ, a=0, s=+∞ Thus, addition over strictly positive or strictly negative affinely extended fields and element-wise scalar multiplication with s∈(0,+∞) are both closed operations.Note that Lemmas <ref> & <ref> do not hold if ISP fields are allowed infinite values of mixed signs. This is why ISP fields are restricted to only positive or only negative affinely extended potential fields.§.§ Subsumption through AdditionThe results of Lemmas <ref> & <ref> are powerful because they imply that the information of arbitrary fields can be added together without losing information about hard constraints. Thus, a control architecture using ISP fields can implement subsumption through addition. § THE POTENTIAL FUNCTION The potential function, which maps image pixel values to potential values, can be defined in arbitrary ways, either with geometric relations, learned relations, or even heuristic methods. In this report, hard constraint information will be derived from a geometric measure over pixels in a temporal image sequences called time-to-contact, or τ. Soft constraint information will be represented by user-specified values meant to bias how strongly directed a chosen control is to a particular goal. The potential function maps these measurement values to a unitless potential value space through the potential transformation defined later in this section. §.§ Obtaining Hard Constraint ValuesAs noted often in literature (e.g. <cit.>), τ can be computed directly from the motion flow of a scene, which is the vector field of motion in an image due to relative motions between the scene and the camera. Unfortunately, it is typically not possible to measure motion flow directly, so it is usually estimated via optical flow, which is defined as the apparent motion flow in an image plane. Historically this has been measured by performing some kind of matching of, or minimization of differences between, pixel intensity values in subsequent image frames <cit.>. More recently deep learning techniques have also been successfully applied to the problem <cit.>.Assuming some reasonably accurate estimation of optical flow vector field exists, τ can be computed directly under certain assumptions <cit.>. A significant advantage of computing τ from optical flow, in particular dense optical flow, is that the computation is independent of the number of objects in a scene. In other words, given optical flow, the scalability issues of object tracking and segmentation can be avoided. In practice, however, the computation of optical flow can be noisy and error prone, so feature- and segmentation-based approaches can also be used <cit.>. The idea of these approaches is to compute τ from the rate of change in detection scale. For a point in time, let s denote the scale (maximum extent) of an object in the image, and let ṡ be its time derivative. When the observed face of the object is roughly parallel to the image plane, and under the assumption of constant velocity translational motion and zero yaw or pitch, it is straightforward to show that <cit.>: τ=s/ṡ As shown in <cit.>, scale has a useful invariance property for these types of calculations that can make τ computations robust to certain types of noise and assumption violations. Lemma <ref> demonstrates this: The scale s of an object on the image plane is invariant to transformations of the object under SE(2) on the XY plane.Let (X_1,Y_1,Z) and (X_2,Y_2,Z) be end points of a line segment on the XY plane in the world space, with XY parallel to the image plane and Z coincident with the camera view axis. Without loss of generality, assume unit focal length. The instantaneous scale s of the line segment in the image plane is given by: s=1/Z√(Δ X^2+Δ Y^2) Thus, any transformation of the line segment on the XY plane for which Δ X^2+Δ Y^2 is constant makes s, and thereby ṡ and τ, independent of the values of (X_1,Y_1) and (X_2,Y_2). By definition, SE(2) satisfies this condition. In addition, the time derivative τ̇ of τ, when available, enables a convenient decision function for whether an agent's current rate of deceleration is adequate to avoid head-on collision or not <cit.>: τ̇↦{[ 1 : τ̇≥-0.5; 0 : τ̇<-0.5 ]. Equation <ref> allows the computation τ for whole regions of the image plane at once given a time sequence of labeled image segmentations, while τ̇ enables decisions to be made about the safeness of the agent's current state. §.§ Obtaining Soft Constraint ValuesWhereas hard constraint values are measurements of geometric properties and intended to prevent undesired physical interactions in the world, soft constraint values are intended only to bias how control actions are chosen. The potential transformation described in the next section guarantees that soft constraint values can never override hard constraints values, which allows soft constraint values to be arbitrarily chosen. In this report, soft constraint values are assume to be user-specified.The following two sections describe how hard and soft constraint values are transformed into the potential space. §.§ The Potential TransformationThe task of projecting sensor measurements and user-defined bias values is accomplished by a potential transformation that transforms pixel-wise measurements that have some semantic meaning into the unitless potential space. Two types of potential transformations are presented: a hard constraint transform and a soft constraint transform. Without loss of generality, assume for the remainder of this report only negative affinely extended potential fields.The hard constraint transform is intended to map values within a specific range to infinite potential values, and values outside the range to finite potential values. In this way sensor measurements corresponding to hard constraint violations are mapped into asymptotic regions, which ensures that information about them is preserved during subsumption. At time t, for a finite input pixel value , finite range [c,c], y-axis translation value t_y, and finite shape parameters α,β>0, the hard constraint transform C_h is given in Equation <ref> and illustrated in Figure <ref>. The time derivative of C_h is given in Equation <ref>. C_h(,[c,c])={[ t_y - α(c-)^-β : <c; -∞ : ∈[c,c]; t_y - α(-c)^-β : >c ]. d/dtC_h={[ αβ(c-)^-β-1d/dt: <c; 0: ∈[c,c]; αβ(-c)^-β-1d/dt: >c ]. Conversely, the soft constraint transform is intended to map all values into a given finite range. In this way all user-given bias values are prevented from becoming part of asymptotic regions, which ensures that hard constraint information is not corrupted. At time t, for a finite input value , finite range [c,c] with midpoint c_mid, x-axis translation value t_x, and finite shape parameters α,β>0, the soft constraint transform C_s is a parameterized logistic function given in Equation <ref> and illustrated in Figure <ref>. The time derivative is given in Equation <ref>. C_s(,[c,c])=c + c-c/(1+ e^-α(-t_x-c_mid))^1/β d/dtC_s=α(c-c)e^-α(-t_x-c_mid)/β(1+e^-α(-t_x-c_mid))^1+1/βd/dtI^t_mn The shape parameters in both Equations <ref> & <ref> allow potential field shape to be manipulated in problem-specific ways. The time derivatives in Equations <ref> & <ref> are useful in the situation that a time derivative of the input pixel value is given. For convenience, let Γ=⟨ t,c,c,α,β⟩ contain all parameters for computing a constraint transform.§ VISION-BASED SUBSUMPTION ARCHITECTUREThis section outlines a subsumption-based control architecture for ISP fields. Subsumption is implemented by formulating the general navigation problem as a guided collision avoidance problem, in which a global guidance controller subsumes a local collision avoidance controller. The navigation problem can be described as below: Navigation: Let A be a set of agents navigating along a 2D manifold and assume that collision is never inevitable in the initial system state. Assume each agent is equipped with cameras, and that each agent has knowledge of the physical dynamic properties of the environment and of other agents. Assume each agent actuates according to a unique decision process and that each agent may assume with certainty that other agents will prefer to avoid collision and to avoid causing collision. How can an agent A∈A navigate toward a goal while remaining collision free? Problem <ref> can be decomposed into a local collision avoidance problem, and a global guidance problem. The local problem is given below: Collision Avoidance: Assume an agent A navigating a workspace W receives some observation input O_t of W over time. Let A be the set of objects and agents that does not include A. For a distance metric μ, threshold ε>0, and a sequence of observations O_i,…,O_t, how can A estimate the rate of change of min_A_i∈Aμ(A,A_i) such that it can compute controls to maintain minμ>ε? The metric μ between any two points is taken as the τ measure between those points, thus, Problem <ref> makes the assumption that maintaining some τ>ε suffices to ensure collision avoidance, which is often a reasonable assumption <cit.>. This assumption implies several other assumptions, namely, that agents maintain controllability at all times, and that agents react within the τ horizon. These assumptions can be loosened in a probabilistically rigorous way by exploiting the Safety-Constrained Interference Minimization Principle <cit.>. Finally, the global guidance controller problem can be defined as: Guidance: For desired goal-directed control u^d, control space metric μ_c, and given a feasible control set 𝒰, choose a control u^⋆ such that:u^⋆=min_u∈𝒰μ_c(u^d,u)Problem <ref> can be solved any number of optimization techniques. In particular, if the solution algorithm to Problem <ref> produces convex sets, many efficient optimization routines become available. §.§ Example AlgorithmsAlgorithms <ref> & <ref> address Problems <ref> & <ref> explicitly with Problem <ref> being straightforward to solve given the input. This solution is intended as a sketch of a control system based on ISP fields, so some details are omitted. The biasing fields in Algorithm <ref> are derived from user-provided values. While Problem <ref> is formulated generally similar to a ground navigation scenario, ISP fields can applied to arbitrary environs so long as Algorithms <ref> & <ref> are modified accordingly.§ CONCLUSIONThis report presented Image Space Potential (ISP) fields, which are a general environment representation for vision-based navigation. ISP fields are constant space complexity with respect to the image, which is crucial for ensuring scalability and running time of algorithms. ISP fields also enable planning and control to occur in a space that is structurally similar to the sensor space, which means that sensor data does not need to go through what are often difficult and noisy projections into a structurally different planning and control space.ISP fields are intended to form the foundation of a vision-based subsumption control architecture, such as that described in the previous section. To enable this use, they allow arbitrary amounts of guidance information to be added into the representation while guaranteeing that hard constraint information will not be lost or corrupted in the process. Thiscapability makes ISP field representation particularly well suited to enabling machine learning to be applied to safety critical applications, such as automated driving, where it is often impractical for machine learning alone to make such guarantees <cit.>.An implementation of the data structures and algorithms described in this report is being developed and maintained under open source license <cit.>. The implementation is expected to change and grow over time, so for any disparity between the implementation and this document, the implementation should be assumed to be authoritative.The implementation itself is developed under ROS <cit.>, and ISP field data structures and operations are implemented using OpenCV <cit.>, a highly optimized, industry standard computer vision library. IEEEtran | http://arxiv.org/abs/1709.09662v1 | {
"authors": [
"Jeffrey Kane Johnson"
],
"categories": [
"cs.RO"
],
"primary_category": "cs.RO",
"published": "20170926220253",
"title": "Image Space Potential Fields: Constant Size Environment Representation for Vision-based Subsumption Control Architectures"
} |
A family of binary inflation rules]Spectral analysis of a family of binary inflation rulesFakultät für Mathematik, Universität Bielefeld,Postfach 100131, 33501 Bielefeld, Germany {mbaake,cmanibo}@math.uni-bielefeld.deSchool of Mathematics and Statistics, The Open University,Walton Hall,Milton Keynes MK7 6AA, United [email protected] family of primitive binary substitutions defined by 1 ↦ 0 ↦ 0 1^m withis investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation (m=1), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous. [ Neil Mañibo Accepted . Received; in original form============================================empty§ INTRODUCTION Due to the general interest in substitutions with a multiplier that is a Pisot–Vijayaraghavan (PV) number, and the renewed interest in substitutions of constant length, other cases and classes have been a bit neglected. In particular, the analysis of non-PV inflations is clearly incomplete, although they should provide valuable insight into systems with singular continuous spectrum. This was highlighted in a recent example <cit.>, where the absence of absolutely continuous diffraction could be shown via estimates of certain Lyapunov exponents. The same method can also be used for substitutions of constant length <cit.> to re-derive results that are known from <cit.> in an independent way.Here, we extend these methods to an entire family of binary inflation rules, namely those derived from the substitutions 1 ↦ 0 ↦ 0 1^m with m∈ by using prototiles of natural length.The inflations are not of constant length, and all have singular spectrum (either pure point or mainly singular continuous), as previously announced in <cit.>. More precisely, we prove the following result, the concepts and details of which are explained as we go along. Consider the primitive, binary inflation rule 1 ↦ 0 ↦ 01^m with m∈, and let γ^_u be the diffraction measure of the corresponding Delone dynamical system that emerges from the left endpoints of the tilings with two intervals of natural length, where u = (u^_0, u^_1) with u^_0 u^_1 0 are arbitrary complex weights for the twotypes of points. Then, one has the following three cases. =2pt* For m=1, this is the well-known Fibonacci chain, which has pure point diffraction and, equivalently, pure point dynamical spectrum.* When m=ℓ (ℓ+1) with ℓ∈, the inflation multiplier is an integer, and the diffraction measure as well as the dynamical spectrum is once again pure point.* In all remaining cases, the inflation tiling is of non-PV type, and the diffraction measure, apart from the trivial peak at 0, is purely singular continuous.The article is organised as follows. We begin with the introduction of our family of inflations and their properties in Section <ref>, where the cases (1) and (2) of Theorem <ref> will already follow, and then discuss the displacement structure and its consequence on the pair correlations in the form of exact renormalisation relations in Section <ref>. This has strong implications on the autocorrelation and diffraction measures (Section <ref>), which are then further analysed via Lyapunov exponents in Section <ref>.The main result here is the absence of absolutely continuous diffraction for all members of our family of inflation systems. One ingredient is the logarithmic Mahler measure of a derived family of polynomials, which we analyse a little further in an Appendix. § SETTING AND GENERAL RESULTS Consider the family of primitive substitution rules on the binary alphabet {0,1} given byϱ_m^:0↦ 01^m,1↦ 0,withm∈ℕ.Its substitution matrix is M_m=([ 1 1; m 0 ]) with eigenvalues λ_m^± =1/2(1±√(4m+1) ), which are the roots of x^2 - x -m=0. Whenever the context is clear, we will simply write λ instead of λ^+_m. Note that one has λ^-_m = -m/λ. For each m∈, there is a unique bi-infinite fixed point w of ϱ^2_m with legal seed 0|0 around the reference point (or origin), and the orbit closure of w under the shift action defines the discrete (or symbolic) hull _m. Then, (_m , ) is a topological dynamical system that is strictly ergodic by standard results; see <cit.> and references therein for background and further details.The Perron–Frobenius (PF) eigenvector of M_m, in frequency-normalised form, isv^_PF= (ν^_0 , ν^_1 )^T=1λ (1,λ -1)^T,where the ν^_i are the relative frequencies of the two letters in any element of the hull, _m. Next, (λ,1) is the corresponding left eigenvector, which gives the interval lengths for the corresponding geometric inflation rule.Up to scale, this is the unique choice to obtain a self-similar inflation tiling of the line from ϱ^_m; see <cit.> for background.This version, where 0 and 1 stand for intervals of length λ and 1, is convenient because [λ] is then the natural -module to work with. The tiling hull _ m emerges from the orbit closure of the tiling defined by w, now under the continuous translation action of . The topological dynamical system (_ m , ) is again strictly ergodic, which can be proved by a suspension argument <cit.>. The unique invariant probability measure on _ m is the well-known patch frequency measure of the inflation rule.§.§ Cases with pure point spectrum Let us begin with the analysis of the case m=1, which defines the Fibonacci chain. Here, the following result is standard <cit.>. For m=1, our substitution defines the well-known Fibonacci chain or tiling system. Both dynamical systems, (^_1, ) and (^_ 1, ), are known to have pure point diffraction and dynamical spectrum.Let us thus analyse the systems for m>1, where we begin with an easy observation. The inflation multiplier λ = λ^+_m is an integer if and only if m=ℓ(ℓ+1) with ℓ∈ℕ, where λ^+_m=ℓ+1 and λ^-_m=-ℓ.In all remaining cases with m>1, the inflation multiplier fails to be a PV number.Let us take a closer look at the cases where λ is an integer, where we employ the concept of mutual local derivability (MLD) from <cit.>. This can be viewed as the natural extension of conjugacy via sliding block maps from symbolic dynamics to tiling dynamics. When m=ℓ(ℓ+1) with ℓ∈ℕ, the inflation tiling hull ^_ m defined by ϱ^_m is MLD with another inflation tiling hull that is generated by the binary constant length substitution ϱ̃^_m, defined by a ↦ a b^ℓ, b ↦ a^ℓ+1, under the identifications a =0 and b = 1^ℓ+1.Consequently, for any such m, the dynamical system(^_ m, ) has pure point spectrum, both in the dynamical and in the diffraction sense.The claim can be proved by comparing the two-sided fixed point w of ϱ^ 2_m, with seed 0|0, with that of ϱ̃^ 2_m, with matching seed a|a, called u say, where we employ the tiling picture and assume that the letters a and b both stand for intervals of length λ=ℓ+1. Clearly, the local mapping defined by a↦ 0 and b↦ 1^ℓ+1 sends u to w. For the other direction, each 0 is mapped to a, while the symbol 1 in w occurs in blocks of length ℓ (ℓ+1), which are locally recognisable. Any such block is then replaced by b^ℓ, and this defines a local mapping that sends w to u. The transfer from the symbolic fixed points to the corresponding tilings is consistent, as the interval lengths match the geometric constraints.The extension to the entire hulls is standard. The constant length substitution ϱ̃^_m has a coincidence in the first position, and thus defines a discrete dynamical system with pure point dynamical spectrum by Dekking's theorem <cit.>. Due to the constant length nature, ^_ m emerges from ^_m by a simple suspension with a constant roof function <cit.>, so that the dynamical spectrum of (^_ m, ), andhence that of (^_ m, ) by conjugacy, is still pure point.By the equivalence theorem between dynamical and diffraction spectra <cit.> in the pure point case, the last claim is clear. Let us mention in passing that all eigenfunctions are continuous for primitive inflation rules <cit.>.For the systems considered in this paper, all eigenvalues are thus topological.So far, we have the following result. Consider the inflation tiling, with prototiles of natural length, defined by ϱ^_m.For m=1 and m=ℓ (ℓ+1) with ℓ∈ℕ, the tiling has pure point diffraction, which can be calculated with the projection method.[For m 1, this works analogously to the case of the period doublingsequence; compare<cit.>.]The corresponding tiling dynamical system (_ m, ) is strictly ergodic and has pure point dynamical spectrum.§.§ Non-PV cases In all remaining cases, meaning those that are not covered by Theorem <ref>, the PF eigenvalue is irrational, but fails to be a PV number. None of the corresponding tilings can have non-trivial point spectrum <cit.>.In particular, the only Bragg peak in the diffraction measure is the trivial one at k=0. If we consider Dirac combs with point measures at the left endpoints of the intervals, which leads to the Dirac comb of Eq. (<ref>) below, this Bragg peak has intensityI^_0=| ν^_0 u^_0 + ν^_1 u^_1|^2 ,where u^_0, u^_1 are the (possibly complex) weights for the two types of points, and ν^_0, ν^_1 are the frequencies from Eq. (<ref>); compare <cit.>. This gives the first part of the following result, the full proof of which willlater follow from Lemma <ref> andProposition <ref>. For all cases of our inflation family that remain afterTheorem <ref>, the pure point part of the diffraction consists of the trivial Bragg peak at 0, with intensity I^_0 according to Eq. (<ref>), while the remainder of the diffraction is purely singular continuous. The first example in our family with continuous spectral component is m=3, where the eigenvalues are 1/2(1 ±√(13) ). This case was studied in detail in <cit.>, where also general methods were developed that can be used for the entire family, as we shall demonstrate below. §.§ Some notation To continue, we need various standard results from the theory of unbounded (but translation bounded) measures on , for instance as summarised in <cit.>. In particular, we use δ_x to denote the normalised Dirac measure at x and δ^_S = ∑_x∈ S^δ_x for the Dirac comb of a discrete point set S. A translation bounded measure μ is simultaneously considered as a regular Borel measure (then evaluated on bounded Borel sets) and as a Radon measure (hence as a linear functional on C_𝖼 (), the space of continuous functions with compact support), which is justified by the general Riesz–Markov representation theorem; compare <cit.>.For a continuous function g, the measure g.μ is defined by μ∘ g^-1 as a Borel measure, while g(.) μ stands for the measure that is absolutely continuous relative to μ with Radon–Nikodym density g. The function g is specified by g (x) = g(-x), which extends to Radon measures by μ (g) = μ (g ); for further details, we refer to <cit.>. § DISPLACEMENT STRUCTURE AND PAIRCORRELATIONS Let m∈ be arbitrary, but fixed, which will be suppressed in our notation from now on whenever reasonable. We will now review some properties of the inflation structure and how this can be used to get exact renormalisation relations for the pair correlation functions.§.§ Displacements and their algebraic structure First, we quantify the relative displacements of tiles in the inflation process by the set-valued matrixT =[ { 0 } { 0 }; S ∅ ],withS := {λ, λ +1, … , λ + m-1} ,where T = (T^_ij)^_0 ⩽ i,j ⩽ 1 with T^_ij being the set of relative positions of tiles (intervals) of type i in supertiles of type j. Here and below, the positions are always determined between the left endpoints of the tiles as markers.From the measure matrix δ^_T := (δ^_T_ij)_0 ⩽ i,j ⩽ 1, with . denoting the standard Fourier transform as used in <cit.>, one obtains the Fourier matrix B of our inflation system asB (k) :=δ^_T (k)= D^_0 +p(k) D^_λwith the trigonometric polynomialp(k) = z^λ ( 1 + z + … + z^m-1) | _z = ^2 π kand digit matrices D_0 = ( [ 1 1; 0 0 ]) and D_λ = ( [ 0 0; 1 0 ]), which are the same matrices for all m ∈. The complex algebragenerated by them is the inflation displacement algebra (IDA) introduced in <cit.>.Invoking <cit.>, one has the following result. For any m∈, the IDAof the inflation defined by ϱ^_m, with intervals of natural length as prototiles, is the full matrix algebra, (2,).This is also the IDA for all powers of the inflation. The matrix function defined by B(k) is analytic in k, and either 1-periodic (whenever λ is an integer) or quasiperiodic with incommensurate base frequencies 1 and λ. The case m=1 is somewhat degenerate in this setting, as we shall explain in more detail later, in Section <ref>.In the genuinely quasiperiodic situation, via standard results from the theory of quasiperiodic functions, one has the representationB(k) =B̃ (x,y)|_x=λ k,y=kwith B̃ (x,y) = ( [11; p̃ (x,y)0 ]) and p̃ (x,y) =^2 π x (1 + z + … + z^m-1)|_z = ^2 π y .Here, both p̃ and B̃ are 1-periodic in both arguments. Our representation is chosen such that we have the correspondencek ↦λ k⟷(x,y) ↦ (x,y) M,where M=M_m is the substitution matrix of ϱ = ϱ^_m. Each such M defines a toral endomorphism on the 2-torus, ^2.§.§ Kronecker products Below, we also need the matrices A(k) = B (k) ⊗B(k) for k∈, which act on the space W:= ^2 ⊗^_^2, and the structure of the -algebragenerated by them. While the is irreducible by Fact <ref>,is not, because each of its elements commutes with the -linear mapping C: WW defined by x ⊗ y ↦y⊗x, where W is considered as an -vector space of dimension 8. Now, W = W_ +⊕ W_ - with W^_± := { w ∈ W : C(w) = ±w }, where ^_ (W_ +) = ^_ (W_ -) = 4 and W_ - =W_ +. These spaces are invariant under , and one has the following result. The -algebrasatisfies ^_ () = 16 and acts irreducibly on each of the four-dimensional invariant subspaces W_ + and W_ - from above.The argument is analogous to that in the proof of <cit.>.In particular, considering (4,) as an -algebra of dimension 32, the subalgebrais conjugate to (4, ) ⊂ (4,) via the conjugation (.) ↦ U (.) U^-1 with the unitary matrixU =1√(2)[ 1 - 0 0 0; 0 1 - 0; 0 - 1 0; 0 0 0 1 - ] ,where U (W^_∓) = 1±/√(2) ^4. One can check that [ U, A (0)] = 0. More generally, one hasA^_U (k) = UA (k)U^-1=[ 1 1 1 1; c(k)+s(k)s(k)c(k) 0; c(k)-s(k)c(k) -s(k) 0; c(k)^2 + s(k)^2 0 0 0 ]withc(k) = ∑_ℓ=0^m-1cos( 2 π (λ+ℓ) k) and s(k) = ∑_ℓ=0^m-1sin( 2 π (λ+ℓ) k).This gives c(k)^2 + s(k)^2 = | p(k) |^2 = ( ∑_ℓ=0^m-1cos((2ℓ+1-m)π k ) )^ 2and A^_U (0) = A (0) = M ⊗ M with the substitution matrix M=M_m.Observe that A(0)^2 is a strictly positive matrix, with determinant (M)^4 > 0. Now, consider A^(2)_U (k) := A^_U( k/λ) A^_U (k), which defines a smooth, real-valued matrix function with lim_k→ 0 A^(2)_U (k) = A (0)^2. Consequently, there is some ε = ε (m) > 0 such that A^(2)_U (k) is strictly positive, with positive determinant, for all | k |⩽ε. We can then state the following result, the proof of which is identical to that of <cit.>. Let k ∈ [0,ε] with the above choice of ε, and consider the iterationw^_n := A^(2)_U( kλ^2n-2) ⋯ A^(2)_U( kλ^2) A^(2)_U (k) w^_0for n⩾ 1 and any non-negative starting vector w^_00. Then, the vector w_n will be strictly positive for all n ∈ and, as n →∞, it will diverge with asymptotic growth c λ^4n w^_PF. Here, c is a constant that depends on w^_0 and k, while w^_PF = v^_PF⊗ v^_PF is the statistically normalised PF eigenvector of M⊗ M, with eigenvalue λ^2 and v^_PF as in Eq. (<ref>).As we shall see later, this growth behaviour will collide with a local integrability condition, and then help to simplify our spectral problem by a dimensional reduction.§.§ Pair correlations To introduce the pair correlation functions, let = _ m be the tiling hull introduced earlier. Any ∈ is built from two prototiles (of length λ>1 and 1, respectively).We now define the corresponding point setvia the left endpoints of the tiles in , so = ^(0) ∪̇ ^(1) with ^(i) denoting the left endpoints of type i. Clearly,andare MLD, as are their hulls; see <cit.> for background.By slight abuse of notation, we usefor both hulls, which meansthat we implicitly identify these two viewpoints.Any two elements ofare locally indistinguishable (LI), soconsists of a single LI class, see <cit.> or <cit.>, which has the following important consequence. For any i,j∈{0,1}, the difference set ^(i) - ^(j) is constant on the hull, which means that it does not depend on the choice of ∈.Given ∈, let ν^_ij (z) denote the (relative) frequency of occurrence of a point of type i (left) and one of type j (right) at distance z, where ν^_ij (-z) = ν^_ji (z).By the strict ergodicity of our system, any such frequency exists (and uniformly so), and is the same for all ∈. One can write the frequency as a limit,ν^_ij (z) = lim_r→∞( ^(i)_r∩ (^(j)_r - z) )/ (^_r)=1 ()lim_r→∞( ^(i)_r∩ (^(j)_r - z ) )/2r ,where ∈, with = ^(0) ∪̇ ^(1) as above. The lower index r indicates the intersection of a set with the interval [-r,r].Moreover, one hasν^_ij (z) > 0⟺z ∈^_ij := ^(j) - ^(i) ,and ^_ ij := ∑_z ∈_ijν^_ij (z)δ^_z defines a positive pure point measure onwith locally finite support. Note that ^_ii is also positive definite. We call the ν^_ij (z) the pair correlation coefficients and the ^_ ij the corresponding pair correlation measures of , where ^_ ij ({ z }) = ν^_ij (z). Our relative normalisation means that we have ν^_00 (0) + ν^_11 (0) = 1, so thatν^_00 (0) =ν^_0andν^_11 (0) =ν^_1are the relative tile (or letter) frequencies from Eq. (<ref>). The pair correlation coefficients ofsatisfy the exact renormalisation relationsν^_ij (z) =1λ∑_k,ℓ∑_r∈ T_ik∑_s∈ T_jℓν^_kℓ( z+r-sλ)for any i,j∈{0,1}, subject to the condition that ν^_mn (z) = 0 whenever z ∉^_mn.In terms of the measures ^_ ij, this amounts to the convolution identity=1λ( δ^_T*⊗δ^_T) * (f. ),where f is the dilation defined by x ↦λ x and *⊗ denotes the Kronecker convolution product,whilestands for the measure vector (^_00, ^_01, ^_ 10, ^_ 11)^T.For m=1, this was shown in <cit.>, while the case m=3 is treated in <cit.>. In general, the underlying observation is that, due the aperiodicity of ϱ and the ensuing local recognisability, each tile lies in a uniquesupertile, and the frequencies for the distance between tiles can uniquely be related to the frequencies of (generally different) supertile distances, after a change of scale. A simple computationthen gives the first relation (a more general version of which willappear in <cit.>).The second identity, in the form of measures, follows from the definition of the ^_ ij by a straightforward calculation; see <cit.> for details in the case m=3.§ AUTOCORRELATION AND DIFFRACTION As above, m∈ is arbitrary but fixed. For ∈, we consider the weighted Dirac combω^_u= ∑_x∈ u^_x δ^_x= u^_0 δ^_^(0)_ + u^_1 δ^_^(1)_ ,where u^_0 and u^_1 are the (possibly complex-valued) weights of the two types of points. The corresponding autocorrelation measure, or autocorrelation for short, is defined by the volume-averaged (or Eberlein) convolutionγ^_u=ω^_u⊛ω^_u= ()∑_iju^_i ^_ ij u^_j ,where we refer to <cit.> for the general setting and to <cit.> for the detailed calculations in the case m=3. Existence and uniqueness (with independence of ) are again a consequence of the strict (and hence in particular unique) ergodicity of our system.Since all the measures ^_ ij = (δ^_ -^(i)⊛δ^_^(j) )/ () are well-defined Eberlein convolutions of translation bounded measures and hence Fourier transformable by <cit.>, we also have the relationγ^_u= () ∑_iju^_i ^_ ij u^_jafter Fourier transform, where γ^_u is a positive measure, for any u ∈^2. Note that each ^_ ij is a positive definite measure on ,and also positive for i=j. Since ^_ ij = ^_ ijby definition, one has^_ ij=^_ ij=^_ ji .This, in combination with Eq. (<ref>), implies the following property. For any bounded Borel set ⊂, the complex matrix ( ^_ ij ())_0 ⩽ i,j ⩽ 1 is Hermitian and positive semi-definite.Note that, since ^_00 and ^_ 11 are positive measures on , the positive semi-definiteness of the matrix ( ^_ ij ()) is simply equivalent to the determinant condition ( ^_ ij ()) ⩾ 0. As a counterpart to Proposition <ref>, withdenoting the vector of Fourier transforms of the pair correlation measures, we get the following result. Under Fourier transform, the second identity of Proposition <ref> turns into the relation=1λ^2 A (.) ·( f^-1 . ) ,where A(k) = B(k) ⊗B(k) with the Fourier matrix B(k) from Eq. (<ref>) and f(x) = λ x. §.§ Pure point part By <cit.>, the identity from Proposition <ref> must hold for each spectral component ofseparately. We write the pure point part as( )_𝗉𝗉= ∑_k∈ K (k)δ^_kwith the intensity vector (k) =({ k }) and K the support of the pure point part, which is (at most) a countable set. Without loss of generality, we may assume that λ K ⊆ K, possibly after enlarging K appropriately. Inserting this into the above identity, one obtains(k) =1λ^2A(k)(λ k).In particular, this gives A(0)(0) = λ^2 (0), which means^_ ij ({ 0 }) = (0) =ν^_i ν^_j=(^(i)) (^(j))/(())^2with the relative frequencies ν^_i from Eq. (<ref>).§.§ Conditions on absolutely continuous part Likewise, if we represent ()^_𝖺𝖼 by the vector h of its Radon–Nikodym densities relative to Lebesgue measure, one obtains <cit.>h(k) =1λ A(k) h(λ k),which holds for a.e. k∈. Here, the different exponent for the prefactor in comparison to Eq. (<ref>) is the crucial point to observe and harvest. Since (B(k)) = - p(k) = 0 holds if and only if k∈ Z_m := 1/m∖, the matrix A(k) is invertible for all k ∉ Z_m, hence for a.e. k∈.For such k, we also haveh (λ k) =λ A^-1 (k) h (k),which is the outward-going counterpart to Eq. (<ref>).If we interpret the vector h as a matrix (h^_ij)^_0 ⩽ i,j ⩽ 1, the iterations from Eqs. (<ref>) and (<ref>) can also be written as( h^_ij(kλ))=λ^-1 B (kλ) ( h^_ij (k) )B^†(kλ)and ( h^_ij (λ k))=λ B^-1 (k) ( h^_ij (k) )( B^†)^-1 (k),where B^† denotes the Hermitian adjoint of B. This suggests a suitable decomposition of h. For a.e. k∈, the Radon–Nikodym matrix ( h^_ij (k)) is Hermitian and positive semi-definite. Moreover, it is of rank at most 1.From Fact <ref>, we know that, given any bounded Borel set , the complex matrix ( ^_ij () ) is Hermitian and positive semi-definite, which also holds for the absolutely continuous part of .By standard arguments, this implies the first claim.For a.e. k∈, the Radon–Nikodym matrix is then of the form H=( [a b+ c; b -cd ]) with a, b, c, d ∈, a, d ⩾ 0 and ad ⩾ (b^2 + c^2)⩾ 0. When (H) = 0, the rank of H is at most 1. On the other hand, when (H) >0, the rank of H is 2, with ad > 0.In general, we have a unique decomposition asH =[ a' b+ c; b -cd ]+ a”[ 1 0; 0 0 ]with a = a' + a” such that each matrix on the right-hand side is positive semi-definite (hence a' ⩾ 0 and a”⩾ 0) and of rank at most 1 (which means a'd = b^2 + c^2).It suffices to prove our second claim for a.e.k∈[ ε/λ, ε] for some ε > 0, as the two iterations in Eq. (<ref>) transport the property to all k>0, and then to k<0 via h^_ij (-k) = h^_ji (k). Here, we choose ε as in Proposition <ref>. Whenever h^_11 (k) = 0, we have ( h^_ij (k) ) = 0 and the Radon–Nikodym matrix has rank at most 1. Otherwise, we define h^''_00 (k) = ( h^_ij (k) )/ h^_11 (k) and h^'_00 (k) = h^_00 (k) - h^''_00 (k), which are measurable functions and achieve the decomposition explained previously.To continue, we switch to the vector notation from Eqs. (<ref>) and (<ref>), and observe that the matrix ( [ 1 0; 0 0 ]) corresponds to the vector w^_0 = (1,0,0,0)^T ∈ W_ + in the notation of Section <ref>. Now, Uw^_0 = 1-/√(2)w^_0, and the iteration of w^_0 under A^_U (k),w^_n= A^_U( kλ^n-1) ⋯ A^_U( kλ)A^_U (k) w^_0 , grows asymptotically as c λ^2n w^_PF by an application of Proposition <ref>, where c>0 depends on k. Observing that U^-1 w^_PF = 1+/√(2)w^_PF and applying a standard argument on the basis of Lusin's theorem as in the proof of <cit.>, we see that h^''_00 (k) > 0 would behave proportional to k^-1 as k↘0, which is impossible for a locally integrable function.Since also h^'_00 (k) ⩾ 0, there cannot be any cancellation with the other term of our decomposition under the inward iteration, and we must conclude that h^''_00 (k) =0 for a.e.k ∈[ ε/λ, ε], and hence for a.e. k∈ as argued above.This implies our claim. §.§ Dimensional reduction and Lyapunov exponents If H∈ (2,) is Hermitian, positive semi-definite and of rank at most 1, there are two complex numbers, v^_0 and v^_1 say, such that H^_ij = v^_i v^_j holds for i,j ∈{ 0,1}; compare <cit.> for more. The main consequence of Lemma <ref> now is that it suffices to consider a vectorv(k) = ( v^_0 (k) , v^_1 (k) )^T of functions from L^2_loc () under the simpler iterationsv ( kλ) =1√(λ)B ( kλ)v (k)andv(λ k) =√(λ)B^-1 (k) v (k),the latter for k ∉ Z_m. In particular, one hasv(λ^n k) =λ^n/2 B^-1 (λ^n-1 k) ⋯ B^-1 (k) v (k),which holds for a.e. k∈∖⋃_ℓ=0^n-1λ^-ℓ Z_m, and thus still for a.e. k∈.There are at most two Lyapunov exponents for this iteration, which agree with the extremal exponents <cit.> defined byχ^_max (k)=log√(λ)+lim sup_n→∞1nlog B^-1 (λ^n-1 k) ⋯ B^-1 (λ k)B^-1 (k) and χ^_min (k)=log√(λ)-lim sup_n→∞1nlog B(k)B(λ k) ⋯ B(λ^n-1 k),where the term log√(λ) emerges from the prefactor on the right-hand side of Eq. (<ref>). Our main concern will be the minimal exponent, for the following reason <cit.>. If χ^_min (k) ⩾ c > 0 holds for a.e. k in a small interval, the diffraction measure γ^_u is a singular measure, for any non-trivial choice of the weights u^_0 and u^_1, which means u^_1, u^_2∈ with u^_1 u^_2 0.§ ANALYSIS VIA LYAPUNOV EXPONENTS Let m∈ be fixed, and λ = λ^+_m as before. Consider the matrix cocycleB^(n) (k) := B(k)B(λ k) ⋯ B(λ^n-1 k), which is motivated by Eq. (<ref>). Note that B^(n) (k) is invertible for k ∉⋃_ℓ=0^n-1λ^-ℓ Z_m.When m=1, one has Z^_1 = ∅ and | (B(k)) | ≡ 1, which makes this case considerably simpler. In general, one has the following result.For a.e. k∈, one haslim_n→∞ 1/nlog|(B^(n) (k) ) |=0.For m=1, one has | ( B^(n) (k) ) | ≡ 1, and the claim trivially holds for all k∈.When m⩾ 2, we invoke Sobol's theorem, as outlined in <cit.>; see <cit.> for a detailed exposition. Clearly, ( B (k)) is a Bohr almost periodic function, but it has zeros for k∈ Z_m. Consequently, log| ( B(k)) | cannot be Bohr almost periodic, because it has singularities at these points. Nevertheless, this function is locally Lebesgue-integrable on , and is continuous on ∖ Z_m, hence locally Riemann-integrable on the complement of Z_m + (-ε, ε) for any ε>0.Then, Sobol's theorem <cit.> (in the periodic case where λ is an integer) or its extension to almost periodic functions <cit.> can be applied as follows.First, recall that the sequence (λ^n k)^_n∈ is uniformly distributed modulo 1 for a.e. k∈. Next, one needs theproperty that Z_m is a Delone set and that, for any fixed ε > 0, and then for a.e. k∈, the inequalitydist (λ^n-1 k, Z_m)⩾ 1n^1+εholds for almost all n∈ (meaning for all except at most finitely many); see <cit.>. Then, again for any fixed ε > 0, it follows from <cit.> that the discrepancy of (λ^n k)^_n∈, for a.e. k∈, is^_N=((log(N))^3/2 +ε/√(N)) asN →∞ . Putting this together, we can apply <cit.> which tells us that the Birkhoff-type averages of the function log| ( B(.) ) |, for a.e. k∈, converge to the mean of this function (see Eq. (<ref>)below for a definition), which gives1nlog |( B^(n) (k) ) | = 1n∑_ℓ=0^n-1log| ( B ( λ^ℓ k) ) |∫_0^1log| ( B(t) ) |t =∫_0^1log| 1 + z + … + z^m-1|_z=^2 π t t = 0,where the last step follows via Jensen's formula from complex analysis (see <cit.> for a formulation that fits our situation) because the polynomial 1+z+… +z^m-1 either equals 1 (when m=1) or has zeros only on the unit circle.When m=ℓ (ℓ+1) with ℓ∈, where λ = ℓ + 1, the result of Proposition <ref> easily follows from Birkhoff's ergodic theorem, because ( B(k)) is then a 1-periodic, locally Lebesgue-integrable function that gets averaged along orbits of the dynamical system defined by x ↦λ x 1 on the 1-torus, . This approach, however, does not extend to the other values of m with m>1, because the sequence (λ^n k)^_n∈, taken modulo 1, is then no longer an orbit of x ↦λ x 1 on ; compare <cit.>.We can now relate the two extremal exponents from Eq. (<ref>) as follows. For a.e. k∈, one has χ^_max (k) + χ^_min(k) = log(λ).Recall that, for any invertible matrix B, one has B^-1 = 1/ (B)B^𝖺𝖽, where B^𝖺𝖽 is the (classical) adjoint of B.The adjoint satisfies (A B)^𝖺𝖽 = B^𝖺𝖽A^𝖺𝖽.Now, in the formula for the extremal exponents, we are free to choose any matrix norm, as this does not affect the limit.For 2× 2-matrices, one has B^𝖺𝖽^_F =B ^_F, where .^_F denotes the Frobenius norm. Our claim now follows from Proposition <ref> after a simple calculation. In view of Lemma <ref>, we defineχ^B (k) :=lim sup_n→∞1nlog B(k)B(λ k) ⋯ B(λ^n-1 k),so that χ^_max (k) = log√(λ) + χ^B (k) and χ^_min (k) = log√(λ) - χ^B (k) holds for a.e. k∈, together with χ^_max (k) ⩾χ^_min (k). We can thus simply analyse χ^B from now on, which clearly is a non-negative function.§.§ Arguments in common Below, we need the mean of a function. If f is a Bohr almost periodic function on(and thus in particular uniformly continuous and bounded), its mean, (f), is defined by(f) = lim_T→∞1T∫_x^x+T f(t)t,where x∈ is arbitrary. By standard results, the mean of such an f exists for all x∈, is independent of x, and the convergence is actually uniform in x. This is also true when f is almost periodic in the sense of Stepanov, which in particular covers some of our later situations; see <cit.> for details. When f is a periodic function, with fundamental period T, the mean is simply given by (f) = 1/T∫_0^T f(t)t.Observing that | p(.)|^2 with p from Eq. (<ref>) is 1-periodic (while p itself need not be), the simplest sufficient criterion for the positivity of all Lyapunov exponents is given bylog (λ) >( log B(.) ^2_F)= ∫_0^1log( 2 + | p (t)|^2)t= ∫_0^1log| q (z) |_z=^2 π t t= (q)where q is the polynomial q (z) = 2 z^m-1 + (1+z+…+z^m-1)^2and the validity of z = z^-1 on the unit circle was used. Here, (q) denotes the logarithmic Mahler measure of q; see <cit.> for background.The integral can now once again be calculated by means of Jensen's formula; see the Appendix for some details.The comparison between log (λ) and (q) is illustrated in Figure <ref>. More generally, one has the following result. For any m ⩾ 18 and then a.e. k∈, all Lyapunov exponents of the outward iteration (<ref>) are strictly positive and bounded away from 0.Since χ^_max (k) ⩾χ^_min (k), we need to show that log√(λ) -c ⩾χ^B (k)holds for some c>0 and a.e. k∈.A sufficient criterion for this is the inequality from Eq. (<ref>). Since (q) is bounded, see Lemma <ref> from the Appendix, it is clear that this inequality holds for all sufficiently large m∈. By Lemma <ref>, this is so for all m⩾ 40, and the slightly better estimate from Remark <ref> improves this to all m⩾ 23.In any case, a (precise) numerical investigation of the remaining cases shows that that our claim is indeed true for all m ⩾ 18; compare Figure <ref>. In order to establish our goal for the remaining values of m, we needto determine a suitable N = N(m) such thatlog (λ) >1N ( log B^(N) (.) ^2_F) =N^-1∫_0^1log B^(N) (k) ^2_F k,if λ∈,N^-1∫_[0,1]^2logB̃^(N) (x,y) ^2_F xy,otherwise.When λ is not an integer, B^(N) (.)^2_F is generally not a periodic, but a quasiperiodic function. In this case, we use the representation as a section through a doubly 1-periodic function according to Eq. (<ref>), which permits the simple expression for the mean in (<ref>). The latter can now be calculated numerically with good precision, and without ambiguity. Note that the choice of the Frobenius norm .^_F does not give the best bounds, but is rather convenient otherwise.The result is given in Table <ref>, with minimal values for N(m). Consequently, we can sharpen Lemma <ref> and complete the proof of Theorem <ref> as follows. For any m∈ and then a.e. k∈, all Lyapunov exponents of the outward iteration (<ref>) are strictly positive and bounded away from 0.§.§ The Fibonacci case Here, the leading eigenvalue is λ = τ, the golden ratio, which is a PV number. Essentially as a consequence of <cit.>, which need some modification and extension to be applicable here, the extremal Lyapunov exponents exist as limits, for a.e. k ∈. Let us look into this in more detail, in a slightly different way that provides an independent derivation of this property.Here, we haveB (k) =[ 1 1; ^2 πτ k 0 ] ,which is τ^-1-periodic. However, this observation doesnot help because alreadyB^(2) (k) = B (k) B (τ k) =[ 1 + ^2 π (τ + 1) k1; ^2 π i τ k ^2 π i τ k ]is genuinely quasiperiodic, with fundamental frequencies τ and 1. In line with our general approach from Eqs. (<ref>)and (<ref>), we now define B̃^(n+1) (x,y) = B̃ (x,y)B̃^(n)( (x,y) M ) with B̃^(1) (x,y) =B̃ (x,y) =[11; ^2 π x0 ]andM =[ 1 1; 1 0 ].Then, B̃^(n) (x,y) defines a matrix cocycle over the dynamical system defined on ^2 by the toral automorphism (x,y) ↦ (x,y) M 1. By Oseledec's theorem, see <cit.>, the Lyapunov exponents for B̃^(n) exist as limits, for a.e. (x,y)∈^2, and are constant.However, what we really need is the existence of the Lyapunov exponents forB^(n) (k) =B̃^(n) (x,y)|^_x = τ k , y = kfor a.e. k∈, which is a statement along the line (τ, 1), respectively its wrap-up on ^2. This is the subspace defined by the left PF eigenvector of M. The problem here is that this defines a null set for Lebesgue measure on ^2, so that the previous argument does not immediately imply what we need. However, for Lebesgue-a.e. starting point on the line (τ, 1), the iteration sequence on this line, taken modulo 1, is also equidistributed in ^2, by standard arguments around Weyl's lemma. This allows for a relation between the result on the line and that on ^2 as follows.Any initial condition for the cocycle B̃^(n) is following an orbit of the toral automorphism that converges, exponentially fast, towards an orbit on this special subspace.This is a consequence of the PV property of τ and the fact that the second eigenvalue of M is 1-τ≈ - 0.618, the algebraic conjugate of τ. Assume that the Lyapunov exponents for B^(n) fail to exist as limits for a subset of (τ, 1) of positive measure. Then, this must also be true of the exponents for B̃^(n) for all initial conditions that lead to orbits which approach the failing orbits on (τ, 1). By standard arguments, these initial conditions would constitute a set of positive measure, now with respect to Lebesgue measure on ^2, in contradiction to our previous finding. We thus have the following result. For m=1 and a.e. k∈, the Lyapunov exponents from Eq. (<ref>) exist as limits, and are constant. One can check numerically that χ^B (k) ≈ 0.16 (3) in this case, and some further analysis with a Furstenberg-type representation should result in a more reliable value.§.§ Integer inflation multipliers In these cases, we know the absence of any continuous spectral components already from Proposition <ref>.Moreover,in view of Lemma <ref>, our treatment in Section <ref> alsoconfirms the absence of absolutely continuous diffraction via the Lyapunov exponents.Here, the exponents also exist as limits for a.e. k∈, by an application of Oseledec's theorem to the matrix cocycle, viewed over the dynamical system defined onby x ↦λ x 1 with λ = ℓ + 1 according to Fact <ref>. When m = ℓ (ℓ+1) for ℓ∈, hence λ = ℓ+1, the Lyapunov exponents from Eq. (<ref>) exist as limits, for a.e. k∈, and are constant. As another way to look at the problem, let us add a quick analysis of the constant length substitutionϱ̃^_m :a ↦ a b^ℓ b ↦ a^ℓ+1with ℓ∈, which defines a hull that is MLD to the one defined via ϱ^_m for m=ℓ(ℓ+1) by Proposition <ref>, so the spectral type of both systems must be the same.The displacement matrix isT =[ 0 { 0, 1, 2, … , ℓ}; { 1,2, … , ℓ} ∅ ] ,which results in the Fourier matrixB(k) =[ 1ψ^_ℓ (z); zψ^_ℓ-1 (z) 0 ]_z=^2 π kwith ψ^_ℓ (z) := 1 + z + … + z^ℓ. One gets an analogue to Eq. (<ref>) in the form( log B(.) ^2_F)=∫_0^1log| s(z)/(z-1)^2|^_z=^2 π t t = (s),with s(z) = z^2ℓ+2 + z^2ℓ+1 + z^ℓ+2 - 6 z^ℓ+1 +z^ℓ + z + 1. As we explain in more detail in the Appendix, we used ( (z-1)^2 ) = 0 in an intermediate step.One could now repeat the general analysis of Section <ref> in this case, with an outcome of a similar kind. However, there is a more efficient way as follows. First, observe that we now have ( B(k)) = - zψ^_ℓ-1 (z)ψ^_ℓ (z) with z=^2 π k, which is a product of a monic polynomial (in the variable z) with two cyclotomic ones.Consequently, the corresponding logarithmic Mahler measures vanish, and we once again getlim_n→∞1nlog| (B^(n) (k) ) | =∫_0^1log|( B(t)) |t =(zψ^_ℓ-1 (z)ψ^_ℓ (z) ) = 0,for a.e. k∈, as in Proposition <ref>.Next, observe that v = (1,1) is a common left eigenvector of B (k) for all k∈, with eigenvalue ψ^_ℓ (z) for z=^2 π k. This giveslim_n→∞1nlog vB^(n) (k) = (ψ^_ℓ)= 0for a.e. k∈. In view of Eq. (<ref>), this implies that both exponents of the cocycle B^(n) vanish in this case. Now, we still have χ^_min + χ^_max = log (λ) as in Lemma <ref>, where λ = ℓ+1, despite the fact that we now consider the substitution from Eq. (<ref>).With this derivation, we have actually shown the following result. The extremal Lyapunov exponents for the outward iteration defined by the constant-length substitution (<ref>) are equal, and given by χ^_min = χ^_max =log√(ℓ + 1).Also this approach implies the diffraction spectrum to be singular. However, as before, this is only a consistency check because Dekking's criterion (see the proof of Proposition <ref>) already gives a stronger result, namely the pure point nature of the spectrum.§ APPENDIX Here, we consider some logarithmic Mahler measures, in particular (q) for the polynomial q from Eq. (<ref>) with m∈. The polynomials q seem to be irreducible over , though we have no general proof for this observation.As follows from a simple calculation, (q) takes the values log (3) for m=1 and log( 2 + √(3) ) for m=2. It is known that one must have (q) = log (ξ) where ξ is a Perron number. A little experimentation shows that ξ is a Salem number for m=3, namely the largest root of z^4 - 3 z^3 - 4 z^2 - 3 z + 1, and a Pisot number for m=4, this time the largest root of z^4 - 4 z^3 - 2 z^2 + 2 z + 1. For m=5, one finds that ξ is the largest root of z^8 - 6 z^7 + 7 z^6 - 3 z^4 + 7 z^2 - 6 z + 1, which is genuinely Perron, as the second largest root of this irreducible polynomial, with approximate value 1.354 > 1, lies outside the unit circle. It would be interesting to know moreabout the numbers that show up here. More generally, expressing | p (t)|^2 in Eq. (<ref>) as (sin (m π t)/sin (π t))^2, one has(q) = ∫_0^1log( 2 + (sin (m π t)/sin (π t))^ 2 )t.Since sin(m π t)^2 ⩽ 1, one gets a simple upper bound as(q) ⩽∫_0^1log1+2 sin(π t)^2/sin (π t)^2 t=log (2) + ∫_0^1log2 - cos (2 π t)/1 - cos (2 π t) t =log (2) + (z^2 - 4 z + 1 ) - ( (z-1)^2 ) =log( 4 + 2 √(3) ) ≈2.010,where the logarithmic Mahler measures of the quadratic polynomials were evaluated via Jensen's formula again.This shows that (q) is bounded for our family of polynomials.A slightly better bound can be obtained as follows.For any m∈, the logarithmic Mahler measure of thepolynomial q from Eq. (<ref>) satisfiesthe inequality(q)<log√(46)≈1.914321.Here, we employ an argument from <cit.> that was also used, in a similar context, in <cit.>. By a simple geometric series calculation, one finds that q (z) = r (z)/(z-1)^2 withr (z) = z^2m + 2 z^m+1 - 6 z^m + 2 z^m-1 + 1 = ∑_ℓ=0^2m c^_ℓz^ℓ.Consequently, we have (q) =(r) - ( (z-1)^2) =(r).Let (r)= exp ( (r)) be the (ordinary)Mahler measure of r;compare <cit.>. By thestrict convexityof the exponential functionand Jensen's inequality,see <cit.> for a suitable formulation, one finds(r) < ∫_0^1| r (z) |_z=^2 π t t = r ^_1 ⩽r ^_2 ,where r = r(t) is considered as a trigonometric polynomial on(with the usual 1-periodic extension to ). In fact, since r is not a monomial, we also have r ^_1 <r ^_2. Assume that m⩾ 2, so that the exponents of r (z) in Eq. (<ref>) are distinct. Consequently, by Parseval's equation, we may conclude thatr ^2_2= ∑_ℓ=0^2m| c^_ℓ|^2= 46,so that (r) < √(46), independently of m. This inequality trivially also holds for m=1, and we get (q) =(r) < log√(46) for all m∈ as claimed. With this bound, one has log (λ) >(q) for all m⩾ 40, where λ = λ^+_m as before. An even better bound can be obtained from Eq. (<ref>) by observing that, as t varies a little, sin (m π t)^2 oscillates quickly when m is large (with mean 1/2), while (sin (π t))^2 remains roughly constant.Under the integral, one can then replace (sin (m π t)/sin (π t))^2 by 1/2(sin (π t))^-2, which still gives an upper bound for (q) because ^2/ t^2log (t) < 0 on _+. Now,(q) ⩽∫_0^1log3 - 2 cos (2 π t)/1 - cos (2 π t) t =(z^2 - 3 z + 1 ) + log (2) =log( 3 + √(5) ) ≈1.655571,which is smaller than log (λ), where λ = λ^+_m as above, for all m ⩾ 23. The values (q), as a function of m∈, seem to be increasing, so that lim_m→∞ (q) would be the optimal upper bound. The limit exists because (q) =(r), and the polynomial r satisfies r (z) = r̃ (z, z^m) withr̃(z,w) = -w (6 - 2 ( z +z^-1) - ( w + w^-1) ) .By a classic approximation theorem for two-dimensional Mahler measures, see <cit.>, one has lim_m→∞( r̃ (z, z^m) ) = ( r̃ (z,w)), where(r̃ )= ∫_^2log( 6 - 2 cos (2 π t_1) - 4 cos (2 π t_2) )t_1t_2= 2 ∫_0^1arsinh( √(2) sin (π t_2) )t_2≈1.550675.So, when (q) is an increasing function (which we did not prove), we immediately get the estimate log (λ) >(q) for all m⩾ 18. The polynomial s from Section <ref> can be analysed in a completely analogous way. Here, one has s (z) = - z^ℓ +1( 6 - ( z + z^-1) - ( w + w^-1) - ( zw + (zw)^-1) ), and the approximation theorem results inlim_ℓ→∞ (s) = ∫_^2log( 6 - 2 cos (2 π t_1) - 2 cos (2 π t_2)- 2 cos (2 π (t_1 + t_2)) )t_1t_2≈1.615.Moreover, various other properties are similar to those of the polynomial q from above. § ACKNOWLEDGEMENTS It is our pleasure to thank Michael Coons, David Damanik, Natalie P. Frank, Franz Gähler, Andrew Hubery, E. Arthur (Robbie) Robinson and Boris Solomyak for discussions.This work was supported by the German Research Foundation (DFG), within the CRC 1283.99BFGR Baake M, Frank N P, Grimm U and Robinson E A, Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction, preprint .BG15 Baake M and Gähler F, Pair correlations of aperiodic inflation rules via renormalisation:Some interesting examples, Topol.& Appl. 205 (2016) 4–27;.BGM Baake M, Gähler F and Mañibo N, Renormalisation of pair correlation measures for primitiveinflation rules and absence of absolutely continuous diffraction, in preparation. TAO Baake M and Grimm U, Aperiodic Order. Vol. 1: A Mathematical Invitation, Cambridge University Press, Cambridge (2013).BG-conf Baake M and Grimm U, Diffraction of a binary non-Pisot inflation tiling, J. Phys. Conf. Ser. 809 (2017) 012026 (4pp); .BHL Baake M, Haynes A and Lenz D, Averaging almost periodic functions along exponential sequences, in Aperiodic Order. Vol. 2: Crystallography and Almost Periodicity,Baake M and Grimm U (eds.),Cambridge University Press, Cambridge (2017), pp. 343–362; .BL Baake M and Lenz D, Spectral notions of aperiodic order,Discr. Cont. Dynam. Syst. S 10 (2017) 161–190; .BLvE Baake M, Lenz D and van Enter A C D, Dynamical versus diffraction spectrum for structures withfinite local complexity,Ergodic Th. & Dynam. Syst.35 (2015) 2017–2043; . Bart Bartlett A, Spectral theory of ^d substitutions, Ergodic Th. & Dynam. Syst., in press; . Borwein Borwein P,Choi S, and Jankauskas J, Extremal Mahler measures and L_s norms of polynomials related to Barker sequences, Proc. Amer. Math. Soc.141 (2013)2653–2663.Clunie Clunie J, The minimum modulus of a polynomial on the unit circle, Quart. J. Math. Oxford 10 (1959) 95–98.Cord Corduneanu C, Almost Periodic Functions, 2nd English ed.(Chelsea, New York, 1989).Dekking Dekking F M, The spectrum of dynamical systems arising from substitutionsof constant length, Z. Wahrscheinlichkeitsth. verw. Geb. 41(1978) 221–239.EW Einsiedler M and Ward T, Ergodic Theory with a View Towards Number Theory, Springer, London (2011).EW2 Everest G and Ward T, Heights of Polynomials and Entropy in Algebraic Dynamics, Springer, London (1999).FSS Fan A-H, Saussol B and Schmeling J, Products of non-stationary random matrices and multiperiodic equations of several scaling factors, Pacific J. Math. 214 (2004) 31–54; .Harman Harman G, Metric Number Theory, Oxford University Press, New York (1998).LMS Lee J-Y, Moody R V and Solomyak B, Pure point dynamical and diffraction spectra, Ann. H. Poincaré 3 (2002) 1003–1018; .LL Lieb E H and Loss M, Analysis, 2nd ed., American Mathematical Society, Providence, RI (2001).Neil Mañibo N, Lyapunov exponents for binary substitutions of constant length, J. Math. Phys. 58 (2017) 113504 (9 pp); . Q Queffélec M, Substitution Dynamical Systems — Spectral Analysis, 2nd ed., Springer, Berlin (2010).K-book Schmidt K, Dynamical Systems of Algebraic Origin, Birkhäuser, Basel (1995).Sobol Sobol I M, Calculation of improper integrals using uniformly distributed sequences, Soviet Math. Dokl. 14 (1973) 734–738.Boris Solomyak B, Dynamics of self-similar tilings,Ergodic Th. & Dynam. Syst. 17 (1997) 695–738 and 19 (1999) 1685 (Erratum).S Simon B, Analysis, Part 1: Real Analysis, Amer. Math. Soc., Providence, RI (2015).Viana Viana M, Lectures on Lyapunov Exponents, Cambridge University Press, Cambridge (2013). | http://arxiv.org/abs/1709.09083v2 | {
"authors": [
"Michael Baake",
"Uwe Grimm",
"Neil Manibo"
],
"categories": [
"math.DS",
"math-ph",
"math.MP",
"37B10, 52C23, 11K70"
],
"primary_category": "math.DS",
"published": "20170926151448",
"title": "Spectral analysis of a family of binary inflation rules"
} |
Why: Natural Explanations from a Robot Navigator Raj Korpan1, Susan L. Epstein1,2, Anoop Aroor1, Gil Dekel21The Graduate Center and 2Hunter College, City University of New [email protected], [email protected], [email protected], [email protected] December 30, 2023 ============================================================================================================================================================================================================================================================== Effective collaboration between a robot and a person requires natural communication. When a robot travels with a human companion, the robot should be able to explain its navigation behavior in natural language. This paper explains how a cognitively-based, autonomous robot navigation system produces informative, intuitive explanations for its decisions. Language generation here is based upon the robot's commonsense, its qualitative reasoning, and its learned spatial model. This approach produces natural explanations in real time for a robot as it navigates in a large, complex indoor environment. § INTRODUCTIONSuccessful human-robot collaboration requires natural explanations, human-friendly descriptions of the robot's reasoning in natural language. In collaborative navigation, a person and an autonomous robot travel together to some destination. The thesis of this paper is that natural explanations for collaborative navigation emerge when a robot controller (autonomous navigation system) is cognitively based. This paper introduces Why, an approach that accesses and conveys the robot's reasoning to provide its human companion with insight into its behavior. The principal results presented here are natural explanations from an indoor robot navigator.Even in unfamiliar, complex spatial environments (worlds), people travel without a map to reach their goals successfully <cit.>. Efficient human navigators reason over a mental model that incorporates commonsense, spatial knowledge, and multiple heuristics <cit.>. They then use the same model to explain their chosen path and their reasons for decisions along the way. Our research goal is an autonomous robot navigator that communicates with its human companions much the way people do.Why explains a navigation decision in natural language. It anticipates three likely questions from a human companion: “Why did you decide to do that?” “Why not do something else?” and “How sure are you that this is the right decision?” Why generates its answers with SemaFORR, a robot controller that learns a spatial model from sensor data as it travels through a partially-observable world without a map <cit.>. SemaFORR's cognitively-based reasoning and spatial model facilitate natural explanations.Why is an interpreter; it uses SemaFORR's cognitive foundation to bridge the perceptual and representational gap between human and robot navigators. Why and SemaFORR could accompany any robot controller to provide natural explanations. More broadly, Why can be readily adapted to explain decisions for other applications of FORR, SemaFORR's underlying cognitive architecture.The next section of this paper reviews related work. Subsequent sections describe SemaFORR and formalize Why. Finally, we evaluate Why-generated explanations and give examples of them as our mobile robot navigates through a large, complex, indoor world.§ RELATED WORKWhen a robot represents and reasons about space similarly to the way people do, it facilitates human-robot collaboration <cit.>. Communication with a robot allows people to build a mental model of how it perceives and reasons, and thereby helps to establish trust <cit.>. A recent approach grounded perceived objects between the robot and a person to build a mutual mental model, and then generated natural language descriptions from it <cit.>. Although that supported natural dialogue, it did not explain the reasoning that produced the robot's behavior. Despite much work on how a robot might understand natural language from a human navigator <cit.>, natural explanations from a robot navigator to a person remain an important open problem. Such work has thus far required detailed logs of the robot's experience, which only trained researchers could understand <cit.>. It is unreasonable, however, to expect people to decipher robot logs.Natural language descriptions of a robot's travelled path have addressed abstraction, specificity, and locality <cit.>. A similar approach generated path descriptions to improve sentence correctness, completeness, and conciseness <cit.>. Those approaches, however, used a labeled map to generate descriptions and did not explain the robot's reasoning. Other work visually interpreted natural-language navigation commands with a semantic map that showed the robot's resulting action <cit.>. Although a person might eventually unpack the robot's reasoning process this way, no natural language explanation was provided.Researchers have generated navigation instructions in natural language from metric, topological, and semantic information about the world <cit.> or rules extracted from human-generated instructions <cit.>. Other work has focused on human spatial cognition <cit.>, or on simplicity and understandability <cit.>. None of these approaches, however, can explain how the instructions were generated, nor can they justify a particular instruction.More generally, researchers have sought human-friendly explanations for systems that learn. Trust in and understanding of a learning system improved when people received an explanation of why a system behaved one way and not another <cit.>. Several approaches to sequential tasks have explained Markov decision processes, but the resultant language was not human-friendly and was not based on human reasoning <cit.>. In summary, although intelligent systems should be able to provide natural explanations during collaborative navigation, to the best of our knowledge no work has focused on explanations for the robot's decisions. Why addresses that gap.§ SEMAFORRSemaFORR is a robot controller implemented in ROS, the state-of-the-art Robot Operating System. SemaFORR selects one action at a time to move the robot to its target location. Instead of a world map, SemaFORR uses local sensor data, learned knowledge, and reactive, heuristic reasoning to contend with any obstacles and reach its target. The resultant behavior is satisficing and human-like rather than optimal.A decision state records the robot's current sensor data and its pose ⟨ x, y, θ⟩, where ⟨ x, y ⟩ is its location and θ is its orientation with respect to an allocentric, two-dimensional coordinate system. As the robot travels, its path to a target is recorded as a finite sequence of decision states. SemaFORR makes decisions based on a hierarchical reasoning framework and a spatial model that it learns while it navigates. Why uses them both to generate its explanations. §.§ Spatial ModelSemaFORR learns its compact, approximate spatial model from experience. The model captures many of the features of a cognitive map, the representation that people construct as they navigate <cit.>. Instead of a metric map, SemaFORR's model is a set of spatial affordances, abstract representations that preserve salient details and facilitate movement. As the robot travels or once it reaches its target, it learns spatial affordances from local sensor readings and stores them as episodic memory. Figure <ref> gives examples. A region is an unobstructed area where the robot can move freely, represented as a circle. A region's center is the robot's location in a decision state; its radius is the minimum distance sensed from the center to any obstacle. An exit is a point that affords access to and from a region, learned as a point where the robot crossed the region's circumference . A trail refines a path the robot has taken. It is an ordered list of trail markers, decision states selected from the robot's path. The first and last trail markers are the initial and final decision states on the path. Trail learning works backward from the end of the path; it creates a new trail marker for the earliest decision state that could have sensed the current trail marker. The resultant trail is usually shorter than the original path and provides a more direct route to the target.A conveyor is a small area that facilitates travel. It is represented in a grid superimposed on the world, where each cell tallies the frequency with which trails pass through it. High-count cells in the grid are conveyors. The spatial model combines affordances to produce more powerful representations. For example, a door generalizes over the exits of a region. It is represented as an arc along the region's circumference. The door-learning algorithm introduces a door when the length of the arc between two exits is within some small ϵ. Once generated, a door incorporates additional exits if they are within ϵ of it. Another example is the skeleton, a graph that captures global connectivity with a node for each region. An edge in the skeleton joins two nodes if a path has ever moved between their corresponding regions. Along with commonsense qualitative reasoning, affordances are used to select the robot's next action. §.§ Reasoning FrameworkSemaFORR is an application of FORR, a cognitive architecture for learning and problem solving <cit.>. FORR is both reactive and deliberative. Reactivity supports flexibility and robustness, and is similar to how people experience and move through space <cit.>. Deliberation makes plans that capitalize on the robot's experience; it is the focus of current workaroor2017toward (Aroor and Epstein, in press).The crux of any FORR-based system is that good decisions in complex domains are best made reactively, by a mixture of good reasons. FORR represents each good reason by a procedure called an Advisor. Given a decision state and a discrete set of possible actions, an Advisor expresses its opinions on possible actions as comments. In a decision cycle, SemaFORR uses those comments to select an action. Possible actions are alternately a set of forward moves of various lengths or a set of turns in place of various rotations. A move with distance 0 is equivalent to a pause. Thus, in any given decision state, SemaFORR chooses only the intensity level of its next move or turn. The resultant action sequence is expected to move the robot to its target.SemaFORR's Advisors are organized into a three-tier hierarchy, with rules in tier 1 and commonsense, qualitative heuristics in tier 3. Tier 1 invokes its Advisors in a predetermined order; each of them can either mandate or veto an action. If no action is mandated, the remaining, unvetoed actions are forwarded to tier 3. (Natural explanations for tier 2, SemaFORR's deliberative layer, are a focus of current work.) Table <ref> lists the Advisors' rationales by tier. Each tier-3 Advisor constructs its comments on the remaining possible actions with its own commonsense rationale. Comments assign a strength in [0,10] to each available action. Strengths near 10 indicate actions that are in close agreement with the Advisor's rationale; strengths near 0 indicate direct opposition to it. For n Advisors, m actions, and comment strength c_ij of Advisor i on action j, SemaFORR selects the action with the highest total comment strength:argmax_j ∈ m∑_i=1^n c_ij.Because ties are broken at random, tier 3 introduces uncertainty into action selection. For further details on SemaFORR, see <cit.>. § APPROACHThis section describes how Why exploits SemaFORRto generate natural explanations. Each of the three questions below focuses on a different aspect of a robot controller. The result is a rich, varied set of natural explanations. §.§ Why did you do that?The first question asks why the robot chose a particular action. Why constructs its answer from the rationales and comments of the Advisors responsible for that choice, with templates to translate actions, comments, and decisions into natural language.Given the robot's current pose, Why maps each possible action onto a descriptive phrase for use in any [action] field. Examples include “wait” for a forward move of 0.0 m, “inch forward” for a forward move of 0.2 m, and “shift right a bit” for a turn in place of 0.25 rad.Algorithm <ref> is pseudocode for Why's responses. Why takes as input the current decision state, target location, and spatial model, and then calculates its response based on the comments from SemaFORR's Advisors. There are three possibilities: tier 1 chose the action, tier 1 left only one unvetoed action, or tier 3 chose the action. SemaFORR only makes a decision in tier 1 if Victory mandates it or AvoidWalls has vetoed all actions but the pause. The applicable templates in those cases are “I could see our target and [action] would get us closer to it” and “I decided to wait because there's not enough room to move forward.”The inherent uncertainty and complexity of a tier-3 decision, however, requires a more nuanced explanation. For a set of m actions, assume tier-3 Advisor D_i outputs comment with strengths c_i1,…,c_im∈[0, 10]. D_i's t-support for action a_k is the t-statistic t_ik = (c_ik - c̅_̅i̅)/σ_i where c̅_̅i̅ is the mean strength of D_i's comments in the current decision state and σ_i is their standard deviation. (This is not a z-score because sampled values replace the unavailable true population mean and standard deviation.) Why can compare different Advisors' t-supports because they have common mean 0 and standard deviation 1. If |t_ik| is large, Advisor D_i has a strong opinion about action a_k relative to the other actions: supportive for t_ik > 0 and opposed for t_ik < 0.Table <ref> provides a running example. It shows the original comment strengths from four Advisors on four actions, and the total strength C_k for each action a_k. Tier 3 chooses action a_4 because it has maximum support. While D_1 and D_2 support a_4 with equal strength, the t-support values tell a different story: D_1 prefers a_4 much more (t_14=1.49) than D_2 does (t_24=0.71). Moreover, D_3 and D_4 actually oppose a_4 (-0.34 and -0.78, respectively).For each measure, we partitioned the real numbers into three intervals and assigned a descriptive natural language phrase to each one, as shown in Table <ref>. This partitioning allows Why to hedge in its responses, much the way people explain their reasoning when they are uncertain <cit.>. Why maps the t-support values into these intervals. For a_4, D_1's t-support of 1.49 is translated as “want” and D_4's -0.78 is translated as “don't want”. Why then completes the clause template “I [phrase] to [rationale]” for each Advisor based on Table <ref> and less model-specific language from Table <ref>. For example, if D_1 were Greedy, then the completed clause template for a_4 would be “I want to get close to the target.” Finally, Why combines completed clause templates into the final tier-3 explanation, but omits language from Advisors with t-support values in (-0.75, 0.75] because they contribute relatively little to the decision. Why concatenates the remaining language with appropriate punctuation and conjunctions to produce its tier-3 explanation: “(Although [language from opposed Advisors], ) I decided to [action] because [language from supporting Advisors]”. The portion in parentheses is omitted if no opposition qualifies. If the Advisors in the running example were Greedy, ElbowRoom, Convey, and Explorer, in that order, and a_4 were move forward 1.6 m, then the natural explanation is “Although I don't want to go somewhere I've been, I decided to move forward a lot because I want to get close to our target.” (Note that D_2's support fails the -0.75 filter and so is excluded.)This approach can also respond to “What action would you take if you were in another context?” Given the decision state and the target location, Why would reuse its current spatial model, generate hypothetical comments, and process them in the same way. The sentence template would substitute “I would [action]” for “I decided to [action]."§.§ How sure are you that this is the right decision?The second question from a human collaborator is about the robot's confidence in its decision, that is, how much it trusts that its decision will help reach the target. Again, Why responds based on the tier that selected the action. Tier 1's rule-based choices are by definition highly confident. If Victory chose the action then the response is “Highly confident, since our target is in sensor range and this would get us closer to it.” If AvoidWalls vetoed all forward moves except the pause, then the explanation is “Highly confident, since there is not enough room to move forward.” Again, tier-3's uncertainty and complexity require more nuanced language, this time with two measures: level of agreement and overall support. The extent to which the tier-3 Advisors agree indicates how strongly the robot would like to take the action. Why measures the level of that agreement with Gini impurity, where values near 0 indicate a high level of agreement and values near 0.5 indicate disagreement <cit.>. For n tier-3 Advisors and maximum comment strength 10, the level of agreement G_k ∈ [0,0.5] on action a_k is defined asG_k = 2 ·[∑_i=1^n c_ik/10n] ·[1-∑_i=1^n c_ik/10n]. In the example of Table <ref>, the level of agreement on a_4 isG_4 = 2 ·[22/40] ·[1-22/40] ≈ 0.50. This indicates considerable disagreement among the Advisors in Table <ref>.The second confidence measure is SemaFORR's overall support for its chosen action compared to other possibilities, defined as a t-statistic across all tier-3 comments. Let μ_C be the mean total strength of all actions C under consideration by tier 3, and σ_C be their standard deviation. We define the overall support for action a_k as T_k = (C_k - μ_C)/σ_C. T_k indicates how much more the Advisors as a group would like to perform a_k than the other actions. In Table <ref>, the overall support T_4 for a_4 is 0.66, which indicates only some support for a_4 over the other actions. Why weights level of agreement and overall support equally to gauge the robot's confidence in a tier-3 decision with confidence level L_k = (0.5 - G_k) · T_k for a_k. It then maps each of L_k, G_k, and T_k to one of three intervals and then to natural language, as in Table <ref>, with implicit labels low < medium < high in order for each statistic. Two statistics agree if they have the same label; one statistic is lower than the other if its label precedes the other's in the ordering.All responses to this question use a template that begins “I'm [L_k adverb] sure because....” If G_k and T_k both agree with L_k, the template continues “[G_k phrase]. [T_k phrase].” For example, “I'm really sure about my decision because I've got many reasons for it. I really want to do this the most.” If only one agrees with L_k, the template continues “[phrase for whichever of G_k or T_k agrees].” For example, “I'm not sure about my decision because my reasons conflict.” Finally, if neither agrees with L_k, it concludes “even though [phrase for whichever of G_k or T_k is lower than L_k], [G_k phrase or T_k phrase that is higher than L_k].” For example, “I am only somewhat sure about my decision because, even though I've got many reasons, I don't really want to do this the most.” For a_4 in Table <ref>, L_4 is near 0, G_4 = 0.50, and T_4 = 0.66. This produces the natural explanation “I'm not sure about my decision because my reasons conflict. I don't really want to do this more than anything else.” §.§ Why not do something else?A human collaborator makes decisions with her own mental model of the world. When her decision conflicts with another team member's, she tries to understand why they made a different decision. Why's approach explains SemaFORR's preference for action a_k over an alternative a_j. If tier 1 chose a_k, the explanation uses Victory's rationale: “I decided not to [action_j] because I sense our goal and another action would get us closer to it.” If AvoidWalls or NotOpposite vetoed a_j, then the natural explanation is “I decided not to [action] because [rationale from Advisor that vetoed it].” The other possibility is that a_j had lower total strength in tier 3 than a_k did. In this case, Why generates a natural explanation with the tier-3 Advisors that, by their comment strengths, discriminated most between the two actions. Why calculates t_ik-t_ij for each Advisor D_i. If the result lies in [-1, 1] then D_i's support is similar for a_k and a_j; otherwise D_i displays a clear preference. The natural explanation includes only those Advisors with clear preferences.The explanation template is “I thought about [action_j] (because it would let us [rationales from Advisors that prefer action_j]), but I felt [phrase] strongly about [action_k] since it lets us [rationales from Advisors that prefer action_k].” The [phrase] is the extent to which SemaFORR prefers a_k to a_j. It is selected based on T_k-T_j, the difference in the actions' overall support, and mapped into intervals as in Table <ref>. The portion in parentheses is only included if any Advisors showed a clear preference for action_j.For “Why didn't you take action a_2?” on our running example, Why calculates the difference in overall support between a_4 and a_2 at 0.38, which maps to “slightly more.” The differences in t-support between a_4 and a_2 are 1.92, 0.44, 0.45, and -2.22. Thus, if D_1 is Greedy and prefers a_4, while D_4 is Explorer and prefers a_2, the natural explanation is “I thought about a_2 because it would let us go somewhere new, but I felt slightly more strongly about a_4 since it lets us get closer to our target.”§ RESULTSImplemented as a ROS package, Why explains Sema-FORR's decisions in real time. We evaluated Why in simulation for a real-world robot (Fetch Robotics' Freight). When the robot navigated to 230 destinations in the complex 60m×90m office world of Figure <ref>, Why averaged less than 3 msec per explanation.Why's many distinct natural explanations simulate people's ability to vary their explanations based on their context <cit.>. Table <ref> provides further details. The Coleman-Liau index measures text readability; it gauged Why's explanations over all three questions at approximately a 6th-grade level <cit.>, which should make them readily understandable to a layperson. For action a_k chosen in tier 3 and every possible alternative a_j, Table <ref> shows how often the values of G_k, T_k, L_k, t_ik-t_ij, and T_k-T_j fell in their respective Table <ref> intervals. The Advisors disagreed (G_k > 0.45) on 67.15% of decisions. Strong overall support (T_k > 1.5) made SemaFORR strongly confident in 2.44% of its decisions (L_k > 0.375) and somewhat confident in 42.64% of them. When asked about an alternative, individual Advisors clearly preferred (t_k-t_j > 1) the original decision 39.50% of the time; SemaFORR itself declared a strong preference (T_k-T_j > 1.5) between the two actions 61.13% of the time.Table <ref> illustrates Why's robust ability to provide nuanced explanations for tier-3 decisions. The target appears as an asterisk and the black box and arrow show the robot's pose. Decision 1 was made when the robot had not yet learned any spatial affordances; decision 2 was made later, when the spatial model was more mature. In decision 3, the Advisors strongly disagreed, while in decision 4 the spatial model-based Advisors disagreed with a commonsense-based Advisor. § DISCUSSIONWhy is applicable more broadly than we have indicated thus far. Any robot controller could have SemaFORR learn the spatial model in parallel, and use it with Why to produce transparent, cognitively-plausible explanations. If the alternative controller were to select action a_j when SemaFORR selected a_k, Why could still explain a_j with any Advisors that supported it, and offer an explanation for a_k as well. Furthermore, once equipped with Advisor phrases and possibly with new mappings, any FORR-based system could use Why to produce explanations. For example, Hoyle is a FORR-based system that learns to play many two-person finite-board games expertly <cit.>. For Hoyle, Why could explain “Although I don't want to make a move that once led to a loss, I decided to do it because I really want to get closer to winning and I want to do something I've seen an expert do.”Because SemaFORR's spatial model is approximate and its Advisors are heuristic, precise natural language interpretations for numeric values are ad hoc. For Table <ref>, we inspected thousands of decisions, and then partitioned the computed values as appeared appropriate. We intend to fine-tune both intervals and phrasing with empirical assessment by human subjects. Because natural explanations have improved people's trust and understanding of other automated systems, we will then evaluate Why with human subjects. SemaFORR and Why are both ongoing work. As heuristic planners for tier 2 are developed, we will extend Why to incorporate plans in its explanations. We also anticipate revisions in Why's phrasing to reflect changes in SemaFORR's possible action set. Finally, Why could be incorporated into a more general dialogue system that would facilitate part of a broader conversation between a human collaborator and a robot. A FORR-based system for human-computer dialogue, could prove helpful there <cit.>.In summary, Why produces natural explanations for a robot's navigation decisions as it travels through a complex world. These explanations are essential for collaborative navigation and are made possible by the robot controller's cognitively-based reasoning. The approach presented here generates explanations that gauge the robot's confidence and give reasons to take an action or to prefer one action over another. As a result, a human companion receives informative, user-friendly explanations from a robot as they travel together through a large, complex world in real time.Acknowledgements. This work was supported in part by NSF 1625843. The authors thank the reviewers for their insightful comments.9pt10ptaaai | http://arxiv.org/abs/1709.09741v1 | {
"authors": [
"Raj Korpan",
"Susan L. Epstein",
"Anoop Aroor",
"Gil Dekel"
],
"categories": [
"cs.AI",
"cs.CL",
"cs.HC",
"cs.RO"
],
"primary_category": "cs.AI",
"published": "20170927213053",
"title": "WHY: Natural Explanations from a Robot Navigator"
} |
firstpage–lastpage Modeling WiFi Traffic for White Space Prediction in Wireless Sensor Networks Indika S. A. Dhanapala^1,*, Ramona Marfievici^1, Sameera Palipana^1, Piyush Agrawal^2, Dirk Pesch^1 ^1Nimbus Centre for Embedded Systems Research, Cork Institute of Technology, Cork, Ireland ^2United Technologies Research Centre, Cork, Ireland ^*Contact Author: December 30, 2023 ========================================================================================================================================================================================================================================================================================================== We investigate the triggering of star formation and the formation of stellar clusters in molecular clouds that form as the ISM passes through spiral shocks. The spiral shock compresses gas into ∼100 pc long main star formation ridge, where clusters forming every 5-10 pc along the merger ridge. We use a gravitational potential based cluster finding algorithm, which extracts individual clusters, calculates their physical properties and traces cluster evolution over multiple time steps. Final cluster masses at the end of simulation range between 1000 and 30000 M_⊙ with their characteristic half-mass radii between 0.1 pc and 2 pc. These clusters form by gathering material from 10-20 pc size scales. Clusters also show a mass - specific angular momentum relation, where more massive clusters have larger specific angular momentum due to the larger size scales, and hence angular momentum from which they gather their mass. The evolution shows that more massive clusters experiences hierarchical merging process, which increases stellar age spreads up to 2-3 Myr. Less massive clusters appear to grow by gathering nearby recently formed sinks, while more massive clusters with their large global gravitational potentials are increasing their mass growth from gas accretion. stars: formation –stars: luminosity function, mass function – globular clusters and associations: general, interstellar medium,galaxies: star formation. § INTRODUCTION Star formation is one of the most important processes in galactic evolution, transforming gas into stars and providing the visible output as well as the chemical and energetic feedback into the galaxy. Current understanding is that at least half of all stars, and potentially all massive stars, form in stellar clusters (). Understanding how stellar clusters form is on ongoing challenge with many observational studies trying to determine the exact initial conditions (). Models for cluster formation show that a turbulent molecular clump that is gravitationally unstable fragments hierarchically into small clusters which eventually merge to form larger stellar systems (, , , , ). Observations of star formation in infrared dark clouds such as <cit.> shows the fragmentation of the filamentary structure (, ) feeding into the formation of a stellar cluster.Numerical simulations have provided significant insight into the dynamical nature of star formation (; ; ; ), showing the importance of turbulence, collapse, fragmentation, interactions and accretion. Although useful in highlighting the physical processes, these numerical simulations suffer from their overly-idealised initial conditions. Our () study showed that a small variation in gravitational boundedness along a cloud results in significantly different stellar populations, clusterings, star formation rates, efficiencies, and stellar IMFs. Initial conditions have a major impact on all star formation properties, and hence using self-consistent initial conditions is crucial to develop realistic models.Recent simulation work, like <cit.>, <cit.>, <cit.>, <cit.> gives results for individual molecular clouds and forming clusters. Observations such as those of <cit.> support the concept that clusters grow hierarchically. However, observations are limited in terms of 3D spatial information and time evolution. Real molecular clouds in spiral galaxies () appear as high density regions in the ISM during the passage through the spiral arms. Works like <cit.>, <cit.>, <cit.> were first attempts to set up more realistic initial conditions in simulations, which could have large effects on how clouds are shaped and how star clusters are forming.There are large numbers of observations being made which contribute towards understanding the processes of stellar cluster formation. Observations are necessarily wide ranging, covering the mass-radius relation(), stellar age spreads (, ), line of sight kinematics (, ), distribution of positions (, ), spatial structure (, , ). However, very few properties can be compared directly between simulations and observations. Simulations are limited by initial conditions and the physics included, while observations are limited by 2D projection in the sky as well as their inability to show us the the past or future evolution of clusters.Star formation is also likely to be affected by feedback from the stars, especially young high mass stars. Feedback may also help to explain the low efficiencies of star formation seen on different scales (e.g. ), but low efficiencies are also possible in the absence of feedback (). Simulations of star formation including feedback generally result in a significant (factor 2) decrease in the star formation efficiencies (e.g. ). Although feedback can decrease the efficiency of star formation, it does not in general stop ongoing star formation. Instead, the feedback findsweak points (lower density) in the surrounding gas through which it can be channelled and escape the dense clump.Gas removal can affect the overall dynamics and lifetimes of forming clusters if it contributes a dominant proportion of the cluster mass. Several N-body studies have investigated the effect of gas removal in young stellar clusters via an assumed potential. These studies, although not fully consistent in terms of the effect of feedback on gas, provide valuable insight into the potential cluster evolution (). For example, <cit.> showed that gas and dissipation free hierarchical mergers have difficulty producing large smooth clusters and that systems, such as R136, NGC3603 and the ONC could have formed from monolithic collapse scenario. However, in these works gas was replaced by static potential, neglecting the dissipational properties of gas dynamics which can greatly decrease the merger timescale (). Indeed, several observational studies find evidence for hierarchical mergers in terms of kinematical subgroups in young stellar clusters (.).R136, NGC3603 and the ONC cases does not necessarily confirm if these clusters has formed from monolithic collapse or internal structure, produced by mergers has been smoothed out by the interaction with gas. In contrast, observational studies find evidence of the highly fragmented nature of cluster formation (), for kinematical subgroups in young stellar clusters () and of clusters that appear in close proximity such that a subsequent merger is possible, pointing to scenarios where hierarchical mergers are a likely process in star cluster formation (). These observational evidences brings support to cluster formation through merging scenario, which we will investigate in details in this work. § METHODS§.§ SPH simulations In a previous work we analysed triggering of star formation during the spiral arm passage <cit.>. Here we use the simulation data of <cit.> which uses assumed spiral potential from <cit.> through which ISM is allowed to flow. We determine physical cluster properties, such as masses, densities, sizes, binding energies and their evolution over time. These key parameters allow us to investigate stellar cluster formation mechanisms and dynamics free from the intrinsic assumptions due to the idealised initial conditions.<cit.> simulation was constructed from a set of nested simulations starting from a full Galactic disc simulation over 350 Myr, with a 50 Myr high-resolution counterpart focusing on the formation of dense clouds in the spiral arms, and a final simulation to follow star formation over 5.8 Myr. The final stage, which we analyse here,included self-gravity and modelled a 250 pc region containing 1.9× 10^6 M_⊙ mass. This simulation used 1.29× 10^7 SPH particles with 0.15 M_⊙ masses. Star formation is followed through the use of sink particles (). A minimum mass for the sink particles corresponds to ≈ 70 SPH particles representing one SPH kernel, or ≈ 11 M_⊙ with a sink radius of 0.25 pc to accrete bound, infalling gas particles while all particles penetrating within 0.1pc were accreted. The sink particles therefore do not represent individual stars but rather a small cluster of stars or star forming region. Gravitational interactions between sinks were smoothed within 0.025 pc.Simulations used for our work, do not have any stellar feedback or magnetic fields, and is hence designed to investigate how stellar clusters are forming in more realistic initial conditions, which includes spiral arm dynamics. We also point out here that before including feedback and magnetic fields, it is essential to understand what properties of forming clusters are being defined purely by the spiral arm dynamics. Stellar feedback could be more relevant for later stages of simulation, once massive stars form, while at early stages only magnetic fields can slow down the formation of the first stars. §.§ Cluster definition In order to follow the early cluster evolution, we require a robust cluster definition and finding algorithm. There are a large number of cluster finding algorithms created to find clusters in datasets. For SPH simulations data, the cluster finding algorithm has to find clusters based on sink particle positions and masses. There are two steps to follow in order to obtain a robust cluster definition and to follow cluster formation history: static cluster definition at a given time and dynamical cluster definition by linking the same cluster from one time step to another.Our cluster definition is based on sink local and enclosed gravitational potentials. Firstly each sink has assigned its local potential from surrounding neighbour sinks within 2 pc. The list of all sinks at a given time step is sorted by these potentials and the deepest potential sink is picked as a starting point. This first and lowest local potential sink is set as a centre of the cluster and enclosed potentials are calculated on all sinks within 2 pc around it. Next sinks are continuously added to the most bound cluster if their local potential is not deeper than twice that of the enclosed potential of the cluster.If no such cluster is found and the sink has its local potential depth below the upper potential threshold (-10^11cm^2s^-2), it starts a new cluster. The search is finished when it reaches the first sink with its local potential above the background level. Clusters, which have fewer than six members are removed after the search is completed. Using the enclosed potential to build clusters results in a bias towards the selection of spherical rather than filamentary structures. We use -10^11cm^2s^-2 potential threshold, as systems below this potential becomes self gravitating and show virialized motions. We limit the cluster algorithm to not add any sinks whose local gravitational potentials are twice or less that of the enclosed potential of the cluster. Radial density profiles show that this ensures that we do not merge multiple clusters that are close to overlapping. In addition we use 0.05 pc softening for potentials in order to define smoother local potentials and remove unwanted fluctuations of potentials in cluster centres, as it can split one cluster into multiple if sharp peaks are detected.We repeat the cluster finding process throughout the entire duration of the simulation at each time step. Clusters can be traced over time by linking two clusters between two neighbouring time steps. A cluster is assumed to be the same if particles representing more than 50% of the cluster mass from the current time step t_i are found in it at the next time step t_i+1. Clusters can also merge. If the mass of the smaller cluster is larger than 30% of the total mass of both parts, then these clusters are major mergers (otherwise they are minor mergers). Mergers can be traced by searching if most of the sinks from two separate clusters at time t_i are found in a single cluster at the next time step t_i+1. Linking clusters over all time steps allows us to create a merger tree for the clusters.In order to remove fluctuations that occur when a cluster is near the cluster definition boundary, we smooth the cluster finding algorithm over neighbouring time steps. Cluster lifetimes are calculated along the merger tree branch. If the lifetime is only 1 time step, clusters are checked to ensure they do not merge again into the same cluster - if so, sinks from the temporary cluster are re-assigned to the main cluster.§ RESULTSWe make use of the <cit.> self-gravity simulation that inherited its initial conditions from a large scale Galaxy simulation. It contains ∼ 12 million SPH particles. The inherited initial conditions set the galactic scale shock, and compresses the ISM to higher densities to start star formation. The gas motions into the shock are predominantly along y-axis of the simulation. Star formation takes place continuously as the shock travels through the gas, forming 2000 sinks and around 20 clusters by the end of the simulation. In addition, the shock reaches one side slightly earlier than another, and thus creates a gradient in stellar ages along the spiral arm.§.§ Cluster statisticsWe apply our cluster finding algorithm to the entire <cit.> self-gravity simulation in order to find clusters of sinks. Clustering statistics based on the gravitational potential definition are shown in Figure <ref>. From the figure we see that at the beginning of the simulation there are no sinks and no clusters. The first sinks form quite early, in the first Myr of the simulation. However, clusters appears only around 3 Myr. This show that the first sinks form individually in the highest density fast collapsing clumps. Clusters appear slightly later when at least 6 sinks assemble in a compact region and meets the definition. Figure <ref> shows that star formation is continuously occurring in the simulation, with individual sinks, and clusters growing in mass, and in numbers throughout the simulation. The number of clusters appears to be growing up to ∼20 at the end of simulation. From ∼4 Myr fluctuations appear in the number of clusters, indicating that clusters also undergo mergers which add to their growth rates but decrease the total numbers of clusters present.§.§ Cluster forming regions Figure <ref> shows the large scale view of the gas, sinks and clusters at the end of simulation (5.6 Myr). The plot, viewed from the plane of the galaxy, shows many clusters lie in or nearby high density regions but do not necessarily match them. In Figure <ref> we plot the sink particles alone within the same limits, colour-coding them by mean stellar age. The plot shows a clear stellar age gradient. Sinks in the right hand side are slightly older because the spiral shock reached that side slightly earlier and triggered star formation there. As the spiral shock passed through, part of the gas was consumed by star formation, while another part moved with the shock or even started to expand as post-shock leaves the region. We can see very high column densities in the Figure <ref> between -30 [pc] < x < -10 [pc], where the shock is moving through the gas. There is very little star formation in this part. However, looking at Figure <ref> we can see that star formation here only starts to happen with visible several very young (< 0.5 Myr) clusters (around [-15; -5]), which may merge in the future and form another big cluster there. Figures <ref> and <ref> gives a support towards both sequential () and triggered star formation models: the sequential star formation appears as the spiral shock continuously propagates to the left but at the same time individual clouds are compressed, where star formation is triggered.Most of the high density star forming regions in Figure <ref> are visible 5-15 pc below the Galactic plane. This is due to the larger scale dynamics that drive the star formation (), where spiral shocks and cloud-cloud collisions can cause some star forming regions to be significantly out of the plane of the Galaxy.Right hand side panel of Figure <ref> shows zoomed in region onto two most massive final clusters. Cluster at left shows some younger stars in the core, but also a few older ones. Stars near the outside are mostly older. Cluster at right shows younger stars in the outer regions, some younger in core but overall a bit older.We take the most massive cluster at the end of the simulation, which is visible on the edge of high density clouds in the middle of Figure <ref>, and trace all its environment, accreted gas and sinks backwards in time. By finding sinks which belong to the cluster, we are also mapping all the gas particles that ultimately contributed to form the cluster. As all accreted particles are found, the cluster mass is assumed to be conserved over all time steps and the centre of mass of the system is well defined. We follow this cluster mass centre to illustrate the formation of the cluster in x-y position maps at four different time steps. Figure <ref> shows the evolution of the gas and sinks forming the most massive cluster, plotted in the cluster's centre of mass frame. Each of 4 panels has a map of surface density derived from all gas particles in the region (left) and a map of surface density calculated from not yet accreted gas particles that contribute mass to the final cluster (right).Initial gas cloud with the size of ∼40 pc is contracting down to several pc size cluster, a contraction of > 10 times over 6 Myr. Comparison between left and right hand side panels show that accreted cluster gas are embedded all the time in the largest density areas of the region. Accreted gas distribution (right panels) well matches the distribution of all gas (left panels). However, most of the particles visible in the left hand panel are not accreted by any sinks and so do not play a part in forming the cluster. Gas particles accreted by sinks not belonging to the cluster are excluded (right panels). The internal geometry of the region is highly fragmented with visible clumps (where the first sinks form) and filaments. Even if the cloud collapses as the whole, the internal structure of clumps and filaments keeps changing over the time. The presence of the galactic spiral shock is visible in the left hand panels, as the shock compresses widely distributed clouds to form a thin ridge through 6 Myr, extending from the top left to bottom right side of the map.Figure <ref> shows the initial regions from which the various clusters form and accrete their final mass. These gas particles are shown in Figure <ref> as coloured particles, with the colour representing the mass of the final cluster. The grey particles in Figure <ref> show the non cluster-forming gas. We note that cluster forming gas clouds are initially embedded in higher density regions. The highest mass clusters form from the central high density region, and there are small mass clusters forming outside this region. We also notice the trend that more massive clusters form from larger clouds. The spiral shock is coming from the bottom-right side of the diagram and there are visible trails of particles, which are coming with the shock to the main star forming region and are also accreted by sinks in clusters (purple trails in the bottom side of the diagram). As the material is incoming with the small pitch angle to the spiral arm, the section of low density inter-arm gas can be seen in the bottom-left side of the diagram.§.§ Accretion histories The Lagrangian nature of the simulation allows us to trace and reconstruct accretion maps in an unprecedented level of detail. The accretion histories for the two most massive clusters are shown in Figure <ref>. Most massive cluster is visible slightly to the right from the centre of the picture, while the second most massive cluster - slightly left and down. Particle positions are plotted relative to the centre of mass frame of all clusters in the simulation. Accreted gas particles are plotted at their final locations, and colour-codedwith the time of their accretion. Sink forming locations are plotted as purple dots. Sink movement paths are shown as grey lines.This figure shows the formation process of the two clusters. The first sinks are forming in relatively isolated regions. They form along filaments, and then flow down the filaments. Additional sink formation forms small clusters which grow through accretion, and mergers. Star formation continues along the filament and down to the intersection points where the filaments flows, and accompanying clusters merge to form the final cluster.§ CLUSTER PROPERTIES In the following sections, we analyse the developing properties of the stellar clusters in our simulation. It should be noted that as the simulations do not include feedback, the properties could be affected (). <cit.> uses <cit.> simulations to show that clusters formed with feedback have only slighlty smaller masses, bigger sizes and larger dynamical times.§.§ Mass merger treeWe trace clusters between multiple time steps in order to follow the evolution of cluster properties over their formation histories. The simplest cluster property is its mass. As we know which sinks are members of which cluster, we obtain cluster masses over multiple time steps. Multiple cluster events can occur over time, such as creation, dissolution, merging and splitting. As our simulation is targeting cluster formation process, we naturally have merging processes of two clusters occurring at particular time steps. Plotting the cluster masses over time produces a mass-merger tree diagram (Figure <ref>). The intersection points show major merger events as red, if the child cluster's mass exceeds 30% of the parent's cluster mass. Minor merger events occur when the child's mass is below 30% of the parent cluster's mass and are plotted as blue points. Colours show the total lifetime of the cluster, from its formation until it merges with the parent cluster.Firstly we notice that cluster mass growth and merging continues throughout the simulation. Clusters continue to grow as long as there is surrounding gas, sinks or smaller clusters to be accreted into larger central clusters. Merger events occur most frequently for larger mass clusters, while low mass systems have small numbers of mergers in their histories. Cluster merging also appears to be a channel of significant mass growth over the time for the more massive systems. Large final mass clusters have also much longer merging histories, well traced over ∼2.5 Myr, while small clusters have lifetimes of only 0.5 - 1 Myr. §.§ Mass-radius relation One of the properties of our simulated clusters is their mass-radius relation. This must reflect the formation process in some way and thus should form a good property with which to compare to real clusters. As cluster definition returns a list of cluster members, the characteristic cluster sizes can be determined. We use twice the half-mass radius in order to characterise full cluster sizes. The half-mass radius for each cluster is found by sorting cluster member sinks by their distances from the cluster's centre of mass. We then use cumulative enclosed mass radial profiles to find at what radius half of the cluster mass is enclosed.We plot twice the half-mass radius as a robust measure of the effective size of the cluster that does not suffer from the variation in position or classification of the outermost cluster members. When cluster masses and sizes are determined we get a cluster mass-radius relation. We then plot cluster mass-radius relation over all clusters at two time steps - blue for early times (5 clusters) and red for late times (16 clusters) (Figure <ref>).The diagram shows that more massive clusters also have larger half-mass radii. Low mass clusters, taken to be those with <1000 M_⊙ have half-mass radii of 0.1 pc. On the other hand high mass clusters have half-mass radii of 0.5 - 2 pc. We found clusters between the -10^11 cm^2s^-2 iso-potential line and 100 M_⊙ pc^-3 iso-density line.On one hand this looks as artificial disadvantage of the definition, as clusters are more likely to be traced as spherical systems. But on the other hand these spherical and centrally condensed systems are virialized and do not change their geometrical shape rapidly in time.Several physical processes are important for the mass-radius relation. These include the gravitational collapse that forms the cluster, subsequent merger events and ongoing gas and sink accretion. Gravitational collapse causes the size of the system to decrease. On the other hand, mergers grow the cluster mass but there is also an increase in the combined cluster size.Blue solid line shows mass radius relation from fitting observational data of star forming clumps by <cit.>. The line appears to be slightly above our data points. This could be because <cit.> mass-radius relation was fitted for clumps, which are still collapsing. If clumps would continue to collapse towards clusters, their radii would decrease and could match our simulated clusters.We also plot a black solid line in Figure <ref> which shows <cit.> theoretically predicted birth mass-radius relation, obtained by using binary populations in clusters. <cit.> mass-radius relation is derived for the densest collapse state of the bulk young stellar population in the cluster, and how these limit the binary statistics. This indirect measurement of the mass-radius relationship is powerful but inherently makes assumptions as to the natal binary properties. Subsequent dynamics and tidal evolution will likely affect the cluster properties ().§.§ Cluster angular momentum Due to the self consistent initial conditions used in this study, we can make a first estimate of the angular momentum of the newly formed clusters. Figure <ref> shows specific angular momenta of these clusters as a function of their masses at early (blue points) and late (red points) time steps. We use all members of the cluster, relative to the centre of mass of the system when calculating angular momenta.We note that larger mass clusters also have larger specific angular momenta. This could be a result of collecting gas and hence angular momentum from larger scales. Merging processes in clusters involve contributions from larger scales. Merging two clusters results in a jump in this diagram towards larger masses and larger specific angular momentum. Even if Figure <ref> shows higher specific angular momentum, the rotational contribution is only several to several ten percent. §.§ Mass growth of stellar clusters Accretion of stars () and gas () from the environment has been investigated in pre-existing clusters. Here we address what contributes mostly in forming different mass clusters - accretion of stars or gas. In order to measure the accretion of gas into the cluster, we follow the change of mass of the sinks that are already in the cluster. In Figure <ref>, we plot this as a fraction of the cluster mass and as a function of the total cluster mass. The first thing we notice is that clusters with low masses appear to have only a small fraction of their mass being due to gas accretion. This is partially by definition in that the cluster first "forms" when six or more sinks are sufficiently close, and hence have a minimum mass to which gas accretion does not contribute. Nevertheless, we see that the fraction of the total mass contributed by gas accretion increases with cluster mass. At high cluster masses, direct accretion of gas is seen to contribute nearly half the total mass of the system.Secondly, we notice that mass gain from gas accretion inside clusters is always less than 40 - 50 % of the total cluster mass (This neglects the gas accretion onto the initial cluster formation consisting of a minimum of several hundred solar masses). The diagram clearly shows that gas accretion on clustered sinks has contributed mostly for large mass clusters.If the cluster is not accreting any sinks but only grows by gas accretion on its existing sinks, then it moves upwards to larger accreted gas mass over cluster mass ratios and also larger cluster masses. This is visible as a forest of parallel trails going upwards in Figure <ref> for low mass clusters at 2-4 Myr. If cluster accretes sinks or other clusters, they "jump" downwards, which is visible for larger clusters at later times (4-6 Myr). §.§ Cluster age spreadsThe merging process leads not only to a growth in mass, but also to a naturally larger age spread as the resultant clusters are formed by mixing different systems formed in different environments. Figure <ref> shows the accretion histories, in terms of accretion time versus the radius at which particle has been accreted. Here we plot only three most massive final clusters and trace their sinks backwards in time to where they formed. The common centre of mass for the system is calculated from all particles which make up the final mass of the cluster.The sinks can be seen to consistently move to small radii, indicating continuous cluster collapse. We see that the cluster collapse is continuous over all time steps as sinks are continuously moving towards smaller radii. We also see that accretion takes place over several Myrs, in the absence of feedback. While feedback is often assumed to halt ongoing star formation, simulations show that star formation can continue due to the channelling of feedback away from the dense star forming gas (). The large dispersion in accretion times of various sinks originating in different regions produces significant spreads in stellar age in the final clusters. Smaller sub-clusters, which are visible as groups of paths, have smaller age dispersions. On the other hand, smaller clusters, which have yet to merge, display smaller age dispersions. In Figure <ref> we plot mean stellar ages for each cluster as a function of cluster mass and time. Stellar ages for each sink are calculated as a mass weighted average of accretion times for already accreted gas particles. This gives, for each sink, its stellar age, which can be different from the time since the sink first formed. The stellar age is smaller than sink age due to more recent accretion events. As we do not fully resolve star formation, we cannot distinguish whether this ongoing accretion could be forming additional stars, or accreting into pre-existing stars. Ongoing accretion onto young stars can also significantly reduce their apparent ages (). We noticed that in our simulation stellar ages are on average 2/3 that of sink ages. Finally the cluster mean stellar age is just an average of stellar ages for all its members. Figure <ref> show the visible trails of individual clusters over different times. At early times, sinks in clusters are accreting intensively and thus, their mean stellar ages increase slowly.Once accretion in the cluster stops (as occurs when most material is accreted), the mean stellar age starts to increase rapidly and cluster mass growth slows down. During this phase, the tracks can be seen to be almost vertical in Figure <ref>. There is also a visible trend with mean stellar ages for early times slightly increasing for higher mass clusters. Stellar age dispersion similarly increases for more massive clusters, as they have experienced more mergers, bringing together stars which have formed at different times in a wider variety of environments.We also show stellar age dispersions in Figure <ref>, which are calculated as an age dispersion of all accretion events over all cluster members. The diagram shows a clear trend that stellar age dispersion increases for higher mass clusters. This is in a good agreement with merging scenario, as more massive clusters have experienced more mergers, bringing together stars of different ages from a wider variety of environments.§ CONCLUSIONS We performed an in depth analysis of how stellar clusters form in large Galactic scale simulations that resolve individual cluster forming regions, but neglect feedback processes. Galactic spiral shocks assemble clouds in the ridge-like structures, where we measured a stellar age gradient as the shock approaches one side of the ridge earlier than another. Older clusters are found in the regions that entered the spiral shock earlier. Younger clusters, and ongoing star formation, are associated with regions that more recently entered the spiral shock and hence are also associated with dense gas clouds.Our analysis relies on a physically based cluster definition, which uses local and enclosed gravitational potentials in order to separate individual clusters. We noticed that a robust physical cluster definition can be one of the most vital steps before determining further physical properties and relations for stellar clusters.The Lagrangian nature of SPH allowed us to trace individual cluster formation and accretion histories over time. We reconstructed accretion maps, showing details of how individual star forming clumps are moving in global gravitational potential, including where and when star formation is occurring. Clusters appear to form in separate local regions, which undergo collapse and experience frequent merging process over the time. We produced a cluster mass merger tree diagram which show that merging is an important process in massive cluster formation. Merging also produces stellar age spreads up to 1 Myr as material comes from different environments.We include predicted cluster mass-radius relation showing how higher mass clusters are expected to be substantially larger. Our smallest resolved 1000 M_⊙ clusters show their half-mass radii of 0.1 - 0.2 pc while 20000 M_⊙ clusters have 1 - 2 pc half-mass radii. In order to resolve smaller mass clusters, higher resolution simulations would be needed. Analysis of angular momenta show that more massive clusters have higher specific angular momentum, which can be attributed to having to accrete from significantly larger volumes and hence higher velocity dispersions. We also address what drives cluster mass growth of different mass clusters. Less massive clusters appear to be growing by assembling locally formed sinks, while more massive clusters have powerful global gravitational potentials, which allow surrounding gas to be efficiently channelled to the cluster centres, accelerating accretion.§ ACKNOWLEDGEMENTS RS and IAB acknowledges funding from the European Research Council for the FP7 ERC advanced grant project ECOGAL. This work used the DiRAC Complexity system, operated by the University of Leicester IT Services, which forms part of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment is funded by BIS National E-Infrastructure capital grant ST/K000373/1 andSTFC DiRAC Operations grant ST/K0003259/1. DiRAC is part of the National E-Infrastructure.We thank William Lucas, Felipe Gerardo Ramon Fox, Duncan Forgan, Rowan Smith and Claudia Cyganowski for helpful discussions and comments.mnras | http://arxiv.org/abs/1709.08948v1 | {
"authors": [
"R. Smilgys",
"I. A. Bonnell"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170926113417",
"title": "Formation of stellar clusters"
} |
NY Her: possible discovery of negative superhumps Sosnovskij, A.^1; Pavlenko, E.^1; Pit, N.^1; Antoniuk, K.^1 Crimean Astrophysical Observatory, Nauchniy, Crimea, Russian Federation, 298409. ^†[email protected], [email protected] NY Her NY Her, dwarf novae, cataclysmic variables, photometry We presents result of CDD photometry for SU UMa dwarf nova NY Her during 6 nights in June 2017 when object was in quiescence. Light curves clearly show strong amplitude variations in a range of 07-11. Time series analysis revealed a period 0.07141(5) d, that we identified as the period of possible negative superhumps of NY Her. § INTRODUCTION Cataclysmic variables (CVs) are composed of a white dwarf (WD) as the primary star and a Roche-lobe filling red (or brown) dwarf as the secondary star which supplies matter from the inner Lagrangian point. This matter forms an accretion disc around the primary star in the case of a non-magnetic white dwarf. The accretion disc is the main source of variability on large time intervals from minutes to hundreds of days. SU UMa-type dwarf novae are a class of CVs showing two types of outbursts: superoutbursts and normal outbursts with amplitudes of 20-80 (Warner, 1995). During superoutburst these objects exhibit light variations called "positive superhumps" (Osaki, 1996). The observed period of the superhumps is a few percent longer than the orbital period of the system. On the other hand, some SU UMa stars show variations shorter than the orbital period, that are called "negative superhumps" (Hellier, 2001), visible mostly in quiescence and in some occasions in the normal outbursts and superoutbursts (Harvey et al. 1995, Pavlenko et al., 2010, Oshima et al. 2014). NY Her (α=17:52:52.60 δ=+29:22:18.8) was originally discovered by Hoffmeister (1949) as a Mira-type variable. Kato et al. (2013a) identified this object as the SU UMa-type dwarf nova with a short supercycle.Using superoutburst data taken by the ASASSN team, Poiner's observations and results of follow-up international campaign, Kato at. al. (2017) revealed an updated positive superhump profile with a period of 0.075525 d and much smaller amplitude (010 mag) than most of SU UMa-type dwarf novae with similar periods of superhumps (or orbital) have. They identified a possible supercycle of ∼63.5 d and that the duration of the superoutbursts was 10 d. The supercycle length of ∼63.5 d is between the supercycle length of the ER UMa-type DN novae subclass (Hellier, 2001; Kato et al., 2013b) that is distinguished by the shortest (20-50d) supercycles and ordinary SU UMa stars which have supercycles longer than 100d. The superoutburst duration of 10 d is much shorter than the duration of superoutbursts seen in the ER UMa-type dwarf novae. Kato et al. (2017) noticed that NY Her may be classified as unique object with a short supercycle and a small superhump amplitude despite the relatively long P_sh and could have the negative superhumps because of infrequent normal outbursts during relatively short supercycle. This motivated us to examine this prediction by photometric investigation of NY Her during quiescence in June 2017. 10cmf1.epsUnfiltred photometry for NY Her for two nights: 19-20 June, 2017. The smaller humps and small dips are marked by red and blue colors correspondently. f1.epsNy Herphotometry § OBSERVATIONS The photometric CCD observations of NY Her were carried out during 6 nights in June 2017 at the Crimean Astrophysical Observatory (CrAO) in unfiltered light, giving a system close to the R_c band in our case, at two telescopes: 2.6–m ZTSh with APOGEE Alta E-47 and 1.25–m AZT-11 with ProLine PL230. Our priority was time series analysis with high time resolution in respect to the multicolor observations. The standard aperture photometry (de-biasing, dark subtraction and flat-fielding) was used for measuring of the variable and comparison star USNO –B1 1193-0272323 (R=17.97) (Monet et al., 2003). The accuracy of a single brightness measurement strongly depended on the telescope, exposure time, weather condition and brightness of NY Her, and reached 001-–003 for 60 s exposure (ZTSH) and 008-015 for 180 s exposure (AZT-11). 9cmf2.epsUpper: Periodogram for combined data from 6 different nights. Position of the positive superhump period (Kato et al., 2017) is shown by blue dotted line. Lower: data folded on the 0.07141 d period. Original data are shown by gray circles. Black squares denote the mean points. f2.epsNy Herperiodogram § DATA ANALYSIS AND DISCUSSION During the quiescent state the brightness of NY Her varied between 185 and 198. The example of two original light curves is shown in Fig.1. As seen in these light curves, the profile changes from night to night. The light curves clearly show variability with a period ∼1.7 h and strong amplitude variations in a range of 07-11.At first night (Fig.1, upper frame) one could see the two humped profile with different height and small dip in bigger hump. At the second night (Fig. 1, lower frame) the light curve profiles become more smoothed, the smaller hump is no longer visible. To search for precise periodicity we have done the periodogram analysis using the Stellingwerf method (Stellingwerf, 1978) implemented in ISDA package (Pel't, 1980). The accuracy of trial periods as well as Abbe statistic, also known as Lafler-Kinman statistic (Lafler and Kinman, 1965)was calculated using ISDA package (Pel't, 1980).Before starting the analysis, we substracted the long term trend. The strongest peak points to the period 0.07141(5) d, surrounded by daily aliased peaks. The periodogram and phase diagram for the most significant period are shown in Fig. 2. Original data show larger scattering in minimum caused by both larger errors and intrinsic variability and smaller one in maximum. The mean light curve displays a flat minimum lasting 0.4 period and amplitude about 07. As empirically established relation shows, all known SU UMa stars with related Porb and Psh are located around equation line: ϵ=P_sh/P_orb – 1 = 0.001(4) + 0.44(6)P_orb (Kato et al., 2009). The measured period (of NY Her in quiescence) cannot be an orbital one, because in this case ϵ=0.057 is situated higher this line (taking into account a scatter of observation around this line). According to this relation, the corresponding orbital period should be slightly larger, and be located in that scattering strip between 0.0722-0.0736 d, withϵ=0.025-0.045. We suggest that 0.07141(5) d period is the period of negative superhumps of NY Her according to Kato's prediction. However a small probability that this period could be interpreted as the orbital one also cannot be neglectedsince the eclipsing SU UMa dwarf nova HT Cas has near the same large epsilon (Kato et al., 2009).Further observations of NY Her aimed at finding the orbital period are necessary for the final identification of the brightness modulation during its quiescence in June 2017. Acknowledgement: We are grateful to Sklyanov A.S. from Kazan Federal University for reading ad discussion the paper and to anonymous referee for valuable comments. Harvey, D., Patterson, J.,1995, PASP,107, 105510.1086/133662 Hellier, C., 2001, Cataclysmic variable stars: how and why they vary, Springer-Verlag London. ISBN 978-1-85233-211-2 Hoffmeister, C., 1949,Erg. Astron. Nachr., 12, 12 Kato, T. et al., 2009, PASJ,61S, 395 10.1093/pasj/61.sp2.S395 Kato, T. et al., 2013a, PASJ,65, 23 10.1093/pasj/65.1.23 Kato, T., Nogami, D., Baba, H., et al. 2013b, arXiv 1301.3202 Kato, T. et al., 2017, arXiv170603870 accepted to PASJ. Lafler, J. and Kinman, T.D., 1965, ApJ Suppl., 11, 216 10.1086/190116 Monet, D. et al., 2003, AJ, 125,98410.1086/345888 Osaki, Y. 1996, PASP, 108, 39 10.1086/133689 Ohshima, T. et al., 2014, PASJ,66, 67010.1093/pasj/psu038 Pavlenko, E. P. et al., 2010, AIPC,1273, 320 10.1063/1.3527832 Pel't, Ya., 1980Frequency Analysis of Astronomical Time Series Tallinn:Valgus Stellingwerf, R.F., 1978, ApJ, 224, 953 10.1086/156444 Warner, B. 1995, Cataclysmic Variable Stars (Cambridge:Cambridge University Press) | http://arxiv.org/abs/1709.08952v2 | {
"authors": [
"A. Sosnovskij",
"E. Pavlenko",
"N. Pit",
"K. Antoniuk"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170926114825",
"title": "NY Her: possible discovery of negative superhumps"
} |
the contact email is [email protected] and is the only one which should appear on the journal version1School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK 2Astrophysics Research Centre, School of Mathematics & Physics, Queen's University Belfast, University Road, Belfast, Northern Ireland, BT7 1NN 3Solar Physics Laboratory (Code 671), Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 4Department of Physics Catholic University of America, 620 Michigan Avenue, Northeast, Washington, DC 20064, USA 5ESA Directorate of Science, Operations Department, c/o NASA/GSFC Code 671, Greenbelt, MD 20071, USA 6ADNET Systems, Inc. In this Letter we report the detection of chromospheric 3-minute oscillations in disk-integrated EUV irradiance observations during a solar flare. A wavelet analysis of detrended Lyman-alpha (from GOES/EUVS) and Lyman continuum (from SDO/EVE) emission from the 2011 February 15 X-class flare (SOL2011-02-15T01:56) revealed a ∼3-minute period present during the flare's main phase. The formation temperature of this emission locates this radiation to the flare's chromospheric footpoints, and similar behaviour is found in the SDO/AIA 1600Å and 1700Å channels, which are dominated by chromospheric continuum. The implication is that the chromosphere responds dynamically at its acoustic cutoff frequency to an impulsive injection of energy. Since the 3-minute period was not found at hard X-ray energies (50–100 keV) in RHESSI data we can state that this 3-minute oscillation does not depend on the rate of energization of non-thermal electrons. However, a second period of 120 s found in both hard X-ray and chromospheric emission is consistent with episodic electron energization on 2-minute timescales. Our finding on the 3-minute oscillation suggests that chromospheric mechanical energy should be included in the flare energy budget, and the fluctuations in the Lyman-alpha line may influence the composition and dynamics of planetary atmospheres during periods of high activity. § INTRODUCTION Quasi-periodic pulsations (QPPs) are widely reported in emission from solar flares. These are regular fluctuations in the flare radiation intensity, which are very clear in hard X-rays (HXRs) and radio waves generated by non-thermal electrons (,and ), but also detected over a wide range of wavelengths[See <http://www2.warwick.ac.uk/fac/sci/physics/research/cfsa/people/valery/research/qpp/> for a comprehensive list of relevant publications, and see <https://aringlis.github.io/AFINO/> for an automatically updated list of GOES SXR QPP observations.]. QPPs in non-thermal signatures are widely interpreted as either revealing the magneto-hydrodynamic (MHD) oscillation modes of the flare's magnetic environment, or reflecting an oscillatory driver for electron acceleration. There have been relatively few reports of quasi-periodic behavior in the emission from the thermal plasmas of the solar atmosphere, including the chromosphere, in response to flare energization. QPPs with periods of ≲1 minute have been found in UV/EUV/SXR flare emission (e.g. , ) perhaps signaling MHD oscillations in post-flare coronal loops, or coronal loop-filling by heated plasma expanding from a periodically-heated chromosphere. <cit.> found a period of 171 s in Interface Region Imaging Spectograph (IRIS; ) observations of chromospheric flare C1 line emission, and interpreted it as chromospheric heating due to quasi-periodic injection of non-thermal electrons. Other studies have found fluctuations of 1–4 minutes in flare chromospheric emission that are interpreted as evidence for episodic reconnection driven by leakage of slow-mode oscillations from an underlying sunspot <cit.>. Different pulsation periods may be present at different phases of the flare. For example, <cit.> found that in the X-class flare SOL2013-10-28 the period of the observed radio and X-ray QPPs changes with short period pulsations (∼40s) dominating during the impulsive, energy-release phase, and longer-period pulsations (∼80s) more prevalent during the gradual, decay phase. Similarly <cit.> found the period of QPPs to change from ∼25 s to ∼100 s during an the X-class flare SOL2013-05-14. This may reflect a change in the dominant driver of pulsations, e.g. from periodic electron acceleration during the impulsive phase, to MHD oscillations of hot post-flare loops in the decay phase.It is well known that the (non-flaring) solar chromosphere oscillates with a dominant period of around 3 minutes. The decrease of the average period of oscillations in the solar atmosphere from about 5 minute in the photosphere to 3 minute in the chromosphere (e.g. ) is due to the strong spatial damping of evanescent waves with height, whereas the 3 minute oscillations at the cut-off frequency are not damped. <cit.>, <cit.>, <cit.>, and <cit.> and others have shown theoretically that any disturbance in the chromosphere, whether impulsive or quasi-periodic, causes it to oscillate at its acoustic cutoff frequency(the “Lamb effect”;).For the chromosphere this cutoff frequency is 5.5 mHz, corresponding to a 3-minute period. Convincing observational evidence for impulsive excitation of oscillations at the acoustic cutoff frequency was recently presented by <cit.> who detected periods of 2.7–3.3 minutes in response to a strong downflow event detected in +0.5Å. This was seen in spatially-resolved chromospheric (Mg2, Ca2) and transition region (C2, Si4) lines measured by IRIS. The authors concluded that these oscillations represent gravity-modified acoustic waves generated by an impulsive disturbance in the chromosphere. In this Letter we present evidence that oscillations with periods around 3 minutes are present in the chromospheric emission during the impulsive phase of an X-class solar flare. We show that the 3-minute period is very prominent in the hydrogen Lyman-alpha line, and the hydrogen Lyman continuum, both characteristic of chromospheric plasma at around 10,000K. However, the 3-minute period is not seen in HXRs (which show a dominant 2-minute signal) indicating that this 3-minute signature is not due to quasi-periodic electron injection. In Section <ref>, the datasets used and the analysis techniques employed are described. Section <ref> outlines the findings, while Section <ref> provides some discussion and interpretation§ OBSERVATIONS AND DATA ANALYSISOne of the most studied solar flares of Solar Cycle 24 is the X2.2 flare that occurred on 2011 February 15 (SOL2011-02-15T01:56). It was the first X-class flare of the cycle and was observed by a number of different instruments. The top panel of Figure <ref> shows the lightcurves of 25–50 and 50–100 keV hard X-ray (HXR) emission from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI; ) at 4 second cadence, and of 1–8 Å SXR emission from the X-Ray Sensor (XRS; ; dashed curve) on the Geostationary Operational Environmental Satellite (GOES15; ) at 2 second cadence, for 30 minutes around the rise and peak of the X-class flare. The middle panel shows the chromospheric response in both the Lyman-alpha line at 1216Å (hereafter, ; red curve) from the E-channel of the EUV Sensor (EUVS-E) on GOES, and the Lyman continuum blue-ward of 912Å (hereafter, LyC; blue curve) from the EUV Variability Experiment (EVE; ) on the Solar Dynamics Observatory (SDO; ) at 10.24 s and 10 s cadence, respectively. For comparison, the time series of 1600Å and 1700Å emission from the Atmospheric Imaging Assembly (AIA; ), also on SDO, areshown in the bottom panel. These data were taken at 24 s cadence. During this event both channels saturated around the peak of the flare. Following <cit.>, a 200”×200” area around the flare site was integrated over in each channel to ensure no loss of counts when deriving the lightcurves. The SDO/EVE MEGS-P photometer is intended to recordirradiance, but <cit.> have identified unexpected behavior in this signal. The EUV Sensor (EUVS) on GOES15 was therefore used instead. The E-channel on EUVS spans theline in a broadband (∼100Å) manner similar to MEGS-P, and was operational during the 2011 February 15 flare. Thelightcurves plotted in the middle panel of Figure <ref> were generated using Version 4 of the data. This version of the data has been corrected for degradation and calibrated using SORCE/SOLSTICEmeasurements.The MEGS-B (Multiple EUV Grating Spectrograph) component of EVE obtains spectra over the 370–1050Å range at 10 s cadence and 1Å resolution. Aside from the numerous spectral lines that occupy this wavelength range, the most prominent feature is that of the free–bound LyC with a recombination edge at 912Å. Time profiles of this continuum emission during flares from EVE were first presented by <cit.>, and followed up by a study of the energetics of this and other chromospheric emissions during the 2011 February 15 flare <cit.>. In order to isolate the continuum emission from the overlying emission lines, <cit.> employed a RANdom Sample Consensus (RANSAC; ) technique that treats the lines as outliers over a chosen wavelength range (see appendix offor more details). Integrating under this fit at each time step allowed the lightcurve of LyC in middle panel of Figure <ref> to be derived. The same technique was applied to Version 5 of the EVE data in this study.To highlight the periodic behavior in theand LyC emissions, the lightcurves were detrended using a Fast Fourier Transform (FFT) filter. A cutoff period of 400 s (2.5 mHz) was chosen for this analysis, but the choice of frequency was not found to affect the derived period (seeSection <ref>). The resulting periodic behavior in bothand LyC emission is shown in Figure <ref> (red and blue curves, respectively). The detrended 1700Å profile is also shown for comparison. There is a remarkable agreement between all three detrended profiles in terms of the phase of each pulsation. The coincidence between profiles taken by three different instruments implies that the bursts are genuinely solar in origin, and that they originate in the chromosphere. The similarity is further evidence that the spikes seen in LyC are not an artifact of the fitting algorithm. The raw lightcurves ofand LyC emission are shown in the top panels of Figure <ref>. Overplotted are the low-pass (400 s cutoff period) Fourier filtered time series in cyan. Having removed the smoothly varying component of the flare time profile for bothand LyC, the resulting detrended profiles are shown in the middle panels of Figure <ref>. The final step was to apply the standard wavelet analysis technique of <cit.> to determine the period(s) of the pulsations during the flare. The corresponding wavelet spectra per unit time are shown in the bottom panels of Figure <ref>.§ RESULTSThe wavelet power spectrum for the detrendedtime series in the bottom left panel of Figure <ref> shows enhanced power over a broad range of periods during the rise and peak of the flare (01:45–02:00 UT). The bulk of this power is evident at periods around 100-200 s. There is also a similar distribution of power in frequency and time in the LyC spectrum (bottom right panel of Figure <ref>). Both spectra also show enhanced power at higher frequencies around the time of the flare onset. The enhanced power around 180 s (5.5 mHz; horizontal dashed white lines in both bottom panels) corresponds to the acoustic cutoff frequency in the chromosphere. The 180 s period is not apparent in the quiescent, full-disk signal from GOES/EUVS-E or SDO/EVE, presumably due to the incoherence of the signal in disk-integrated emission, although longer period (300 s) oscillations are apparent in non-flaring regions of AIA 1600Å and 1700Å images (see below). The flare therefore seems to either initiate the oscillation itself, or it amplifies or enhances a pre-existing oscillation. In the latter case, the flare may either drive a pre-existing dynamical behavior, or it changes the properties of the radiating gas so that the variations in intensity become more visible (see Section <ref> for further discussion). It is also worth noting that this 3-minute power is readily apparent in wavelet analysis of the rawdata with no detrending applied.The two UV channels on SDO/AIA - 1600Å and 1700Å - image the solar chromosphere at 24 s cadence. While this emission is largely continuum <cit.> rather than hydrogen line emission, it is worth including given that any flare emission (ribbons) should come from the same spatial location as theand LyC emission, although they may originate at different depths in the flaring atmosphere. These data were again detrended using an FFT filter (top panels of Figure <ref>). The bottom two panels of Figure <ref> show the resulting wavelet power spectra for the two channels. The 3-minute oscillation is apparent, as well as even stronger power at ∼120 s. Evidence for 3-minute oscillations was also found in theline from MEGS-B, although no such oscillations were detected in the higher order Lyman lines (, , ).A common explanation for many QPP observations is that they are simply due to bursty energy release and particle acceleration on the measured time scales (e.g. ). However, performing a wavelet analysis on the 50–100 keV time profiles from RHESSI for this event did not reveal any power at 3-minute time scales. This not only strengthens the case that the chromospheric pulsations are a genuine oscillatory response that is independent of the energy injection rate, but it also does not support the argument by <cit.> and others that the underlying sunspot oscillation could have been responsible for initiating the energy release and particle acceleration in the first place. Had this been the case then the 3-minute pulsations would have been evident at each step of the energy transport process, including in the HXRs. This is demonstrated by comparing the global wavelet power spectrum (integrated over the duration of the flare; 01:30–02:30 UT) of the HXRs with those of the chromospheric emission, as shown in Figure <ref>. Theprofile shows a strong signature at 180 s (which is independent of the choice of Fourier filter frequency as denoted by the different colored curves), as does LyC to a lesser extent. Both AIA UV channels show enhanced power at around 200 s, while the 50-100 keV emission shows no such power. The two AIA channels, LyC, andalso all show enhanced power at 120 s to some degree. This does correspond to the peak in the the HXR power, indicating that the chromospheric response on these timescales is more likely due to accelerated particles. The 3-minute period is a separate phenomenon which is taken to be an oscillatory response of the chromosphere at its acoustic cutoff frequency.§ CONCLUSIONSObservational evidence for 3-minute oscillations inand LyC emission from full-disk irradiance measurements during an X-class solar flare is presented. This study supports the notion that when the chromosphere is impulsively disturbed, compressible waves with periods around the acoustic cutoff are generated. It is the recurring compression and expansion of these waves that leads to the oscillation in intensity. The impulsive disturbance may have been caused by the injection of energy, probably in the form of non-thermal electrons, and the amount of the injected energy is likely to have been much larger than that required for sustaining oscillations in non-flaring regions. The oscillation in the flaring region could be identified from the full-disk Lyman alpha and continuum data since the emission from the flaring region is much stronger than the non-flaring regions. <cit.> reported a similar phenomenon in the chromosphere in response to an impulsive downflow event as observed in chromospheric emission lines by IRIS. Such oscillatory responses have been predicted for decades (e.g. ) and this may be the first report of such a disturbance during a major solar flare. Numerical models by <cit.> and <cit.> demonstrate that any impulsive disturbance to a quiescent chromosphere can be found to generate acoustic oscillations around the cutoff frequency; a pre-existing oscillation is not required for a flare-driven wave to exist. However, it is known that such oscillations do exist around non-flaring active regions as observed in AIA 1600Å and 1700Å images by <cit.>. Therefore the possibility that the flare somehow “amplifed” the quiet-Sun 3-minute oscillations cannot be excluded based on the available data.It is remarkable that these oscillations show up with a significant amplitude in the full-sun irradiance observations. This may provide a clue to estimate the energy contained in the pulses, assuming that this oscillatory signal comes only from a limited area. Although the 3-minute oscillations appear to be independent of the rate of electron injection, the acoustic waves may transport a significant amount of mechanical energy. <cit.> claimed that theline alone can radiate away ∼10% of the non-thermal energy deposited in the chromosphere.is also a known driver of disturbances in planetary atmospheres (e.g. the ionospheric D-layer on earth; ), and such oscillations may play a role in changing the atmospheric composition and dynamics during periods of high activity. Withphotometers currently on SDO, GOES13-15, SORCE, the Mars Atmosphere and Volatile EvolutioN (MAVEN; ), and scheduled for the next generation of GOES satellites, as well as theimager on Solar Orbiter, knowledge of the behavior of this emission during flares could be important when interpreting future science results. The authors would like to thank Drs. Mihalis Mathioudakis, Hugh Hudson, and Andrew Inglis, and Laura Hayes, for many insightful discussions on this subject, as well as the anonymous referee for clarification that greatly improved the quality of this manuscript. ROM would like to acknowledge support from NASA LWS/SDO Data Analysis grant NNX14AE07G, and the Science and Technologies Facilities Council for the award of an Ernest Rutherford Fellowship (ST/N004981/1). LF would like to acknowledge support from STFC Consolidated Grants ST/L000741/1 and ST/P000533/1.36 natexlab#1#1[Allred et al.(2005)Allred, Hawley, Abbett, & Carlsson]allr05 Allred, J. C., Hawley, S. 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Edited by Fineschi, 6689, 66890K[Woods et al.(2012)Woods, Eparvier, Hock, Jones, Woodraska, Judge, Didkovsky, Lean, Mariska, Warren, McMullin, Chamberlin, Berthiaume, Bailey, Fuller-Rowell, Sojka, Tobiska, & Viereck]wood12 Woods, T. N., et al. 2012, , 275, 115 | http://arxiv.org/abs/1709.09037v1 | {
"authors": [
"Ryan O. Milligan",
"Bernhard Fleck",
"Jack Ireland",
"Lyndsay Fletcher",
"Brian R. Dennis"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170926142633",
"title": "Detection of 3-Minute Oscillations in Full-Disk Ly$α$ Emission During A Solar Flare"
} |
.tifpng.png`convert #1 `basename #1.tif`.png 1.5cm #1Eq. (<ref>)#1Fig. <ref>= footnoteIPIR1cm a]E. Gotsman,a,b] and E. Levin[a]Department of Particle Physics, School of Physics and Astronomy, Raymond and Beverly SacklerFaculty of Exact Science, Tel Aviv University, Tel Aviv, 69978, Israel[b]Departemento de Física, Universidad Técnica FedericoSanta María, and Centro Científico-Tecnológico de Valparaíso, Avda. Espana 1680, Casilla 110-V,Valparaíso, [email protected]@post.tau.ac.il, [email protected] In the paper we discuss the angular correlationpresent inhadron-hadron collisionsat largerapidity difference (y_12 ≫ 1). We find thatin theCGC/saturation approachthelargest contribution stems fromthe density variation mechanism. Ourprincipal resultsare that the odd Fourierharmonics(v_2n+1), decrease substantially as function of y_12, while theeven harmonics (v_2n ), increaseconsiderablywith a growth ofy_12. TAUP - 000/17 December 30, 2023 Azimuthal angle correlations at large rapidities: revisiting density variation mechanism [ December 30, 2023 ===========================================================================================§ INTRODUCTIONIn this paper we address the problem of the azimuthal angle correlationsof two hadrons with transverse momenta p⃗_T1 andp⃗_T2 and rapidities y_1 and y_2, at large valuesof y_12≡ | y_1 - y_1| ≫ 1/. Our main theoreticalassumption is that these correlations stem frominteractionsin the initial state.We are aware that, unlike rapidity correlationswhich at large rapidities are originated from the initial stateinteractions due to causality reasons<cit.>, a substantialpart of these correlations couldbe due to the interactions in thefinalstates<cit.>. On the other hand, it has been demonstrated that atsmall rapiditydifferencey_12 < 1 the interactions in the initialstate <cit.>gives the value of the correlations, whichdescribe the major partof the experimentally observed correlations<cit.>. In this paper we concentrate our efforts,oncalculating the long range rapiditypart of angularcorrelationswith large value of the rapidity difference y_12.All previous calculations,assumedthat y_12<1<cit.>. It turns out, that in this kinematic region, the mainsource of the azimuthal angle correlations, is the Bose-Einsteincorrelationsofidentical gluons, which corresponds to the interference diagram intheproduction oftwo partonic showers.Intuitively, we expect thatthe correlations in the process, where two different gluons areproduced from twodifferent partonic showers,should not depend on the difference ofrapidities (y_12), as well as on the values ofy_1 and y_2. Using the AGK cutting rules <cit.>[In the frameworkof perturbativeQCDfor theinclusive crosssections, the AGK cutting rules were discussed and proven inRefs.<cit.>. However, inRef.<cit.> itwas shown that the AGK cutting rules are violated fordouble inclusiveproduction. This violation is intimately related to the enhanced diagrams<cit.> and to the production of gluon from triple Pomeron vertex.It reflects the fact that different cuts ofthe triple BFKLPomeronvertexwith produced gluon,lead to different contributions. Recall, that we do notconsider such diagrams.]one can prove that the two gluon correlations can be calculatedusing the Mueller diagrams<cit.> of becor. The diagrams of becor lead tocorrelations which donot depend on y_1 and y_2, butonly for y_12 ≪ 1.For large y_12 the contributionsof becordecrease.The main goal of this paper to find thecontributions which survive at large y_12 (y_12 ≫ 1).At large y_12, we have to take into account the emission gluons,with rapidities y_2< y_i < y_1 which transform the Muellerdiagram of becor-bto more general diagrams of mudia. Thegeneral features of becor-bis that the lower Pomerons carrymomenta Q⃗_T + p⃗_12 and -Q⃗_T - p⃗_12 with p⃗_12 = p⃗_T1 - p⃗_T2.Q⃗_Tdenotes the momentum along the BFKL Pomeron. After integration overQ_T,we obtain p_12∼ 1/R_h, where R_h is thesize of the target(projectile), which has a non-perturbative origin. Roughly speaking, the correlation function turns out to be proportionalto G p_12, where G denotes the non-perturbative form factor ofthetargetor projectile <cit.>. This conclusionstems from the value of the typical Q_T for the BFKLPomeron, which is determined by the size of the largest dipoles in thePomeron. mudia does not have these features. We will show that theazimuthal angle correlationsoriginatefrom the integration overQ⃗_T(see mudia), due to the structure of the vertices ofemission of the gluons with p⃗_T1 and p⃗_T2, whichhave contributions proportional to (p⃗_T1·Q⃗_T)^n(p⃗_T2·Q⃗_T)^n. Recall, that thesekind ofvertices,are the only possibilities to obtainangular correlations in theclassical Regge analysis<cit.>. This mechanism forazimuthalangular correlations was suggested in Ref.<cit.> (see alsoRefs.<cit.>), and in thereviewof Ref. <cit.>, was called, the densityvariation mechanism. The paper is organized as follows. In the next section we discussthe contribution of the diagram of mudia in the momentumrepresentation. In theremainder of the paper, we will use themixedrepresentation: the dipole sizes and momentum transferred (Q_T),which will be introduced in section 3. Section 4 is devoted to thediscussion of the single inclusive production in the Colour GlassCondensate (CGC)/saturation approach.The double inclusive productionis considered in section 5, in which the rapidity dependence of the masterdiagram of mudia will be calculated. In section 6, we estimate the angularcorrelationfunction and Fourier harmonics v_n, andwe present ourprediction for dependence of v_n on the differenceof rapidities(y_12). In section 7 we draw our conclusions and outline problems for future investigation.§ CORRELATIONS IN THE MOMENTUM REPRESENTATION The double inclusive cross section of mudia takes the following formd^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2mudia=2 C_F /(2 π)^2 ^2∫d^2 k_T/( 2 π )^2 d^2 k'_T/( 2 π )^2 d^2 Q'_T/( 2 π )^2 d^2 Q_T/( 2 π )^2d^2 Q”_T/( 2 π )^2k^2_T( - )^2 ×N Q'_T ϕ^G_H -k⃗_T + Q⃗'_T, k⃗_T; Y - y_1 ϕ^G_Hk⃗_T - Q⃗_T, -k⃗_T + Q⃗_T - Q⃗'_T; Y -y_1Γ_ν -k⃗ + Q⃗_T, p⃗_T1 Γ_νk⃗ - Q⃗_T - Q⃗'_T, p⃗_T1 ×ϕ-k⃗_T , -k⃗_T + Q⃗_T; k⃗'_t + p⃗_T2,-k⃗'_T - p⃗_2T - Q⃗_T; y_12 ϕ- k⃗_T + Q⃗'_T+ p⃗_T1, k⃗_T - p⃗_T1- Q⃗_T - Q⃗'_T; k⃗'_T - Q⃗”_T +Q⃗_T, k⃗'_T- p⃗_T2; y_12 ×N Q”_T ϕ^G_Hk⃗'_T - Q⃗”_T + Q⃗_T,- k⃗'_T - Q⃗_T; y_2 ϕ^G_H -k⃗'_T - Q⃗”_T + Q⃗_T ,k⃗'_T;y_2 Γ_μ -k⃗'_T-p⃗_T2 + Q⃗”_T ,p⃗_T2 Γ_μk⃗'_T - p⃗_T 2,p⃗_T2where ϕ^G_Hk⃗_T,-k⃗ +Q⃗_T', as well as allotherfunctions ϕ of this type,are the correlation functions which atQ'_T=0, give the probability to find a gluon withtransversemomentumk⃗_T in the hadron(nucleus) of the projectile (target). ϕk⃗_T,-k⃗ +Q⃗_T;k⃗'_T,-k⃗'_T+Q⃗_T describes the interaction of two gluons with momentak⃗_T and k⃗'_T,which scatter at momentum transferredQ"_T. N Q'_T is a pure phenomenological form factor thatdescribes the probability to findtwo Pomerons in the projectileor target, with transferred moment Q⃗'_T and -Q⃗'_T.C_F=N^2_c - 1/2 N_c where N_c is the number ofcolours. The Lipatov vertexΓ_μ k_T,p_T1 has the following formΓ_μ k_T, p_T1=1/p_12^2 k^2_Tp_T1,μ - k_T,μ p^2_T1; Using LV we obtain 2 Γ_ν -k⃗ + Q⃗_T, p⃗_T1 Γ_νk⃗ - Q⃗_T - Q⃗'_T, p⃗_T1= 1/p^2_T1-k⃗ + Q⃗_T ^2 k⃗_T - p⃗_T1 - Q⃗_T - Q⃗'_T^2 + -k⃗ +p⃗_T1 - Q⃗_T ^2 k⃗_T - Q⃗_T - Q⃗'_T ^2 - Q'^2_T; 2 Γ_μ -k⃗'_T-p⃗_T2 + Q⃗”_T ,p⃗_T2 Γ_μk⃗'_T - p⃗_T 2 - Q⃗_T,p⃗_T2 =1/p^2_T2 -k⃗'_T-p⃗_T2 + Q⃗"_T ^2k⃗'_T - Q⃗_T ^2+-k⃗'_T + Q⃗”_T + Q⃗_T ^2k⃗ - p⃗_T 2- Q⃗”_T ^2 - Q”^2_T ; We can simplify the master equation (see MAEQ byobserving,that dependence on Q'_T and Q”_T is determinedbythe non-perturbative scale of the projectile(target) structure,whichin MAEQ, is absorbed in the phenomenological formfactors N(Q'_T) and N(Q”_T).Therefore, the typical Q'_Tand Q”_T turn out to be of the order of the soft scaleμ_ soft, which is much smaller that the other typicalmomenta in MAEQ, assuming that P_T1 and P_T2 arelarger than μ_ soft. Introducingμ^2_ soft=∫d^2 Q'_T/(2 π)^2 N Q'_T we can neglect Q'_T and Q”_T in the BFKL Pomeron Green's functions andre- write MAEQ in the form: d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2mudia=2 C_Fμ^2_ soft/(2 π)^2 ^2∫d^2 k_T/( 2 π )^2 d^2 k'_T/( 2 π )^2d^2 Q_T/( 2 π )^2 k^2_T( - )^2 ×ϕ^G_H -k⃗_T , k⃗_T; Y - y_1 ϕ^G_Hk⃗_T - Q⃗_T, -k⃗_T + Q⃗_T;Y - y_1 Γ_ν -k⃗ + Q⃗_T, p⃗_T1 Γ_νk⃗ - Q⃗_T , p⃗_T1 ×ϕ-k⃗_T , -k⃗_T + Q⃗_T; k⃗'_t + p⃗_T2,-k⃗'_T - p⃗_2T - Q⃗_T;y_12 ϕ- k⃗_T + p⃗_T1, k⃗_T - p⃗_T1- Q⃗_T ; k⃗'_T+Q⃗_T, k⃗'_T- p⃗_T2; y_12 ×ϕ^G_Hk⃗'_T+ Q⃗_T,- k⃗'_T - Q⃗_T; y_2 ϕ^G_H -k⃗'_T + Q⃗_T ,k⃗'_T;y_2 Γ_μ -k⃗'_T-p⃗_T2, p⃗_T2 Γ_μk⃗'_T - p⃗_T 2,p⃗_T2 with KERN which takes the following form:2 Γ_ν -k⃗ + Q⃗_T, p⃗_T1 Γ_νk⃗ - Q⃗_T, p⃗_T1=1/p^2_T1-k⃗ + Q⃗_T ^2 k⃗_T - p⃗_T1 - Q⃗_T ^2 + -k⃗ +p⃗_T1 - Q⃗_T ^2 k⃗_T - Q⃗_T^2 - Q^2_T; 2 Γ_μ -k⃗'_T-p⃗_T2 ,p⃗_T2 Γ_μk⃗'_T - p⃗_T 2- Q⃗_T,p⃗_T2 =1/p^2_T2 -k⃗'_T-p⃗_T2^2k⃗'_T- Q⃗_T ^2+-k⃗'_T^2k⃗ - p⃗_T 2 - Q⃗_T ^2 - Q^2_T;At high energies the parton densities ϕ(…; Y)in MAEQ and in MAEQ1, are proportional to expΔ_ BFKL Y for the BFKL Pomeron, whereΔ_BFKL = 2.8 is the intercept of the BFKL Pomeron. Bearing this in mind, one can see, that the interference diagram forthe double inclusive cross sectiondoes not depend on y_1, y_2 or ony_12. The main diagram of becor-a, also does not depend on rapidities,and its expressionhas the following form:d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2becor-a=2 C_F /(2 π)^2 ∫ d^2 Q_T N^2 Q_T^2∫d^2 k_T/( 2 π )^2 d^2 k'_T/( 2 π )^2 ×ϕ^G_H -k⃗_T , k⃗_T; Y - y_1 ϕ^G_Hk⃗_T - p⃗_T1, - y_2 Γ_ν -k⃗ , p⃗_T1 Γ_νk⃗ , p⃗_T1 ×^G_H-l⃗_T , l⃗_T ; Y - y_2 ^G_Hl⃗_T + p⃗_T2, -l⃗_T - p⃗_T2; y_2 Γ_μl⃗_T , p⃗_T2 Γ_μ- l⃗_T ,p⃗_T2§ BFKL POMERON IN THE MIXED REPRESENTATIONFora more convenientpresentation, it turns out that the mosteconomicalway ofcalculating the diagram of mudia, is to use the mixedrepresentation of the BFKL Pomeron Green's function, G, ,,Y, where r and R are the sizes of twointeracting dipoles , Q_T denotes the momentum transferred bythe Pomeron, andYthe rapidity between the two dipoles.This Green's function is well known<cit.> and we discussit here for the completeness of presentation, referring toRefs.<cit.> for all details.It has the following form:G, ,; Y=rR/16∑^∞_n=- ∞∫^∞_-∞ d ν1/ν^2 +(n-1)^2 ν^2 + (n+1) V_ν,n, V^*_ν,n, e^ων,n Y whereων,n = 2Reψ + |n| + ν - ψ 1 ; ων,0 = 2Reψ ++ ν - ψ 1 Δ_ BFKL-D ν^2;where ψ(z)is the Euler ψ-function (see Ref.<cit.>formulae8.36) and Δ_ BFKL = 4 ln 2 , D =14 ζ(3), ξ = ln r^2_1/r^2_2. Each term in BFKLGF has a very simple structure, being thetypical contribution of the Regge pole exchange: the product of twovertices and Regge-pole propagator. From OMEGA one can seethat at large Y the main contribution comes from the term withn = 0, and in what follows we will concentrate on this particular term. The vertices with n=0 have beendetermined inRefs.<cit.>,and they have an elegant form in the complex number representationfor the point on the two dimensional plane: viz.(x,y): ρ = x + i y; ρ^*= x - i y; (Q_x, Q_y): q = Q_x + i Q_y; q^*= Q_x - i Q_y;Usingthis notation the vertices have the following structure:V_ν, =Q^2_T^i ν Γ^2 1 - i ν{ J_-i ν q^* ρ J_- i ν q ρ^*- J_i ν q^* ρ J_ i ν q ρ^*} At Q_T → 0 this vertex takes the form: 2^6 iν V_ν,(r^2/2^6)^-i ν((ν +i)8(·)^4 - 8(·)^2 Q^2_Tr^2 + 5 Q^4_T r^4+(2 i +ν)Q^4_T r^4 )/64^2 (ν +2 i) (1-i ν )^2+i(ν +i)(2 ( ·)^2 - Q^2_T r^2/32 (1-i ν )^2+1)+ (Q^2)^i ν(Q^2 r^2/2^6)^i ν((ν -2 i) 8 (·)^4 - 8(·)^2 Q^2_T r^2 + 5 Q^4_T r^4)/2^12 ((2+i ν ) (1+i ν ))^2+2 (1+i ν ) (2 ( ·)^2 - Q^2_T r^2/2^6 (1+i ν )^2-1) For small values of ν (which are related to the region oflarge Y≫ 1), VSQ can be simplified and reduced to theform:2^6 i ν V_ν,(r^2/2^6)^-i ν( (·)^4- (·)^2 Q^2_Tr^2 + 9/16 Q^4_T r^4)/2^8 -2 ( ·)^2 - Q^2_T r^2/2^5 +1)-(Q^2)^i ν(Q^2 r^2/2^6)^iν( (·)^4 - (·)^2 Q^2_T r^2 + 9/16 Q^4_T r^4)/2^8- 2 ( ·)^2 - Q^2_T r^2/2^5+ 1)Usingthat J_-i ν z sinπ + z + i πν√(2/π)√(1/z)atν≪ 1 we obtain for Q^2_T r^2 ≫ 1V_ν, Q^2_T^i ν Γ^2 1 - i νcos· 4i ν/Q_TrThe contribution of the first term in BFKLGF can be reducedto the following form for the scattering amplitude of two dipoleswith sizes r_1 and r_2:N r_1,r_2; Y = r_1 r_2/16∫ d ν1/ν^2 + 1/4^2 V_ν r_1,Q_t → 0 V^*_ν r_2,Q_T → 0 e^ων,0 Y= r_1 r_2/16∫ d ν1/ν^2 + 1/4^2 e^ων,0 Y{(r^2_1)^- i ν - Q^4_T r^2_1/2^12^ i ν} {(r^2_2)^ i ν - Q^4_T r^2_2/2^12^ - i ν}= r_1 r_2/16∫ d ν1/ν^2 + 1/4^2 e^ων,0 Y 2 r^2_2/r^2_1^i ν 2r_1 r_2 ∫ d νexp 4 ln 2-14 ζ(3) ν^2 Yr^2_2/r^2_1^i ν=r_1r_2√(2 π/ D Y)expΔ_ BFKL Y - ξ^2/4 D Y where Δ_ BFKL and D are defined in OMEGA.Inthe derivation of N12 we neglected the contributionsthatare proportional to Q^4_T r^2_2r^2_1/2^12^- i ν since this contribution will be the same as in N12,but with, ξ = lnQ^4_T r^2_2r^2_1/2^12 ≫ 1. To integrate over ν, we use the method of steepest descent, andthe expansion of ων,0 at small ν (diffusion approximation, see the second equation in OMEGA).N r_1,r_2; Y denotes the imaginary part of the dipole-dipolesacttering amplitude at Q_T=0, which is related to the cross section. One can check that N12 has the correct dimension.§ SINGLE INCLUSIVE PRODUCTION IN A ONE PARTON SHOWER§.§ BFKL Pomeron: the simplest approach fora one parton shower The single inclusive cross sectionresulting from the one BFKLPomeron is known,and it is equal tod^2 σ/d y d^2 p_T=2 C_F /(2 π)^2∫d^2 d^2 k_T/(2 π)^2ϕ^G_Hk⃗_T, Q_T=0; Y -yϕ^G_Hk⃗_T - p⃗_T, Q_T=0; yΓ_νk⃗_T,p⃗_T Γ_ν -k⃗_T,p⃗_T The relation between the parton densities ϕ and the Green'sfunction of the BFKL Pomeron has beengiven in Ref.<cit.>:N^ BFKL r, r_1; y, Q_T=0=/2∫ d^2 k_T 1 - e^i k⃗_T ·r⃗ ϕ^G_Hk⃗_T, Q_T=0; y /k^2_Twhere N^ BFKL r, r_1; Y is given byBFKLGFor by N12, in the high energy limit. SINCL1 can be re-written as followϕ^G_Hk⃗_T , Q_T=0; y =2/∫ d^2 e^i k⃗_T ·r⃗ ∇^2_rN^ BFKL r, r_1; y, Q_T=0We have Γ_νk⃗_T,p⃗_T Γ_ν -k⃗_T,p⃗_T=k^2_T k⃗_T - p⃗_T^2/p^2_TPlugging in SINCL1 and SINCL2 into SINCL we obtain <cit.>d^2 σ/d yd^2 p_T=8 C_F / (2 π)^21/p^2_T∫ d^2 re^i p⃗_T ·r⃗ ∇^2 _r N^ BFKL_ prr, r_1; Y- y, Q_T=0 ∇^2 _r N^ BFKL_ trr, r_2; y, Q_T=0where N_ pr and N_ trdenote the probability tofind a dipole in the projectile and target, respectively. r_1 andr_2 are the typical dipoles sizes in the projectile and target.Ascan be seen from MAEQ we need to generalize SINCL3 forthe case Q_T ≠ 0. SINCL has to be replaced byd^2 σ/d y d^2 p_T Q_T ≠ 0= 2 C_F /(2 π)^2∫ d^2 k_T/(2 π)^2ϕ^G_Hk⃗_T, Q_T, Y - yϕ^G_Hk⃗_T - p⃗_T, Q_T; yΓ_νk⃗_T,p⃗_T Γ_ν -k⃗_T + Q⃗_T,p⃗_TTaking into account SINCL1 for Q_T ≠ 0 andΓ_νk⃗_T,p⃗_T Γ_ν -k⃗_T + Q⃗_T,p⃗_T ={1/p^2_T[k⃗_T - Q⃗_T^2 k⃗_T - p⃗_T^2 +k⃗_T^2 k⃗_T - p⃗_T - Q⃗_T^2]-Q^2_T} we re-write SINCL3 in the form d^2 σ/d yd^2 p_T Q_T ≠ 0 = 4 C_F / (2 π)^21/p^2_T∫ d^2 re^i p⃗_T ·r⃗× {-∇^2 _r N^ BFKL_ prr, r_1; Y - y, Q_T ( - i ∇_r- Q⃗_T)^2 N^ BFKL_ trr, r_2; y, Q_T +( - i ∇_r- Q⃗_T)^2 (-∇^2_r)N^ BFKL_ prr, r_1; Y- y, Q_TN^ BFKL_ trr, r_2;y, Q_T} -Q^2_T 4 C_F / (2 π)^2∫ d^2 re^i p⃗_T ·r⃗ N^ BFKL_ prr, r_1; Y - y, Q_T N^ BFKL_ trr, r_2; y, Q_T §.§ General estimatesIt should be stressed that the single inclusive production has theform of SINCL3 and SINCL6aswas shown in Ref.<cit.>for the general structure of the single parton shower.For example,for theprocessshown in becor-c. We needonlytosubstituteN^G_ trr, r_2;y, Q_T for 2 N^ BFKL_ trr, r_2; y, Q_T where2 N^ BFKL_ trr, r_2;y, Q_T→ N^G_ trr, r_2;y, Q_T=2 N_ trr, r_2;y, Q_T -∫ d^2 Q'_TN_ trr, r_2;y, Q⃗_T - Q⃗'_T N_ trr, r_2;y,Q⃗'_TN_ trr, r_2;y, Q_T is a solution to thenon-linear evolution equation.For the case ofinclusiveproduction, we can considerably simplify estimates noting that∇^2_rN_ trr, r_2;y, Q_TN^ BFKL_ trr, r_2;y, Q_T≪1; ∇^2_rN_ trr, r_2;y, Q_T0; where Q_s(y) denotes the saturation momentum. In other words, the main contribution toinclusive productioncomes from the vicinity of the saturation scale, where r^2 Q^2_s≈ 1. Fortunately, the behavior of N in this kinematic regionis determined by the linear BFKL evolution equation <cit.>and has the following form<cit.> N_ trr, r_2;y, Q_T = 0∝ r^2 Q^2_s(y)^1 - γ_cr Q^2_s =(1/r^2_2) expωγ =+ i ν = γ_cr/1 - γ_cr y = (1/r^2_2)e^κ y where γ_cr = 0.37. We have seen in VLQ that for Q_T ≠ 0 ,the scattering amplitude decreasesat Q^4_Tr^2r^2_2 ≫ 1. Therefore, we need to considerrather small values of Q_T: Q^4_Tr^2r^2_2 ≤ 1.The product of vertices that determines the amplitude has two terms ( see VSQ )which are proportional tor^2/r^2_2^i νandto Q^4_Tr^2r^2_2^I ν. Therefore, the maximum of ∇^2_r Ncan be reached ifr^2/r^2_2 e^κ y ∼1 and Q^4_T r^2 r^2_2 e^κ y ∼ 1 and the amplitudethen has the following formN_ trr, r_2;y, Q_T∝ c_1 r^2/r^2_2 e^κ y^1 - γ_cr+c_2Q^4_Tr^2r^2_2 e^κ y^1 - γ_cr The first term does not depend on Q_T and, therefore, the upper limit of the integralover Q_T, goes up to Q^max_T^2 ≈ 1/(r r_2). The second term both for Q^2_Tr r_2 < e^- κ y and for Q^2_Tr r_2 > e^- κ y turns out to be small. Indeed,in the first region the amplitude issmall,while inthe second region we are deep in the saturation domainwhere∇^2_r N → 0.Hence, we expect that in the integral over Q_T,the first term gives a larger contributionthan the second term and we will keep only this contribution in our estimates.§ DOUBLE INCLUSIVE CROSS SECTIONFOR TWO PARTON SHOWERPRODUCTION§.§ The simplest diagramIn this section we calculate the simplest diagram of mudia. We need to integratetheproduct of two BFKL Pomerons over Q_T (see MAEQ1):I=∫ d^2 Q_T V_ν_1r⃗_1,Q⃗_T V^*_ν_1r⃗_2,Q⃗_TV_ν_2r⃗'_1,Q⃗_T V^*_ν_2r⃗'_2,Q⃗_TFrom MAEQ1 in the momentum representation, we see that r_1 ≠r'_1(r_2 ≠r'_2) but they are close to each other,being determined by the same momentum k_T. We assume that p_T1< k_T, since k_T ∼ Q_s Y - y_1≫μ_ soft. Consideringr_1≈ r'_1≪ r_2 ≈ r'_2 we will show that in the integralover Q_T, the typical Q_T ∼ 1/r_2. In other words, the dependenceof Q_T is determined by the largest of interacting dipoles.From VLQ we see that for large Q_T, when r^2_1 Q^2_T≫ 1and r^2_2 Q^2_T ≫ 1, the integrand is proportional to 1/Q^4_T,and converges.The mainregionof interest isr^2_2Q^2_T ≫ 1andr^2_1 Q^2_T ≪ 1. In this kinematic region for verticesV_ν_1r⃗_1,Q⃗_TandV_ν_2r⃗'_1,Q⃗_T, we can use VSQ1, whilethe conjugated vertices are still in the regime of VLQ.DISD1then takes the form:I = 2^6i (ν_1 + ν_2) - 16ν_1 ν_2π ×∫_1/r^2_2 d Q^2_T {Q^2 r^2_1/2^6^-iν_1 - Q^2 r^2_1/2^6^iν_1}{Q^2 r'^2_1/2^6^-iν_2 - Q^2 r'^2_1/2^6^iν_2}cos^2·r⃗_1/Q^2_Tr^2_2Assuming that both ν_1 and ν_2 are small, we see thatall four terms are equal to each other, and the integral can be written as follows.I= 2^6i (ν_1 + ν_2) - 2^6ν_1 ν_2π1/i (ν_1 + ν_2)r^2_1/r^2_2^i (ν_1 + ν_2)1/r^2_2The appearance of the pole ν_1=-ν_2 indicates that the contributionfrom this kinematic region is large.Closing the contour of integration on ν_2 over the pole, we obtain I=2^6 π ν^2_11/r^2_2Actually, the double inclusive cross section depends on ∇ ^2 N as we argued in the previous section. Repeating the procedure forI= ∫ d^2 Q_T ∇^2_r_1 r_1 V_ν_1r⃗_1,Q⃗_T ∇^2_r'_1 r'_1 V^*_ν_1r⃗_2,Q⃗_T ∇^2_r_2 r_2 V_ν_2r⃗'_1,Q⃗_T ∇^2_r'_2 r'_2 V^*_ν_2r⃗'_2,Q⃗_T we obtain for small ν_1 and ν_2:I=2^6 π ν^2_11/r_1 r'_1 r^2_2 r'^2_2 Taking the integral over ν_1, using the method of steepest descent,we obtain the following contribution:I=2^51/r_1 r'_1 r^2_2 r'^2_2√(π/ 2D y_12^3)e^ 2 Δ_ BFKL y_12 where Δ_ BFKL and D are defined in OMEGA. Rewriting MAEQ1 in the coordinate representation we obtain: d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2mudia=2 C_Fμ^2_ soft/ (2 π)^2 ^21/p^2_T1 p^2_T2 ∫d^2 Q_T/(2 π)^2 ×∫d^2 r_1 d^2r'_1d^2 r̃ _1 d^2 r̃'_1 e^-i p⃗_T1·r⃗̃⃗'_1δ^(2)r⃗_1 + r⃗'_1 - r⃗̃⃗_1 - r⃗̃⃗'_1 ∇^2_r_1 N_ pr r_1; Y - y_1 e^i Q⃗_T ·r⃗̃⃗'⃗_1 ∇^2_r̃'_1 N_ prr̃'_1; Y - y_1 ×∇^2_r̃_1∇^2_r̃_2 Nr̃_1; r̃_2, Q_T; y _12∇^2_r̃'_1∇^2_r̃'_2 Nr̃'_1; r̃'_2,Q_T; y _12 ×∫d^2 r_2 d^2 r'_2d^2 r̃ _2 d^2 r̃'_2 e^- i p⃗_T2·r⃗̃⃗_2 δ^(2)r⃗_2 + r⃗'_2 - r⃗̃⃗_2 - r⃗̃⃗'_2 ∇^2_r_2 N_ tr r_2; y_2 e^i Q⃗_T ·r⃗̃⃗'⃗_2 ∇^2_r̃'_2 N_ trr̃'_2; y_2In MAEQC1 we neglected the terms which are proportional toQ^2_T (see MAEQ1)as the typical Q_T aresmall as we have argued, and becausethese terms do not leadto additional correlations in the azimuthal angles. We have discussed the integral over Q_T, andit has the form ofDISD7.The extra e^i r⃗'_1 ·Q⃗_Tgive an additional numerical factor, replacing 2^5 by 2^7 inDISD7. Tointegrate over k_T and k'_T we replace ∫∏ d ϕ_i e^-i p⃗_T1·r⃗̃⃗'_1δ^(2)r⃗_1 + r⃗'_1 - r⃗̃⃗_1 - r⃗̃⃗'_1 →2 π^4 ∫ k_T d k_T J_0 k_T r_1 J_0 |k⃗_T + p⃗_T1| r̃'_1 J_0 k_T r̃_1 J_0 k_T r̃_1Now we can take the integrals over r_i bearing in mind DISD7 and N_pr r_i, Y - y_1= ∫d ν/2 πμ^2_ soft r^2_i^ + i ν_i e^ων_i,0 Y - y_1The integrals over r̃_1and r̃'_1 have thefollowing form (see Ref.<cit.> formulae6.511(6)) ∫^r̃_2_0 J_0 k_T r̃_1 d r̃_1=r̃_2 J_0 kr̃_2 + πr̃_2 ( J_1kr̃_2H_0kr̃_2 - J_0kr̃_2H_1kr̃_2)= {[r̃_2 if kr̃_2 ≪1; 1/k if kr̃_2 ≫1; ]. Using DISD9we obtain∫^∞_0 r_i d r_i J_0 k_Tr_i,∇^2_r_i N_pr r_i, Y - y_1=1/k4 μ^2_ soft/k^2^iν_i e^ων_i,0Y - y_1 Collecting DISD9,DISD10 and DISD11 we see thatthe main contribution stems from the region kr̃_2 ≪1and the integral over k_T has the form r̃_2 r̃'̃_2 ∫^1/r̃_2^2_p^2_T1d k^2_T/k^2_T4 μ^2_ soft/k^2^i(ν_1 +ν'_1) e^(ων_i, 0 +ω 0,ν'_1)Y - y_1= r̃_2 r̃'̃_21/i(ν_1 +ν'_1)1/r̃_2^2 p^2_T1^i(ν_1 +ν'_1) e^(ων_i, 0 +ω 0,ν'_1)Y - y_1 √(π/2D (Y - y_1)) e^ 2 Δ_ BFKL Y - y_1 The integral over k'_T has the same structurewhile theintegration in DISD10goes to infinity. As the result wecan reducethe integral to the form;∫_p^2_T2d k^2_T/k^4_T4 μ^2_ soft/k^2^i(ν_1 +ν'_1) e^(ων_i, 0 +ω 0,ν'_1)Y - y_1 =1/1 + i (ν_1+ν'_1)1/4 μ^2_ soft4 μ^2_ soft/p^2_T2^1+i(ν_1 +ν'_1) e^(ων_i, 0 +ω 0,ν'_1)Y - y_1π/ D y_2 1/p^2_T2Finally, d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2mudia=4 π2 C_F/ (2 π)^2 ^21/p^2_T1 p^4_T2 √(1/2D (Y - y_1))√(1/ 2D y_12^3) 1/ D y_2 e^ 2 Δ_ BFKLY §.§ The CGC/saturation approach The integral over k'_T in DISD13has an infra-red singularitywith a cutoffat p_T2 since weassume thatp_T2 is the smallest momentum. This reflects the principle featureof the BFKL Pomeron parton cascade, which has diffusion, both in theregion of small and large transverse momenta. On the other hand we knowthat the CGC/saturation approach suppressed the diffusion in the smallmomenta<cit.>, providing the naturalcutofffor theinfrared divergency. We expect that such a cutoff will be the value of the smallestsaturation momenta: Q_s Y - y_1 or Q_s y_2, which will replace one of p^2_T2 in the dominator of DISD14.Therefore, we anticipate that for therealistic structure of the one partonshower cascade, (see becor-c for example),the contribution forthedouble inclusive cross section will be different. We need to specify the behavior of the scattering amplitude in thevicinity of the saturation scale. We have discussed the basicformulae<cit.>of VICQS,but for integration over the dipolesizes we need to know the size of this region.The scattering amplitude can be writtenin the form: N r_1,r_2; Y=∫^ϵ + i ∞_ϵ - i ∞d γ/2 π n_inγ e^ωγ, 0 Y - ( 1 - γ) ξ where ω(γ,0) is given by OMEGA,replacing + iν≡γ and ξ = ln r^2_1/r^2_2. The saturationscale is deterring by the line on which the amplitude is a constant (C) of the order one. This leads to the following equation for thesaturation scale<cit.>: ωγ_cr, 0 Y - ( 1 - γ_cr) ξ_s = 0; ω'_γγ, 0 Y -ξ_s = 0;which results in the value of γ_cr given by the equation: ωγ_cr, 0/1 - γ_cr=ω'_γγ, 0which gives γ_cr= 0.37 and the equation for the saturationmomentum:ξ_s≡ln Q^2_s r^2_2 = κ Y = ωγ_cr/1 - γ_cr YExpanding the phase ωγ, 0 Y - ( 1 - γ) ξ in the vicinity Δξ = ξ - ξ_s and Δγ =γ - γ_crwe obtainN r_1,r_2; Y= ∫^ϵ + i ∞_ϵ - i ∞d γ/2 πr^2_1 Q^2_s^1 - γ_cr∫d Δγ/2 π i e^ω”_γγγ, 0 Y(Δγ)^2 +ΔγΔξ=r^2_1 Q^2_s^1 - γ_cr√(π/D Y) e^ - (Δξ)^2/4 D YAt first sight AM2 shows thatthe amplitude has a maximum at τ = r^2_1Q^2_s=1. However, this is not correct. AM2gives the correct behavior for τ < 1 while for τ >1 we need totake into account the interaction of the BFKL Pomerons and the non-linearevolution, generatedby these interactions. The generalresult of this evolution is the fact that the amplitude depends on onevariable<cit.>τ, i.e.Nτ (as it has geometricscaling behavior).The peak at τ=1 appears in∇^2_r_1 N r_1,r_2; Y =4 Q^2_s(Y) 1/τd/d ττd/d τNτ From AM3 we can conclude that the width of the distributionin r^2_1 is of the order of Q^2_s, but dependscruciallyon the modelfor the Pomeron interaction. In DN-a weplot this value for the behavior of the scattering amplitude deepin the saturation domain (see Ref.<cit.>).This approach is not correct for τ→ 1 and - ∇^2 N = 1.58at τ=1,but it starts to be small at τ > 2, which could belarge enough to trust the formulae of Ref.<cit.>. At least such aconclusion can be justified considering the fit of the DIS data in thesaturation model of Refs.<cit.>, which based on the idea ofRef.<cit.>, and has thecorrect behavior of the scatteringamplitude,bothdeep in the saturation domain, and near τ = 1. Hence, weexpectthat ∇^2 N decreases faster than we can see from AM2. Bearingthese conclusions in mind, we will calculated the contribution ofmudia, keeping all N in MAEQC1 in the vicinities ofthe saturation scales, by replacing∫^∞_0d τ (-∇^2N)= -∫^1_0d τ (-∇^2 N).We willshow in the folowing, that we cannottake all sixPomerons in thevicinities of the saturation scale.We have to take two of them,either deep in the saturation domain, or in the perturbative QCD region.We choose totake two Pomerons between rapidities y_1 and y_2,(see DN-b) i.e. in the perturbative QCD region. Unfortunately,we cannot use the AGK cutting rules<cit.>, which state that thesePomerons will be not affected by the Pomeron interaction, andthe contributionsof these interactions (see red Pomeron in DN-b) are canceled.Indeed, it has been proven that for the double inclusiveproduction<cit.>, that they do not work in perturbativeQCD. On the otherhand, these Pomerons carrytransverse momentum Q_T ,unlike the others one in the diagram, which islarger than the saturation scale Q_s y_2 and therefore,their contributions are suppressed in comparison with the other Pomerons in mudia.In addition our choice provides the natural matching with the region y_12 < 1.The integration over Q_T will produce the same result asDISD7, as in the previous section. We re-write the integrationover ϕ_i (see DISD8) in the following way:∫∏ d ϕ_i e^-i p⃗_T1·r⃗̃⃗'_1δ^(2)r⃗_1 + r⃗'_1 - r⃗̃⃗_1 - r⃗̃⃗'_1 →2 π^4∫ d ϕ e^i p⃗_T1·r⃗∫ k_T d k_T J_0 k_T r J_0 k_T r_1 J_0 k_T r̃_1 J_0 k_T r̃'_1 We see that the integrals over r'_1 and r'_1 leads tor_1 ∼ 1/Q_s(Y - y_1) and r'_1 ∼ 1/Q_s(Y - y_1).The sameholds for the integrals overr'_2 andr'_2,leading tor_2 ∼ 1/Q_s(y_2) and r'_2 ∼ 1/Q_s(y_2).Assuming that Q_s Y - y_1 > Q_s y_2 we concludethat r_i and r'_i are much smaller than r_2 and r'_2.Replacing∇^2_r_1 N_ pr r_1; Y - y_1 e^i Q⃗_T ·r⃗̃⃗'⃗_1 ∇^2_r̃'_1 N_ prr̃'_1; Y - y_1 → 2^8 γ̅^4/r_1 r'_1 r^2_1 Q^2_s Y - y_1^γ̅ r'^2_1 Q^2_s Y - y_1^γ̅ where γ̅ = 1 - γ_cr, we obtain from INTR that integration over r takes the form 1/Q_s1/1 + 2γ̅ ∫^1_0 d τJ_0k_T/Q_s√(τ) τ^γ̅d τ/2 √(τ) = 1/Q_s1/1 + 2γ̅ _1F_2{ + γ̅},{1, 3/2 + γ̅}, - k^2_T /4Q^2_sRecall that we consider Q_s = Q_s(Y - Y_1) in AM5. Forp_T1≪ Q_s Y - y_1 we can replacee^-i p⃗_T1·r⃗̃⃗'_1 = 1 in INTR.In this case the integralhas the form1/Q^2_s1/(1 + 2γ̅)^2 ∫^1_0d τ' ( 1/Q_s1/1 + 2γ̅ _1F_2{ + γ̅},{1, 3/2 + γ̅}, - 1 /4τ')^2d τ'/τ' = 0.18/Q^2_swhere τ' = k^2/Q^2_s.The integral overr in thelower part of the diagram takes theform: ∫d^2 r'/r'^2 J_0 k_T r = π ln k^2_T/(4 μ^2_ soft) Using AM6 for p_T2≪ Q_s y_2 the integralover k'_T can be reduced to 1/(1 + 2γ̅)^2 ∫^1_0d τ” (_1F_2{ + γ̅},{1, 3/2 + γ̅}, - 1 /4τ”)^2 lnτ”/τ”^2 = 3.50 Finally, collecting all numerical coefficients, we obtain d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2DN-b=^4 2^3 π^3 2 C_F / (2 π)^2 ^21/p^2_T1 p^2_T20.18 3.5/Q^2_s Y - y_1 2 γ̅^8√(1/ 2D y_12^3)e^2 Δ_ BFKL y_12whereconstantis the value of the amplitude at τ=1. This contribution is proportional to ∝ e^2 Δ_ BFKL y_12/ Q^2_s Y-y_1 for p_T1≪ Q_s Y - y_1 and p_T2≪ Q_sy_2.Note that Q^2_s Y - y_1 >Q^2_s y_2.We need to estimate the diagram of becor-a (see MAIN).This diagram can be re-written asd^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2becor-a = μ̃^2_ softd^2 σ/d y_1 d^2 p_T1 Q_T = 0; SINCL3 d^2 σ/d y_2 d^2 p_T2 Q_T = 0; SINCL3 μ̃^2_ soft = ∫ d^2 Q_TN^2 Q_T; Examining SINCL3, one can see that in general casewhen Y - y_1 ≠ y_1 and Y - y_2 ≠ y_2 all fourPomerons cannot be in the vicinity of the saturation scale.Actually we have two kinematic regions which give the maximalcontributions (assuming Q_s Y - y_1 > Q^2_s y_1):*r^2Q^2_s Y - y_1 ≈ 1 but r^2 Q^2_s y_1→ Q^2_s Y - y_1/Q^2_s y_1 ≪ 1;*r^2Q^2_sy_1 ≈ 1 but r^2 Q^2_s y_1→ Q^2_sy_1/Q^2_s Y - y_1 ≫ 1; In the region 1 the upper Pomeron is in the vicinity of the saturationscale, while the lowerPomeron is in the perturbative QCD region. In region 2 the lower Pomeron is in the vicinity of the saturation scale,and the upper Pomeron is deep inside the saturation domain. As we have discussed (see DN-a) ∇^2 N decreases in thesaturation region much faster than in the perturbation QCD region and,therefore, we bf assume that the kinematic region 1 gives the largestcontribution.Hence, for p_T1 ≪ Q^2_s y_1 we obtain d^2 σ/d y_1 d^2 p_T1 Q_T = 0; SINCL3 =8 C_F / (2 π)^21/p^2_T∫ d^2 re^i p⃗_T ·r⃗ ∇^2 _r N^ BFKL_ prr, r_1; Y- y, Q_T=0 ∇^2 _r N^ BFKL_ trr, r_2; y, Q_T=0 =8 C_F / (2 π)^21/p^2_T^2 (4γ̅^2)^2exp - ln^2 Q^2_s Y - y/Q^2 y /4 DyIn DISD17 we usedbackward evolution, from the saturationboundarywhere N =. The ratio of two contributions takes the following form: R = d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2DN-b/d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2becor-a = 1/N^2_c - 1μ̃^2_ soft/Q^2_s y_2 8 π^52 γ̅^4 0.183.5× √(π/ 2D y_12^3)e^2 Δ_ BFKL y_12expln^2 Q^2_s Y - y_1/Q^2 y_1 /4 Dy_1expln^2 Q^2_s Y - y_2/Q^2 y_2 /4 Dy_2One can see thatDISD18 demonstrates the additional suppression in comparison with the calculation of the simplest diagram, due to infrared cutoffat Q_s y_2 instead of p_T2. The factor exp2 Δ_ BFKL y_12 reflects the fact that two BFKL Pomerons between rapidities y_1 and y_2 aretaken in the perturbative QCD region. It should be stressed that we can trust our estimates onlyfor values of y_12 at which the exchange of the BFKL Pomeron with rapidity y_12 give the contribution smaller than C. This condition means that1/ 2D y_12e^2 Δ_ BFKL y_12< CTakingΔ_ BFKL = 0.25 and Q^2_s(Y) ∝expλ Ywith λ = 0.25 (these valuescorrespond to the BFKL phenomenology) we obtain that the l.h.s. of DISD19is smaller than 0.15 for y_12 ≤ 7. Therefore, we can trust our estimatesshown in R for C>0.15. We are used to takeC=0.3 which lead to the contribution of the shadowing corrections of the order of30%. Two last factors in DISD18 stem from the perturbative QCD nature of two Pomerons in DISD16 ( see DISD17). In R we plotted ratio R as function of y_12for y_12≤ 7 (see DISD19. One can see thatthe ratio increases for large y_12 . § AZIMUTHAL ANGLE CORRELATIONS The azimuthal angle correlations stem from terms Q⃗_T·r⃗_i^n in the vertices (see VSQ and VSQ1).Indeed, after integrating over r_i these terms transform toexpressions of the following type<cit.>: Q⃗_T·p⃗_T1^m_1 Q⃗_T ·p⃗_T2^m_2,which lead to term of p⃗_T1·p⃗_T2^m.Wehave illustrated in VSQ and VSQ1 how theseoriginatefromthe general form of BFKL the Pomeron vertices in the coordinaterepresentation. From VSQ and VSQ1 onlyterms proportionalto Q⃗_T ·r⃗_i^n with even n appear in theexpansion. Therefore, the azimuthal angle (ϕ)correlation functioncontains only terms cos^2 nϕ, and it is invariant withrespect to ϕ→π - ϕ. Inother words, v_n with odd n,turnout to be zero. Hence, we have the first prediction: the value v_n withodd n should decreases with y_12, and their dependence shouldfollowthe dotted lines in R-a. We return to DISD1 and integrate over Q_T, collectingterms that depend on the angles between Q⃗_T and r⃗_i,whichwe have neglected in the previous section. As we have learned the typicalvalues of Q_T ∝ 1/r_2 ∼ 1/r'_2 where r_2 and r'_2 are largerthan r_1 and r'_1. In other words , we showed thatthe maincontributions stem from the kinematic regions: r^2_1 Q^2_s Y -y_1∼ 1 ( r'^2_1 Q^2_s Y - y_1∼ 1 ) andr^2_2 Q^2_sy_2∼ 1 ( r'^2_2 Q^2_sy_2∼ 1 ).Assuming that Q_s Y - y_1≫ Q_s y_2 we conclude thatr_1(r'_1)≪ r_2(r'_2). The typical Q_T is determined by thelargest dipoles and, therefore, we expectQ_T ≈ 1/r_2(1/r'_2), as has beendemonstrated above. Bearing these estimate in mind, we can replace vertices V_ν_1r⃗_1,Q⃗_T andV_ν_2r⃗'_1, Q⃗_T in DISD1by VSQ1in which weput Q_T = 1/r_2and Q_T = 1/r'_2, respectively.Taking into account thatr_1/r_2 ≪ 1 (r'_1/r'_2 ≪ 1) we obtain V_ν_1r⃗_1, Q⃗_T V_ν_2r⃗'_1,Q⃗_T ={(r^2_1/2^6)^-i ν_1- Q^4_T r^2_1/2^6^ i ν_1} ( 1- 1/2^4Q⃗_T ·r⃗_1^2+ 1/2^8Q⃗_T ·r⃗_1^4 )×{(r'^2/2^6)^-i ν_2- Q^4_T r'^2_1/2^6^ i ν_2} ( 1- 1/2^4Q⃗_T ·r⃗'_1^2 + 1/2^8Q⃗_T ·r⃗'_1^4 ) At first sightANGLECOR1 should enter two angles between Q⃗_T and r⃗_1 and r⃗'_1, respectively.However, in the integrandfor integration over r_i (see DISD8)depends only on one vector p⃗_T1 . Therefore, after integrationover all angles, wefindthat the angle ϕ in ANGLECOR1 istheangle between Q⃗_T and p⃗_T1.For vertices V^*_ν_1r⃗_2, Q⃗_T and V^*_ν_2r⃗'_2, Q⃗_Tin DISD1weuse VLQ.Finally, we need toevaluate the integral I_Q = -16ν_1 ν_2∫ Q_T d Q_T{V_ν_1r⃗_1, Q⃗_T V_ν_2r⃗'_1, Q⃗_T}^r_1 = r'_1=1/Q_s Y - y_1_ANGLECOR1 Q^2_T ^ -i (ν_1 +ν_2)cos^2·r⃗_2/Q^2_Tr^2_2 cos·r⃗_2with better accuracy that we did in section 5.1, keepingthe dependence on the angle between Q⃗_T and r⃗_2.Note, thatthe factor cos·r⃗_2 comes from expi·r⃗_2 in MAEQC1. Takingthis integral wesubstitute for the terms in parentheses inANGLECOR1,|Q_T| = 1/r^2_2 (1/r'^2_2).The integral is equal to I_Q= 2^6i (ν_1 + ν_2) - 2^7ν_1 ν_2 r^2_1/r^2_2^i (ν_1 + ν_2)1/r^2_2×( 1- 1/2^4n⃗·r⃗_1^2 /r^2_2 + 1/2^8n⃗·r⃗_1^4 /r^4_2)( 1- 1/2^4n⃗·r⃗'_1^2 /r'^2_2 + 1/2^8n⃗·r⃗'_1^4 /r'^4_2) × {1/ i (ν_1 + ν_2) - 9/32 cos 2 ϕ_2 + 3/16cos 4ϕ_2} where n⃗ = Q⃗_T/Q_T, andϕ_2 is the angle betweenn⃗ and n⃗_2 = r⃗_2/r_2.InANGLECOR3 the terms in …… stemfrom the expansion with respect to r^2_1/r^2_2≪ 1.However, for the terms in {…} there are no suchsmall parameters, and we expand thefunction of ϕ_2 in the Fourier series. Integrating over n⃗ one obtains ……{…} = 1/ i (ν_1 + ν_2) + 3/ 2^10r^2_1/r^2_2n⃗_1 ·n⃗_2^2 +n⃗'_1 ·n⃗_2^2 + 3/2^12r^4_1/r^4_2n⃗_1 ·n⃗_2^4+ n⃗'_1 ·n⃗_2^4 where n⃗_1 = r⃗_1/r_1,n⃗'_1 =r⃗_1/r_1 and n⃗_2 = r⃗_2/r_2. Deriving ANGLECOR4 we neglected the extra powers ofr^2_1/r^2_2 , which are small.FinallyI_Qr⃗_1, r⃗',r⃗_2; ν_1, ν_2 = 2^6i (ν_1 + ν_2) - 2^7ν_1 ν_2 r^2_1/r^2_2^i (ν_1 + ν_2)1/r^2_2× {1/ i (ν_1 + ν_2) + 9/ 2^10r^2_1/r^2_2n⃗_1·n⃗_2^2 +n⃗'_1 ·n⃗ _2^2 + 3/2^12r^4_1/r^4_2n⃗_1 ·n⃗_2^4+ n⃗'_1 ·n⃗_2^4 } From MAEQC1 we can see that the integration over r_i can be written in the formd^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2mudia=2 C_Fμ^2_ soft/ (2 π)^2 ^21/p^2_T1 p^2_T2×∫d^2 r_1 d^2r'_1d^2 r̃ _1 d^2 r̃'_1 e^-i p⃗_T1·r⃗_1δ^(2)r⃗_1 + r⃗'_1 - r⃗̃⃗_1 - r⃗̃⃗'_1 ∇^2_r̃_1r̃_1 V_ prr̃_1∇^2_r̃'_1r̃'_1 V_ prr̃'_1 × ∫d^2 r_2 d^2 r'_2d^2 r̃ _2 d^2 r̃'_2 e^-i p⃗_T2·r⃗_2δ^(2)r⃗_2 + r⃗'_2 - r⃗̃⃗_2 - r⃗̃⃗'_2 ∇^2_r̃_2 r_2 V_ trr̃_2∇^2_r̃'_2r̃'_2 V_ trr̃'_2 ×∇^2_r_1∇^2_r_2∇^2_r'_1∇^2_r'_2 (r_1 r_2 r'_1 r'_2 I_Qr⃗_1, r⃗',r⃗_2; ν_1, ν_2) × 2 π{1/ i (ν_1 + ν_2) +9/ 2^10r^2_1/r^2_2n⃗_1 ·n⃗_2^2 +n⃗'_1 ·n⃗_2^2 + 3/2^12r^4_1/r^4_2n⃗_1 ·n⃗_2^4+ n⃗'_1 ·n⃗_2^4 }Each term in ANGLECOR6 can be factorized as a product oftwo functionswhich depend on r^i_1 and on r^i_2.Bearing this feature in mind wecalculate eachterm going to the momentum representationusing INTR. We obtain a product offunctions of k_T. Each ofthese function has the following general form: ∫ d^2 re^ i k⃗_T ·r⃗ ∏_i=1^jr_μ_i F r=(- i∇⃗_k_T)^j∫ d^2 re^ i k⃗_T ·r⃗F r=2 π (-i∇⃗_k_T)^j∫ d^2 rJ_0 k_T rF rAs we have seen the dependence on r⃗_i stem fromthe integration over Q_T or, in other words, from I_Q In I_Q dependence on r_1 and r'_1can be extracted explicitly, leading to F r∝ 1/r.Hence the momentum image for ANGLECOR7 has a simple form:∫ d^2 re^ i k⃗_T ·r⃗ ∏_i=1^jr_μ_i F r= 2 π (-i∇⃗_k_T)^j 1/k_TFor j =2 and j=4 which we need to calculate ANGLECOR6we have (-i∇⃗_k_T)^2 1/k_T = {3/k^5_Tk_T,ik_T,i'- 1/k^3_Tδ_i,i'};(-i∇⃗_k_T)^4 1/k_T = {105/k^9_Tk_T,ik_T,i'k_T,jk_T,j'- 15/k^7_Tδ_i jk_T,i'k_T,j' + δ_i i'k_T,jk_T,j' + δ_i' jk_T,ik_T,j' + δ_j i'k_T,ik_T,j' + δ_i' j'k_T,jk_T,i + δ_j j'k_T,ik_T,i' +3/k^5δ_i i'δ_j j' + δ_i jδ_i' j' + δ_i jδ_i' j'};Note that for integration over r⃗_1, ANGLECOR8 takes the form ∫ d^2 r_1e^ i (k⃗_T + p⃗_T1) ·r⃗_1 ∏_i=1^jr_1,μ_i F r_1= 2 π (-i∇⃗_k⃗_T + p⃗_T1)^j 1/√((k⃗_T + p⃗_T1 )^2) The term r^2_1n⃗_1 ·n⃗_2^2+ r'^2_1n⃗'_1 ·n⃗_2^2 can be re-written as r_1,μ r_1,ν + r'_1,μ r'_1,ν r_2,μ r_2,ν and in the momentumrepresentation it looks as ∫ d ϕ{(3/k^5_Tk_T,ik_T,i' - δ_i i'/k^3_T) 1/√(k^2_T + p^2_T1 + 2cosϕ k_T p_T1) +(3/√(k^2_T + p^2_T1 + 2cosϕ k_T p_T1)^5( k⃗_T + p⃗_T1)_i( k⃗_T + p⃗_T1)_i' - δ_i i'/√(k^2_T + p^2_T1 + 2cosϕ k_T p_T1)^3)1/k_T} =Ap_T1, i p_T1, i'/p^2_T1 +Bδ_i i'The expressions for A and B can bewritten in a general form. Assuming that both p_T1 andp_T2 are smallerthan Q_s y_2, we can expand the answer, only taking intoaccountterms that are proportional to p^2_T1/k^2_Tand p_T2^2/k'^2_T. We obtainA k_T, p_T1 = 3p^2_T1/4 k^8_T -13 k^2_T + 50p^2_T1 ; B k_T, p_T1 = 1/8 k^4_T 8 k^4_T + 65 k^2_T p^2_T1 - 150 p^4_T1 ; The integrations over r'_2 and r_2differ from the integrationsover r_1 and r'_1, due to extra factor 1/r^2_2 which comesfrom the integration over Q_T in DISD2 and DISD3.Since r^2_2 ≈ 1/Q^2_s(y_2) we replace it by 1/r^2_2= Q^2_s y_2. In the case the integral over k'_T takes thesame form asthe integral over k_T, leading to the followingexpression which isproportional to cos^2ϕ, where ϕ isthe bangle between p⃗_T1 and p⃗_T2: d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2mudia∝ Q^2_s y_2A k_T,p_T1 A k'_T,p_T2 cos^2ϕ which is responsible for the appearance ofv_2,2 and v_2.Usingthe secondexpression inANGLECOR12 we can calculate theterm whichis proportional to cos^4ϕandhas the form d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2mudia∝ Q^2_s y_2A^(4) k_T,p_T1 A^(4) k'_T,p_T2 cos^4ϕ withA^(4) k_T,p_T1=15 573/81/k^6_T p^2_T1/k^2_T The values of v_2 and v_4 can be determined from the following representation of the double inclusive cross section d^2 σ/d y_1d y_2d^2 p_T1 d^2 p_T2∝1+2 ∑_n v_ n,np_T1, p_T2 cos n φwhere φ is the angle betweenp⃗_T1 andp⃗_T2.v_nis calculated fromv_n,n p_T1, p_T21. v_n p_T=√(v_n,n p_T, p_T) ; 2. v_n p_T=v_n, n p_T,p^ Ref_T/√(v_n,n p^ Ref_T, p^ Ref_T) ;vn-1 and vn-2depict two methodsof how thevaluesof v_n have been extracted from the experimentally measured v_n,n p_T1, p_T2, where p^ Ref_T denotes themomentum of the reference trigger.These two definitions are equivalent ifv_n, n p_T1,p_T2 can be factorized as v_n, n p_T1, p_T2 = v_n p_T1 v_n p_T2. In this paper we usethe definition in vn-1. Introducing the angular correlation function asC p_T, ϕ≡d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2DN-b/d^2 σ/d y_1 d^2 p_T1 d y_2 d^2 p_T2becor-awe obtainv_n,n=∫^2 π_0d ϕ C( p_T, ϕ) cos n ϕ/2 π + ∫^2 π_0dϕ C(p_T, ϕ); v_n = √(v_n,n) ;In DISD18 we have calculatedthe part of C p_T, ϕwhich does not depend on ϕ, which coincides with C p_T,ϕ=0 = R of DISD18for Q_s Y - y_1 ≫Q_s y_2. To calculate the contribution to C, which dependson ϕ, we needto take the separateintegrals over ν_1 andν_2 since the terms, which are proportional to cos^2ϕand cos^4ϕ, do not have a pole at ν_1 = -ν_2 (seeANGLECOR5). These integrations lead to the following extra factorin C p_T,ϕ - C p_T, ϕ=0C p_T,ϕ - C p_T, ϕ=0∝R p^2_T/Q^2_s Y-y_1 p^2_T/Q^2_s Y-y_1C p_T, ϕ=0;R = 2 ξ^2√(1/ 2 D y_12^3) exp - 2 ξ^2/ 4D y_12 where ξ = ln Q^2_s Y - y_1/Q^2_s y_2.We took factors proportional to p_T from the expression forA k_T, p_T1 and A^(4) k_T, p_T1 puttingp_T1 = p_T2 = p_T. To find the final correlation function and v_2,2 andv_4,4, we need to collectall numerical factorsthat come from A k_T, p_T1 , A^(4) k_T, p_T1and ANGLECOR6, and to integrate over ϕ, asgiveninC. Note, that in the symmetric kinematics, whereY - y_1 = y_2= Y - y_12, ξ = 0and ANGLECOR14 vanishes.In this case, we have to useVSQ instead of VSQ1, keeping track of the corrections,which are proportional to ν_i. As the result, we can considerξ = 0 in ANGLECOR14, but we need to replace factor ξ^2 by 1. DISD18 andANGLECOR14 suffer lanumerical uncertainties, which stem both fromthe values of soft parameters μ̃_ soft and μ_ soft as well as the values of the saturation scale at low energies, and from theintegration in AM5 andAM6, which were taken neglectingcontribution from the regionτ'< 1. On the other hand, thecontribution to the double inclusive cross sections of the diagram ofmudia at y_12 ≪ 1 coincide with the contributionof becor-b, d^2 σmudia/d y_1d y_2d^2 p_T1 d^2 p_T2d^2 σbecor-b/d y_1d y_2d^2 p_T1 d^2 p_T2Therefore, to obtain the realistic estimate we use the followingprocedure of matchingv_2 p_T = 5GeV,y_12= 2|_mudia = v_2p_T = 5GeV|_becor-b ; v_4 p_T = 5GeV, y_12 = 2|_mudia = v_4p_T = 5GeV|_becor-b ; where v_2p_T = 0.5GeV|_becor-b andv_4p_T = 0.5GeV|_becor-b are takenfrom Ref.<cit.> where the estimates were performed basedon the model for soft interaction which describes all featuresof soft interaction at high energy and provides the interfacewith the hard processes. v shows the p_T and y dependence of the v_2 and v_4using ANGLECOR15 for normalization. In addition we takeΔ_ BFKL = 0.25 and Q^2_s( y) ∝ expλ y with λ = 0.25. These values correspondto the BFKL Pomeron phenomenology. We believe that this figureillustrates the scale of rapidity dependence and will beinstructive for future experimental observations.§ CONCLUSIONSIn this paper we generalize the interference diagram, that described the Bose-Einstein correlation for small rapidity difference y_12 ≪ 1, to include the emission of the gluons with rapidities (y_i)between y_1 and y_2 (y_1 , y_i < y_2). We calculate the resulting diagram in CGC/saturation approach and make two observationswhich we consider as the main result of this paper.The first one is a substantial decrease of the odd Fourierharmonics v_2 n + 1 as a function of the rapidity differencey_12 ( see R-c). The second result is, that even Fourierharmonic v_2 n has a rather strong dependence on y_12,showing a considerable increase in the region of large y_12(see v). We believe that our calculations, that have beenperformed both for the simplest diagrams and for the CGC/saturationapproach, will be instructive for further development of the approachespecially in the part that is related to the integration of themomenta transferred by the BFKL Pomerons.We demonstrated in this paper the general origin of the densityvariation mechanism, whose nature does not depend on the techniquethat has been used. This mechanism has to be taken into account,since it leads to the values of Fouriers harmonics that are largeand of the order of v_n that have been observed experimentally. We hope that the paper will be usefulin the clarification of theoriginof the angular correlation, especially for hadron-hadron scattering athigh energy.We firmly believe that the experimentalobservation of both phenomena: the sharp decrease of v_n with odd n and the substantial increase of v_n with even n as a function of y_12,will be a strong argument forCGC/saturation nature of the angular correlations.§ ACKNOWLEDGEMENTSWe thank our colleagues at Tel Aviv university and UTFSM forencouraging discussions. Our special thanks go toCarlos Contreras, Alex Kovner and Misha Lublinsky for elucidating discussions on thesubject of this paper. This research was supported by the BSF grant 2012124, by Proyecto Basal FB 0821(Chile) ,Fondecyt (Chile) grant1140842, and by CONICYT grant PIA ACT1406. 99CAUSALITY A. Dumitru, F. Gelis, L. McLerran and R. Venugopalan,Nucl. Phys. A810 (2008) 91 [arXiv:0804.3858 [hep-ph]]. FINSTATE E. V. Shuryak,Phys. Rev. C76 (2007) 047901 [arXiv:0706.3531 [nucl-th]]; S. 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Rev.D70, 114017 (2004) [Erratum-ibid.D71, 079901 (2005), [arXiv:hep-ph/0405266]. GOLELAST E. Gotsman and E. Levin,Bose-Einstein correlations in perturbative QCD: v_n dependence on multiplicity, Phys. Rev. D( in press)[ arXiv:1705.07406 [hep-ph]]. | http://arxiv.org/abs/1709.08954v1 | {
"authors": [
"E. Gotsman",
"E. Levin"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170926115746",
"title": "Azimuthal angle correlations at large rapidities: revisiting density variation mechanism"
} |
Picard-Fuchs operatorsfor octic arrangements I(The case of orphans) Slawomir Cynk and Duco van Straten========================================================================= Current Filamentation Instability (CFI) is capable of generating strong magnetic fields relevant to explain radiation processes in astrophysical objects and lead to the onset of particle acceleration in collisionless shocks. Probing such extreme scenarios in the laboratory is still an open challenge. In this work, we investigate the possibility of using neutral e^- e^+ beams to explore the CFI with realistic parameters, by performing 2D particle-in-cell simulations. We show that CFI can occur unless the rate at which the beam expands due to finite beam emittance is larger than the CFI growth rate and as long as the role of competing electrostatic two-stream instability (TSI) is negligible. We also show that the longitudinal energy spread, typical of plasma based accelerated electron-positron fireball beams, plays a minor role in the growth of CFI in these scenarios.§ INTRODUCTIONThe fireball is a promising model for the generation of γ-ray bursts (GRBs) <cit.>. The model relies on the dissipation of kinetic energy of an ultrarelativistic flow, which emits γ-rays via synchrotron or synchrotron self-Compton emission. As a result, dense radiation and e^- e^+ pair fluids are produced, known as a fireball <cit.>. The interaction of the fireball beam, characterised by relativistic factors ranging from 10^2-10^6, with the external medium can drive field structures that accelerate particles to high energies. As particles accelerate, they will also emit strong radiation bursts, with wavelengths ranging from γ-rays to radio waves. Astrophysical observations indicate that the main process leading to radiation emission is synchrotron radiation, which requires large amplitude magnetic fields on the order of Gauss to operate <cit.>. The origin of magnetic fields, and their amplification to these extreme values is a pressing challenge in astrophysics <cit.>. There has been an extensive effort, based on theoretical and numerical advances, with the objective of understanding the mechanisms by which strong magnetic fields are formed in astrophysical scenarios <cit.>. Medvedev, Waxman and Loeb <cit.>proposed that the current filamentation/Weibel Instability (CFI/WI) is a leading mechanism allowing for the growth of magnetic fields in the astrophysical context. The corresponding growth rates range from a few microseconds to a few tenths of a second, consistent with the time scales of GRBs <cit.>. These instabilities arise due toan anisotropic velocity distribution in the plasma (WI) or due to a counter-streaming flow of plasma slabs (CFI). Numerical calculations have shown that these instabilities are capable of generating strong magnetic fields with 10^-5 - 10^-1 of the energy density equipartition <cit.>.Similar observational evidence of electron-positron (e^- e^+) pair production has been found in TeV Blazers <cit.>, where the interaction of TeV photons with the extragalactic background light produces ultra-relativistic e^- e^+ beams. As these e^- e^+ beams stream through the intergalactic medium (IGM), the collective beam-plasma instabilities can be relevant. The impact of beam-plasma instabilities upon γ-rays emission of bright TeV sources and their subsequent cosmological consequences have been previously investigatedtheoretically and numerically using realistic parameters <cit.>. The goal of this work is to identify conditions to explore such instabilities in laboratory conditions by using realistic finite size fireball beams. Leveraging on fully kinetic one-to-one particle-in-cell (PIC) simulations, we define the criteria for probing the Oblique Instability (OBI) and the CFI experimentally.Exploring laboratory surrogates capable of reproducing these mechanisms under controlled conditions is a promising path to gain physical insights that would be otherwise inaccessible. One of the configurations that have been identified towards this goal is the study of the propagation of quasi-neutral relativistic fireball beams in the plasma <cit.>. A globally neutral fireball beam is composed of equal amounts of electrons and positrons with identical density and spectral distributions. Recent experimental developments <cit.> promise to make this exploration possible. The generation of quasi-neutral electron-positron fireball beams, with maximum energy ≃ 400 MeV (average γ∼ 15), has been achieved in a laser-plasma accelerator. These beams have large energy spreads, they have a finite length and transverse size, and have limited charge. Another method for generating a fireball beam is to superimpose an electron e^- and a positron e^+ bunches as could be done for example at SLAC <cit.>. Numerical simulations show that this extremely relativistic fireball beam γ = 40000 is also subject to CFI <cit.>. Note that CFI of a mildly relativistic e^- bunch γ = 112 was observed showing filamentation and its coalescence <cit.>. Thus, although there have been efforts to understand the generation of magnetic fields through the Weibel/CFI under ideal conditions (i.e. infinitely wide planar plasma slabs) <cit.>, the role of realistic beam parameters in the growth of these instabilities remains to be understood in detail. In this work, we perform a detailed numerical and theoretical study of the interaction of a realistic fireball beam (with a length comparable or shorter than the plasma wavelength) in an uniform plasma using ab initio two-dimensional PIC simulations with the PIC code OSIRIS <cit.>. We examine in detail the temporal growth of the magnetic field that arises during the interaction between the fireball beam with the plasma. We then find that the growth of electrostatic modes, associated with competing instabilities, can be suppressed as long as the ratio between the beam density and the plasma density is sufficiently high. To make connection with recent experiments <cit.>, we also investigate the role of the finite beam emittance in the beam dynamics, and find a threshold beam emittance for the occurrence of CFI. In addition, we found that the beam energy spread will not affect the growth of the CFI significantly. We consider ultra-relativistic fireball beams, with Lorentz factor γ ranging between ∼10^3-10^4, propagating in the plasma with densities ranging between 10^15-10^17 cm^-3. These are parameters that can be explored in the laboratory. Our results show that the physics of OBI or CFI could be tested in the laboratory using presently or soon to be available electron-positron fireball beams.§ SIMULATIONS OF THE CURRENT FILAMENTATION INSTABILITYThe onset of CFI occurs when the ratio between the transverse beam size (σ_y) and the plasma skin depth (k_p^-1=c/ω_p), is k_p σ_y ≥ 1, where ω_p = √(4 π n_0 e^2 /m_e) is the plasma frequency, n_0 the background plasma density, m_e the mass of the electron, e the charge of the electron and c the speed of light. When the transverse beam size is larger than the plasma skin depth the plasma return currents can flow through the beam leading to the growth of CFI. If this condition does not hold, i.e when (σ_y ≤ c/ω_p), the CFI does not grow <cit.>.In order to illustrate the generation of magnetic fields through the CFI, we start by describing the results from 2D OSIRIS PIC simulations <cit.>. The simulations use a moving window travelling at c. The simulation box has absorbing boundary conditions for the fields and for the particles in the transverse direction. The globally neutral fireball beam is initialized at the entrance of a stationary plasma with n_0=10^17 cm^-3. The initial density profile for the electron and positron fireball beam is given by n_b = n_b0exp(-x^2/σ_x^2 - y^2/σ_y^2) where n_b0= n_0=10^17 cm^-3, σ_x=0.99 c/ω_p=10.2 μm and σ_y=2 c/ω_p=20.4 μm are the bunch peak density, length and transverse waist, respectively. The beam propagates along the x-axis with Lorentz factor γ_b = 5.6 × 10^4, with transverse velocity spread v_th/c = 1.7 × 10^-5 and with no momentum spread in the longitudinal direction. The simulation box dimensions are L_x= 8.02 c/ω_p and L_y= 20.0 c/ω_p with a moving window travelling at c along x. The box is divided into 128 × 512 cells with 2×2 particles per cell. Figure <ref> depicts the growth of the transverse magnetic field energy (panel a), the beam filaments due to the CFI (panel b), and the typical electromagnetic field structure (panels c and d). Figure <ref>a shows that the growth of the magnetic field energy as a function of the propagation distance is exponential as expected from the CFI. In Fig. <ref>a, the electromagnetic field is normalized with respect to the initial kinetic energy of the particles ϵ_p = (γ_b - 1)V_b, where V_b = (πσ_xσ_y) is the volume of the beam. Simulations reveal that the field energy grows at the expense of the total kinetic energy of the fireball beam. The linear growth rate of the CFI measured in the simulation is Γ_CFI/ω_p≃ 6.0 × 10^-3, in good agreement with Silva et al 2002 <cit.>. As a consequence of the instability, the beam breaks up into narrow (with a width on the order of 0.5 c/ω_p, which correspond to 5μ m for our baseline paramters) and high current density filaments. Figure <ref>(b) shows that these electron-positron filaments are spatially separated from each other. Each filament carries strong currents which lead to the generation of strong out-of-the-plane (i.e. azimuthal) magnetic fields with amplitudes beyond 20 T. The azimuthal magnetic fields are also filamented, as shown in Figure <ref> (c). Because of their finite transverse momentum, simulations show that current filaments can merge. As merging occurs, the width of the filaments increases, until beam breakup occurs. At this point, the CFI stops growing, and no more beam energy flows into the generation of azimuthal magnetic fields. Simultaneously, radial E-fields above 10 GV/m are also generated (Fig. <ref>d).§ ROLE OF THE PEAK BEAM DENSITY AND BEAM DURATION IN THE GROWTH OF CURRENT FILAMENTATION INSTABILITYIn the previous section, and for illustratation purposes only, we have considered that the total beam density was twice the background plasma density. In this section, we will investigate the propagation of beams with lower peak densities. In order to keep the number of particles constant, we then increase the beam length, such that σ_x ≥λ_p. In these conditions, the OBI competes with the CFI <cit.>. The OBI can growwhen the wave-vector is at an angle with respect to the flow velocity direction, and it leads to the generation of both electric and magnetic field components. The maximum growth rate for the CFI and the OBI are given by Γ_CFI∼√(α/γ_b) β_b0 and Γ_OBI∼√(3)/2^4/3 (α/γ_b)^1/3 respectively <cit.>, where α is the beam (n_b) to plasma density (n_0) ratio and β_b0 = v_b/c is the normalized velocity of the beam. Thus, the ratio between the CFI growth rate and the OBI growth rate, which is given by: Γ_OBI/Γ_CFI = √(3)/2^4/31/β_b( γ_b/α)^1/6Equation (3.1) provides the range of parameters for which each instability will dominate. The OBI dominates over the CFI when α is smaller 1. For the number of particles we considered in the simulations, this implies that only beams with σ_y < 2 λ_pe will be subject to the CFI.In order to verify this hypothesis, we have carried out additional two-dimensional OSIRIS PIC simulations using the initial set up described in Sec. <ref>, varying σ_x between 2λ_p and 10λ_p, for which α varies between 0.0026 and 1.0. In all these cases, our results have consistently shown the evidence of the OBI growth.In Figure <ref>, we show an illustrative simulation result considering σ_x = 2λ_p, with n_b = 1.274×10^15 cm^-3, for which α= 0.01274. In order to describe the propagation of a longer beam, we have increased the simulation box length. We then increased the longitudinal box length to L_x = 63 c/ω_p (L_y = 20 c/ω_p remains identical to that of Sec. <ref>). The box is now divided into 1024×512 cells with 2×2 particles per cell for each species.Figure <ref> (a) illustrates the evolution of the longitudinal and transverse electric and transverse magnetic energy (normalized to ϵ_p = (γ_b - 1)V_b, where V_b = (πσ_x σ_y) is the volume of the beam). The emergence of oblique modes can be seen in Fig. <ref> (b), which shows tilted beam filaments. In a multi-dimensional configuration, the oblique wave-vector couples the transverse (filamentation) and longitudinal (two-stream) instabilities resulting in the electromangetic beam plasma instability. Unlike Fig. <ref>, the simulation results in Fig. <ref> show that the transverse electric field (E_y) component provides the dominant contribution to the total field energy. The plasma is only weakly magnetized ω_c/ω_p = 0.01, much lower than in Fig. <ref>c where ω_c/ω_p≃ 0.6, and the CFI does not play a critical role in the beam propagation. The longitudinal and transverse electric fields grow exponentially, as predicted by the linear analysis of the OBI, matching well the simulation results. The growth rate measured in the simulations is Γ_max/ω_p≃Γ_OBI≃ 2.1 × 10^-3, while the theoretical growth rate is ≃ 2.0 × 10^-3. The OBI generates plasma waves with strong radial electric fields in excess of 500 MV/m [Fig. <ref> (d)]. After 20 cm, the OBI saturates. Despite being limits of the same instability, the electromagnetic beam plasma instability, we will refer to the CFI and the OBI has manifestition of qualitative different behaviour of the same instability.§ EFFECTS OF FINITE BEAM WAIST AND EMITTANCE Theoretical and numerical studies performed to identify the effect of beam emittance on the growth of plasma instabilities and their saturation <cit.> typically assume that the beam is infinitely wide. In this section, we will investigate the role of the beam emittance considering finite beam size effects, in order to make closer contact with laboratory conditions. To study the influence of the beam emittance on the propagation, we first consider the equation for the evolution of the beam waist σ_y in vacuum <cit.>1/c^2d^2 σ_y/dt^2 = ϵ_N^2/σ_y^3 γ_b^2,where σ_y is the beam radius, ϵ_N ≃Δ p_⊥σ_y is a figure for the beam emittance (corresponding to the area of the beam transverse phase space), and Δ p_⊥ is the transverse momentum. According to Eq. (<ref>), the evolution for σ_y and for sufficiently early times is given by:σ_y ≃σ_y0(1 + ϵ_N^2 t^2 c^2/2 σ_y0^4 γ_b^2)^1/2 , where σ_y0 is the initial beam radius. Hence, according to Eq.(4.1), the rate at which σ_y increases is:1/σ_ydσ_y/dt = t c^2 ϵ_N^2/σ_y0^2 γ_b^2 ,Equation (4.3) indicates that the beam expands in vacuum due to its transverse momentum spread. As the beam expands, n_b decreases as n_b/n_0 ∼ (σ_y0/σ_y)^2, in 3D, and as (σ_y0/σ_y), in 2D. Because of the reduction of n_b/n_0, the growth rates for the CFI and for the OBI will also decrease. We then estimate that these instabilities (i.e CFI and OBI) are suppressed when the rate at which n_b/n_0 decreases is much higher than the instability growth rate. Matching the rate at which the beam density drops, which in 2D is given by (1/σ_y) (dσ_y/d t), to the growth rate of the instability (Γ) gives an upper limit for the maximum beam divergence θ = Δ p_⊥/γ_b (and emittance ϵ_N ≈σ_r (<p_⊥^2>)^1/2) allowed for the growth of the CFI/OBI:θ = ( Γσ_y0^2/L_growth c)^1/2,where we have considered that t∼ L_growth/c in Eq. (4.4), being L_growth the growth length of the CFI/OBI instability. Equation (4.4) then gives the threshold beam divergence, beyond which the CFI/OBI will be suppressed. It indicates that beams with higher energy can support higher divergences and still be subject to the growth of the CFI because the beam expands slowly in comparison to lower energy beams. Similarly, beams with higher σ_y0 also support higher emittance than narrower beams because of the slower expansion rate.To confirm our theoretical findings, we performed additional two-dimensional simulations using fireball beams with relativistic factors γ_b =700, 1050, 1400 (the lower γ_b factors used now, in comparison to Sec. 2, minimize the computational requirements). We use σ_x=0.22 c/ω_p=11.7 μm and σ_y=10 c/ω_p=530 μm with peak density n_b0=10 n_0=10^15 cm^-3. For each case, we varied the transverse temperature Δ p_⊥ = γ_b θ_ze0 = 1, 3, 5, 7, 10 and 20 in order to determine the threshold beam spread for the occurrence of instability. We note that we have used the classical addition of velocities in the beam thermal spread initialization in order to more clearly identify the dependence of evolution of the instabilities with emittance.Figure <ref>(a) shows that the magnetic field energy decreases with increasing transverse momentum spread. Fig. <ref> (a) also shows a transition in the evolution of the magnetic energy between Δ p_⊥ = 10, where the B-field still grows at the end of the simulation, and Δ p_⊥ = 20, where the B-field decreases with propagation distance. According to Eq. (<ref>), using L_growth∼ 0.06 m and Γ_CFI∼ 1.657 × 10^11 s^-1, we obtain the threshold θ∼ 0.12 for the shutdown of the instability. This is in good agreement with Fig. <ref> (a). Figure <ref> (b) depicts the dependence of the threshold beam emittance with the fireball beam energy. Figures <ref> (c)-(d)show the positron density for two simulations, where all the parameters are kept constant, except for the beam emittance. In particular, in Fig. <ref> (c) a beam emittance of Δ p_⊥ = 1, much smaller that the threshold value given by Eq. (3), has been considered. In this case the CFI develops, leading to the filamentation of the beam (see Fig. <ref>c) and to the exponential growth of the magnetic field energy (see Fig. <ref> (a), red curve). However, in the second case (Fig. <ref> (d)) a higher beam emittance Δ p_⊥ =10is considered. This suppresses the growth of the magnetic field energy (see Fig. <ref> (a), black curve). As a result, the beam expands before the development of the CFI. These results show that the growth of CFI can only be achieved if the beam emittance is sufficiently small.§ EFFECT OF BEAM ENERGY SPREAD In typical laboratory settings <cit.>, electron-positron fireball beams can contain finite energy spreads. It is, therefore, important to evaluate the potentially deleterious role of the energy spread in the growth of CFI. In this section, we then present simulation results with finite longitudinal momentum spreads. We consider that the central beam relativistic factor is γ_b = 700, and compare two simulations with Δ p_x/γ_b = 0.13 and Δ p_x/γ_b = 0.29 (Δ p_x is the longitudinal momentum spread). All other simulation parameters are similar to those described in Sec. 4.Figures (4) (a)-(b) show the temporal evolution of the beam electrons density for Δ p_x/γ_b = 0.13 (Fig. 4(a)) and for Δ p_x/γ_b= 0.29 (Fig. 4(b)). The initial energy spectra of these two beams are shown in Fig. 4(c). Figure 4 (d) shows the comparison of the magnetic field energy evolution.The blue curve shows the growth of magnetic field energy generated by the fireball beam with energy spread Δ p_x/γ_b = 0.29, while the red curve is associated with the lower energy spread Δ p_x/γ_b = 0.13. Simulation results demonstrate that the CFI grows in all cases, in agreement with analytical calculations <cit.>. The green line in Fig. 4(d) is the theoretical growth rate Γ/ω_p ≃ 2.0× 10^-2, which is in good agreement with the simulation growth rate (shown by the red line in Fig. 4b). § SUMMARY AND CONCLUSIONS In summary, the growth and saturation of a ultra-relativistic beam propagating through a plasma have been investigated using particle-in-cell (PIC) simulations. We have shown that short fireball beams, i.e beams shorter than the plasma wavelength, interacting with uniform plasmas lead to the growth of the CFI. For typical parameters available for experiments, the instability can generate strong transverse magnetic field on the order of the MGauss. The instability saturates after 10 cm of propagation in a plasma with n_0 ∼ 10^17 cm^-3.We have demonstrated that the beam density needs to be higher than the background plasma density to suppress the growth of the competing OBI instability, which leads to the growth of electrostatic modes (instead of electromagnetic). Beams with lower peak densities will then drive the OBI, which results in tilted filaments and the generation of mostly electrostatic plasma waves. We have also showed that the beam emittance needs to be minimized, reducing transverse beam defocusing effects, which can shutdown the CFI or the OBI if the beam defocuses before these instabilities grow. We have also extended our numerical studies to investigate the effect of finite fireball energy spreads on the growth of CFI, and showed that the energy spreads of currently available fireball beams allow for the growth of CFI in the laboratory.In conclusion, we have identified the factors for the generation of strong magnetic fields via CFI. We expect that the results will influence our understanding of astrophysical scenarios, by revealing the laboratory conditions where these effects can be studied.This work was partially supported by the European Research Council (ERC-2016-InPairs 695088). Simulations were performed at the IST cluster (Lisbon, Portugal). J. V. acknowledges the support for FCT (Portugal). jpp | http://arxiv.org/abs/1709.09747v2 | {
"authors": [
"N. Shukla",
"J. Vieira",
"P. Muggli",
"G. Sarri",
"R. Fonseca",
"L. O. Silva"
],
"categories": [
"physics.plasm-ph"
],
"primary_category": "physics.plasm-ph",
"published": "20170927215657",
"title": "Conditions for the onset of the current filamentation instability in the laboratory"
} |
SURGE: Continuous Detection of Bursty Regions Overa Stream of Spatial Objects Kaiyu Feng^1, 2, Tao Guo^2, Gao Cong^2, Sourav S. Bhowmick^2, Shuai Ma^3^1 LILY, Interdisciplinary Graduate School. Nanyang Technological University, Singapore^2 School of Computer Science and Engineering, Nanyang Technological University, Singapore ^3 SKLSDE, Beihang University, China {kfeng002@e., tguo001@e., gaocong@, assourav@}ntu.edu.sg, [email protected] 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================= With the proliferation of mobile devices and location-based services, continuous generation of massive volume of streaming spatial objects (i.e., geo-tagged data) opens up new opportunities to address real-world problems by analyzing them. In this paper, we present a novel continuous bursty region detection () problem that aims to continuously detect a bursty region of a given size in a specified geographical area from a stream of spatial objects. Specifically, a bursty region shows maximum spike in the number of spatial objects in a given time window. The problem is useful in addressing several real-world challenges such as surge pricing problem in online transportation and disease outbreak detection. To solve the problem, we propose an exact solution and two approximate solutions, and the approximation ratio is 1-α/4 in terms of the burst score, where α is a parameter to control the burst score. We further extend these solutions to support detection of top-k bursty regions. Extensive experiments with real-world data are conducted to demonstrate the efficiency and effectiveness of our solutions.§ INTRODUCTIONPeople often share geo-tagged messages through many social services like Twitter and Facebook. Each geo-tagged data is associated with a timestamp, a geo-location, and a set of attributes (e.g., tweet content). In this paper, we refer to them as spatial objects. With the proliferation of GPS-enabled mobile devices and location-based services, the amount of such spatial objects (e.g., geo-tagged tweets and trip requests using Uber) is growing at an explosive rate. Their real-time nature coupled with multi-faceted information and rapid arrival rate in a streaming manner open up new opportunities to address real-world problems. For example, consider the following problems.The world regularly faces the challenge of tackling a variety of virus epidemics such as sars, mers, Dengue, and Ebola. Most recently, the outbreak of mosquito-borne Zika virus started in Brazil in 2015.Hence, the Center for Disease Control and Prevention needs to continuously monitor different areas for possible Zika outbreak and issue alerts to people who are traveling to or livingin regions affected by Zika. Since early detection of such outbreak is paramount, how can we identify potential Zika-affected region(s) in real time?One strategy to address this issue is to continuously monitor geo-tagged tweets (i.e., spatial objects) coming out of a specific area (e.g., Florida) and detect regions where there are sudden bursts in tweets related to Zika (e.g., containing Zika-related keywords) in real time. Observe that these “bursty regions” are dynamic in nature. However, it is computationally challenging to continuously monitor massive streams of spatial objects and detect bursty regions in real time.Online transportation network companies such as Uber, Lyft, and Didi Dache have disrupted the traditional transportation model and have gained tremendous popularity among consumers[In 2017, Uber is available in over 81 countries and 570 cities worldwide.]. Consumers can submit a trip request through their mobile apps. If a nearby driver accepts the request, he will pickup the consumer.Although this disruptive model has benefited many drivers and consumers, the latter may have to wait for a long time for a car when the number of car requests significantly surpasses the supply of nearby drivers. Clearly, it is beneficial to both passengers and drivers if we can notify idle drivers in real time whenever there is a sudden burst in consumer demand in areas of interest to them.An additional benefit to the drivers is that the trip fare may be increased due to the “surge pricing” policy [ For example, the price increased 10X on new year's eve in 2016 in the United States (<www.geekwire.com/2016/customers-complain-uber-prices-surge-near-10x-new-years-eve/>)] where thecompanies may increase a trip price significantly when demand is high. For instance, consider Figure <ref>, which shows the trip requests in two time windows [t_1, t_2] and [t_2, t_3]. Suppose a driver is only interested in the area shown by dashed rectangle to pick up passengers. Observe that there is a burst of trip requests in regions r_1 and r_2 (both increased by 3). If the app can notify the driver in real time about these two regions, then he can move in there to pickup potential passengers. Note that such soaring demand is not always predictable as it may not only occur during holidays or periodic events (e.g., new year's eve) but also due to unpredictable events such as subway disruption, concerts, road accident, inclement weather, and terrorist attack.There are two common themes in the two examples. First, we need to continuously monitor a large volume of spatial objects (e.g., trip requests and geo-tagged tweets) to detect in real time one or more regions that show relatively large spike in the number of spatial objects (i.e., bursty region) in a given time window. Second, a user needs to specify as input the size a× b of rectangular-shaped bursty region that one wishes to detect. For instance, in Example <ref> different drivers may prefer bursty regions of different sizes according to their convenience.In this paper, we refer to the problem embodied in the aforementioned motivating examples as continuous bursty region detection () problem. Specifically, given aregion size a× b and an area A, the aim of the problem is to continuously detect a region of the specified size in A that demonstrates the maximum burstiness from a stream of spatial objects. To model the burstiness of a region, we propose a general function based on the sliding window model. We also extend our problem to detect top-k bursty regions as in certain applications one may be interested in a list of such regions. The problem and its top-k variant are challenging as we need to handle rapidly arriving spatial objects in high volumes to efficiently detect and maintain bursty regions. For example, 10 million geo-tagged tweets are generated each day in Twitter[<https://www.mapbox.com/blog/twitter-map-every-tweet/>].As we shall see later, it is prohibitively expensive to recompute bursty regions frequently. In this paper, we first propose an exact solution called cell-cSpot to keep track of the bursty region over sliding windows. Specifically, we first reduce the problem to continuous bursty point detection () problem. Then we propose a cell-based algorithm to continuously detect the bursty point.It takes O(|c_max|^2 + log n) time to process a new arriving spatial object on average, where |c_max| is the maximum number of objects that we search inside a cell, and n is the number of indexed rectangle objects.Although cell-cSpot can address the problem efficiently in several scenarios, it becomes inefficient as |c_max| increases (e.g., the sliding windows get larger, the region size gets larger, or the arrival rate of the spatial objects increases). To address this we further propose two approximate solutions, namely gap-surge and mgap-surge, with an O(log n) time complexity to process a spatial object. The approximation ratio is bounded by 1-α/4, where α∈ [0,1) is a parameter used in the burst score function. Last, we show that our proposed solutions can be elegantly extended to continuously detect top-k bursty regions. Our experiments reveal that our proposed solutions can handle streams with up to 10 millions spatial objects arrived per day. In summary, this paper makes the following contributions:(1) We propose a novel continuous bursty region detection () problem for continuously detecting bursty regions in a specified area from a stream of spatial objects. (Section <ref>)(2)We present an exact solution (cell-cSpot) and two approximate solutions (gap-surge and mgap-surge) to address the problem (Sections <ref> and <ref>). We further extend these solutions to keep track of top-k bursty regions efficiently (Section <ref>).(3)We conduct experiments with real-world datasets to show the efficiency of our proposed solutions. All solutions are efficient in real time. Moreover, gap-surge and mgap-surgescale well w.r.t. high arrival ratewhile the returned regions have competitive burst scores. The extended versions can also detect top-k bursty regions efficiently in real time. (Section <ref>). The proofs of lemmas and theorems are given in Appendix <ref>. § RELATED WORKBurst detection. Our problem is related to the problem of detecting bursty patterns and topics. A host of work has been done to detect temporal bursts <cit.>. A collection of proposals focus on detecting bursty features (represented by probability distribution of words)<cit.>. The other work focuses on detecting a timespan over the stream such that its aggregate is larger than a threshold <cit.>. All these burst detection problems are different from our problem as they disregard the spatial information when detecting the temporal bursts.Most germane to our work are efforts on exploring spatial-temporal bursts <cit.> albeit from different aspects. Mathioudakis et al. <cit.> study the problem of identifying notable spatial burst out of a collection of user generated information. They divide the space into cells, and recognize two states for each cell, namely “bursty” and “non-bursty”.additive cost function. Our problem differs from it in two key aspects. First, the spatial burst is identified as a cell in the grid whereas the bursty region in can be located at any position. Second, the solution developed in <cit.> is designed for data stored in a data warehouse, and it cannot be deployed or adapted to solve the problem.Lappas et al. <cit.> study the problem of identifying a combination of a temporal interval and a geographical region with unusual high frequency for a term from a set of geo-tagged text streams. Its problem setting is different from ours: Lappas et al.<cit.> takes as input a set of text streams with fixed geographical locations, while in our problem, spatial objects arrive as a stream and an object can be located in any location of the given space. In addition, the proposed solution can only handle a small number of text streams (tens to hundreds) due to its high computational complexity.Given a geo-tagged tweet stream, Zhang et al. <cit.> aim to continuously detect real-time local event. Specifically, a local event is defined as a cluster of tweets that are semantically coherent and geographically close. For each keyword in a tweet, its burstiness is a linear combination of its temporal burstiness and its spatial burstiness with a balance parammeter η. The spatiotemporal burstiness of a cluster of tweets is the aggregation of the burstiness of all the keywords in the cluster.Our problem differs from it in the following aspects. First, the bursty event is identified as a cluster of geo-tagged tweets, while our problem aims to detecting a spatial region. Second, the proposed framework is built over geo-textual stream. The textual content serves as an important feature in their system. Our is applicable to any kind of spatial stream.Dense region search. Our problem is also related to dense region search over moving objects <cit.>. Given a set of moving objects, whose positions are modeled as linear functions in Euclidean space, the dense region search problem aims to find all dense regions at query time t. Jensen et al. <cit.> constraint dense regions to be non-overlapping square-shaped regions of given size, whose density is larger than a user-specified threshold.Ni et al. <cit.> propose a new definition of dense regions, which may have arbitrary shape and size.In the dense region search problem, the positions of the moving objects are modeled as linear functions. Thus the position of each moving object can be computed at any time.In contrast, in the problem, the number of the newly-arriving spatial objects and their positions are unknown a priori. Moreover, the density function is different from our burst score function, requiring different techniques to compute the burst score of a given region. Region search. Our problem is also related to the region search problem. A class of studies aims to find a region of a given size such that the aggregation score of the region is maximized<cit.>. Given a set of spatial objects, the max-enclosing rectangle () problem <cit.> aims to find the position of a rectangle of a given size a× b such that the rectangle encloses the maximum number of spatial objects. This problem is systematically investigated as the maximizing range sum (MaxRS) problem <cit.>.Feng et al.<cit.> further study a generalized problem of the MaxRS problem, in which the aggregate score function is defined by submodular monotone functions, which include sum. Liu et al. <cit.> study the problem of finding subject oriented top-k hot regions, which can be considerd as a top-k version of the MaxRS problem. Cao et al.<cit.> study the problem of finding a subgraph of a given size with the maximum aggregation score from a road network. All these aforementioned region search problems focus on static data. Moreover, the idea of invoking the approach designed for the region search problem whenever a object enters or leaves the sliding windows is prohibitively expensive (We will elaborate on this in Section <ref>). Our work is closely related to the recent efforts on continuous MaxRS problem <cit.>. Amagata et al. <cit.> propose the problem of monitoring the MaxRS region over spatial data streams. Specifically, given a stream of weighted spatial objects, the continuous MaxRS problem aims to monitor the location of a rectangle of a size a× b such that the sum of the weights of the objects covered by the rectangle is maximized. In the proposed algorithm, a grid is imposed over the space, whose granularity is independent from the size of the query rectangle. For each spatial object in the stream, they generate a rectangle of a size a× b whose center is located at the spatial object. The generated rectangle is mapped to the cells with which it overlaps. For each cell, they maintain a graph where each node in the graph is a rectangle mapped to this cell, and two nodes are connected by a directed edge if they overlap with each other. The graph is used to handle the updates of the stream. For each rectangle in the cell,they maintain an upper bound to determine when to invoke the sweep-line algorithm <cit.> to find the most overlapped region inside the rectangle. With the maintained upper bounds, they use a branch-and-bound algorithm to reduce the search space.The difference of the problem from the continuous MaxRS problem is that the burst score of the problem is defined over two consecutive sliding windows, and spatial objects in different windows contribute differently to the burst score. Though their solution cannot be directly applied to solve the problem, we can adapt their solution with some modifications for the problem. The details of the modification are reported in Appendix <ref>. One issue of this solution is that they need to maintain a graph for each cell with a space cost of O(n^2), where n is the number of rectangle objects that are mapped to the cell. When the number of objects mapped to a cell is large, the space cost could be extremely high. We will show in Section <ref> that our proposed solutions outperform the aG2 algorithm for the problem. Hussain et al. <cit.> investigates the MaxRS problem on the trajectories of moving objects. Given the trajectories of a set of moving points, they aim to maintain the result of the MaxRS problem at any time instant. Its problem setting is different from ours: it takes as input the trajectories of a set of fixed number moving objects, while in our problem, the number of spatial objects in the sliding windows may vary with time and the positions of the new arrived objects are unknown a priori. Data stream management. Our work is also related to data stream management. There has been a long stream of work on various aspects ofdata streams since the last decade. Some examples are stream clustering <cit.>, stream join processing <cit.>, and stream summarization <cit.>. Since most of these studies focus on general data streams, we only review the work that involves spatial information. Given a stream of spatial-textual objects, <cit.> aims to estimate the cardinality of a spatial keyword query on objects seen so far. A host of work has also been done to study content-based publish/subscribe systems <cit.> over spatial object streams. In these systems, streaming published items are delivered to the users with matching interests. However, none of these studies consider the problem of detecting bursty regions. Spatial outlier detection. Lastly, our work is also related to spatial outlier detection <cit.>.Lu et al. <cit.> investigate the spatial outlier detection problem over point data. Specifically, given a set of weighted spatial points,the spatial outlier detection problem aims to identify top m points such that their weight is greatly different from the average weight of its k nearest neighbors. Zhao et al.<cit.> further investigate region outliers detection over meteorological data. All these aforementioned problems focus on static data. Moreover, the outliers are selected from the data points in the spatial outlier detection problem. In contrast, the location of the bursty region in our problem can be located at any position in the space. In addition, the spatial outlier detection problems use a totally different function to evaluate how much a data point is different from its neighbors. Due to these differences, their proposed solutions cannot be adapted to address the problem.§ PROBLEM STATEMENT We formally define the ContinuouS BUrsty ReGion DEtection () problem. We begin by defining some terminology.§.§ Terminology A spatial object is represented with a triple o=⟨ w, ρ, t_c⟩, where w is the weight of o, ρ is a location point with latitude and longitude, and t_c is the creation time of object o. In this paper, we consider a stream of spatial objects. For example, geo-tagged tweets in Twitter can be viewed as a stream of spatial objects arriving in the order of creation time. The weight of a tweet could be the relevance of its textual content to a set of query keywords. The car requests in Uber can also be viewed as a stream of spatial objects arriving in the order of calling time. In this case, the weight could be the passenger number or travel fare.We next introduce two consecutive time-based sliding windows, namely current and past windows. Given a window size |W|, the current window, denoted by W_c is a time period of length |W| that stretches back to a time point t - |W| from present time t. The past window, denoted byW_p is a time period of length |W| that stretches back to a time point t - 2 |W| from the time point t - |W|. Given a region r and a sliding window W, let O(r, W) be the set of spatial objects which is created in W and located in region r, i.e., O(r,W) = {o|o.ρ∈ r ∧ o.t_c ∈ W}. Let f(r, W) be the summation of weights of objects in O(r, W) normalized by W's length, i.e., f(r, W) = ∑_o∈ O(r,W)o.w/|W|, which is the score of a region r w.r.t. the sliding time window W. Note that in this paper, for the sake of simplicity, we assume the current window and the past window have the same length |W|. However, our proposed solution is equally applicable when the two sliding windows have different lengths.§.§ Burst Score Intuitively, the burst score of a region r reflects the variation in the spatial objects in r in recent period. This motivates us to design the burst score based on the current and past windows.We first discuss the intuition in designing the burst score using Example <ref>. In this scenario, Uber drivers are interested in regions in which they have a higher chance to pick up a passenger.Obviously, a driver is more likely to find a passenger in a region that contains a large number of requests in the current window, which represents the significance of the region. On the other hand, if a region suddenly experiences a surge of requests, which represents the burstiness of the region, then it is highly likely that existing drivers in that region may not be able to fulfill this sudden increase in demand. Consequently, a driver will have a higher chance to find a passenger there. Thus, we consider the following two factors in our burst score: (a) The score of the region w.r.t. the current window, i.e. f(r, W_c), which measures the significance, and (b) the increase in the score of the region between the current window and the past window, i.e., max(f(r, W_c) - f(r, W_p), 0), which measures the burstiness. Note that we use the max function to guarantee that the increase in the score between the current and past windows is always non-negative since we are only interested in increase in the score. We now formally define the burst score as follows.. Given a region r, we define its burst score 𝒮(r) as:𝒮(r) = αmax(f(r, W_c) - f(r, W_p), 0) + (1-α) f(r, W_c),whereα∈ [0, 1) is a parameter that balances the significance and the burstiness.§.§ Continuous Bursty Region Detection (SURGE)Problem We are now ready to formally define the problem. Consider a stream of spatial objects 𝒪. Letq = ⟨ A, a× b, |W|⟩ be a query where A is a preferred area, a× b is the size of the query rectangle, and |W| is the length of the current and past windows. Given such a query q, the aim of the problem is to continuously detect the position of the region r of size a× b in A with the maximum burst score. The region r is referred to as the bursty region. § AN EXACT SOLUTION The problem is challenging to address due to the following reasons. First, given a snapshot of the stream, we are required to locate the bursty region in the preferred area A. Intuitively, this bursty region can be located at any point and it is prohibitively expensive to check the region located at every point, which is infinite. Second, whenever a spatial object enters or leaves the sliding windows, the burst score of any region which encloses this object will change. This implies that the location of the bursty region may change as well and we need to recompute the new bursty region. With the high arrival rate of the stream, it demands an efficient strategy to update the bursty region. In this section, we present a solution to address the problem. We first introduce the continuous bursty point detection () problem in Section <ref>. We show that by reducing the problem to the problem, for any snapshot of the stream, we convert the challenge of selecting a point from infinite points in the preferred area A to selecting a bursty point from O(n^2) disjoint regions. To address the second challenge, we present a cell-based algorithm to continuously update the bursty point in Section <ref>.§.§ The cSPOT ProblemWe next define the problem and present how to reduce the problem to the problem. Firstly, we introduce some terminology that will be used to define the problem. A rectangle object, denoted with a triple g=⟨ w, ρ, t_c⟩, is a rectangle of size a× b, where g.w is its weight, g.ρ is the location of its left-bottom corner, and g.t_c is the creation time of g.Given the stream of spatial objects 𝒪, each spatial object o in 𝒪 can be mapped to a rectangle object g by using o as the left-bottom corner, i.e., g.w=o.w, g.ρ = o.ρ, and g.t_c = o.t_c. Let 𝒢 denote the stream of rectangle objects that are mapped from 𝒪. Let G(p, W) be the set of rectangle objects which covers point p and is created in window W, i.e., G(p, W) = {g|g.t_c∈ W ∧ p ∈ g ∧ g ∈𝒢}. Next, we define the burst score of a point by following the definition of burst score of a region in Section <ref> . With a slight abuse of notation, we continue to use f(p, W) and 𝒮(p) to denote the score of a point p w.r.t. the window W, and the burst score of p, respectively. Consider a stream of rectangle objects 𝒢. The burst score 𝒮(p) of point p is defined as 𝒮(p) = αmax (f(p, W_c) - f(p, W_p), 0) + (1-α) f(r, W_c)where W_c and W_p are the current and past windows, and for a sliding window W, score f(p, W) is the summation of weights of rectangle objects in G(p, W), i.e., f(p, W) = ∑_g∈ G(p, W)g.w/|W|, which is the score of a point p w.r.t. the sliding time window W. We are now ready to formally define the problem. Consider a stream of rectangle objects 𝒢, a parameter α, as well as the current window W_c and past window W_p. The Continuous Bursty PointDetection () problem aims to keep track of a point p in the space, such that its burst score 𝒮(p) is maximized. A point p with the maximum score is referred to as bursty point. In order to reduce the problem to the problem, for each spatial object o in the problem, if o is in the preferred area A, i.e., o.ρ∈ A, we generate a rectangle object g of size a× b with o as the left-bottom corner such that o.t_c=g.t_c and g.ρ = o.ρ. We illustrate this reduction with the example in Figure <ref>. Assume that o_1, …, o_3 are all in A. For each spatial object o_i, i∈ [1,3], a corresponding rectangle object g_i is generated. We next show the relationship between the bursty region and the bursty point of the corresponding and problem.Let p_m be a bursty point for the reduced problem given a snapshot. The rectangular region r_m of size a× b whose top-right corner is located at p_m is a bursty region for the original problem for the snapshot.Note that the reduction is inspired by the idea of transforming the max-enclosing rectangle problem to the rectangle intersection problem <cit.>. The rectangle intersection problem aims to find the most overlapped area given a set of rectangles. Since our problem has a different burst score function, the techniques designed for the rectangle intersection problem cannot be utilized to search for the bursty point at a snapshot. We address the problem by solving the corresponding problem. Observe that in the problem, the edges of the rectangle objects divide the space into many disjoint regions. Consider the example in Figure <ref>. The shaded area is one of the disjoint region which is the overlap of g_1, g_2, and g_3. All points in a disjoint area are covered by the same set of rectangles. Thus they have the same burst score. Next we present a theorem which justifies the reason behind the reduction. Given a snapshot of the stream of rectangle objects in the problem, there are at most O(n^2) disjoint regions, where n is the number of rectangle objects in windows W_c and W_p.<cit.>. Since all points in a disjoint region have the same burst score, Theorem <ref> tells us that we only need to consider O(n^2) disjoint regions, which addresses the first challenge of the problem, i.e., locating the bursty region from infinite possible locations. Consider a snapshot of the stream shown in Figure <ref>. Assume that o_1, o_2 and o_3 are three spatial objects in the current window W_c in the problem, and o_i.w=1 for i∈[1,3]. According to the reduction process, g_1, g_2 and g_3 are three rectangle objects in the current window in the problem, and g_i.w=1 for i∈ [1,3]. Assume that |W_c|=1. The shaded area is the intersection of g_1, g_2 and g_3. Thus, any point p in the shade area has the maximum burst score, i.e., 𝒮(p) = 3. The point p in the figure is a bursty point at the given snapshot. The solid line rectangle, whose top-right corner lies in p, is the bursty region as it encloses three spatial objects and its burst score is 3. We next present an exact solution to address the problem efficiently. Specifically, given the stream of rectangle objects, we use a grid to divide the space into cells, and maintain the upper bounds of burst score for the points in each cell. Several optimization techniques are proposed to avoid redundant recomputation. If the upper bound of any cell is larger than the score of the current bursty point, we invoke a sweep-line based algorithm to search the cell to update the location of the bursty point.In the rest of this section, we first introduce the sweep-line based algorithm, which finds the bursty point given a set of rectangle objects ( Section <ref>). Then we present the cell-based lazy update strategy, which determines whether we should invoke the sweep-line algorithm to recompute the bursty point (Section <ref>).§.§ Detecting Bursty Point on a SnapshotTo address the first challenge, i.e., detecting the bursty point given a snapshot of the stream, we propose a sweep-line based algorithm called sl-cSpot in this subsection. The high level idea of the sl-cSpot algorithm is as follows. We use a horizontal line, referred to as the sweep-line, to scan the space top-down. The sweep-line is divided into 2n + 1 intervals at most by the vertical edges of the n rectangle objects. For instance, in Figure <ref>, the vertical edges of the three rectangles divide the sweep-line into 7 intervals, {I_0, …, I_6}. For each interval I, we use I.f_c and I.f_p to denote the score w.r.t. the current and past windows, respectively for the points on the interval I. We use I.𝒮 to denote the burst score of such points.For any interval I_i, the set of rectangles which can cover interval I_i changes when the sweep line meets the top or bottom edge of a rectangle which can cover I_i, and its burst score I_i.𝒮 is updated accordingly. A point with the maximum burst score during the sweeping process is returned as the bursty points.We next illustrate the algorithm with an example. Figure <ref> shows a snapshot of the stream. Rectangle g_1 is in the past window W_p (marked in blue), while g_2 and g_3 are in the current window W_c (marked in red).As shown in Figure <ref>, when the sweep-line meets the top edge of g_3, any point, such as p_1, which is beneath the overlapped intervalsI_3, I_4 and I_5 and above the next horizontal line, will be covered by g_3. Since g_3 is in the current window, the score of p_1 w.r.t. W_c will be increased by g_3.w/|W_c| = 2, resulting in an increase of its burst score. We set I_i.f_c = 2 and I_i.f_p = 0 for i∈ [3,5], and thus I_i.𝒮 = 0.5 ·max(I_i.f_c - I_i.f_p, 0) + 0.5 · I_i.f_c =2 for i∈ [3,5]. We select p_1 as the current bursty point. Then the sweep-line meets the top edge of g_1 and g_2, consecutively. The two edges are processed similarly, and we have I_4.𝒮 = 3. Thus p_3 is selected as the new bursty point. When the sweep-line meets the bottom edge of the rectangle g_3, any point, such as p_4, which is beneath the overlapped intervals and above the next horizontal line, will no longer be covered by g_3. Thus, the scores w.r.t. W_c of the overlapped intervals I_3, …, I_5 are decreased. We have I_i.f_c = 1 for i∈ [3,4], and I_5.f_c=0. Their burst scores are updated as: I_3.𝒮 = 1 - α, I_4.𝒮 = 1 and I_5.𝒮 = 0. We repeat this process until the whole space is scanned. Point p_3 has the maximum burst score during the sweeping process. Thus p_3 is returned as the bursty point.Algorithm <ref> outlines this procedure. It takes as input a set of rectangle objects G, and outputs a bursty point p with the maximum burst score in the space. Result point p is initialized as null. The algorithm uses a sweep-line to scan the space (lines 2–7). When it meets an horizontal edge of a rectangle r, it first locates the intervals that are covered by r (line 3). Then it updates I.𝒮 for each interval I one by one (line 5). The point p is updated if any interval has a larger burst score (lines 6–7).Time Complexity. Let n be the number of rectangles in the space. The sweep-line scans 2· n edges (each rectangle has two horizontal edges). In the worse case, when the sweep-line meets an horizontal edge, 2· n + 1 intervals are all affected. As a result, the time complexity of Algorithm <ref> is O(n^2).§.§ Handling the Stream We have presented Algorithm sl-cSpot to detect a bursty point given a snapshot of the stream. But how to continuously detect the bursty point? Recall that the burst score of a point is determined by the set of rectangle objects that cover it. The bursty point is likely to change when a rectangle object enters or leaves the sliding windows. Specifically, any of the following events may change the bursty point: (1) a new rectangle object enters the current window, (2) an existing rectangle object leaves the current window and enters the past window, and (3) an existing rectangle object leaves the past window. We refer to these three events as a new event, a grown event, and an expired event, respectively. We use a tuple e=⟨ g, l ⟩ to denote an event, where g is the rectangle object, and l is one status from {New, Grown, Expired} to indicate the type of the event.Intuitively, a naïve idea is whenever an event happens, we invoke Algorithm <ref> to detect a bursty point on the snapshot of the stream. However, this idea does not address the problem efficiently. First, it is not necessary to search the whole space. When an event happens, it only affects the burst score of the points inside the rectangle object of the event. Second, frequent recomputation of the bursty point is computationally expensive. To address the two issues, we next present a cell-based algorithm called Cell-cSpot. §.§.§ Cell-based Lazy UpdateAn event only affects the burst scores of the points inside the rectangle of the event. This locality property motivates us to divide the space into cells, and develop approaches to handle the cells that are affected by an event. We first define the grid that we use as follows. We consider a grid as a set of vertical and horizontal lines defined by x = i · b, y = i · a for all integers i ∈ [-∞, +∞]. For each cell c, we maintain a list of rectangle objects which overlap with the cell over the two sliding time windows W_c and W_p, denoted by c.G. We have the following lemma based on obvious observations. A rectangle object of size a× b overlaps with at most four cells of the grid in Definition <ref>.For each cell in the grid, we maintain a burst score upper bound for the points inside the cell (to be discussed in Section <ref>). When an event happens, the corresponding rectangle can only affect at most four cells. Instead of searching the affected cells immediately after an event happens, we propose a lazy update strategy by utilizing the maintained upper bound: Whenever an event happens, we first update the upper bounds of the affected cells. Then, we invoke Algorithm <ref> to search the cells iteratively in the descending order of their upper bounds. In each iteration, we always search the cell with the maximum upper bound. We terminate the process when there is no upper bound larger than the current maximum burst score. Hence, when an event happens, if the upper bounds of the affected cells are less than the current maximum burst score, these cells will not be searched. Thus the lazy update strategy significantly reduces the number of times that Algorithm <ref> is invoked to search affected cells. In addition, to reuse the result of Algorithm <ref> from previous iterations, we record the point returned by Algorithm <ref> for each cell which is called candidate point. The status of each candidate point is either valid or invalid. If the candidate point of a cell is guaranteed to have the maximum burst score in the cell, its status is valid. On the other hand, the status is set to invalid if it is unknown whether the candidate point has the maximum burst score. We do not need to invoke Algorithm <ref> to search a cell if its candidate point is valid. By exploiting the candidate points, we can further avoid searching in some cells (discussed in Section<ref>).Algorithm <ref> presents an overview of our algorithm called Cell-cSpot (cell-based cSpot). It takes as input an event e=⟨ g, l⟩, and reports a bursty point in the space. The algorithm first locates the set C_g of cells that overlap with g (line 1). Then for each cell c in C_g, it updates its upper boundbased on Equations <ref>, and <ref> (to be introduced in Section <ref>), and determine the status of the candidate point c.p based on Lemma <ref> (to be introduced in Section <ref>) (line 3). Then it accesses the cells in descending order of their upper bounds U(c) iteratively (lines 4–8). In each iteration, if the candidate point c.p is invalid, we invoke Algorithm <ref> to search the cell and update c.p (line 6) and the upper bound (line 7). Otherwise c.p is valid, and this indicates that c.p has the maximum burst score in cell c and c has the maximum burst score as there is no cell whose upper bound is larger than the current maximum burst score. Therefore we terminate the process and report point c.p as the result. Time Complexity. According to Lemma <ref>, at most four cells are affected by an event rectangle g. Thus, it takes O(1) time to update the upper bounds and candidate points. A cell will not be searched unless it is overlapped with a rectangle object. Thus, O(1) cells are searched in processing a rectangle object. In our implementation, we use a heap to maintain the cells based on their upper bounds. Let |c_max| be the maximum number of rectangle objects in a cell. Let n be the number of rectangle objects created in W_c and W_p. It takes O(log n) time to get the cell c and O(|c_max|^2) time to search the cell. Putting these together, the complexity of Algorithm <ref> is O(|c_max|^2+log n).Space Complexity. Each rectangle object is stored in at most four cells. Thus, the space cost of Algorithm <ref> is O(n). §.§.§ Upper Bound Estimation Next, we present the details about estimating the upper bound for a cell.Static Upper Bound. We first consider a simple strategy to estimate an upper bound for a cell. According to the definition of the burst score, rectangle objects in the current window have a positive impact on the burst score, while the rectangle objects in the past window have a non-positive impact. Hence, we can estimate an upper bound burst score for a cell by only utilizing the objects in the current window. We refer to this upper bound as static upper bound. For a cell c, its static upper bound is computed as follows:U_s(c) = ∑_g∈ c.G ∧ g.t_c ∈ W_cg.w/|W_c|where c.G is a set of rectangle objects overlapped with c. Next, we show the correctness of the static upper bound. For any point p in a cell c, we have 𝒮(p) ≤ U_s(c). Consider the example shown in Figure <ref>. The solid-line rectangle is a cell in the grid. After event e_1 happens, there are three new rectangle objects overlapped with the cell c. The static upper bound of cell c is U_s(c) = 3. Dynamic Upper Bound. Next, instead of just using objects in the current window, we introduce another way to estimate the upper bound by using both the event and information from the previous computation. Specifically, when an event happens, we dynamically update the upper bound computed from previous upper bound. We refer to such upper bound as dynamic upper bound. Let p_m be the point with the maximum burst score in cell c at a snapshot i when event e_i arrives. Apparently 𝒮(p_m) is an upper bound burst score for cell c at snapshot i. Thus, whenever we search a cell c with Algorithm <ref> on a snapshot i, the dynamic upper bound U_d^i(c) can be set as U_d^i(c) = 𝒮(p_m).Let U_d^i(c) be the upper bound of cell c on snapshot i when event e_i arrives, and U_d^i+1(c) be the upper bound when e_i+1 arrives. Let g be the corresponding rectangle object of e_i+1, i.e., e_i+1=⟨ g, l⟩. Then we haveU_d^i+1(c)= U_d^i(c) + g.w/|W_c| U_d^i(c)U_d^i(c) + αg.w/|W_p|We next show the correctness of the dynamic upper bound with the following lemma. Consider a cell c. For any point p in c, we have 𝒮(p) ≤ U_d(c) after e happens. Consider the example shown in Figure <ref>. We first consider an event e_1=⟨ g_3, New⟩, i.e., a new rectangle object enters the current window. Assume before e_1 happens, we have searched the cell and the point p_1 has the maximum burst score in c. The dynamic upper bound is set as U_d^0(c) = 1. After e_1 happens, we update the dynamic upper bound as U_d^1(c) = U_d^0(c)+g_3.w/|W_c|=2. Then we consider an event e_2=⟨ g_1, Grown⟩, i.e., and existing rectangle object g_1 exits the current window and enters the past window. According to Eqn <ref>, the dynamic upper bound remains the same, i.e., U_d^2(c) = 2, since p_2 remains to have the maximum burst score in cell c. We have presented the static upper bound and the dynamic upper bound. We now combine them for a tighter upper bound. For a cell c, we define its upper bound U(c) as U(c) =min(U_s(c), U_d(c)).§.§.§ Candidate Point MaintenanceAn expensive operation of Algorithm <ref> is to invoke Algorithm <ref> to find a point with the maximum burst score for a cell. To reuse the computation, for each cell c, we maintain a candidate point, denoted by c.p, to record the point returned by Algorithm <ref>. The candidate point has two possible status as introduced in Section <ref>. We next present Lemma <ref>, which is employed to determine the status of a candidate point.Let c.p be a point with the maximum burst score in cell c currently. Consider an event e=⟨ g, l⟩. After e happens, if either (1) e is either new or expired, g can cover c.p, and f(c.p, W_c) - f(c.p, W_p) > 0, or (2) e is grown object and g cannot cover c.p, then the point c.p still has the maximum burst score.We determine the status of a candidate point based on Lemma <ref>. Consider a cell c and an event e which can affect c. If c.p is valid and the conditions in Lemma <ref> hold, then c.p remains to be valid. Otherwise, c.p is invalid after e happens. Reconsider the example shown in Figure <ref>. We consider the event e_1=⟨ g_3, New⟩, where a new rectangle g_3 arrives. Before e_1 happens, assume that we have invoked Algorithm <ref> to search the cell and p_1 is the point with the maximum burst score. When e_1 happens, since e_1 is new and g_3 cannot cover p_1, p_1 is invalid after e_1 happens. In fact, points in the shaded area have the maximum burst score after e_1 happens. § APPROXIMATE SOLUTIONS Although our exact solution can continuously detect the bursty region efficiently in real time, we observe that its runtime performance degrades when the number of spatial objects created in time windows W_c and W_p increases significantly (e.g., the sliding windows get larger, the region size gets larger, or the arrival rate of the spatial objects increases). Since a slight imprecision is acceptable in most cases in real life, to tackle this challenge, we propose two algorithms to solve the problem approximately. We prove that the burst score of the region returned by our proposed approximate algorithms is always bounded by a ratio 1-α/4 compared to the exact result.§.§ A Grid-based Solution The key idea behind our grid-based approximate solution is as follows: We use a grid to divide the space into cells of size a× b. Each cell is a candidate region. By maintaining the burst score for each cell, we continuously report the cell with the maximum burst score to users as an approximation to the bursty region. A nice feature of this idea is that it is intuitive while it has performance guarantees.Algorithm <ref> outlines our proposed algorithm called gap-surge (Grid-based APproximate ). Here we abuse the notation e=⟨ o, l⟩ to denote an event of spatial object o enters or leaves the sliding windows. It first locates the cell that the spatial object o lies in (line 1). The burst score of the cell c is updated accordingly (lines 2–5). The cell with the maximum burst score is returned as an approximate result (line 6). Before we show that the region returned by Algorithm <ref> has a burst score with an approximation guarantee, we present some interesting properties of the burst score function. For any two region r_1 and r_2, r_1⊆ r_2, we have 𝒮(r_2) ≥ (1-α)𝒮(r_1). Let r_1, r_2 be two non-overlapping regions. We have 𝒮(r_1) + 𝒮(r_2) ≥𝒮(r_1∪ r_2). Now we are ready to prove the approximate ratio of Algorithm <ref>. Given a snapshot of the stream, let r be the region returned by Algorithm <ref>, and r_opt be the bursty region returned by our exact solution. We have 𝒮(r) ≥1-α/4𝒮(r_opt).The approximation ratio is tight. Time Complexity. In Algorithm <ref>, it takes constant time to locate the cell and update the burst score. In our implementation, we use a heap to maintain all cells according to their burst scores. Let n be the number of spatial objects created in W_c and W_p. Since there are O(n) non-empty cells, it takes O(log n) time to report the cell with the maximum burst score.§.§ A Multi-Grid-Based SolutionThe burst score of the region returned by Algorithm <ref> is highly dependent on the position of the grid. In this subsection, we adopt multiple grids to further improve the result quality. In the grid-based solution, we use a grid defined by linesGrid 1: x = i · b, y = i · afor all integers i ∈ [-∞, +∞]. By shifting the grid, we generate three additional grids for all integers i∈ [-∞, +∞]: Grid 2:x = 0.5b + i · b, y = i · a, Grid 3:x = b + i · b, y = 0.5a + i · a, Grid 4:x = 0.5 b + i · b, y = 0.5a + i · a,The multi-grid-based solution (called the mGap-surge algorithm) invokes Algorithm <ref> four times by using the four different grids. Among the four returned regions, the one with the maximum burst score is returned to users. The pseudocode of the mGap-surge algorithm is reported in Algorithm <ref> in Appendix <ref>. The approximate ratio of the mGap-surge algorithm is 1-α/4. Time Complexity. mGap-surge invokes Algorithm <ref> four times, and its complexity is O(log n), where n is the number of spatial objects created in W_c and W_p. § TOP-K BURSTY REGION DETECTION Recall that in Example <ref>, it is paramount to monitor regions with outbreak of diseases. Intuitively, monitoring only the most bursty region is not sufficient. In fact, it is reasonable to be interested in a small list of such bursty regions. Specifically, given the size of a region, we need to continuously monitor the top-k regions of the given size with highest burst scores. In this section, we present how we can elegantly extend our proposed solutions to continuously detect top-k regions with highest burst scores. We begin by formally defining the top-k bursty regions.§.§ Definition Although at first glance it may seem that it is easy to definetop-k bursty regions, in reality it is tricky. First of all, are the top-k regions allowed to overlap? It may seem that detecting k non-overlapping regions is a good choice. However, the non-overlapping requirement may lead us to overlooking some highly bursty regions. Hence, it is beneficial to allow the top-k bursty regions to be overlapping instead of disjoint in nature.Next, how do we define the burst scores for two overlapped regions? For example, if a spatial object lies at the intersection of two overlapping regions, which region's burst score should it contribute to? A naïve idea is to consider it in both regions. However, this may result in k regions that are highly similar to one another. To resolve this issue, we ensure that a spatial object contributes only to the burst score of at most one region.The aforementioned considerations lead us to a greedy strategy for defining the top-k bursty regions. Specifically, given the first i bursty regions, the (i+1)-th bursty region is the region with maximum burst score in the space but excluding all spatial objects that are already covered by the first i bursty regions. Given k rectangular regions r_1, …, r_k such that each has a size of a× b, we say r_1, …, r_k are the top-k bursty regions if and only if for any region r of size a× b, we have 𝒮(r_i∖r_[1, i-1]) ≥𝒮(r ∖r_[1, i-1]) for i ∈ [1, k], where r_[1, i-1] is union of regions r_1, …, r_i-1.In order to address the top-k bursty regions problem, we reduce the top-k bursty regions problem to k problems following the reduction in Section <ref>. The (i+1)-th problem aims to detect the (i+1)-th bursty point from the space that excludes the set of rectangles that cover the top-i bursty points.Observe that Definition <ref> essentially paves the way to a greedy approach for selecting top-k bursty regions.Whenever an event happens, we can first detect a region with the maximum burst score by invoking Algorithm <ref>. Then we remove the spatial objects covered by the region. After that, we detect a region with the maximum burst score over the remaining objects. We repeat this process until k regions are selected.However, the naïve strategy is inefficient as there are too many redundant computations, i.e., it is possible that we search a cell in all the k reduced problems. To address the k problems efficiently, we want to share the common computations among the k problems.§.§ Extension of the Exact Solution In the extension of our exact solution, for each cell c, we maintain k upper bounds and k candidate points in order to solve the k problems by following the idea of Algorithm <ref>. For each problem, we adopt the lazy update strategy to access the cells in descending order of their upper bounds. If the candidate point of the top cell is not valid, we search the cell by invoking Algorithm <ref>.We develop two ideas of sharing computation among the k problems. Firstly, if a rectangle object can cover the i-th bursty point, it will not be considered in the problems with order higher than i. For the extension, we maintain a level, denoted by g.lvl, for each rectangle object g. To select the i-th bursty point in response to a new event, we consider the set of rectangles G[i:k] whose levels are no smaller than i, i.e., G[i:k] = {g|g.lvl ≥ i}. When the i-th bursty point is selected, the levels of all the rectangles that cover the i-th bursty point are set as i, and these rectangles will not be considered by the problems with a higher order than i. Meanwhile, if a rectangle covers the old i-th bursty point, but not the new i-th point, its level is reset to k so that it will be considered in all the k problems.Secondly, if no rectangle in a cell covers any of the k detected bursty points, all the rectangles in the cell will be considered in all k problems. Thus, the upper bounds and the candidate points w.r.t. the k problems for the cell are the same. That is, once the upper bound and the candidate point for the cell are computed for one problem, we do not need to recompute them again for other problems.Algorithm <ref> presents the detail of our extension. It takes as input an event e=⟨ g, l ⟩, and output the top-k bursty points, denoted by p[1:k]. It uses V to denote the set of objects that need to be handled subsequently, and is initialized as {g} (line 1). It then solves the k problem iteratively (lines 2–17). In each problem, it first locates the set of cells affected by the objects in V (line 4). For each cell c∈ C, the upper bound U(c)[j] and candidate point c.p[j] w.r.t. the j-th problem are updated for j∈ [i, k] (lines 5–6). Then it accesses the cells in descending order of their upper bounds w.r.t. the i-th problem (lines 8–14). The upper bound and candidate point are updated as in Algorithm <ref> (lines 9–10). If no rectangle in cell c covers any of the k detected bursty points, its k upper bounds and candidate points are set to the same (lines 11-12). When a new bursty point is found, we reset the levels for the affected objects as discussed earlier (lines 15–16): The rectangles that cover the old bursty point p_old but not the new bursty point p[i] are newly visible to all the k problems, while the rectangles that cover the new bursty point p[i] are newly invisible to the problems with a higher order than i. The two types of rectangle objects comprise V, which will be processed in the next problem (line 17). After k iterations, it returns the top-k bursty points p[1:k] as the result. Time Complexity. A cell is searched if its upper bound is either changed by an event or by a detected bursty point. Thus, the algorithm searches O(1+k) = O(k) cells on average when processing a rectangle. The complexity of Algorithm <ref> is O(|c_max|^2 · k), where |c_max| is the maximum number of objects that we search in a cell.§.§ Extension of the Approximate Solutions We also extend our approximate solutions in Section <ref> to find k regions with relatively high burst score. Extending the gap-surge Algorithm. Consider the grid-based solution.We use a heap to maintain all cells with their burst scores. Thus, we can simply return top-k cells with highest burst scores. In our implementation, we use a heap to maintain the cells. Thus, its complexity is O(log n).(Algorithm <ref> in Appendix <ref>). Extending the mGap-surge Algorithm. We extend the multi-grid-based solution similarly. Note that one cell in a grid may overlap with at most four cells in another grid. Thus, for each grid, we maintain the top-4k cells. Then we merge the 16· k cells and return the top-k non-overlapping cells. Its time complexity is O(log n + k).(Algorithm <ref> in Appendix <ref>).§ EXPERIMENTAL STUDY We investigate the performance of our proposed techniques.All algorithms are implemented in C++ complied with GCC 4.8.2. The experiments are conducted on a machine with a 2.70GHz CPU and 64GB of memory running Ubuntu. §.§ Experimental Setup Datasets. We conduct experiments on three public real-life datasets as reported in Table <ref>.consists of 1,000,000 geo-tagged tweets posted in UK. consists of 1,000,000 geo-tagged tweets posted in US and has a higher arrival rate. [<crawdad.org/roma/taxi/20140717>] consists of mobility traces of taxi cabs obtained from the gps in Roma, Italy. It contains 1,000,000 records over 5 days. For each dataset, the weight of each spatial object is randomly chosen from from [1, 100] with a uniform distribution.Algorithms. We evaluate the performances of the three proposed algorithms, namely the exact method Cell-cSpot (denoted by ), the grid-based approximation algorithm gap-surge (denoted by ), and the multi-grid-based technique mGap-surge (denoted by ).We denote the top-k extensions of these algorithms as,, and, respectively. To evaluate the usefulness of our proposed method of upper bound estimation, we comparewith an approach that only utilizes the static upper bound, denoted by , and a baseline approach that does not use any upper bound estimation technique, denoted by .To the best of our knowledge, there is no existing technique that address the problem. Hence we are confined to compare our proposed algorithms with <cit.>, which is designed for continuously monitoring the MaxRS problem.In our experiments, we use a modified version of . With a slight abuse of notation, we still useto denote the modified . The details of theandare reported in Appendix <ref>. Parameters. By default, we set the size of the past window W_p and the current window W_c as 1 hour forand , and 5 minutes for . We set the size of the query rectangle as 1/1000 of the range of each dataset by default, denoted by q. We set the preferred area A as the whole space. For thealgorithm, we set the size of a cell to 10q.Stream Workload. We start the simulation when the system becomes stable, i.e., there exists an expired object from the past sliding window. We continuously run each algorithm for 1,000,000 new arriving spatial objects over the two sliding windows. The average processing time per object is reported. §.§ Evaluation of the Exact SolutionWe first evaluate the runtime performance of ,andon each dataset. Then we study the usefulness of the upper bound in . Runtime Performance. The aim ofthe first set of experiments is to evaluate the efficiency of our exact solution the sliding window size and the query rectangle size. Forand , we vary the sliding window with the following sizes: 30 minutes, 1 hour, 2 hours, 5 hours, and 12 hours. For , we use the following five sizes for sliding windows: 1 minute, 5 minutes, 10 minutes, 20 minutes, and 30 minutes. We use the following four sizes for the query rectangle: 0.5q, q, 2q, and 3q. Figures <ref>(a)–(c) report the average runtime of the three methods for processing one spatial object as we vary the size of sliding windows. Note that the y-axis is in logarithmic scale. We find thatruns efficiently andoutperforms . For example, for , it takes about 3× 10^-4 seconds to process an object when the current and past windows are both set to 30 minutes, whiletakes 7× 10^-3 seconds. In addition, we find thatrun out of the 64 GB memory onwhen the current window and past window are both set as 12 hours, as there are too many spatial objects in the two sliding windows. Moreover, we observe that the processing time per object of all algorithms increases as the size of window increases. This is due to the need to consider a larger number of spatial objects when we search for the bursty region with the increase in size of the sliding window.Consequently, the runtime per object increases.Figures <ref>(d)–(f) report the average runtime for processing one spatial object as we vary the size of the query rectangle. Similarly, the average runtime increases as size of the rectangle increases. Usefulness of Upper Bound. Next, we evaluate the usefulness of the method for upper bound estimation in . In this set of experiments, we process 1,000,000 new objects and report how many rectangles trigger a search.The results are reported in Table <ref>. Clearly, only a small portion of rectangle messages (2%-5% for , and less than 1% forand ) trigger a search incompared with . This is becausecan estimate a much tighter upper bound for cells. Thus, many cells are eliminated from further checking. This also explains whyis much more efficient than .As shown in Figure <ref>, we observe thatis more efficient than the other two methods. The runtime ofis more than one order of magnitude faster thanand , respectively. Moreover, we observe thatis only marginally better than , which indicate thatonly using the static upper bound cannot effectively avoid unnecessary recomputation. This is because the static upper bound is too loose, especially when the weights of the objects are randomly chosen from 1 to 100. §.§ Evaluation of the Approximate Solutions The detailed results on approximation ratios are reported in Appendix <ref>. A short summary of the results is that the burst score of the region detected by(resp. ) is about 73% – 92% (resp. 85% – 94%) of that of the optimal region. We next report the runtime of the approximate solutions. Runtime Performance. We evaluate the efficiency of our approximate solutions the sliding window size and the query rectangle size under the same setting as for the exact solution. Figures <ref> (a)–(c) report the average runtime for processing one spatial object usingandas we vary the sliding window. Figures <ref> (d)–(f) report the average runtime for processing one spatial object as we vary the size of the query rectangle. We find that the runtime ofis about 2-5 times of , which is expected asinvokesfour times. Moreover,andare about three orders of magnitude faster thanby comparing Figure <ref> and Figure <ref>.§.§ Effect of α In the definition of burst score, we use a parameter α to balance the significance and the burstiness.In this set of experiments, we evaluate the impact of the parameter α on the efficiency and approximation ratio of our proposed algorithms on thedataset. We use 1 hour for the sliding windows and q for the size of the query rectangle. Impact on Runtime Performance. We evaluate the efficiency of our exact and approximate solutions w.r.t. the balance parameter α. Figure <ref> reports the average runtime for processing one spatial object as we vary α from 0.1 to 0.9. We observe that the efficiency is hardly affected by the parameter α for both our exact solution and approximate solutions. Impact on Approximation Ratio. In this set of experiments, we evaluate the approximate ratio of the burst scores of regions detected byandby varying α. The results are reported in Table <ref>. We find that the approximate ratios of the two algorithms decrease as α increases. This is because their theoretical approximate ratio 1-α/4 decreases as α increases.§.§ Scalability We now investigate the scalability of our proposed techniques by varying the arrival rate of the spatial objects. Specifically, we use 1 hour for both the current window and past window, and q for the size of the query rectangle on all three datasets. We stretch the stream to change its arrival rate from 2 million per day to 10 million per day. For example, in , 1 million spatial objects arrived in 174 hours. Hence, we shrink the arrival time of each object to make all objects arrive in 24 hours. Then the arrival rate of the stream is 1 millions per day. We only report the average time for processing the objects arrived in one hour (denoted by t_h) ofandin Figure <ref>. Formally, t_h = runtime/|𝒪|_time, where runtime is the runtime of the algorithm, and |𝒪|_time is the total timespan of the stream.We observe that it takes several hours forto process the objects arrived in an hour for thedataset, which means that itdoes not scale well and cannot handle streams with high arriving rate. On the other hand, our approximate solutions,and , scale well with the increase in arrival rate. They can process the objects arrived in an hour within seconds.§.§ Finding Top-k Bursty Regions We next evaluate the performance of the extensions of our three algorithms for continuously detecting top-k bursty regions. We study the effect of k and the size of sliding windows. Runtime Performance. This set of experiments aims to evaluate the efficiency of these algorithms the sliding window size. We adopt the same setting as in Section <ref>. Figures <ref>(a)–(c) report the average runtime per object of,, and for different sliding windows. We observe that as the sliding window gets larger,does not scale well and cannot process the top-k queries efficiently. Meanwhile,andcan find top-k bursty regions efficiently.We also compare the naïve solution for finding top-k bursty regions with these algorithms. Recall from Section <ref>, in the naïve solution, we detect the top-k bursty regions for each newly-arrived object. Clearly, the naïve solution is prohibitively expensive. Hence, we only run it with a small sliding window on , and its runtime per object is about 100X more than . Effect of k. Next, we study how the value of k affect the runtime performance of the three extensions. We use the following 4 values for k: 3, 5, 7 and 9. The runtime performance is depicted in Figures <ref>(d)–(f). We observe the runtime per object ofincreases as k increases. This is because we divide the top-k bursty region detection problem into k instances of bursty region detection problems. Each bursty region detection problem takes O(n_c^2) time to find a bursty region, where n_c is the number of spatial objects in the cells that we actually searched. In addition, we also observe thatandare less affected by k.§.§ Case Study In order to give a better view of our problem, we conduct a case study on the region monitored by our cell-cSpot algorithm. Due to the space constraints, the detailed results are reported in Appendix <ref>.§ CONCLUSIONS The work reported in this paper is motivated by new opportunities brought by the massive volumes of streaming geo-tagged data (i.e., spatial objects) generated by location-enabled mobile devices. Specifically, we have studied a new problem called the problem to continuously detect the bursty region in a given area in real time. The problem is important as it can underpin various applications such as disease outbreak detection.We have proposed an exact solution and two approximate solutions for . We have also extended these solutions to find top-k bursty regions. Finally, our experiment study with real-world datasets has demonstrated the efficiency of our framework. As part of future work, we intend to explore the problem in the context of road network.abbrv § PROOFS§.§ Proof for Theorem <ref>Let p be any point in the problem, and r be the rectangular region of size a× b whose top-right corner is located at p. A spatial object o is in r iff the corresponding rectangle object g can cover p. Since the corresponding o and g have the same creation time and weight, we can derive that f(r, W_c) = f(p, W_c), f(r, W_p) = f(p, W_p), and thus r and p have the same burst score. As a result, if the point p_m has the maximum burst score in the problem, then r_m also has the maximum burst score in the problem. §.§ Proof for Lemma <ref> We have 𝒮(p) = αmax(f(p, W_c) - f(p, W_p), 0) + (1-α)f(p, W_c) ≤α f(p, W_c) + (1-α)f(p, W_c) = f(p, W_c) = U_s(c)§.§ Proof for Lemma <ref>Let Δ𝒮(p), Δ f(p, W) be the increase of 𝒮(p) and f(p, W) after e happens, respectively. We discuss the following three cases.Case 1:e is new. For any point p that is covered by g, its current score is increased by Δ f(p, W_c) = g.w/|W_c|. We have Δ𝒮(p) ≤Δ f(p, W_c) = g.w/|W_c|.Case 2: e is grown. For any point p that is covered by g, its current score is decreased, i.e., Δ f(p, W_c) = -g.w/|W_c|, and its past score is increased, i.e., Δ f(p, W_p) = g.w/|W_p|. Thus, we can easily get Δ𝒮(p) ≤ 0.Case 3: e is expired. For any point p covered by g, its current score is not affected, and its past score is decreased, i.e., Δ f(p, W_p) = -g.w/|W_p|. Thus, we have Δ𝒮(p) ≤α (-Δ f(p, W_p)) = αg.w/|W_p|.Since Δ𝒮(p) ≤Δ U_d(c), we still have 𝒮(p) ≤ U_d(c).§.§ Proof for Lemma <ref>We use Δ to denote the increase of the score. We consider the following three cases.Case 1: e.l is new.We have Δ𝒮(c.p) = g.w/|W_c| if and only if g can cover c.p and f(c.p, W_c) - f(c.p, W_p) > 0. In this case, c.p still has the maximum burst score as Δ𝒮(p) ≤g.w/|W_c| for any p in g (Lemma <ref>). Otherwise, it is possible that there exists a point p' in g with a larger increase such that p' has a larger burst score than c.p after g arrives.Case 2: e.l is grown.For any point p in g, the increase Δ𝒮(p) < 0. If g does not cover c.p, c.p's burst score does not change and it still has the maximum burst score. Otherwise, c.p's burst score is decreased and could be exceeded by a point outside g.Case 3:e.l is expired.As shown in the proof for Lemma <ref>, Δ𝒮(p) ≤αg.w/|W_c| for any p in g. We have Δ𝒮(c.p) = αg.w/|W_c| if and only if g can cover c.p and f(c.p, W_c) - f(c.p, W_p) > 0. In this case, c.p still has the maximum burst score. Otherwise, similar to Case 1, it is possible that there exists a point p' in g with a larger increase of burst score. Putting these together, the lemma is proved.§.§ Proof for Lemma <ref> According to the definition of the burst score, we have𝒮(r_2)=αmax(f(r_2, W_c) - f(r_2, W_p), 0) + (1-α) f(r_2, W_c)≥(1-α) f(r_2, W_c) ≥ (1-α) f(r_1, W_c) ≥ (1-α) 𝒮(r_1) §.§ Proof for Lemma <ref> Since r_1 and r_2 are non-overlapping, according to the definition of burst score, we havef(r_1, W_c) + f(r_2, W_c) = f(r_1∪ r_2, W_c) f(r_1, W_p) + f(r_2, W_p) = f(r_1∪ r_2, W_p)Then we can easily getmax(f(r_1∪ r_2, W_c) - f(r_1∪ r_2, W_p), 0)≤ max(f(r_1, W_c)-f(r_1, W_p), 0) + max(f(r_2, W_c)-f(r_2, W_p), 0)Thus, we have 𝒮(r_1∪ r_2) ≤𝒮(r_1) + 𝒮(r_2). §.§ Proof for Theorem <ref> Since the sizes of r_opt and any cell are both a× b, then r_opt either overlaps with a cell or intersects with four cells. We consider the following two cases.Case 1: r_opt overlaps one cell. Since we return the candidate with maximum burst score, we will return the bursty region to users. The approximate ratio is 1.Case 2: r_opt intersects with 4 cells. Consider the example shown in Figure <ref>. Let the solid line rectangle be r_opt. The four dashed line rectangles are four cells which intersect with r_opt. According to Lemma <ref>, we have (1-α) 𝒮(r_opt) ≤𝒮(c_1∪…∪ c_4). According to Lemma <ref>, we can derive that 𝒮(c_1∪…∪ c_4) ≤∑_i∈[1,4]𝒮(c_i). Since we report the cell with the maximum burst score, i.e., 𝒮(r) ≥𝒮(c_i) for any i∈[1,4]. Thus, we have 1-α/4𝒮(r_opt) ≤𝒮(r).§.§ Proof for Lemma <ref> We show the approximation ratio is tight by giving an example. Consider an instance in Figure <ref>, where c_1, c_2, c_3 and c_4 are cells in the grid, and the solid-line rectangle r_opt is the bursty region with the maximum burst score. The white nodes are the spatial objects in window W_c and the black nodes are in W_p. We assume that o.w/|W_c| = o.w/|W_p| = 1 for each object o. The burst score for the region r_opt is 𝒮(r_opt) = αmax(4-0, 0)+ (1-α) 4 = 4. The burst score of cell c_i is 𝒮(c_i) = αmax(1-1, 0) + (1-α) = 1-α, for any i∈ [1, 4]. Thus, the approximation ratio is tight. §.§ Proof for Theorem <ref> Since the mGap-surge returns the best result of found by Algorithm <ref>, its approximation ratio is 1-α/4. § PSEUDOCODE FOR MGAP-SURGE The pseudocode for the mGap-surge Algorithm is presented in Algorithm <ref>.§ PSEUDOCODE OF TOP-K BURSTY REGIONS DETECTION ALGORITHMS§ ADDITIONAL EXPERIMENTS§.§ Details of the evaluated algorithms We evaluate the performances of the three proposed algorithms, namely the exact method Cell-cSpot (denoted by ), the grid-based approximation algorithm gap-surge (denoted by ), and the multi-grid-based technique mGap-surge (denoted by ).We denote the top-k extensions of these algorithms as,, and, respectively. To evaluate the usefulness of our proposed method of upper bound estimation, we comparewith an approach that only utilizes the static upper bound. We denote this baseline method by . We also comparewith a baseline approach that does not use any upper bound estimation technique, denoted by . Specifically, inwe divide the space into cells, and we search all the cells that overlap with the rectangle object when an event happens. To the best of our knowledge, there is no existing technique that address the problem. Hence we are confined to compare our proposed algorithms with<cit.>, which is designed for continuously monitoring the MaxRS problem. Obviously, we cannot directly apply it to solve the problem. In our experiments, we use a modified version of . Specifically, the modified algorithm inherits the grid index structure and the branch-and-bound strategy from the original algorithm. The main difference between the modified and the original algorithms is how we search a rectangle object given a snapshot of the stream. In the original algorithm, they invoke the sweep-line algorithm <cit.> to search a rectangle object to find a region with maximum sum score, while in the modified algorithm, we use our proposed sl-cSpot algorithm instead.§.§ Approximate Ratio Approximate Ratio. In this set of experiments, we vary the sliding window to assess the approximate ratio of the burst scores of region detected byand . The detailed results are reported in Table <ref>. Though the theoretical approximate ratio is 1-α/4, in practice it is much better, especially for . We observe that for , the burst score of the region detected byis about 70%–90% of the burst score of the optimal region.The region detected byis about 85%–95% of the burst score of the optimal region. Sinceandare much more efficient than(about three orders of magnitude faster), they are good alternatives towhen a slight imprecision is acceptable.§.§ Case Study To evaluate the result quality of our cell-cSpot algorithm, we conduct a case study on the region monitored by the algorithm. We run the cell-cSpotalgorithm on the tweets posted in United States from 2012 April to 2012 October. Note that since the algorithm continuously reports the location of bursty regions, we only present two examples of the detected bursty region and explain the connection between the region and real life events. In the first example, we present detecting bursty regions about “concert”. Specifically, we only consider tweets containing keyword “concert” and continuously report the detected bursty region. On July 8, 2012, our algorithm detected a region as shown in Figure <ref>. The frequent keywords in this region during this time are “Walt” and “Concert”. By checking the events that happened in July 2012, we find that there was a concert performed by Ketherine Eason with Inner City Youth Orchestra of Los Angeles in Walt Disney Concert Hall in the detected region. In the second example, we present detecting bursty regions about “parade”. On May 19, 2012, our algorithm detected a region as shown in Figure <ref>. The frequent keywords in this region are “annual”, “dance”, and “parade”. By checking the events that happened in May 2012, we noticed that the dance parade is an annual parade and festival in New York. Specifically, in 2012 the parade took over Broadway Street on May 19th. | http://arxiv.org/abs/1709.09287v2 | {
"authors": [
"Kaiyu Feng",
"Tao Guo",
"Gao Cong",
"Sourav S. Bhowmicks",
"Shuai Ma"
],
"categories": [
"cs.DB"
],
"primary_category": "cs.DB",
"published": "20170926235829",
"title": "SURGE: Continuous Detection of Bursty Regions Over a Stream of Spatial Objects"
} |
empty[0.6] Analysis of structured Markov processes Ivo Adan, Johan van Leeuwaarden, Jori Selen version September 26, 2017 CHAPTER: PREFACE Markov processes are popular mathematical models, studied by theoreticians for their intriguing properties, and applied by practitioners for their flexible structure. With this book we teach how to model and analyze Markov processes. We classify Markov processes based on their structural properties, which in turn determine which analytic methods are required for solving them. In doing so, we start in each chapter with specific examples that naturally lead up to general theory and general methods. In this way the reader learns about Markov processes on the job.By studying this book, the reader becomes acquainted with the basic analytic methods that come into play when systems are modeled as structured Markov processes. These basic methods will likely prove useful, in real-time when studying the examples at hand, but more importantly for future encounters with Markov processes not covered in this book. Methods are more important than examples. The methods have a large scope of application, even outside the scope of Markov processes, in areas like probability theory, industrial engineering, mechanical engineering, physics and financial mathematics.This book arose from various courses taught in the last decade at master level and postgraduate level. We thank the students and colleagues that participated in these courses for their valuable feedback.[chapter]toc *CHAPTER: INTRODUCTION Markov processes provide essential instruments for modeling and analyzing a large variety of systems and networks, including manufacturing systems, communication networks, traffic networks and service systems such as clinics or hospitals. This book provides the basic tools you need to build models that are detailed enough to capture the essential system dynamics, but are simple enough in terms of mathematical structure to be amenable for theoretical analysis and efficient numerical evaluation. The first two parts of this book assume only prior exposure to stochastic processes, linear algebra and basic analysis at the undergraduate level. The third part is meant for graduate students, researchers and practitioners, and requires more background in probability theory and complex analysis.Markov processes fall under the umbrella of Stochastics, the branch of mathematics that aims to establish rigorous statements about systems that are inherently uncertain, and therefore subject to some degree of randomness. A classical example is a queue, in which jobs need to wait for service. The queue grows when new jobs arrive and shrinks when jobs complete service. Queues occur virtually everywhere and can be seen as stochastic systems that are subject to variability in arrivals and services. Under certain assumptions, a queueing system can be modeled as a Markov process and analyzed using the techniques described in this book. This analytic treatment of a queue then leads to explicit formulas or algorithms for performance measures such as the mean queue length or the probability that the queue grows beyond a certain level. Such performance measures often reveal critical dependencies between the system performance and the system utilization. In fact, many real-life systems operate in regimes that dwarf the trade-off between high system utilization and short queues, two confliction goals. The analysis of Markov processes therefore also serves the purpose of dimensioning, with the objective to balance the system capacity and demand so as to achieve a certain target performance standard or optimize a certain cost criterion.§ A BALANCE ACT This book deals with obtaining the equilibrium distribution that characterizes the long-term fractions of time that the Markov process spends in each of the possible states. Think of a queue that evolves in time. What is the long-term probability that the queue is empty? If we denote this probability by p(0), we could estimate it by simply observing the queue for a very long time and divide the total time that the queue is empty by the total time we have observed the queue. We could similarly estimate the probability p(i) of seeing a queue of size i.Instead of this brute-force approach to estimate p(i) for all possible i, we will use the structure that is governed by the interaction between states. For a simple queue in which one job could leave or join, it is clear that p(i) should be related to p(i - 1) and p(i + 1). And indeed, under some further assumptions, we could argue that the probabilities p(i) should satisfy certain balance equations. A famous example is the simplest possible queue that serves jobs at an exponential rate μ and to which new jobs arrive at exponential rate . Because of the exponential rates, at any moment in time only one event can happen: a new arrival or a service completion. The Markov process that describes the queue size evolves on the state space { 0, 1, 2, …} according to the rates displayed in <ref>. <ref> is called a transition rate diagram and displays the states of the Markov process with the arrows depicting the rates at which the process transitions from one state to the other. Rateshould be smaller than μ, otherwise the queue will grow to infinity, and under this assumption, the balance equations are given byp(0)= μ p(1), ( + μ) p(i)=p(i - 1) + μ p(i + 1),i ≥ 1. You can interpret these equations as what goes out should equal what comes in (either from the left or the right). These balance equations together can be written as the system of linear equations Q = with = [ p(0) p(1) p(2)⋯ ] and Q the transition rate matrix given by Q = [ -; μ -( + μ); μ -( + μ); ⋱ ⋱ ⋱ ]. All Markov processes considered in this book can be brought into the matrix form (<ref>). For the readers familiar with linear algebra, this makes available a powerful toolbox for numerically solving foras the null space of the kernel Q. But this is not the road we will pursue in this book. Instead, we will try to exploit additional structures that are hidden in the general matrix equation (<ref>). For the simple queue we know for instance that Q is extremely sparse and contains only elements on three diagonals. Moreover, the state space in <ref> and the form of (<ref>) may allow an iterative solution. Indeed, using p(0) = μ p(1) one gets for i = 1 from (<ref>) that p(1) = μ p(2), and more generally, p(i) = μ p(i + 1),i ≥ 0. Iteration then gives p(i) = p(0) ρ^i with ρ/μ. Since ∑_i ≥ 0 p(i) = 1 we conclude that p(0) = 1 - ρ to arrive at the elegant solution p(i) = (1 - ρ) ρ^i,i ≥ 0. This is our very first product-form solution! And if you like it, many more will follow for more advanced, yet structured, Markov processes.We call (<ref>) a product-form solution, because of the term ρ^i, the product of i times ρ. Most of the Markov processes in this book are multi-dimensional, in which case we encounter multi-dimensional product forms, for instance of the types ^i ^j or R^i with R some matrix (instead of a scalar ρ). While in most cases, finding these product forms will be less straightforward than in the case of (<ref>), we will often use ways to exploit recurring structures. A second technique we will often use is that of making an educated guess. Through many examples we learn the reader when to expect a product form (and when not). If we return to (<ref>) and we would guess that p(i) is of the form c ^i with c andsome unknown constants, we could simply substitute p(i) = c ^i in (<ref>) to obtain ( + μ) c ^i =c ^i - 1 + μ c ^i + 1, or equivalently, ( + μ)=+ μ^2, from which we conclude that = ρ and c = 1 - ρ. Although this guessing technique appears naive at first sight, it is a mathematical rigorous way of proving that (<ref>) uniquely characterizes the equilibrium distribution. The substitution of product forms in difference equations like (<ref>) is a well-known analytic technique, but when the difference equation is in fact a balance equation there are some specific features that can be exploited. For instance, we know from the start that all p(i) are nonnegative and that ∑_i ≥ 0 p(i) = 1. The latter condition we have used in (<ref>) to conclude that = ρ is the unique solution, and not the other candidate solution of (<ref>). Indeed, only when < 1 the infinite series ∑_i ≥ 0 p(i) converges to a finite constant. The final step concludes that (1 - )^-1 = c^-1 = p(0)^-1, which can be interpreted as solving the boundary condition ∑_i ≥ 0 p(i) = 1. For this simple Markov process this gives p(0) ∑_i ≥ 0^i = 1. In this case this boundary condition gives one additional equation for solving p(0). For the more advanced Markov processes in this book the boundary conditions give rise to an additional system of equations from which equally many remaining equilibrium probabilities need to be determined.§ WHY THIS BOOK? Many books have been written about general stochastic processes and Markov processes in particular. This book views Markov processes as continuous-time processes, and studies their equilibrium or long-term behavior. Finding the equilibrium distributions requires solving a system of difference or difference-differential equations. Each Markov process in this book comes with its own system of equations, and its own specific challenges. We classify the Markov processes by the analytic methods required to solve the system of equations, which in turn depends strongly on the underlying structure of Markov processes. The reader will learn to recognize these structures, and hence choosing the adequate methods for analysis. While this book puts much emphasis on basic real and complex analysis, less attention goes to the more formal or probabilistic aspects of Markov processes, for instance related to operators, function spaces, martingale characterizations, stability, and weak convergence to limiting processes. Excellent books exist that cover these topics in much depth, for example Ethier and Kurtz <cit.>, Feller <cit.> and Whitt <cit.>.This book is not just about queueing theory. While queueing theory generates intriguing questions that can be answered using the theory of Markov processes, this book only introduces queueing models that ask for a different analytic method. More advanced queueing models—that arise for instance when relaxing Markovian assumptions—are not treated just for the sake of generalization or enhancing the scope of applicability. Books with more theory and examples of queues are for instance Cohen <cit.>, Gross and Harris <cit.>, Kleinrock <cit.>, Prabhu <cit.>, Robert <cit.> and Takács <cit.>. Parts of the material covered in this book can also be found in text books on applied probability or Markov chains, such as Asmussen <cit.>, Chung <cit.>, Grimmett and Stirzaker <cit.>, Karlin and Taylor <cit.>, Liggett <cit.>, Norris <cit.>, Resnick <cit.> and Ross <cit.>, although the same topics are often presented in a different manner. This book is complementary, again because of the dominant role of exact analysis, product-form solutions, and structure of Markov processes.§ OVERVIEW In Part I we cover the essential theory of continuous-time Markov processes and some basic methods. We furthermore introduce some common queues with their associated Markov processes and present how transforms are useful in the analysis of such Markov processes.<ref> covers the foundations needed to build a Markov process. Essential ingredients are the exponential distribution and its memoryless property, which makes that after each event that takes place in the Markov process, we can forget about the past and only use the current information. We introduce notions like irreducibility, positive recurrence and regularity. Brief consideration is given to the evolution of Markov processes as a function of time, but we will focus mostly on the long-term of equilibrium behavior.<ref> introduces the Laplace-Stieltjes transform and the probability-generating function. Both transforms play a crucial role in the analysis of the equilibrium distribution and other related quantities. We demonstrate the use of the transforms together with other important results by analyzing single-server queues that are at the heart of queueing theory. Numerical inversion algorithms are provided to retrieve the underlying probability distributions from their transforms. In Part II we focus on several classes of widely studied structured Markov processes, including birth–and–death processes, queueing networks, quasi-birth–and–death processes and quasi-skip-free processes. Each chapter is dedicated to one class of processes and introduces the techniques required to obtain their equilibrium distributions.<ref> is devoted entirely to birth–and–death (BD) processes, a highly structured class of Markov processes. The distinguishing feature of BD processes is that the state space can be ordered on a line and that transitions occur only between neighboring states. The queue in <ref> is an example of a BD process. Like that queue, all BD processes have product-form equilibrium distributions that can be solved iteratively. The class of BD processes contains many classical Markov processes that occur in queueing theory or in epidemics.<ref> extends the one-dimensional BD processes to multi-dimensional network models, and hence multi-dimensional Markov processes. Although these Markov processes have multiple dimensions, the equilibrium distribution can often be derived by making an educated guess.<ref> again extends the BD processes of <ref>, but now by including a finite second dimension. Here we encounter product-form solutions that involve matrices instead of the scalars that we have seen in this introduction. We discuss the matrix-geometric, matrix-analytic and spectral expansion method. Deriving an explicit expression for the matrices in the product-form solution proves to be difficult in many cases, so also numerical algorithms are provided to obtain these matrices.<ref> considers Markov processes on the same state space as the QBD processes of <ref>. The difference is that in <ref> we allow the process to have larger jumps in one direction. The structure of the solution for the equilibrium distribution is similar to the one for the QBD process, but calculating the matrices of interest is more involved. In Part III we tackle specific models that require advanced techniques to obtain the equilibrium distribution. Each chapter in this part is devoted to a specific model and for each model we develop multiple techniques to calculate the equilibrium distribution. The models serve as a vehicle through which we can demonstrate various techniques and allow the reader to compare methods. While applying the methods, we regularly exploit structural properties of the Markov process to obtain explicit expression for the equilibrium probabilities.<ref> considers a queueing system consisting of a single server and two priority classes, where low-priority jobs are only served when there are no high-priority jobs in the system. We model this system as a Markov process with two dimensions, where the dimensions keep track of the number of jobs of each class in the system. We demonstrate a difference equations approach, the generating function approach and two approaches related to QBD processes to obtain the equilibrium distribution.<ref> describes a single-server queue where waiting jobs are only allowed into the system when the system empties. We present three methods to obtain the equilibrium distribution of the associated two-dimensional Markov process: the generating function approach, the matrix-geometric approach and the compensation approach.<ref> covers three different production systems that give rise to two-dimensional Markov processes. The first two models are QBD processes and the third model has two countably infinite dimensions. For each system we present a tailor-made solution method to obtain the equilibrium distribution.<ref> analyzes a system consisting of two single-server queues where an arriving job joins the shortest of the two queues. The dynamics of this model are described by a Markov process that takes values in the positive half-plane. We use the compensation approach to determine the equilibrium probabilities.PART:Basic methodsCHAPTER: MARKOV PROCESSES Markov processes are stochastic processes whose future behavior only depends on the present and not on the past. This special property makes Markov processes mathematically tractable. Markov processes therefore serve as widely applied models in areas as diverse as biology, physics, chemistry, logistics, economics and social sciences.In this book we analyze a host of Markov processes. In this chapter we present the mathematical notions that are required to define Markov processes. To that end we start with a discussion of the exponential distribution, which is essential in the construction of Markov processes. We then show how to build Markov processes and discuss some of the basic properties. We will study Markov processes as functions of time, but our main focus in this chapter and throughout the remainder of this book will be on the analysis of the long-term or equilibrium behavior.§ EXPONENTIAL DISTRIBUTION The continuous random variable X follows an exponential distribution with parameter > 0, denoted by X ∼, if its probability density function is given by f_X(t) = ^- t,t ≥ 0, and the associated cumulative distribution function is F_X(t) = ∫_0^t f_X(u) u = 1 - ^- t,t ≥ 0. It readily follows that the expectation and variance of X are X = 1/X = 1/^2.The exponential distribution enjoys the so-called memoryless property or Markov property, which is arguably the most important property for analytic tractability of stochastic processes in this book. The property reads X > s + t | X > s = X > s + t, X > s/ X > s = X > s + t/ X > s= ^-(s + t)/^- s = ^- t = X > t. Think of X as the lifetime of some component. Then, the memoryless property states that the remaining lifetime of X, given that X is still alive at time s, is again exponentially distributed with the same mean 1/. In other words, the probability that X dies in the next t time units is independent of the current age s of X. The exponential distribution is the only continuous distribution that satisfies this memoryless property. Denote by X_1,X_2,…,X_n independent exponentially distributed random variables with parameters _1,_2,…,_n. Define now the minimum over these random variables as Y_n min(X_1,X_2,…,X_n). We have Y_n ≥ t = min(X_1,X_2,…,X_n) ≥ t =X_1 ≥ t, X_2 ≥ t, …, X_n ≥ t =X_1 ≥ t X_2 ≥ t ⋯ X_n ≥ t = ^-_1 t^-_2 t⋯^-_n t = ^-(_1 + _2 + ⋯ + _n)t, where the third equality follows from the independence of the random variables. We have just proved the second important property of the exponential distribution: the minimum of n exponential random variables is again an exponential random variable with parameter the sum of the n parameters. A printer can fail due to power outages, paper jams or ink shortages. Let these events be independent and occur after exponential times with rates _, _ and _. The up-time of the printer is the minimum time until any failure occurs and hence the up-time is exponentially distributed with parameter _ + _ + _ with mean up-time 1/(_ + _ + _). Next consider the probability that an exponential random variable turns out to be the minimum among n exponential random variables: X_n = min(X_1,X_2,…,X_n)X_n = Y_n. The event { X_n = Y_n } is the same as the event { X_n ≤ Y_n - 1} since it implies that the n-th exponential random variable is the minimum. Using Y_n - 1∼μ, where we abbreviated μ∑_m = 1^n - 1_m, and conditioning on the length of X_n, X_n = Y_n = X_n ≤ Y_n - 1= ∫_0^∞Y_n - 1≥ t f_X_n(t) t = ∫_0^∞^-μ t_n ^-_n tt = _n/μ + _n = _n/_1 + _2 + ⋯ + _n. A similar reasoning shows that for any k = 1,2,…,n, X_k = min(X_1,X_2,…,X_n) = _k/_1 + _2 + ⋯ + _n.Combining the two previous properties, one can even show that one particular X_k being equal to min(X_1,X_2,…,X_n) is independent of the value of min(X_1,X_2,…,X_n). This will prove to be a very useful property when constructing Markov processes. Returning to <ref>, this means that the printer fails due to an ink shortage with probability _/(_ + _ + _).§ POISSON PROCESSES Before we introduce Markov processes in greater detail, we describe a specific type of Markov process called a Poisson process, a counting process that counts how many events have occurred in a time interval. For a Poisson process these events occur randomly in time and the time between two events is exponentially distributed with parameter . Denote by {N(t) }_t ≥ 0 the Poisson process where N(t) is the number of events that have occurred in the interval [0,t] and set N(0) = 0.We model the Poisson process as a collection of states representing the cumulative number of events that have occurred, and transitions between states that model the time needed to go to the next state, see <ref>. A transition is marked with the rate at which it occurs. To be more precise, given that the process is in state i, a transition from state i to state i + 1 occurs after an exponential amount of time with parameter , see <ref> again. When modeling the Poisson process in this way, we have constructed a Markov process description of the Poisson process!Suppose that X_1,X_2,… are independent and identically exponentially distributed random variables with parameter . From <ref> we find that the time to reach state n is then X_1 + X_2 + ⋯ + X_n. The probability that there are at most n events in [0,t] can be expressed as N(t) ≤ n = X_1 + X_2 + ⋯ + X_n + 1 > t,n ≥ 1, t ≥ 0. In other words, the event to be in any of the states in { 0,1,…,n } at time t is equivalent to the event that the time it takes to reach state n + 1 is larger than t. The probability on the right-hand side of (<ref>) can be characterized further in terms of the Erlang distribution. If S_nX_1 + X_2 + ⋯ + X_n, then S_n follows an Erlang-n distribution with parameterdenoted as n. The density of the n distribution is given by f_S_n(t) = ( t)^n - 1/(n - 1)!^- t, which we can prove using induction. For n = 1 we have f_S_1(t) = ^- t. Assume that f_S_n(·) satisfies (<ref>). Then, f_S_n + 1(t)= f_S_n + X_n + 1(t) = ∫_0^t f_S_n(u) f_X_n + 1(t - u) u = ∫_0^t ^- u( u)^n - 1/(n - 1)!^-(t - u)u = ^- t^n/(n - 1)!∫_0^t u^n - 1u = ( t)^n/n!^- t , showing that (<ref>) is correct. The cumulative distribution function we give without proof: F_S_n(t) = 1 - ∑_m = 0^n - 1( t)^m/m!^- t,n ≥ 1, t ≥ 0. Returning to (<ref>) armed with (<ref>), we find N(t) ≤ n = ∑_m = 0^n ( t)^m/m!^- t, and therefore N(t) = n = ( t)^n/n!^- t,n ≥ 0, t ≥ 0. The distribution of N(t) is called a Poisson distribution with parameter t. Some quick calculations show that N(t) =t N(t) =t. The expected number of events in [0,t] is thus the rateat which events occur multiplied by the length of the interval t.The Poisson process is vital for modeling practical applications: to model the occurrence of software errors or machine breakdowns, the arrival of jobs at a processor, or the arrival of orders at a production system. It is empirically found that in many conditions the real-world processes can be well approximated by a Poisson process. We next establish a theoretical result that supports the assumption of Poisson processes in practical settings. Let X follow awith parameters n and p, that is X = k = nk p^k (1 - p)^n - k,k = 0,1,…,n. Let p → 0 as n →∞ such that n p =t, then X = k→( t)^k/k!^- t,k ≥ 0, t ≥ 0.Let k be a fixed integer. Then we have lim_n →∞X = k = lim_n →∞nk p^k (1 - p)^n - k= lim_n →∞n!/(n - k)!k!(t/n)^k ( 1 - t/n)^n - k= ( t)^k/k!lim_n →∞n (n - 1) (n - 2) ⋯ (n - k + 1)/n^k( 1 - t/n)^n - k= ( t)^k/k!lim_n →∞( 1 - 1/n) ( 1 - 2/n) ⋯( 1 - k - 1/n) ( 1 - t/n)^n ( 1 - t/n)^-k= ( t)^k/k! 1 · 1 ⋯ 1 ·^- t· 1 = ( t)^k/k!^- t, proving the statement. Many real-world arrival processes fit into the framework of <ref>. To see this, consider n potential voters each having a small probability p of arriving at a particular polling station in a small interval [0,t]. The probability that k out of the n voters show up in [0,t] is binomially distributed: there are nk groups of size k in a population of size n and exactly k voters arrive with probability p^k (1 - p)^n - k. If n is large and p is small, then the expression in terms of the Poisson distribution closely approximates the actual probability and is moreover easy to work with. In other words, if a large number n of arrivals can occur in a time interval [0,t] with a small probability p and we can constructsuch that ≈ np/t, then the Poisson process with rateclosely approximates the arrival process at the polling station.We next mention two important properties of a Poisson process. Suppose that N_1(·),N_2(·),…,N_n(·) are independent Poisson processes with rates _1,_2,…,_n. Define N(t)N_1(t) + N_2(t) + ⋯ + N_n(t) for all t ≥ 0. The time until a next event occurs for the counting process N(t) is, by the memoryless property of the exponential distribution, the minimum over n independent exponential random variables with parameters _1,_2,…,_n. So, by (<ref>) we have that the time until a next event is exponentially distributed with parameter _1 + _2 + ⋯ + _n and { N(t) }_t ≥ 0 is a Poisson process with rate _1 + _2 + ⋯ + _n. This property is called the merging property of independent Poisson processes.For the second property, consider a Poisson process { N(t) }_t ≥ 0 with ratewhere X_1,X_2,… are the times between events and each event is given a label out of n possible labels. For each arrival, label k is given with fixed probability p_k > 0 and p_1 + p_2 + ⋯ + p_n = 1. Denote the number of events with label k in the interval [0,t] as N_k(t). We determine the time T until the next event occurs for the counting process { N_k(t) }_t ≥ 0. To that end, we require the total number of events that occur until the first time an event is given the label k (this counts the last event with label k as well). This is exactly a random variable K with a geometric distribution, that is, K = i = (1 - p_k)^i - 1 p_k,i ≥ 1. We can now express the time T in terms of the time between events of the original Poisson process and K as T ∑_i = 1^K X_i ∼ p_k. We will show (<ref>) in <ref> of <ref>, since the proof requires Laplace-Stieltjes transforms. For now, we can conclude that { N_k(t) }_t ≥ 0 is an independent Poisson process with parameter p_k. The second property thus says that under probabilistic splitting, a Poisson process remains a Poisson process.§ GENERAL MARKOV PROCESSES A continuous-time stochastic process { X(t) }_t ≥ 0 is called a Markov process if it takes values in a countably infinite or finite state spaceand satisfies the Markov property. Let 𝔉(s) be the history of the process until and including time s at which X(s) = x. A process satisfies the Markov property if for all x,y ∈ and t,s ≥ 0, X(t + s) = y |𝔉(s) = X(t + s) = y | X(s) = x. The Markov property states that the future state at time t + s does not depend on the past states, but only on the current state at time s. The right-hand side of (<ref>) is called a transition function. A Markov process for which X(t + s) = y | X(s) = x does not depend on s is said to have stationary transition functions, an assumption we shall make throughout this book.A Markov process is a jump process. This means that the Markov processstays in a state x ∈ a certain amount of time and after that time, makes a transition to a different state y ∈, y ≠ x. A transition alters the state of the process in a sudden and radical way, hence the name jump process. Both the time spent in a state and the possible transitions (and the probabilities with which they occur) are allowed to depend on the state. Because of the jumps a sample path of a Markov process is assumed continuous from the right and having a limit from the left.For an elaborate and technical description of Markov jump processes we refer the interested reader to Asmussen <cit.>.Assume that the Markov process is currently in state x ∈. The event that causes a transition from state x to y, where x ≠ y, takes places after an exponential amount of time with parameter x,y≥ 0 (where 0 indicates a transition is not possible). Let us call this the transition time from x to y and refer to x,y as the transition rate from x to y. Clearly, the time spent in state x until a transition occurs (the sojourn time H_x) is the minimum over all end states y of the transition times from x to y. According to the properties of exponential random variables, we have that a Markov process obeys two basic rules (see also <ref>):* The sojourn time H_x in state x is exponentially distributed with parameter x∑_y ≠ xx,y;* After the sojourn time the Markov process jumps from state x to y ≠ x with probability x,y/x. We require the sojourn time in each state x to be positive. This means that we restrict our analysis to Markov processes that satisfy 0 ≤x < ∞ for all x ∈. A state x is called absorbing if x = 0. An absorbing state is a state from which the Markov process cannot leave: once it reaches this state, it will stay there indefinitely.[Browsing the internet] The internet browsing behavior of a user is tracked for the purpose of ranking websites. From numerous previous observations, the behavior of this particular user has become apparent. The user starts his session at some website. He stays at each website x an exponential amount of time with mean 1/x. After that time, the user proceeds to a different website that he picks from a set of websites n_x, which is allowed to depend on the current website since the user might want to visit a website on a related topic. The set n_x can also contain an element representing the end of the browsing session.The browsing behavior is a Markov process { X(t) }_t ≥ 0. The states of the Markov process are the websites and state 0 is the end of the browsing session (with 0 = 0). Then, X(t) is the website the user is on at time t. The sojourn time H_x in state x is exponentially distributed with rate x and after the sojourn time, the Markov process transitions to a different website y ∈ n_x with some probability b_x,y that can be determined from previous browsing behavior. Notice that we require ∑_y ∈ n_x b_x,y = 1. We next discuss regularity,A rigorous treatment of regularity and explosion can be found in Resnick <cit.> and Norris <cit.>. the property that states that the Markov process makes a finite number of transitions in a finite length of time with probability 1. If a Markov process is not regular, we call it an explosive process. Explosive processes have the property that within a finite amount of time, an infinite number of transitions can occur. We assume throughout the book that all Markov processes are regular. This will always hold for Markov processes with a finite state space, or when sup_x ∈x < ∞. If sup_x ∈x = ∞, the Markov process might still be regular, however. Unless mentioned otherwise, we will henceforth assume that sup_x ∈x < ∞, since handling the other case introduces technical hurdles that detract from the book's main storyline.[An explosive process] Consider a Markov process labeled { X(t) }_t ≥ 0 with initial state X(0) = 1, transition rates x,x + 1 = x^2, x ≥ 1 and all other transition rates are zero. Clearly, the Markov process proceeds through the numbered states 1,2,… and resides in each state x an exponential amount of time with mean 1/x^2. Let T_∞ be the time until the process reaches state ∞. Then T_∞ = ∑_x ≥ 1 1/x^2 = π^2/6 and T_∞ < ∞ = 1, showing that with probability 1 infinitely many transitions occur in a finite interval. The transition rates are the basic ingredients of the Markov process. We therefore introduce the transition rate matrix Q of dimension || × ||, with as the elements the transition rates. A row of Q indicates the state the process is currently in and the column is the target state. The diagonal elements are different in the sense that in row x, the element on the diagonal is -x. This makes the row sums equal to zero. For = _0, the transition rate matrix is then Q = [ - 0 0,1 0,2 0,3 ⋯; 1,0 - 1 1,2 1,3 ⋯; 2,0 2,1 - 2 2,3 ⋯; 3,0 3,1 3,2 - 3 ⋯; ⋮ ⋮ ⋮ ⋮ ⋱ ]. In general one needs to order the state space to be able to characterize the transition rate matrix Q.The transition rate matrix Q can be visualized in a transition rate diagram. This diagram depicts the states of the Markov process, the possible transitions between the states and the rates at which they occur. The transition rate diagram can be incredibly helpful in recognizing the underlying structure of the transition rates of the Markov process. See <ref> for an example. Both the description of a Markov process in terms of the transition rate matrix Q and the transition rate diagram are sufficient to fully characterize the Markov process.A useful concept for Markov processes are stopping times. Namely, a Markov process before a stopping time is independent of the Markov process after the stopping time. This property is called the strong Markov property.Establishing the strong Markov property in continuous time is actually more technical than we make it seem in <ref>. We have opted for the current description to bring across the main idea of the strong Markov property without going into too much technical detail. For a technical and precise treatment of the strong Markov property in continuous time we refer the reader to Norris <cit.>. It essentially applies the Markov property at a `random time' with a clear definition of when this time stops. We briefly describe these two concepts.A random variable T is called a stopping time if its realization depends only on the history of the Markov process 𝔉(T) until and including time T and whose value is the time at which the process meets a `stopping rule'. A good example of a stopping time is the time T it takes for the Markov process to go from state x to state y. If asked to stop at time T, you only need to observe when the Markov process enters state y for the first time. An example that is not a stopping time is the time T at which the Markov process exits the set of states A for the last time. Clearly, the future states of the Markov process are needed to determine if it actually was the last time the process exits the set of states A. So, in general, a last exit time is not a stopping time. The Markov process evaluated at a stopping time T, conditional on { T < ∞}, starts anew from the state X(T). More precisely, a Markov process { X(t) }_t ≥ 0 satisfies the strong Markov property, which says that for each stopping time T, conditioned on the event { T < ∞}, we have that for each t ≥ 0, X(T + t) only depends on X(T). As an example, say we have the time T it takes to go from state x to state y. Conditioning on the event that T is finite, X(T + t) = z | X(0) = x = X(T + t) = z | X(T) = y= X(t) = z | X(0) = y, since T is a stopping time and the Markov process has stationary transition functions.§ CLASSIFICATION OF STATES We now discuss the notions of irreducibility, recurrence and transience. A state y is said to be accessible from state x if there is a positive probability of ever reaching state y given that the process starts in state x. If x is also accessible from y, the states x and y are said to communicate and is denoted by x ↔ y. Furthermore, if x ↔ y and y ↔ z, then also x ↔ z.The accessibility and communication properties are treated in many classical books, usually for Markov chains in discrete-time, see Feller <cit.>, Karlin and Taylor <cit.> or Ross <cit.>. The concept is identical for Markov processes, however.States that communicate are said to be in the same equivalence class, or class for short. This indicates that the state space of a Markov process can be partitioned into separate classes. If all states communicate with each other, then there is only one class and the Markov process is called irreducible. Alternatively, a Markov process is irreducible if X(t) = y | X(0) = x > 0, for all states x,y ∈ and t > 0, indicating that there is a positive probability that the process is in state y at time t given it started in x. So, state y is accessible from state x. Irreducibility is a direct property of the transition rate matrix Q, but a transition rate diagram such as the one in <ref>, can also be helpful in assessing if a Markov process is irreducible.A state is said to be recurrentLiggett <cit.> and Norris <cit.> both have an excellent treatment of recurrence and transience for Markov processes. if the Markov process returns to that state infinitely many times with probability 1. Otherwise the state is called transient. So, a recurrent state is always visited a next time, but there exists a time at which a transient state is visited for the last time. State x is recurrent if xX(t) = xfor arbitrary large t = 1 and transient otherwise, where the notation xf(X) and xf(X) are the expectation and probability of a functional of a process { X(t) }_t ≥ 0 given X(0) = x.Consider a Markov process with state space = { 1,2,3,4 }. Transitions can occur between these states. If the Markov process is in state 1, it transitions to state 2 after an exponentially distributed time with rate 1 and to state 3 with rate 2. From 2 the process transitions to state 3 with rate 2 and to state 4 with rate 3. With rate 1 the process transitions from state 3 to state 2 and with rate 4 from state 4 to state 3. This explanation is rather verbose and can be condensed by simply giving the transition rate matrix Q = [ -3120;0 -523;01 -10;004 -4 ]. Another concise description of the behavior of the Markov process is the transition rate diagram shown in <ref>. The states are represented by the labeled circles and the transitions with their rates are described using the arrows. If we inspect the transition rate diagram in <ref>, we see that the process cannot return to state 1 since there are no transitions leading to this state and therefore state 1 is transient. The communicating class { 2,3,4 } is recurrent. Recurrence has a number of equivalent definitions. To that end we need the total time spent by the Markov process in a state and hitting-time random variables. Define T_y ∫_0^∞X(t) = yt to be the total time spent in state y. Taking the expectation with respect to the initial state x yields xT_y = x∫_0^∞X(t) = yt = ∫_0^∞xX(t) = yt. Introduce the hitting time random variables x,yinf{ t > 0 : lim_s ↑ t X(s) ≠ X(t) = y | X(0) = x}, with the convention inf∅ = ∞. Note that x,x is the time it takes the process to return to state x. The hitting time x,y is a stopping time.Now, the first equivalent condition of recurrence is then as follows. A state x is recurrent if xT_x = ∞ and transient otherwise. A second equivalent condition of recurrence is x,x < ∞ = 1, which indicates by the strong Markov property that the process returns to state x unboundedly many times with probability 1. State x is transient if x,x < ∞ < 1. Both conditions can be understood from the viewpoint of the number of visits to a state. If state x is transient and X(0) = x, then the number of visits to state x follows a geometric distribution with failure probability xthe process returns to state x = x,x < ∞. Given that the process starts in state x, the expected number of visits to state x is 1/1 - x,x < ∞. Each time the process visits state x it stays there, in expectation, 1/x time. Naturally, the expected total time spent in a transient state x is finite, since x,x < ∞ < 1. To be more precise, the total time T_x spent in a transient state x conditional on X(0) = x is a sum of i.i.d. exponential random variables with distribution H_x where the number of terms in the summation is an independent geometric random variable with failure probability x,x < ∞. We conclude that for a transient state x, T_x conditional on X(0) = x is an exponential random variable. A recurrent state x is visited infinitely often and thus the expected total time spent in state x is infinite.Recurrent states can be classified even further. A state is said to be positive recurrent if the expected return time is finite and null recurrent if the expected return time is infinite. Recurrent states in a Markov process with a finite number of states are always positive recurrent. A recurrent state x is positive recurrent iff x,x < ∞ and null recurrent otherwise. In <ref> it is easy to see that the expected returns times for the recurrent states 2, 3 and 4 are finite, which makes them positive recurrent.Recurrence and transience are class properties. If any one state in an equivalence class is (positive or null) recurrent, than all states in that class are (positive or null) recurrent. Equivalently, a transient state implies that all states in that class are transient. There are many more ways to characterize recurrence and transience, but the current level of discussion is sufficient for this book.Most Markov processes are one of three types: (i) all states communicate and are recurrent; (ii) some transient classes and some recurrent classes and the Markov process eventually enters one of the recurrent classes; or (iii) all states in the countably infinite state space of the Markov process are transient. In this book we focus mostly on type-(i) Markov processes.§ TIME-DEPENDENT BEHAVIOR By the law of total probability the probability mass function of X(t) satisfies X(t) = y = ∑_x ∈X(t) = y | X(0) = xX(0) = x and is thus uniquely characterized by the transition functions p_x,y(t) X(t) = y | X(0) = x and the matrix of transition functions P(t)[p_x,y(t)]_x,y ∈. The transition functions satisfy the Chapman-Kolmogorov equations, which state that each transition can be split at any intermediate time. The proof of this theorem can be found in many textbooks, e.g., <cit.>. For all t,s ≥ 0, P(t + s) = P(t) P(s), or, in scalar form with x,y ∈, p_x,y(t + s) = ∑_z ∈ p_x,z(t) p_z,y(s). The transition functions satisfy two sets of differential equations called the Kolmogorov backward and forward equations. The Kolmogorov backward equations are derived from the Chapman-Kolmogorov equations by conditioning on the state at time h. We have p_x,y(t + h) = ∑_z ∈ p_x,z(h) p_z,y(t) and subtracting p_x,y(t) from both sides, dividing by h and taking h ↓ 0 yields lim_h ↓ 0p_x,y(t + h) - p_x,y(t)/h= lim_h ↓ 0∑_z ≠ xp_x,z(h)/h p_z,y(t) - lim_h ↓ 01 - p_x,x(h)/h p_x,y(t). By definition, the left-hand side of (<ref>) equals / t p_x,y(t). On the right-hand side we have two limits. Since the transition functions satisfy p_x,x(t) = 1 - x t + (t),p_x,y(t) = x,y t + (t),y ≠ x, which is proved in, e.g., <cit.>, these limits can be simplified. In case the state spaceis finite, the interchange of the limit and the finite summation is clearly allowed. If the state space is countably infinite, the interchange is also allowed (see, e.g., <cit.>) and we obtain the Kolmogorov backward equations. For all t ≥ 0, / t P(t) = Q P(t), or, in scalar form with x,y ∈, / t p_x,y(t) = ∑_z ≠ xx,z p_z,y(t) - x p_x,y(t) and initial conditions p_x,x(0) = 1 and p_x,y(0) = 0, y ≠ x. The Kolmogorov forward equations are obtained by conditioning on the state at time t. We have p_x,y(t + h) = ∑_z ∈ p_x,z(t) p_z,y(h) and subtracting p_x,y(t) from both sides, dividing by h and letting h ↓ 0 gives lim_h ↓ 0p_x,y(t + h) - p_x,y(t)/h= lim_h ↓ 0∑_z ∈ p_x,z(t) p_z,y(h)/h - p_x,y(t) lim_h ↓ 01 - p_y,y(h)/h . In this case, the interchange of limit and summation is not always allowed. For example, an explosive process does not satisfy the Kolmogorov forward equations as they are formulated in the following theorem, where we did interchange the limit and the summation. However, these equations do hold for all birth–and–death processes (see <ref>) and for all Markov processes with a finite state space . We state the following theorem without a proof, since all results follow from the definition of the derivative and (<ref>), assuming that the limit and summation can be interchanged. For all t ≥ 0 and under suitable regularity conditionsThe conditions for which the Kolmogorov forward equations hold are formulated either fairly restrictive, as in the case in Gikhman and Skorohod <cit.>, or difficult to check, as in Liggett <cit.>. The former conditions are sup_x ∈x < ∞ and the latter conditions are ∑_z ∈ p_x,z(t) z < ∞. If the former conditions hold, than they imply the latter conditions: ∑_z ∈ p_x,z(t) z≤sup_x ∈x∑_z ∈ p_x,z(t) = sup_x ∈x· 1 < ∞./ t P(t) = P(t) Q, or, in scalar form with x,y ∈, / t p_x,y(t) = ∑_z ≠ x p_x,z(t) z,y - p_x,y(t) y with initial conditions p_x,x(0) = 1 and p_x,y = 0, y ≠ x. The Kolmogorov forward equations are often easier to solve, since these equations express the transition functions in terms of a common initial state X(0) = x. Still, obtaining explicit expressions for the transition functions is notoriously difficult, and can generally be done only for toy models or Markov processes with a pronounced structure in the transition rate matrix. Let us consider such an example.[Star gazing] We study the visibility of a star. Statistical analysis shows that the light source is visible for an exponential amount of time with parameter μ and remains invisible for an exponential amount of time with parameter . We denote by X(t) if the star is visible or not at time t. Under this description, X(t) has a finite state space = { 0,1 } and transition rates 0,1 = and 1,0 = μ (all other rates are 0). The transition functions with X(0) = 0 satisfy the Kolmogorov forward equations, so / t p_0,0(t)= μ p_0,1(t) -p_0,0(t),/ t p_0,1(t)=p_0,0(t) - μ p_0,1(t). We can solve this system of equations by noting that at time t the star has to be either visible or invisible, or, symbolically, p_0,0(t) + p_0,1(t) = 1. From the first equation we derive / t p_0,0(t) = μ - ( + μ) p_0,0(t), which can be turned into a separable equation by / t( ^( + μ)t p_0,0(t) ) = p_0,0(t) / t^( + μ) t+ ^( + μ) t/ t p_0,0(t) = μ^( + μ)t. Integrating the above equation and using the initial condition p_0,0(0) = 1 finally gives p_0,0(t) = μ/ + μ + / + μ^-( + μ)t. The transition functions for all initial states are derived in an identical way. The result is P(t) = [ p_0,0(t) p_0,1(t); p_1,0(t) p_1,1(t) ] = 1/ + μ[μ + ^-( + μ)t- ^-( + μ)t; μ - μ^-( + μ)t + μ^-( + μ)t ], which has a nice symmetrical form.For finite state spaces, a solution to the Kolmogorov backward and forward equations always exists and it is given byThere are some technical hurdles that one needs to overcome to be able to formulate P(t) = ^Qt as the solution to both the Kolmogorov backward and forward equations. In particular, the derivative needs to be interchanged with an infinite series. Norris <cit.> shows that this indeed can be done by showing that ^Qt has an infinite radius of convergence. For an infinite state space, the same author derives in <cit.> that the minimal non-negative solution of the backward equation is also the minimal non-negative solution of the forward equation. However, the solution P(t) is not characterized. P(t) = ^Qt, where the matrix exponential is defined as ^Qt∑_n ≥ 0(Qt)^n/n! with (Qt)^0 = and therefore P(0) =. Indeed, (<ref>) is a solution to both the Kolmogorov backward equation / t P(t)= Q + t Q^2 + t^2/2! Q^3 + t^3/3! Q^4 + ⋯= Q (+ t Q + t^2/2! Q^2 + t^3/3! Q^3 + ⋯) = Q P(t) and the forward equation / t P(t)= Q + t Q^2 + t^2/2! Q^3 + t^3/3! Q^4 + ⋯= (+ t Q + t^2/2! Q^2 + t^3/3! Q^3 + ⋯) Q = P(t) Q. Computing the matrix exponential is difficult, especially since the matrix Q has both negative and positive elements and subtractions can cause loss of significant digits. Since the state spaceis finite, one can truncate the series to a finite sum to obtain a numerical approximation of P(t). In <ref> we have seen that the transition rate matrix Q is the primary ingredient for constructing a Markov process. Coming to the end of this section, we have shown that the matrix Q governs the time-dependent behavior of the Markov process as well. In the next section we show that Q again plays an important role in determining probabilities of interest when t →∞ and the Markov process reaches an equilibrium.§ EQUILIBRIUM BEHAVIOR With time, Markov processes that are irreducible and positive recurrent converge to an equilibrium. This means that the probability distribution of X(t) (which depends on X(0)) tends to some other probability distribution that does not depend on X(0) as t tends to infinity.An interesting object to study is the long-term fraction of time that the Markov process occupies a state y ∈ given some initial state x ∈, which is given by lim_t →∞1/t∫_0^t X(s) = y | X(0) = xs,x,y ∈. It seems likely, and is indeed true, that if the Markov process is irreducible and positive recurrent, then this long-term fraction of time does not depend on the initial state x. If we label p(x) = lim_t →∞1/t∫_0^t X(s) = x | X(0) = ys,x,y ∈, then it is easy to see that p(x) > 0 for each x ∈ by positive recurrence and ∑_x ∈ p(x) = 1 since we are talking about fractions of time. The distribution p(x), x ∈ in (<ref>) is called the occupancy distribution. Now, one can prove that the occupancy distribution is uniquely given by p(x) = 1/xx,x > 0,x ∈. The proof of this statement uses a renewal-reward process, but we will only give an intuitive explanation. Due to the strong Markov property, we can just look at paths (or cycles) of the Markov process that start and end at state x. These cycles occur infinitely often because the Markov process is positive recurrent. The expected time of such a cycle is x,x < ∞. Within each cycle, the expected time spent in state x is 1/x. Dividing these two quantities as in (<ref>) exactly gives the fraction of time spent in state x in the long run.[Occupancy distribution in a complete digraph] Consider a Markov process with N + 1 states, where from each state every other state is reachable in one transition. The transition rate diagram of this Markov process constitutes a complete digraph; every state is connected to every state. We furthermore make the simplifying assumptions that the sojourn time in each state is exponentially distribution with mean 1 and the probability of making a transition to a particular state is 1/N.The described Markov process is irreducible and positive recurrent since the number of states is finite. It is moreover symmetric and so we already know that the occupancy distribution p(x) = 1/(N + 1). We verify this by deriving the expected return times and using (<ref>). Fix the initial state as 1 and abbreviate R_x = x,1. By a one-step analysis we derive R_1= 1 + 1/N∑_y ≠ 1 R_y, R_x= 1 + 1/N∑_y ≠ 1,x R_y,x ≠ 1. Add R_x/N to both sides of (<ref>) to get R_x ( 1 + 1/N) = 1 + 1/N∑_y ≠ 1,x R_y + 1/N R_x = 1 + 1/N∑_y ≠ 1 R_y = R_1. Now, sum over all x ≠ 1 to obtain ( 1 + 1/N) ∑_x ≠ 1 R_x = N R_1 ⇒∑_x ≠ 1 R_x = N^2/N + 1 R_1. Substituting (<ref>) into (<ref>) gives R_1 = 1 + 1/NN^2/N + 1 R_1 ⇒ R_1 = N + 1 and so p(1) = 1/(1 R_1) = 1/(N + 1). Since we fixed an arbitrary state and the Markov process is symmetric, all expected return times are N + 1 and the occupancy distribution follows. So far, we derived that an irreducible and positive recurrent Markov process has a unique occupancy distribution expressed in terms of the expected sojourn times and expected return times. The expected return times are usually difficult to determine. We wish to have an easier way of computing the occupancy distribution. To that end, we introduce two concepts and relate these to the occupancy distribution. A probability distribution p(x), x ∈ with ∑_x ∈ p(x) = 1 is said to be a stationary distribution for the Markov process if it satisfies p(y) = ∑_x ∈ p(x) p_x,y(t),y ∈, t ≥ 0. In light of (<ref>), the above definition should be interpreted as follows: if the initial state is distributed according to a stationary distribution , then the distribution of X(t) is independent of t and equal to the stationary distribution . Moreover, in that case, { X(t) }_t ≥ 0 is called a stationary process.A more natural and intuitive distribution is the limiting distribution. A probability distribution p(x), x ∈ with ∑_x ∈ p(x) = 1 is said to be a limiting distribution for the Markov process if it satisfies lim_t →∞ p_x,y(t) = p(y),x,y ∈, when the limits exist. Taking expectations on both sides of (<ref>) shows that the occupancy distribution can be expressed in terms of transition functions:In this case, taking expectations of (<ref>) requires some work. It requires the use of the dominated convergence theorem and Tonelli's theorem, but we do not show it here. lim_t →∞1/t∫_0^t p_x,y(s) s = p(y),x,y ∈. So, the existence of a limiting distribution implies the existence of an occupancy distribution. More importantly, the three distributions mentioned in this section are equivalent. We present this fact here without proof, see <cit.> for an elaborate discussion and the proof. An irreducible and positive recurrent Markov process has a unique occupancy distribution, a unique stationary distribution and a unique limiting distribution and all three distributions are identical. To calculate the occupancy distribution, we require the expected return times and to calculate the stationary and limiting distributions we require the transition functions. In most cases, this is prohibitively difficult. Thankfully, we can work with another distribution that is the unique solution to a system of linear equations called the balance equations. An irreducible and positive recurrent Markov process has a probability distribution = [ p(x) ]_x ∈ with = 1 which is the unique solution of theQ = , or, in scalar form, p(y) y = ∑_x ≠ y p(x) x,y,y ∈. This distribution is called theand is equal to the occupancy, stationary and limiting distribution. Solving the balance equations proves to be very useful since it also ensures positive recurrence of the Markov process. The following theorem is a continuous-time version of Foster's theorem <cit.>. If there exists a non-zero solution of the balance equations and this solution is absolutely convergent, then the Markov process is positive recurrent and the solution can be normalized to obtain the equilibrium distribution. [Star gazing] We consider again the star of <ref>. Recall the transition functions in (<ref>). The Markov process is irreducible and positive recurrent. We derive that the occupancy, stationary, limiting and equilibrium distribution are identical. The occupancy distribution is given by p(0)= lim_t →∞1/t∫_0^t p_0,0(s) s = 1/ + μlim_t →∞1/t( μ t + 1 - ^-( + μ)t/ + μ) = μ/ + μ and similarly for p(1) to obtain p(1) = /( + μ). Let us verify that the occupancy distribution is a stationary distribution. We have P(t)= [ p(0) p(1) ][ p_0,0(t) p_0,1(t); p_1,0(t) p_1,1(t) ]= [ p_1,0(t) + p(0)(p_0,0(t) - p_1,0(t)) p_0,1(t) + p(1)(p_1,1(t) - p_0,1(t)) ]= [ p_1,0(t) + p(0) ^-( + μ)t p_0,1(t) + p(1) ^-( + μ)t ]= [ p(0) p(1) ] = , where we used p(0) + p(1) = 1. The limiting distribution is found by taking the limit t →∞ for the transition functions: lim_t →∞ P(t) = lim_t →∞[ p_0,0(t) p_0,1(t); p_1,0(t) p_1,1(t) ] = [ p(0) p(1); p(0) p(1) ] = 1/ + μ[ μ; μ ]. Finally, the balance equations read p(0)= p(1) μ, p(1) μ = p(0) , which is a dependent system of linear equations, as is required. Using p(0) + p(1) = 1 we also obtain p(0) = μ/( + μ) and p(1) = /( + μ). So, for this simple two-state example the four probability distributions indeed agree, in line with <ref>. One can think of the balance equations as the result of taking t →∞ in the Kolmogorov forward equations of <ref>. Intuitively, an irreducible and positive recurrent Markov process reaches an equilibrium in which the transition functions do not change anymore and we heuristically argue that / t p_x,y(t) → 0 for t →∞. Since an irreducible and positive recurrent Markov process has a limiting distribution, we have lim_t →∞ p_x,z(t) = p(z), and the interchange of the limit and infinite summation is allowed by the regularity conditions that were assumed in <ref>.A possibly more intuitive and natural interpretation of the balance equations is the following. If the Markov process is in an equilibrium, we require that the rate at which the process leaves a (set of) state(s) is equal to the rate at which the process enters the (set of) state(s). If this would not be the case, the Markov process is not in an equilibrium. Let us consider a countable set A⊂. Now, given that the Markov process is in state y ∈A, the Markov process transitions to states outside A with rate ∑_x ∈A^cy,x. The probability that in equilibrium the Markov process is in state y is given by the equilibrium distribution and is therefore equal to p(y). Similarly, one derives the rate at which the Markov process transitions to states inside A from a state x ∈A^c. Balancing the two produces ∑_y ∈A∑_x ∈A^c p(y) y,x = ∑_x ∈A^c∑_y ∈A p(x) x,y. The balance equations Q = follow from the above formula by taking A = { y }. The set of equations Q = is also called the global balance equations, see also <ref>(a). Sometimes the set A can be chosen in a way such that p(y) y,x = p(x) x,y, for all x,y ∈. These equations are called the local balance equations, see <ref>(b). Local balance equations are ideal to work with. These equations make it far easier to determine the equilibrium probabilities since it allows one to express each equilibrium probability p(y) in a specific other equilibrium probability, say (0), and p(0) follows from the normalization condition. Local balance equations do not exist in general, but they do exist for Markov processes with a specific type of structure in the transition rate matrix Q, such as the birth–and–death processes that we study in <ref>, and for processes that are time-reversible. The topic of time-reversibility and its implications is studied in <ref>.Choosing the set A in a smart way and then invoking the balance principle is something that requires intuition, which can be trained through seeing and analyzing a variety of different Markov processes. This will be one of the goals of this book.[Star topology] Consider a Markov process on the state space = _0. State 0 is central: from state 0 the process transitions to state x with rate ^x, but from state x the process can only transition to state 0 with rate μ^x, see <ref>. Since we want all x to be finite, we require < 1, otherwise the process leaves state 0 instantaneously. This gives 0 = ∑_x ≥ 1^x = /(1 - ).The Markov process is irreducible and recurrent. It remains to see if the states are null recurrent or positive recurrent. The process transitions from state 0 to state x with probability (1 - )^x - 1. If μ > 1 (μ < 1) the process resides in expectation a longer time at the states with a low (high) index.We know that if an equilibrium distribution exists, the Markov process is positive recurrent, see <ref>. We therefore investigate if a solution exists to the balance equations. This system of linear equations is given by p(0) /1 -= ∑_x ≥ 1 p(x) μ^x, p(x) μ^x= p(0) ^x,x ≥ 1. Summing over all x ≥ 1 on both sides of (<ref>) produces (<ref>) and the system of equations is dependent. Armed with the relation p(x) = p(0) ( /μ )^x and the normalization condition the equilibrium distribution can be obtained, if it exists. The normalization condition reads 1 = ∑_x ∈ p(x) = p(0) ∑_x ≥ 0( /μ)^x. We immediately see from the above equation that < μ is necessary for an equilibrium distribution to exist. Under this condition, the Markov process is indeed positive recurrent. Assuming < μ, we find p(0) = 1 - /μ and all p(x) = (1 - /μ) ( /μ )^x. Equation (<ref>) allows us to determine the expected return times from the occupancy distribution: 0,0 = μ(1 - )/(μ - ), x,x = μ/^x (μ - ),x ≥ 1. Since < 1, the expected return times grow unboundedly with increasing x, but for each state x the expected return time is indeed finite. A technique called censoring can also be instrumental in calculating the equilibrium probabilities by allowing for the derivation of a different set of balance equations. We will use <ref> as a visual guide. Censoring a process to a set A means that we only observe the process while it resides in this set. Practically it means that we can draw a new transition rate diagram: all transitions from states inside A to states in A^c are redirected to states within A. This redirection is done in a natural way, which we describe with an example. Say that state x ∈A has a single transition with rateto a state y outside A. With probability r_x,y,z the process returns to z ∈A for the first time after leaving A with a transition from state x to state y. The transition with rateis then split in many transitions according to these return probabilities: each new transition occurs with rate r_x,y,z for all states z ∈A. Notice that the potential transition from x to x does not need to be drawn, since it does not have any effect. Once the new transition rate diagram has been drawn, we can write down a different set of balance equations in the same way that we have described earlier.[Censoring] Consider the Markov process with three states as shown in <ref>(a). The balance equations that we can derive from <ref>(a) are p(1)= p(2) , p(2) ( + θ)= p(1)+ p(3) μ, p(3) μ = p(2) θ. Let us censor the process to the set A = { 1,3 }. So, we need to redirect all transitions that lead to state 2 to a state in A, since state 2 is outside this set. From state 1 the process can transition to state 2 with rateand it returns to A in state 1 with probability / ( + θ) (but we do not need to draw that transition since it returns to the same state) and it returns to A in state 3 with probability θ / ( + θ). So from state 1 we need to draw a transition to state 3 with rate θ / ( + θ). The same reasoning for state 3 leads to the transition rate diagram in <ref>(b). From <ref>(b) we derive another balance equation: p(1) θ/ + θ = p(3) μ/ + θ, which gives us p(1)= p(3) μ / (θ) and therefore by (<ref>) shows that p(2) = p(3) μ / θ. The normalization condition p(1) + p(2) + p(3) = 1 then gives us that p(1) = μ/θ + μ + μ,p(2) = μ/θ + μ + μ,p(3) = θ/θ + μ + μ. This simple example demonstrates how you can use censoring to derive new balance equations. This technique will prove useful when tackling more advanced processes. § MANUFACTURING EXAMPLES We now apply our knowledge of Markov processes to some realistic manufacturing examples.[A failing component] We assume that the quality of a component deteriorates through a total of N phases where in each phase the component resides for an exponential amount of time with parameter θ. After N phases the component fails completely. So, the lifetime of a component has an Nθ distribution. A lower quality component has a negative influence on the production capacity of the machine it resides in and therefore an operator visits the machine to check the quality of the component and replaces or repairs it whenever it is below perfect condition. The time between two visits of the operator is approximated by an exponential distribution with parameter . Both replacing and repairing a component is assumed to take no time as it is short compared to the time between two successive visits of the operator.Denote the quality of the component at time t as X(t). The process { X(t) }_t ≥ 0 is a Markov process with state space { 0,1,…,N } and transition rate matrix Q = [-θ θ; -(θ + ) θ; -(θ + ) θ; ⋮ ⋱ ⋱; -(θ + ) θ; -; ], where unspecified elements are zero. This Markov process is irreducible and positive recurrent because its state space is finite. So, the Markov process has an equilibrium distribution that we denote by [ p(i) ]_0 ≤ i ≤ N.The global balance equations Q = read p(0) θ = ∑_n = 1^N p(n), p(i) (θ + )= θ p(i - 1),i = 1,2,…,N - 1, p(N)= θ p(N - 1). With the help of the normalization condition ∑_n = 0^N p(n) = 1 we are able to derive p(0) from (<ref>) as p(0) θ =(1 - p(0)) ⇒ p(0) = /θ + . The remaining balance equations are iterated to obtain p(i)= ( θ/θ + )^i p(0) = ( θ/θ + )^i /θ + ,i = 0,1,…,N - 1, p(N)= θ/( θ/θ + )^N - 1/θ + . From these equilibrium probabilities we see that if θ is large in comparison to γ, then p(N) is large, which means that the component has deteriorated through all of its phases and has now completely failed. From these equilibrium expressions, an operator can, e.g., determine how often on average he needs to inspect the component so that with 99% certainty it does not reach deterioration phases 5 and higher. [Multiple failing components] A machine naturally consists of multiple components that can be replaced or repaired if they are not in perfect condition. Let us consider a situation in which there are two components with each their own failure process. The behavior of the operator is the same as before, but now he replaces or repairs all components that are not in mint condition. Replacing or repairing both components at the same time makes the two failure processes dependent: if we know that one of the two components is in phase 0, then it is probable that both components were replaced or repaired recently, which shows that we also have information on the failure process of the other component. The time until failure for component 1 is N_1θ_1 and N_2θ_2 for component 2. Let X_1(t) and X_2(t) denote the quality level of component 1 and component 2 at time t and let X(t)(X_1(t),X_2(t)) describe the configuration of quality levels at time t. { X(t) }_t ≥ 0 describes an irreducible and positive recurrent Markov process with finite state space { (i,j) ∈_0^2 : 0 ≤ j ≤ N_1, 0 ≤ j ≤ N_2 }. A transition rate diagram for a specific instance of N_1 and N_2 is shown in <ref>.Equilibrium probabilities of a two-dimensional Markov process are denoted as p(i,j). For ease of exposition, we assume that N_1 and N_2 are large (so as to not worry about boundary behavior), but this approach works for any N_1 and N_2. The equilibrium probabilities p(i,j) with (i,j) ∈ can be solved in a recursive fashion. To start, p(0,0) (θ_1 + θ_2) = ∑_(i,j) ∈∖{ (0,0) } p(i,j), which implies by the normalization condition ∑_(i,j) ∈ p(i,j) = 1 that p(0,0) = /θ_1 + θ_2 + . Now that we have the equilibrium probability of state (0,0) we can exploit the structure of the transition rate diagram in <ref>. In particular, we proceed along diagonals: the equilibrium probabilities of states (1,0) and (0,1) are expressed in terms of (0,0) as p(1,0)(θ_1 + θ_2 + )= p(0,0) θ_1, p(0,1)(θ_1 + θ_2 + )= p(0,0) θ_2. Along the next diagonal, the equilibrium probabilities of states (2,0), (1,1) and (0,2) are expressed in terms of the states on the previous diagonal: p(2,0)(θ_1 + θ_2 + )= p(1,0) θ_1, p(1,1)(θ_1 + θ_2 + )= p(1,0) θ_2 + p(0,1) θ_1, p(0,2)(θ_1 + θ_2 + )= p(0,1) θ_2. Clearly, the equilibrium probabilities of the states on one diagonal can be expressed in terms of the equilibrium probabilities of the states on the preceding diagonal. When proceeding in this manner the complete equilibrium distribution can be obtained explicitly.The recursive calculation of the equilibrium probabilities is not restricted to a system of two components, but can actually be applied to a system with an arbitrary number of components. For example, for a system with three components we can first determine p(0,0,0) and from that find p(1,0,0), p(0,1,0) and p(0,0,1) which leads to p(1,1,0), p(1,0,1) and p(0,1,1) and ultimately gives p(1,1,1). For the three-component example the sets of states are not diagonals but rather triangles. [Production capacity] We now study the impact of a single deteriorating component on the production capacity of a machine. Products arrive at the machine according to a Poisson process with rateand are served in order of arrival. If the machine is already occupied, the products wait in a queue. The rate at which the machine serves a product depends on the quality level of the deteriorating product: if the component is in phase n then the service rate is μ_n for n = 0,1,…,N with μ_1 ≥μ_2 ≥⋯≥μ_N = 0. The operator behaves the same as before and replaces or repairs the component after an γ amount of time and the lifetime of the component has an Nθ distribution.The Markov process associated with this system is two-dimensional: X_1(t) denotes the number of products in the system at time t and X_2(t) is the quality level of the component at time t and X(t)(X_1(t),X_2(t)) is the state of the system at time t. The state space of this irreducible Markov process is { (i,j) ∈_0^2 : 0 ≤ j ≤ N } and the transition rate diagram is given in <ref>. We note that the state space of this Markov process is countably infinite. A convenient way to partition the state space is by introducing levels. A level is a vertically aligned set of states. Specifically, level i is i{ (i,0),(i,1),…,(i,N) },i ≥ 0, so that = 0∪1∪2∪⋯At this point we will not determine the equilibrium distribution since it requires the theory of <ref>. Rather, we derive the condition for which the process is positive recurrent. Intuitively, the states are positive recurrent if the Markov does not diverge `towards infinity', by which we mean that X_1(t) does not grow without bound. For X_1(t) to not grow without bound, we require that the average transition rate from level i to level i - 1 (to the left) is greater than the average transition rate from level i to level i + 1 (to the right). We can make this statement without specifying the exact level i since the transition rate behavior is the same for any level greater than level 0. The average transition rate to the left is sum over i of the the proportion of time spent in phase i multiplied by μ_i. We can similarly calculate the average transition rate to the right. Clearly the proportions sum to 1, so that the average transition rate to the right is exactly . So, to determine the average transition rate to the left we require to determine the fractions of time spent in each of the phases.If we only observe transitions in the vertical direction, then we end up with exactly the Markov process of the failure process of a single component. Let us denote the equilibrium distribution of the phase process by π(i), i = 0,1,…,N (we reservefor the equilibrium distribution of the Markov process). From our earlier analysis of the single component we know that π(i)= ( θ/θ + )^i /θ + ,i = 0,1,…,N - 1,π(N)= θ/( θ/θ + )^N - 1/θ + . The average transition rate to the right is therefore ∑_i = 0^N π(i) μ_i = /θ + ∑_i = 0^N - 1( θ/θ + )^i μ_i. Under the stability condition < /θ + ∑_i = 0^N - 1( θ/θ + )^i μ_i, X_1(t) does not grow without bound and therefore the Markov process is positive recurrent. Compare this with the single-server system of <ref>, where the stability condition is < μ. This inequality also says that the average transition rate to the left is greater than the average transition rate to the right. § TAKEAWAYS Markov processes can describe the evolution in time of many systems. This chapter discussed some of the prerequisites needed to define Markov processes in a mathematical way. For analyzing Markov processes, in order to quantify their behavior, we discussed three basic systems of equations: the Kolmogorov backward and forward equations, and the balance equations. The Kolmogorov equations capture the dynamics of the Markov process, over all time, while the balance equations describe long-term behavior. The focus of this book lies primarily with balance equations, although for all Markov processes discussed in the subsequent chapters one could state the Kolmogorov equations and study these as well. We do this in <ref>, where we treat birth–and–death processes that have an exceptionally nice structure, leading to analytic solutions for both the balance and the Kolmogorov equations. In general, however, solving the Kolmogorov equations is more challenging than solving the balance equations. Solving the balance equations alone is challenging enough to write an entire book about.From the theory side, much more can be said about the mathematics of Markov processes. While this chapter is restricted to the bare minimum needed to work with the mathematics in this book, there is a wealth of mathematical theory for Markov processes to be discovered. We encourage the interested reader to study for instance the books of Brémaud <cit.>, Chung <cit.>, Ethier and Kurtz <cit.>, Feller <cit.>, Jacod and Shiryaev <cit.>, Karlin and Taylor <cit.>, Liggett <cit.>, Norris <cit.>, Resnick <cit.>, and Rogers and Williams <cit.>.From the practical side, much more can be said about the applications of Markov processes. Throughout the book we give examples of practical flavor, but these examples only serve the purpose of illustrating and practicing the mathematical methods. Those who want to learn more about modeling real-world applications as Markov processes can find many inspirational examples in books like Asmussen <cit.>, Bruneel and Kim <cit.>, Buzacott and Shantikumar <cit.>, Harchol-Balter <cit.>, Kelly and Yudovina <cit.>, Kiss, Miller and Simon <cit.> and Van Mieghem <cit.>.If there is one thing we have learned from this chapter is that defining the Markov process in terms of its transition rate matrix or diagram is only the beginning. In order to study the Markov process, we are confronted with solving systems of equations. This challenge does not only require basic analysis or linear algebra, but should be combined with recognizing the structure hidden in the transition matrix. It is only then that the Markov process will reveal its beautiful properties, most notably the product-form solutions for the balance equations we encountered in <ref>. Many chapters now will follow, about classes of Markov processes, each with their specific structures and specific mathematical challenges. In all but a few cases we will be able to construct product-form solutions. CHAPTER: QUEUES AND TRANSFORMSThis book is centered around analytic methods for finding the equilibrium distribution of a Markov process. So far, we have discussed methods targeted at directly solving the balancing equations, for instance by exploiting recursive structures or by substituting product forms. Transforms arise as an alternative method when an infinite system of linear equations—such as the balance equations—is converted into a single functional equation for the transform. The mathematical challenge then becomes to find the transform as the solution of the functional equation, which in some cases might prove the easiest or only method to tackle the problem. Once a transform is obtained, all information about the underlying distribution can be extracted from it. Taking derivatives of the transforms readily gives all moments. The underlying distribution can be retrieved by more advanced algorithms that invert the transform. This chapter covers the basics of transforms. For discrete random variables we introduce the probability generating function and for continuous random variables the Laplace-Stieltjes transform. We then learn how to work with these transforms by applying transform techniques to several classical queueing systems. We also introduce several numerical algorithms for transform inversion, which are largely based on Cauchy's formula. The transform technique and associated algorithms introduced in this chapter have a large scope of application, not only in later chapters in this book on more advanced Markov processes, but also in probability theory <cit.>, combinatorics <cit.> and digital signal processing <cit.>.§ BASIC TRANSFORMS We introduce basic properties of the probability generating function (PGF) for discrete random variables and the Laplace-Stieltjes transform (LST) for continuous random variables. We also give a first demonstration of how to use these transforms in the context of the basic single-server queue covered in <ref>. §.§ Probability generating functions The PGF of a non-negative random variable X that takes values in the set { 0,1,2,…} is defined as X^X = ∑_i ≥ 0 p(i) ^i, where p(i) = X = i is the probability mass function of X. A PGF of a random variable with a countably infinite support gives rise to an infinite series. Since we know that p(·) is a probability distribution and therefore ∑_i ≥ 0 p(i) = 1, we can conclude for || ≤ 1 that |X| = | ∑_i ≥ 0 p(i) ^i | ≤∑_i ≥ 0 p(i) ||^i ≤∑_i ≥ 0 p(i) = 1 and therefore the PGF converges for anythat is inside the closed unit disk. Depending on the form of p(i) the PGF might converge for other values ofas well. Specifically, there exists an r ≥ 1 such that the PGF converges absolutely for all || < r and diverges for all || > r. This r is called the radius of convergence of the PGF.[Geometric distribution] The probability mass function of the geometric distribution with failure probability ρ is given by p(i) = (1 - ρ)ρ^i,i ≥ 0 and therefore its PGF is = ∑_i ≥ 0 p(i) ^i = (1 - ρ) ∑_i ≥ 0 (ρ)^i = 1 - ρ/1 - ρ. The last equality only holds if || < 1/ρ, which ensures that the series converges. Note that the radius of convergence r is 1/ρ. [Poisson distribution] The probability mass function of the Poisson distribution with parameteris given by p(i) = ^i/i!^-,i ≥ 0 and therefore its PGF is = ∑_i ≥ 0 p(i) ^i = ^-∑_i ≥ 0()^i/i! = ^-(1 - ). The last equality holds for all ∈. So, the radius of convergence of the PGF of a random variable with a Poisson distribution with parameteris infinite.A PGF · is said to have radius of convergence r whenis an analytic function for all ∈ satisfying || < r and has at least one singularity on the circle || = r. A function that is analytic in a region A⊂ is a function that is complex differentiable at every ∈A, or equivalently, if it has a convergent series expansion in an open disk around every ∈A. For the mathematical definition of these terms we refer the reader to <cit.>; we will only use the property thatis analytic for || < r. Returning to <ref>, we see thatin (<ref>) is an analytic function for all ∈ satisfying || < 1/ρ. This function has a pole (a simple singularity) at = 1/ρ. The PGF in (<ref>) is called an entire function because it is analytic for all ∈. The probability mass function can be retrieved from the PGF X· through p(0) = X0 and p(i) = 1/i!^i/^iX|_ = 0,i ≥ 1. All probabilities { p(i) }_i ≥ 0 thus follow by taking derivatives of the PGF at = 0. This observation leads to one of the most important properties of a PGF, which is that if X = Y, then XY, and vice versa, if XY, then X = Y. Moreover, since the derivatives are evaluated at = 0, we conclude that if two PGFs are equal on any real interval containing the value 0, then the underlying probability mass functions are equal. We should mention that taking derivatives can become computationally cumbersome, either because of the complexity of the symbolic expressions of the derivatives, or because of numerical inaccuracies, particularly for p(i) with i large. We therefore also present an alternative method for PGF inversion in <ref> based on contour integrals.One of the great advantages of using PGFs is that the moments of the random variables are easy to determine. For example, /X|_ = 1 = /∑_i ≥ 0 p(i) ^i |_ = 1 = ∑_i ≥ 0 p(i) /^i |_ = 1 = ∑_i ≥ 0 i p(i) = X, where the interchange of derivative and summation is allowed because the series converges uniformly. More generally, the factorial moments are given by X(X - 1) ⋯ (X - k + 1) = ^k/^kX|_ = 1,k ≥ 1. A PGF is also useful when considering sums of random variables. For example, if we set ZX + Y and X and Y are independent, then Z = ^Z = ^X + Y = ^X^Y = XY.§.§ Laplace-Stieltjes transforms The LST of a non-negative random variable X is defined as X^- X = ∫_0^∞^- tF(t). When the random variable X has a density f(·), then the transform simplifies to X = ∫_0^∞^- t f(t) t. The region of convergence of an LST is at least the complex numbersthat satisfy > 0, but in most cases this region is larger. Notice that |X| ≤ 1 for > 0.[Exponential distribution] The exponential distribution with ratehas the probability density function f(t) = ^- t for all t ≥ 0. The LST of this distribution is therefore = ∫_0^∞^- t f(t) t = ∫_0^∞^-( + ) tt = / + . The last integral is finite if > -. So, the region of convergence of the LST associated with the exponential distribution with rateis described by all ∈ satisfying > -. An LST uniquely determines the underlying distribution just as a PGF does: if X = Y, then XY and vice versa if XY, then X = Y. In <ref> we show how to retrieve the probability distribution function f(·) using the Bromwich line integral or using an algorithm.An LST satisfies many useful properties; some of the most important ones include X0 = 1, /X|_ = 0 = -X, ^k/^kX|_ = 0 = (-1)^k X^k. Furthermore, if ZX + Y and X and Y are independent, then Z = XY. Now that we have introduced LSTs, we are able to prove the second property of Poisson processes: under probabilistic splitting, a Poisson process remains a Poisson process. We are required to prove (<ref>). Taking the LST on the right-hand side of (<ref>) and conditioning on K, ^- ∑_i = 1^K X_i = ∑_j ≥ 1^- ∑_i = 1^K X_i| K = j K = j= ∑_j ≥ 1^- ∑_i = 1^j X_i K = j = ∑_j ≥ 1^-X_1^jK = j= ∑_j ≥ 1( / + )^j (1 - p_k)^j - 1 p_k =p_k / + ∑_j ≥ 0( (1 - p_k)/ + )^j =p_k/ ++ / +- (1 - p_k) =p_k/ p_k + , which is exactly the LST of an exponential random variable with parameter p_k.§.§ Applying the transforms to a simple queue In <ref> we have introduced the simple queue where jobs arrive according to a Poisson process with rateand are served by a single server with exponential rate μ. This queueing system is denoted in Kendall's notation as the M/M/1 system. Here, M stands for Markovian or memoryless (so exponentially distributed). In later sections we will also encounter the letter G, which stands for general. The order in which the letters appear in Kendall's notation matters: the first entry describes the distribution of the inter-arrival times, the second entry the distribution of the service times and the third entry the number of servers in the system.In <ref> we have demonstrated how to obtain the equilibrium distribution of the M/M/1 system in two ways. We now demonstrate a third way using transforms. Recall that the balance equations are given by p(0)= μ p(1), ( + μ) p(i)=p(i - 1) + μ p(i + 1),i ≥ 1. We aim to find an expression for ∑_i ≥ 0 p(i) ^i by manipulating the balance equations (<ref>).Multiply both sides of (<ref>) by ^i and sum on both sides over all i ≥ 1 to obtain ( + μ) ∑_i ≥ 1 p(i) ^i = ∑_i ≥ 1 p(i - 1) ^i + μ∑_i ≥ 1 p(i + 1) ^i. By appropriately adding and subtracting terms on both sides of (<ref>) and multiplying by , we obtain ( + μ)(- p(0) ) = ^2+ μ (- p(1)- p(0) ). Use (<ref>) to express p(1) in terms of p(0) and obtain the relation (1 - (1 + ρ)+ ρ^2 )= (1 - ) p(0). Noticing that 1 - (1 + ρ)+ ρ^2 = (1 - )(1 - ρ) gives = p(0)/1 - ρ. Since 1 = ∑_i ≥ 0 p(i) = 1, we find that p(0) = 1 - ρ and = 1 - ρ/1 - ρ. From the geometric series ∑_i ≥ 0 x^i = 1/(1 - x) if |x| < 1, we deduce that = (1 - ρ) ∑_i ≥ 0 (ρ)^i = ∑_i ≥ 0 (1 - ρ) ρ^i ^i, and hence p(i) = (1 - ρ) ρ^i. The generating function approach is a powerful approach that works well even if an explicit expression for p(i) is difficult to obtain. In fact, if the expression of the PGF was not as nice as in (<ref>), then we could have stopped at that point and used algorithms that can numerically invert the PGF to calculate values for any p(i), see <ref>.For queueing systems with Poisson arrivals, so for M/·/· systems, the unusual property holds that arriving jobs find on average the same situation as an outside observer looking at the system at an arbitrary point in time. More precisely, the fraction of jobs finding on arrival the system in some state i is exactly the same as the fraction of time the system is in state i. This is called the Poisson arrivals see time averages (PASTA) property <cit.>. This property is only true for Poisson arrivals, and can be explained intuitively by the fact that Poisson arrivals occur completely random in time. If we label the probability that an arriving job sees i jobs in the system (excluding itself) as a(i), then we conclude that a(i) = p(i).The PASTA property can be used to determine the distribution of how much time a job spends in the system, which is also called the sojourn time S. With probability a(i) an arriving job finds i jobs in the system. Since the service times are exponentially distributed, we know that the sojourn time of the arriving jobs is the sum of i + 1 exponential phases, each with rate μ. By conditioning on the number of jobs seen on arrival, we therefore find that S = ∑_i ≥ 0 a(i) ( μ/μ + )^i + 1 = μ(1 - ρ)/μ + ∑_i ≥ 0( μρ/μ + )^i = μ(1 - ρ)/μ + 1/1 - μρ/μ += μ(1 - ρ)/μ(1 - ρ) + . Since we have learned earlier that an LST uniquely determines the distribution of a random variable, we conclude that the sojourn time is exponentially distributed with rate μ(1 - ρ).§ SINGLE-SERVER QUEUE WITH GENERAL SERVICE TIMES Consider a single-server queueing system where jobs arrive according to a Poisson process with rate —so with exponentially distributed inter-arrival times—and service times that are i.i.d. copies of some random variable B. Assume that B has a cumulative distribution function F_B(·) and a probability density function f_B(·). We require for stability that ρB < 1. This queueing system is denoted in Kendall's notation as the M/G/1 system. §.§ Departure distribution The state of the queueing system can be described by (i,t) with i the number of jobs in the system and t the service time already received by the job in service. This state description is then two-dimensional with one discrete dimension and one continuous dimension. The continuous dimension makes the analysis prohibitively difficult, so we will look for another state description. If we observe the number of jobs in the system at the instant just after a job departs, then we know that t = 0, which essentially removes the second continuous dimension in the state description. In equilibrium, we denote by d(i) the probability that a departing job leaves behind i jobs. In other words, d(i) is the fraction of departing jobs that leaves behind i jobs.From one departure instant to the next the number of jobs in the system reduces by one, but increases by the number of jobs that have arrived during its service time. We specify the probability r_i that a change of size i occurs in the number of jobs from one departure instant to the next. By conditioning on the length of the service time and using that the number of arrivals within the interval [0,t] is Poisson distributed with parameter t, we establish that r_i = ∫_0^∞( t)^i + 1/(i + 1)! ^- t f_B(t) t,i ≥ - 1. A departing job can leave behind zero jobs. In that state, we first wait for a job to arrive and depart before observing the number of jobs in the system. This means that from state 0, we return to state 0 with probability r_-1 and move to state i ≥ 1 with probability r_i - 1.By specifying the states and the transition probabilities r_i, we have in fact constructed an embedded Markov chain. It is called embedded because we only observe the process at embedded points in time (at departure instants) and the term `chain' indicates that it has transition probabilities instead of transition rates and that the time spent in each state is equal. The transition probability diagram of this Markov chain is presented in <ref>.Each state has incoming transitions from states below itself, from itself, and one incoming transition from one state higher. This gives the following balance equations: d(i)=r_i d(0) + r_i - 1 d(1) + ⋯ + r_0 d(i) + r_-1 d(i + 1)= ∑_j = 0^i + 1 r_i - j d(j). We manipulate the balance equations (<ref>) by making use of PGFs. Define d∑_i ≥ 0 d(i) ^i, r∑_i ≥ 0 r_i - 1^i,|| ≤ 1. Multiply both sides of (<ref>) by ^i and sum over all i to obtain d = ∑_i ≥ 0∑_j = 0^i + 1 r_i - j d(j) ^i= ∑_i ≥ 0 r_i d(0) ^i + ∑_i ≥ 0∑_j = 1^i + 1 r_i - j d(j) ^i = d(0) r + ∑_i ≥ 0∑_j = 1^i + 1 r_i - j d(j) ^i. By changing the order of the double summation and writing ^i = ^j ^i - j, we get d = d(0) r + ∑_j ≥ 1 d(j) ^j ∑_i ≥ j - 1 r_i - j^i - j. Changing the summation index of the inner summation to k = i - j + 1 yields d = d(0) r + ∑_j ≥ 1 d(j) ^j ∑_k ≥ 0 r_k - 1^k - 1= d(0) r + ∑_j ≥ 1 d(j) ^j r/= d(0) r + ( d - d(0) ) r/, so that d = d(0) r1 - 1//1 - r/ = d(0) r1 - /r - . It remains to determine d(0) and r. We first find an expression for r: r = ∑_i ≥ 0 r_i - 1^i = ∑_i ≥ 0∫_0^∞( t)^i/i! ^- t f_B(t) t^i = ∫_0^∞∑_i ≥ 0( t)^i/i! ^- t f_B(t) t = ∫_0^∞^-(1 - )t f_B(t) t = B(1 - ), where B· is the LST of the service time B. Substituting (<ref>) into (<ref>) yields d = d(0) B(1 - )1 - /B(1 - ) - , where d(0) follows from lim_→ 1d = 1. If we apply this limit to the right-hand side of (<ref>), then we get an indeterminate form. By taking → 1 in (<ref>) and applying l'Hôpital's rule to the fraction on the right-hand side, we obtain 1 = d(0) lim_→ 11 - /B(1 - ) -= d(0) lim_→ 1-1/-B(1 - ) - 1 = d(0)/1 - ρ, so that d(0) = 1 - ρ. We finally obtain d = (1 - ρ) (1 - )B(1 - )/B(1 - ) - , which connects the PGF of the departure distribution to the LST of the service time distribution. This formula is referred to as the Pollaczek-Khinchin formulaThe Pollaczek-Khinchin formula comes in several variants. Accounts of this formula are published by Pollaczek <cit.> in German and two years later by Khinchin in Russian (see <cit.> for a translation). Equation (<ref>) can be found in <cit.> or <cit.>, whereas the variant for the LST of the sojourn time distribution presented in (<ref>) can be found in the classical textbooks <cit.>, <cit.> or <cit.>.. By differentiating (<ref>) we can determine the moments of the number of jobs in the system at a departure instant. To find its distribution, however, we have to invert (<ref>), which under general conditions is not straightforward. If the LST B is a rational function—which means that it is a quotient of polynomials in —then the right-hand side of (<ref>) can be decomposed into partial fractions and the inverse transform can be easily determined. We now show this by example.[Erlang services] Assume that the service times B follow an Erlang distribution consisting of two exponential phases with rate μ in each phase. Label the two exponential phases as B_1 and B_2. The LST of B is given by B = ^- B = ^- (B_1 + B_2) = ^- B_1^- B_2 = ( μ/μ + )^2, and when evaluated in (1 - ) we can write B(1 - ) = ( 1/1 + ρ/2(1 - ))^2, where in this case ρ = 2/μ. Substituting this expression into (<ref>) gives d = (1 - ρ) ( 1/1 + ρ/2(1 - ))^2 1 - /(1/1 + ρ/2(1 - ))^2 - . Multiplying the numerator and denominator of the second fraction by the term (1 + (ρ/2)(1 - ))^2 and simplifying gives d = 4(1 - ρ)/4 - (4ρ + ρ^2)+ ρ^2 ^2. If we now pick a value for ρ, then we can easily decompose d into partial fractions. For example, let us choose ρ = 1/3 to obtain d = 8/3/4 - 13/9 + 1/9^2 = 24/36 - 13+ ^2 = 24/(4 - )(9 - )= 6/54/4 -- 8/159/9 -= 6/51/1 - /4 - 8/151/1 - /9= 6/5∑_i ≥ 01/4^i^i - 8/15∑_i ≥ 01/9^i^i = ∑_i ≥ 0( 6/51/4^i - 8/151/9^i) ^i. From this expression for d we conclude that d(i) = 6/51/4^i - 8/151/9^i,i ≥ 0. Notice that (<ref>) agrees with d(0) = 1 - ρ = 2/3, since 6/5 - 8/15 = 2/3. We have determined the PGF of the departure distribution. However, as usual we are interested in the equilibrium probability p(i) of having i jobs in the system. We know from the PASTA property for M/·/· systems that a(i) = p(i) for all i. We now argue that d(i) is also equal to a(i). Taking the number of jobs in the system as the state of the queueing system, the changes in state are of a nearest-neighbor type: if the system is in state i, then an arrival of a job leads to a transition to state i + 1 and a departure of a job leads to a transition to state i - 1. Now, if the system is in equilibrium, then the number of transitions per unit time from state i to i + 1 is equal to the number of transitions per unit time from state i + 1 to i. The former transitions correspond to jobs finding upon arrival i jobs already in the system, which occurs at rate a(i). The latter transitions correspond to departing jobs leaving behind i jobs in the system, which occurs at rate d(i) (under the stability condition ρ < 1, jobs depart at rate ). Since these two transition rates are equal, we establish that a(i) = d(i) and therefore d(i) = p(i). Notice that in the argument establishing a(i) = d(i) we did not use the distribution of the inter-arrival times or service times nor the number of servers; we only used that jobs depart one by one. So, the equality a(i) = d(i) even holds for systems such as G/G/c queues. In some case, we are able to write the Pollaczek-Khinchin formula (<ref>) as the ratio d = N()/D() with both N() and D() polynomials without any common roots. Let _1,_2,…,_k be the roots of D() = 0. Since the radius of convergence of d is at least 1, we know that all |_j| > 1. We can write D() as D() = ( - _1)( - _2) ⋯ ( - _k), which means that we can use a partial fraction decomposition to write d = ∑_j = 1^k n_j/_j - , where we still need to determine the coefficients n_j. If we restrictto || < |_j| for all _j, then we can write d = ∑_j = 1^k n_j/_j1/1 - /_j = ∑_j = 1^k n_j/_j∑_i ≥ 0( /_j)^i = ∑_i ≥ 0( ∑_j = 1^k n_j/_j^i + 1) ^i. Comparing this expression to (<ref>) we conclude that d(i) = ∑_j = 1^k n_j/_j^i + 1. The coefficients n_j follow from n_j= lim_→_j (_j - ) d = lim_→_j (_j - ) N()/D()= lim_→_j - N()/( - _1) ⋯ ( - _j - 1)( - _j + 1) ⋯ ( - _k) = - N(_j)/D'(_j), where D'(·) is the derivative with respect toof D(·).§.§ Sojourn time distribution We now ask how much time a job spends in the system and we show that there is a nice relationship between the transforms of the time spent in the system and the departure distribution.Consider a job arriving to the system in equilibrium. Denote the sojourn time of this job by the random variable S with cumulative distribution function F_S(·) and probability density function f_S(·). If we assume that jobs are served in first-come first-served order, then we know that a departing job leaves behind exactly those jobs that arrived during its sojourn time. By conditioning on the length of the sojourn time, we can construct the departure distribution: d(i) = ∫_0^∞( t)^i/i! ^- t f_S(t) t. Multiply both sides of (<ref>) by ^i and sum over all i to retrieve the PGF of the departure distribution on the left-hand side and the LST S· of the sojourn time on the right-hand side (similar to the derivation in (<ref>)): d = S(1 - ). Substituting this relation into (<ref>) and introducing = (1 - ), we finally arrive at S = (1 - ρ) B/B +- , which is, like (<ref>), a form of the Pollaczek-Khinchin formula.[Erlang services] Consider again the model described in <ref>, where the service times B follow an Erlang distribution consisting of two exponential phases with rate μ in each phase. We determine S and invert it to obtain F_S(·). From (<ref>) we find that S = (1 - ρ) ( μ/μ + )^2 / (μ/μ + )^2 +- Multiplying the numerator and denominator of the second fraction by the term (μ + )^2 and simplifying gives S = (1 - ρ)/1 - ρ + 1/μ(2 - ρ/2)+ 1/μ^2^2. Choose μ = 6 and ρ = 1/3 so that S = 2/3/2/3 + 1/6(2 - 1/6)+ 1/36^2 = 2/3/2/3 + 11/36 + 1/36^2= 24/24 + 11+ ^2 = 24/(8 + )(3 + ) = 8/8 + 3/3 + . From the LST of S we can deduce that S is the sum of two exponential random variables with rates 3 and 8, denoted by X_1 and X_2, respectively. Obtaining the cumulative distribution function F_S(·) requires some more work: F_S(t) = S ≤ t = X_1 + X_2 ≤ t = ∫_0^t X_1 ≤ t - x f_X_2(x) x. Solving the integral finally givesF_S(t) = 8/5(1 - ^-3t) - 3/5(1 - ^-8t),t ≥ 0. where we recognize the cumulative distribution functions of X_1 and X_2 multiplied by some weights.§.§ Distributional Little's law The relation (<ref>) between the PGF of the number of jobs left behind upon departure and the LST of the sojourn time is a special case of distributional Little's lawThe distributional variant of Little's law is due to <cit.>. For the distributional law in the context of Poisson arrivals, we refer the reader to <cit.>. A discussion of this law under milder conditions can be found in <cit.> and the references therein. Many of the results presented in <ref> on distributional Little's law are adapted from <cit.>.. This fundamental law holds under a number of conditions, namely* All arriving jobs enter the system one at a time, remain in the system until served and leave one at a time;* Jobs leave the system in the order of arrival;* Jobs that arrive later in time do not affect the time spent in the system of jobs that arrived earlier in time. Here, a system can be used to mean only the queue, only the server, or the complete queueing system. To formulate distributional Little's law, define N(t) as the number of arrivals up to time t, where the first inter-arrival time is distributed as a residual inter-arrival time and all other inter-arrival times are distributed according to the stationary inter-arrival time. The residual inter-arrival time is the time between any given time t and the next arrival epoch of the arrival process. We describe distributional Little's law in terms of the equilibrium number of jobs in the system L and the equilibrium sojourn time S. Let F_S(·) denote the cumulative distribution function of S. Under the conditions mentioned above and under the further assumption that L and S exist, LN(S), or, in terms of the s of L and N,, L = ∫_0^∞N,tF_S(t), where L∑_n ≥ 0L = n^n, N,t∑_n ≥ 0N(t) = n^n.This proof can be found in <cit.>. Define t to be a random observation epoch and let t_n be the arrival time of the n-th job still in the system at time t and S_n its sojourn time in the system. The order in which we number the jobs is important. Job 1 is the job that arrived most recently in time with respect to the random observation time t and is therefore at the end of the queue. The job with the highest index is the one currently in service. So, the job with index n departs the system at time t_n + S_n. The t_n and S_n are ordered in reverse time direction. Next, define the inter-arrival times as A_1^*t - t_1 and A_nt_n - 1 - t_n, n ≥ 1. We note that A_1^* is a residual inter-arrival time. <ref> displays the notation and indexing used.If, at the random observation time t, the observer sees at least n jobs in the system, then the n-th most recently arrived job is still in the system at the observation time t. In particular, this means that the departure time t_n + S_n of the n-th job is larger than t. So, L ≥ n if and only if S_n > t - t_n. This indicates that L ≥ n = S_n > t - t_n = S_n > A_1^* + ∑_m = 2^n A_m, where we used a telescoping sum to derive the last equality. In equilibrium S_nS, so that conditioning on the length of the sojourn time leads to L ≥ n = ∫_0^∞t > A_1^* + ∑_m = 2^n A_mF_S(t). Finally, the probability inside the integral is exactly the probability that at least n arrivals occur in [0,t] where the first inter-arrival time is distributed according to the residual inter-arrival time and the other inter-arrival times are distributed according to the stationary inter-arrival time. Therefore, L ≥ n = ∫_0^∞N(t) ≥ nF_S(t) = N(S) ≥ n, proving the first statement of the theorem. The probability that there are exactly n jobs in equilibrium easily follows from (<ref>) as L = n = L ≥ n - L ≥ n + 1= ∫_0^∞N(t) ≥ nF_S(t) - ∫_0^∞N(t) ≥ n + 1F_S(t) = ∫_0^∞N(t) = nF_S(t). Multiplying both sides of (<ref>) by ^n, summing over all n ≥ 0 and applying Tonelli's theorem to interchange the summation and integral on the right-hand side produces the second statement of the theorem. For a Poisson arrival process, both the residual and stationary inter-arrival times are exponentially distributed with parameter . The PGF N,t then reads N,t = ∑_n ≥ 0^n ( t)^n/n!^- t = ^- (1 - ) t. Substituting this simplification into (<ref>) yields L = ∫_0^∞^- (1 - ) tF_S(t) = S(1 - ), which we have seen before in (<ref>).Note that <ref> does not hold in general for the number of jobs in an M/G/c system with c > 1 servers and a FCFS service discipline, since jobs may overtake other jobs and therefore violate the second condition. On the other hand, it does hold for the number of jobs in an M/D/c system with a FCFS service discipline, since being taking into service guarantees a certain departure time.§ SINGLE-SERVER QUEUE WITH GENERAL INTER-ARRIVAL TIMES The dual of the M/G/1 system discussed in <ref> is the G/M/1 system, which is a single-server queueing system with generally distributed inter-arrival times and exponential service times with rate μ. We assume that the inter-arrival times have a cumulative distribution function F_A(·), a probability density function f_A(·) and have mean 1/. For stability we require that ρ/μ < 1.The state of the G/M/1 system can be described by a pair (i,t) with i the number of jobs in the system and t the elapsed time since the last arrival. As we have argued for the M/G/1 systen, this state description leads to complications and the analysis simplifies considerably if we focus on special points in time. In this case, we look at the system at arrival instants so that in the state description t is always 0 and we only keep track of the number of jobs in the system at an arrival instant. We denote by a(i) the equilibrium probability that an arriving job encounters i jobs in the system (excluding itself).Unfortunately, since the arrivals do not follow a Poisson process, we cannot use PASTA to relate a(i) to the equilibrium distribution p(i) of the number of jobs in the system at arbitrary times. Nonetheless, we are still able to derive the distribution of the sojourn time using a(i). §.§ Arrival distribution We now derive the equilibrium probability a(i) of encountering i jobs in the system just before the arrival of a job. From one arrival instant to the next the number of jobs in the system increases by one, but decreases by the number of jobs that have arrived during its inter-arrival time. The number of jobs cannot decrease by more than the one plus the number of jobs present at the previous arrival instant. So, from state i we can transition to any of the states 0,1,…,i + 1. Denote by r_i the probability that a change of size i occurs, under the assumption that this change does not bring us to state 0 (state 0 requires special treatment). We reuse the notation r_i from the M/G/1 system because this probability has the same interpretation. By conditioning on the length of the inter-arrival time, we find that r_i = ∫_0^∞(μ t)^1 - i/(1 - i)!^-μ t f_A(t) t,i ≤ 1. The transition probability from state i to 0 is denoted by q_-i. Since the transition probabilities for each state sum to 1, it is easy to see that we must have q_-i = 1 - ∑_j = -1^i - 1 r_-i.By specifying the states and the transition probabilities, we have constructed the Markov chain associated with the G/M/1 system embedded at arrival instants. The transition probability diagram of this Markov chain is presented in <ref>The balance equations of this Markov chain are a(0) = a(0) q_0 + a(1) q_-1 + a(2) q_-2 + ⋯ = ∑_j ≥ 0 a(j) q_-j, and for i ≥ 1, a(i) = a(i - 1) r_1 + a(i) r_0 + a(i + 1) r_-1 + ⋯ = ∑_j ≥ 0 a(i - 1 + j) r_1 - j. It appears that the generating function approach does not work here. Instead, we guess that the solution to these balance equations is of the form a(i) = ^i,i ≥ 0. Substitution of (<ref>) into (<ref>) and dividing by ^i - 1 yields = ∑_j ≥ 0^j r_1 - j. By also substituting (<ref>) for r_1 - j we obtain = ∑_j ≥ 0^j ∫_0^∞(μ t)^j/j!^-μ t f_A(t) t = ∫_0^∞∑_j ≥ 0(μ t)^j/j!^-μ t f_A(t) t = ∫_0^∞^-μ(1 - ) t f_A(t) t. The last integral can be recognized as the LST of the inter-arrival time and we obtain the equation = Aμ(1 - ). Since A0 = 1, it is easy to see that = 1 is a root of (<ref>). However, this root is of no interest, since it does not produce a solution that can be normalized to obtain the equilibrium distribution. We show that you can obtain another root ∈ (0,1), that does lead to a solution that can be normalized. Define f() Aμ(1 - ). We derive some properties of f() to show that it must intersect with the function g() = for ∈ (0,1). First, it is easy to see that f(0) = Aμ = ∫_0^∞^-μ t f_A(t) t > 0 and f(1) = A0 = 1, as we have already established. The derivative f'() of f() is given by f'()= /∫_0^∞^-μ(1 - ) t f_A(t) t = ∫_0^∞( ∂/∂^-μ(1 - ) t) f_A(t) t = μ∫_0^∞^-μ(1 - ) t t f_A(t) t, where the interchange of the derivative and the integral is allowed in this case by Leibniz's integral rule (see <cit.>) if we assume that ρ < 1. We will not discuss this interchange here. Substituting = 1 in (<ref>) gives f'(1) = 1/ρ > 1 if ρ < 1. Pick _1 and _2 such that 0 ≤_1 < _2 ≤ 1, so that ^-μ(1 - _1) t < ^-μ(1 - _2) t,t > 0. By using this inequality, we see that f'() is increasing infor ∈ [0,1], we then say that f() is strictly convex for ∈ [0,1]. The properties of f() and g() are shown in <ref>. Combining these properties we conclude that (<ref>) has a single root ∈ (0,1), which satisfies (<ref>) for a(i) = ^i. Notice that the remaining balance equation (<ref>) is also satisfied, since the balance equations are dependent and one equation can therefore be omitted. We finally normalize the proposed solution to arrive at a(i) = (1 - ) ^i,i ≥ 0. Hence, the equilibrium number of jobs in the system just before arrival instants follows a geometric distribution with parameter , whereis the unique root of (<ref>) in the interval (0,1).[Erlang arrivals] Suppose that the inter-arrival times A follow an Erlang distribution consisting of two phases, where each exponential phase has rate . So, A = 2/ and ρ = /(2μ), where we assume that ρ < 1. The LST of A is given by A = ( / + )^2, and (<ref>) becomes = ( / + μ(1 - ))^2, which can be rewritten as ( + μ(1 - ))^2 - ^2 = 0. Since we know that = 1 is a solution of this equation, we can write ( - 1)( ^2 μ^2 -(μ^2 + 2μ) + ^2 ) = 0. If we choose = 3 and μ = 4, then we arrive at the solutions = 1/4, = 1 and = 9/4, so that a(i) = 3/4( 1/4)^i,i ≥ 0, for this specific G/M/1 system.§.§ Sojourn time distribution Since the arrival distribution is geometric, it is easy to determine the distribution of the sojourn time of a job. With probability a(i) an arriving job finds i jobs in the system. Because the service times are exponentially distributed, we know that the sojourn time of the arriving jobs is the sum of i + 1 exponential phases, each with rate μ. By conditioning on the number of jobs seen on arrival, we therefore find that S = ∑_i ≥ 0 a(i) ( μ/μ + )^i + 1 = μ(1 - )/μ + ∑_i ≥ 0( μ/μ + )^i = μ(1 - )/μ + 1/1 - μ/μ += μ(1 - )/μ(1 - ) + . So the sojourn time is exponentially distributed with rate μ(1 - ): F_S(t) = S ≤ t = 1 - ^-μ(1 - )t,t ≥ 0.§ A REFLECTED RANDOM WALK In this section we introduce the reflected random walk, which can be seen as an extension of the embedded Markov chains associated with the M/G/1 and G/M/1 system. This reflected random walk can be modeled by a Markov chain with state space the non-negative integers _0. The term reflected refers to the fact that the Markov chain is reflected in state 0 back to the positive values. Let X_n be the position of this random walk after n steps with X_00 and satisfying the recursion X_n + 1 = max(0,X_n + A_n),n ≥ 0, with { A_n }_n ≥ 0 a sequence of i.i.d. discrete random variables that share the same distribution as some common random variable A. If we allow A to take values in { -1,0,1,2,…} then the Markov chain described by the recursion (<ref>) has the same transition structure as the embedded Markov chain associated with the M/G/1 system (see <ref>) If we allow A to take values in {…,-2,-1,0,1 } then it has the same transition structure as the embedded Markov chain associated with the G/M/1 system (see <ref>). To demonstrate some important techniques, we instead focus on the case where A ∈{ -s,-s + 1,…,-1,0,1,2…} with s a positive integer and A = -s > 0. Notice that we can also write A = B - s so that B has as support the non-negative integers _0. The PGF of A is therefore given by A = ^A = ^B - s = B/^s.Assuming A < 0, which is equivalent to ρB/s < 1, the Markov chain is positive recurrent and we can study the equilibrium distribution. Denote by X the equilibrium version of X_n. In equilibrium, the recursion (<ref>) becomes X max(0,X + A) = max(0,X + B - s). From this relation we deduce that X = 0 = max(0,X + B - s) = 0 = ∑_i = 0^s X + B = i and for k ≥ 1, X = k = max(0,X + B - s) = k = X + B = k + s. Multiplying (<ref>) by ^k and summing over all k produces an expression for the PGF of X: X = ∑_i = 0^s X + B = i + 1/^s∑_k ≥ 1X + B = k + s^k + s= ∑_i = 0^s X + B = i + 1/^s[ ∑_i ≥ 0X + B = i^i - ∑_i = 0^s X + B = i^i ]. Recognizing the PGF of X + B, we can rewrite (<ref>) as X = ∑_i = 0^s - 1X + B = i(^s - ^i)/^s - B. This expression still involves the s unknowns X + B = i, 0 ≤ i ≤ s - 1. Factorize the polynomial inof degree s in the numerator of (<ref>) as ∑_i = 0^s - 1X + B = i(^s - ^i) = γ∏_k = 1^s ( - _k), where _k are the s roots of the polynomial and γ is a constant. The values of the roots are still unknown, but we return to this issue later in <ref>. However, it is immediate that one of the roots, say _s, takes the value 1, so that we obtain ∑_i = 0^s - 1X + B = i(^s - ^i) = γ ( - 1) ∏_k = 1^s - 1 ( - _k). What remains is to determine the constant γ. Taking derivatives with respect toand substituting = 1 on both sides of (<ref>) yields ∑_i = 0^s - 1X + B = i(s - i) = γ∏_k = 1^s - 1 (1 - _k). Now the function ^s - B (sometimes called the kernel) comes into play. Since we know that X1 = 1, we can apply l'Hôpital's rule to (<ref>) to find ∑_i = 0^s - 1X + B = i(s - i) = s - B1, which shows that γ = s - B1/∏_k = 1^s - 1 (1 - _k). Returning to (<ref>) we finally obtain X = (s - B1)( - 1)/^s - B∏_k = 1^s - 1 - _k/1 - _k. §.§ Finding the roots _k The roots _k in (<ref>) are still unknown. We do not directly study _k, but instead focus on the properties of the PGF X. In particular, we show that ^s - B has s roots in the closed unit disk and invoke the general properties of the PGF to conclude that these roots must coincide with the _k in the numerator: otherwise X would tend to infinity at those points, invalidating the analyticity of the function.Recall from <ref> that B is an analytic function for all ∈ satisfying || < 1 and is moreover continuous up to the unit circle. We introduce Rouché's theorem to show that ^s - B = 0 has s roots in the closed unit disk. Consider a bounded region L with continuous boundary ∂L and two complex-valued functions f(·) and g(·) that are analytic on L. If |f()| > |g()|, ∈∂L, then f(·) and f(·) + g(·) have the same number of zeros in the interior of L. When the radius of convergence of B exceeds 1, we can prove the following result concerning the number of zeros on and within the unit circle of ^s - B by using Rouché's theorem. Let B be a PGF that is analytic in || ≤ 1 + ν, ν > 0. Assume that the condition B1< s for positive recurrence is satisfied. Then the function ^s - B has exactly s zeros in || ≤ 1.Define the functions f() ^s and g()- B. Notice that both functions are analytic for || ≤ 1 + ν. It is clear that f() has s roots within the closed unit circle. We aim to show that |f()| > |g()| along the circle || = 1 + ϵ for 0 < ϵ < ν so that by Rouché's theorem f(·) + g(·) has s zeros inside the circle || = 1 + ϵ. Then, finally letting ϵ↓ 0 proves the statement.Observe that |f()| = f(||) and |g()| = |B| ≤B|| by the triangle inequality. So, instead we prove f(||) > B|| for || = 1 + ϵ. The Taylor series of f() and B|| at = 1 evaluated in the point = 1 + ϵ are f(1 + ϵ)= 1 + ϵ s + (ϵ),B1 + ϵ = 1 + ϵB1 + (ϵ). From the assumption B1< s and these Taylor expansions we conclude for sufficiently small ϵ satisfying 0 < ϵ < ν that f(1 + ϵ) > B1 + ϵ. Letting ϵ tend to zero yields the proof. Note that the application of <ref> is limited to the class of functions B with a radius of convergence larger than 1, so random variables B of which all moments (derivatives of B at = 1) exist.X is an analytic function for at least all || < 1. However, from <ref> we see that term ^s - B in the denominator of (<ref>) approaches zero for s values inside the closed unit disk. An analytic function in the region || < 1 does not have singularities in that region, so at the s values at which ^s - B = 0, the numerator of (<ref>) must also approach zero. It is clear that one of the s roots is = 1 and the other s - 1 roots must equal the _k present in the numerator of (<ref>).When B is assumed to not equal zero for all || ≤ 1, we know that the s roots of ^s = B in || ≤ 1 satisfy = B^1/s, where ^s = 1 are the roots of unity. For each unit root , (<ref>) can be shown to have a single root inside the closed unit disk || ≤ 1. One could try to solve (<ref>) by successive substitutions as _k^(n + 1) = _k B_k^(n)^1/s,k = 1,2,…,s, with starting values _k^(0) = 0 and _k = ^2π k/s. Under the additional condition that for || ≤ 1, the derivative | /B^1/s | < 1, it can be shown that (<ref>) indeed converges to the desired roots _k for n →∞.[Poisson distribution]Let us assume that B follows a Poisson distribution with rate < s so that B = ^-(1 - ). It is readily seen that B does not equal zero anywhere and | /B^1/s | < 1 for || ≤ 1, so that the successive substitutions (<ref>) converges to the correct root _k. In <ref> we show for variousand s = 10 the iterates _k^(100), k = 1,2,…,s and the curve on which they lie.When B has a bounded support, i.e., B ≤ s + m with m ≥ 1, we know that B is a polynomial of degree s + m. From <ref> it immediately follows that ^s = B has m roots outside the closed unit disk, to be denoted by _s + 1,_s + 2,…,_s + m. Write ^s - B = ξ∏_k = 1^s + m ( - _k) with ξ a constant. Substituting this expression in (<ref>) provides an alternative expression for X in terms of the roots outside the closed unit disk: X = (s - B1)/ξ∏_k = s + 1^s + m ( - _k)∏_k = 1^s - 11/1 - _k. The constant ξ is determined by setting = 1 and using X1 = 1, which finally yields X = ∏_k = s + 1^s + m1 - _k/ - _k. This expression is amenable for explicit inversion. In particular, using partial fraction expansion gives X = ∏_k = s + 1^s + m1 - _k/ - _k = ∑_l = s + 1^s + mx_l/ - _l, where x_l = lim_→_l ( - _l) ∏_k = s + 1^s + m1 - _k/ - _k = ∏_k = s + 1^s + m (1 - _k)/∏_k = s + 1, k ≠ l^s + m (_l - _k). Dividing the numerator and denominator in (<ref>) by -_l, we get X = - ∑_l = s + 1^s + mx_l/_l1/1 - /_l = - ∑_l = s + 1^s + mx_l/_l∑_k ≥ 0( /_l)^k, which gives X = k = - ∑_l = s + 1^s + mx_l/_l^k + 1,k ≥ 0. For k large enough, the sum on the right-hand side of (<ref>) is dominated by the pole of X with the smallest modulus, to be denoted without loss of generality by _s + 1. Omitting all fractions in (<ref>) other than the one that corresponds to _s + 1 gives the following approximation for the tail probabilities: X = k≈ - x_s + 1( 1/_s + 1)^k + 1,k →∞.By expressing the PGF of X in terms of the roots outside the closed unit disk, we are able to obtain an explicit product-form solution for the equilibrium distribution.Armed with the values of _k inside the closed unit disk, we can return to (<ref>) to construct a linear system of equations for the s unknown boundary probabilities X + B = i. In particular, we can substitute _k for k = 1,2,…,s - 1 into (<ref>) to obtain the s - 1 equations ∑_i = 0^s - 1X + B = i(_k^s - _k^i) = 0,k = 1,2,…,s - 1, which, together with the normalization condition (<ref>), constitutes a system of s linear equations for the s unknowns X + B = i.§ NUMERICAL INVERSION OF TRANSFORMS In some cases it is difficult or even impossible to explicitly retrieve the probability mass function from a PGF or the probability density function from an LST. In this section we describe numerical inversion algorithms that approximate these probability mass and density functions to an arbitrary precision. §.§ Inverting univariate generating functions Recall that we denote the PGF by ∑_k ≥ 0 p(k) ^k, wherecan be complex-valued, p(k) ≥ 0 and ∑_k ≥ 0 p(k) = 1. To retrieve the probabilities p(k) from , we use the fact thatis an analytic function for at least all ∈ satisfying || < 1 (see <ref>), which allows us to apply the Cauchy contour integral. The Cauchy contour integral reads p(k) = 1/2π∮_C_(k)/^k + 1 withthe complex unit and C_(k) a circle of radius (k) ∈ (0,1) that depends on k. We make the change of variables = (k) ^πθ so that the contour integral (<ref>) can be written as p(k) = 1/2π(k)^k∫_0^2π(k)^θ ^- k θ θ. Use ^- = cos() - sin() and =+ to rewrite the integral as p(k)= 1/2π(k)^k∫_0^2π[ ((k)^θ +(k)^θ) ·( cos(k θ) - sin(k θ) ) ]θ. = 1/2π(k)^k[ ∫_0^2π( cos(k θ) (k)^θ + sin(k θ) (k)^θ)θ + ∫_0^2π( cos(k θ) (k)^θ - sin(k θ) (k)^θ)θ]. The last integral in (<ref>) equals zero because cos(·) is an even function, sin(·) is an odd function, = - and =, whereis the complex conjugate of .It remains to determine the other integral in (<ref>). We follow the approach outlined in <cit.>, which ultimately leads to an approximation p(k) and a bound on the error e(k), see <cit.>. We can use the trapezoidal rule to approximate the integral. If we use a step size of π/k, then we can write p(k) ≈p(k) = 1/2 k (k)^k∑_l = 1^2k (-1)^l (k)^πl/k. By using the inherent symmetry, we finally arrive at the following expression for the approximation, for k ≥ 1, p(k) = 1/2 k (k)^k( (k) + (-1)^k -(k)+ 2 ∑_l = 1^k - 1 (-1)^l (k)^πl/k), where (k) ∈ (0,1) is actually a tunable parameter that controls the error term e(k) = p(k) - p(k), since |e(k)| ≤(k)^2k/1 - (k)^2k≈ r(k)^2k. The approximate equality is valid if r(k)^2k is small. Observe that p(0) does not need to be approximated, since it easily follows from p(0) = 0. With (k) = 10^-d/(2k) we find that |e(k)| ≤ 10^-d and therefore the approximation p(k) in (<ref>) is accurate until at least the d-th decimal.For reference in later chapters, we present in full the algorithm to numerically invert PGFs.[Gamma distributed service times] Consider the M/G/1 system with arrival rateand service times B that are distributed according to a gamma distribution with shape parameter > 0 and rate parameter > 0. Specifically, the probability density function of B is given by f_B(t) = ^ t^ - 1/Γ()^- t,t ≥ 0, where Γ() is the complete gamma function. The mean is given by B = / and the LST is B = ( / + )^. The Pollaczek-Khinchin formula (<ref>) says that the PGFof the equilibrium number of jobs in the system can be calculated from = (1 - ρ) (1 - ) ( / + (1 - ))^/( / + (1 - ))^ - , where ρ = B. It is not immediate how we can explicitly invert this expression to obtain the equilibrium probabilities p(k), especially ifis not an integer. To demonstrate the numerical inversion algorithm, we take = 2√(2) and = √(2) and invert the PGF to derive the equilibrium distribution. Notice that the load is given by ρ = 2. We select d = 8 in <ref> and obtain for various values ofthe equilibrium distribution, see <ref>.§.§ Inverting bivariate generating functions A bivariate PGF is a PGF of the joint probability mass function of two random variables and therefore takes two arguments. We encounter bivariate PGFs in some of the more advanced chapters, where we would like to numerically invert them. So, we present a numerical inversion algorithm for PGFs of two variables. The bivariate PGF is defined as x,y∑_k ≥ 0∑_l ≥ 0 p(k,l) x^k y^l, where x and y can be complex-valued, p(k,l) ≥ 0 and ∑_k,l ≥ 0 p(k,l) = 1. The bivariate PGF satisfies 1,1 = 1 and converges for at least all |x|,|y| ≤ 1 and is therefore analytic for at least all x,y ∈ satisfying |x|,|y| < 1.One of the standard numerical inversion algorithms is described in <cit.>. Here we present a version of that algorithm with specific parameter choices so that it resembles the univariate case. The algorithm approximates p(k,l) by p(k,l) = p(k,l) - e(k,l). The approximation is given by p(k,l)= 1/4 j_1 j_2 _1(k)^k _2(l)^l·∑_m = - j_1^j_1 - 1∑_n = - j_2^j_2 - 1^- π (k m/j_1 + l n/j_2)_1(k)^πm/j_1, _2(l)^πn/j_2, where j_1,j_2 ∈, and 0 < _1(k),_2(l) < 1 are tunable parameters that control the error: |e(k,l)|≤_1(k)^2 j_1 + _2(l)^2 j_2 - _1(k)^2 j_1_2(l)^2 j_2/(1 - _1(k)^2 j_1)(1 - _2(l)^2 j_2)≈_1(k)^2 j_1 + _2(l)^2 j_2, where the approximate equality is a valid approximation if both _1(k)^2 j_1 and _2(l)^2 j_2 are small. When we are interested in p(k,l) for k,l ≥ 1, then we can set j_1 = k and j_2 = l to simplify the approximation and the bound on the error term. Moreover, if we then choose _1(k) = 10^-d/(2k) / 2 and _2(l) = 10^-d/(2l) / 2, then the resulting approximation is accurate until at least the d-th decimal.<ref> summarizes the numerical scheme for inverting bivariate PGFs. §.§ Inverting univariate Laplace-Stieltjes transforms Most of the continuous random variables that we consider in this book are non-negative and have a continuous probability density function. With these characteristics the LST is given by ∫_0^∞^- t f(t) t,> 0, where f(·) is a probability density function that we often wish to retrieve from ·. An integral formula for the inverse Laplace transform called the Bromwich integral provides an expression for f(t) in terms of a contour integral: f(t) = 1/2π∮_C_r^ t , where C_r is the vertical line in the complex plane with constant real part equal to r. The value of r must be chosen such that all singularities of · are to the left of the vertical line. Since we are dealing with LSTs, we can safely pick any positive value for r. Notice that (<ref>) establishes that an LST uniquely defines the underlying probability distribution function.One of the standard inversion algorithm for LSTs is called the Euler method and is presented in <cit.>. The derivation of the approximation resembles the derivation of the approximation for the univariate PGF presented in <ref>, so we omit it here. The algorithm approximates f(t) by f(t). To construct the approximation f(t) we require the definition s_n(t) ^/2/2tL( /2t) + ^/2/t∑_k = 1^n (-1)^k L( /2t + π k/t), where we still need to choose . In <cit.> it is explained that s_n(t) is an approximation of a more accurate infinite series expression for f(t) by truncating the infinite series to n terms. By increasing n in (<ref>), the accuracy of the approximation increases. Euler summation can be used to accelerate convergence of the approximation (to get more accurate results with fewer computations): f(t) = ∑_k = 0^m mk 2^-m s_n + k(t). Since ∑_k = 0^m mk 2^-m = 1 and the summands are positive, we see that f(t) is the weighted average of the terms s_n(t),s_n + 1(t),…,s_n + m(t). More specifically, it is the binomial average of those terms, since the weights are in terms of binomial coefficients.It still remains to choose , m and n. Typically, m = 11 and n = 15 produce accurate results. If more accurate results are required, the value of n can be increased, but m can usually remain fixed. There are various types of errors that decrease the quality of the approximation. One of those errors is the discretization error, which occurs when we replace an integral by a series, as was done here. The value ofdirectly influences the magnitude of this discretization error e_(t), since |e_(t)| ≤^-/1 - ^-≈^-, where the approximate equality holds if ^- is small. If we choosetoo large, then we can run into computational difficulties, such as loss of significant digit, or roundoff errors. There is no exact error bound on the approximation (<ref>), but in most cases, we can select = d log10 to get d - 1 correct decimals. We often select d = 8 and use = 8 log10≈ 18.4.For reference in the following chapters, we present in full the algorithm to numerically invert univariate LSTs.The inversion algorithm also works for distributions that have discontinuities, but the results might be distorted due to some oscillations around the points of discontinuity. By increasing the accuracy of the method by, e.g., increasing m and n in <ref>, one can damp these oscillations. We treat an example to show how this works in practice.[Uniform services] Consider the M/G/1 system with arrival rate = 0.35 and service times B that are distributed according to a uniform distribution on the interval [1,3] and mean 2. Specifically, the probability density function of B is given by f_B(t) = 1/2,t ∈ [1,3] and the LST is B = ^- - ^-3/2.The LST of the sojourn time of an M/G/1 queue is given in (<ref>) and is in this case S = (1 - ρ) ^- - ^-3/2/^- - ^-3/2 +- , where ρ = B = 0.7. Explicitly inverting this LST to obtain f_S(·) proves to be difficult due to the exponential functions. We therefore turn to the numerical inversion techniques presented in <ref>. We will see that the uniform service time distribution causes numerical inaccuracies due to the discontinuities of f_B(t) at t = 1 and t = 3. For the algorithm settings, we will fix = 18.4 and show the influence of m and n.<ref> shows that at the points of discontinuity, the approximation obtained from <ref> oscillates. This oscillations is damped when the values of m and n increase. It is important that the inverted function is checked for irregularities such as the one we encounter now. In <ref> we display the time required to compute f_S(t) for each t from 0.9 to 10 in steps of size 0.001 (so 9101 times) for each combination of algorithm settings. § TAKEAWAYS Transforms are powerful tools that can simplify and facilitate calculating with distributions. Transforms enjoy the property that they uniquely characterize probability distributions. Once the PGF or LST of a random variable is known, all moments and the probability distribution often readily follows. Another advantage of transforms, of particular use in this book, is that an infinite system of linear balance equations can be converted into a single functional equation for the PGF; see <ref>.Transforms need to be inverted. This can be done by differentiation orintegration. Both methods can be useful and will be applied in later chapters. Sometimes a PGF can be written in the form of an infinite sum involving powers of . In those cases, the coefficients of ^k together constitute the probability mass function. In this chapter we have embedded the M/G/1 queue at departure instants and the G/M/1 queue at arrival instants. Both approaches lead to a state space _0 with a particular transition structure for each queue. The linear systems of balance equations associated with these embedded Markov chains are amenable to transform analysis and lead to some canonical relations such as the Pollaczek-Khinchin formula and distributional Little's law. The embedding technique is not restricted to the M/G/1 or G/M/1 queue and can be used for many stochastic models.The embedded Markov chains associated with the M/G/1 and G/M/1 system are skip-free to the left and right, respectively. In <ref> we introduce processes that also possess the skip-free property, but each state is replaced by a finite set of states. For the skip-free to the right variant of these processes, the transform analysis that was used in this chapter can be extended to determine the equilibrium distribution. For the other variant we turn to matrix-analytic methods.In this chapter we have encountered various product-form solutions. For the Erlang service time distribution, (<ref>) shows that the departure distribution of the M/G/1 queue has a product-form solution. If we are able to write the PGF of the departure distribution in an M/G/1 as a ratio of polynomial without any common roots, then the departure distribution is given by a sum of product-form solutions, see (<ref>). For any inter-arrival time distribution, the arrival distribution is given by the product-form solution (<ref>). In case of bounded jumps in both directions in the random walk setting, we find the product-form solution (<ref>) for the equilibrium distribution. PART:Basic processes CHAPTER: BIRTH–AND–DEATH PROCESSES In this chapter we introduce a structured class of Markov processes called the birth–and–death processes. This structure allows for local balance equations to be used in the derivation of the equilibrium distribution.§ GENERAL BIRTH–AND–DEATH PROCESSES We start by defining the birth–and–death process. A birth–and–deathprocess is a Markov process on the state space = {0,1,…,S} with S possibly infinite, where transitions are between adjacent states: from state i to state i + 1 (a birth) and to state i - 1 (a death). Unless stated otherwise, we focus on BD processes that have an infinite state space = _0 and all transition rates are strictly positive, leading to an irreducible Markov process. Birth rates are commonly denoted as _i and death rates as μ_i. This leads to the following transition rate matrix of the BD process: Q = [ -_0_0; μ_1 -(_1 + μ_1)_1; μ_2 -(_2 + μ_2)_2; μ_3 -(_3 + μ_3)_3; ⋱ ⋱ ⋱ ], where unspecified elements are zero. A BD process with rates _i = and μ_i = μ is called homogeneous and inhomogeneous otherwise. The transition rate diagram of the BD process is depicted in <ref>.The sojourn time in state i is the minimum of the time to transit to state i + 1 and the time to transit to state i - 1. Since both of these times are exponentially distributed, the time spent in state i until a transition occurs is exponentially distributed with parameter _i + μ_i. Given that a transition occurs, we have a birth with probability _i/(_i + μ_i) or a death with probability μ_i/(_i + μ_i).The above reasoning indicates that for simulation purposes one needs to repeat these two steps: sample a sojourn time and flip a biased coin to determine to which state the process transitions. This simple procedure is summarized in <ref>.Using <ref>, we simulate one sample path each for three different homogeneous BD processes. These sample paths are depicted in <ref>. Notice that for - μ < 0 the process seems to have a drift towards zero. On the other hand, if - μ > 0, X(t) seems to increase as time passes. For the case - μ = 0 no clear conjectures can be made. Intuitively these three statements make sense,is the rate at which the process transitions upwards and μ is the rate at which the process transitions downwards. So if > μ there is a net rate upwards and vice versa for < μ. We formalize this intuition and extend it to inhomogeneous BD processes in <ref>. We will see that this net rate decides if the Markov process is transient or recurrent.§ TIME-DEPENDENT BEHAVIOR Analyzing time-dependent behavior of BD processes is difficult. Explicit expressions for the transition functions p_i,j(t)X(t) = j | X(0) = i exist, but only for special cases and often involve special functions related to orthogonal polynomials (see Karlin and McGregor <cit.> and Karlin and Taylor <cit.>). Nonetheless, we review some of the techniques used.The transition functions satisfy both the Kolmogorov forward and backward equation, see <ref>. The Kolmogorov forward equation in case of a BD process reads in scalar form / t p_i,0(t)= -_0 p_i,0(t)_ + μ_1 p_i,1(t)_,/ t p_i,j(t)= _j - 1 p_i,j - 1(t) + μ_j + 1 p_i,j + 1(t)_ - (_j + μ_j) p_i,j(t)_,j ≥ 1, with the initial condition p_i,i(0) = 1.[Poisson process] The homogeneous Poisson process can be seen as a BD process with _i =, μ_i = 0 and X(0) = 0. This pure birth process will drift off towards infinity since all states are transient. The transition function p_0,j(t) is the probability that j births have occurred in the interval [0,t]. Obviously, the number of births in the interval [0,t] is distributed according to a Poisson distribution with parameter t. We will verify this statement through (<ref>)–(<ref>), which for p_0,j(t) read / t p_0,0(t)= - p_0,0(t),/ t p_0,j(t)=p_0,j - 1(t) -p_0,j(t),j ≥ 1. Together with p_0,0(0) = 1 this leads to p_0,0(t) = ^- t. Equation (<ref>) is separable using / t( ^ t p_0,j(t) ) = p_0,j(t) / t^ t+ ^ t/ t p_0,j(t) = ^ t p_0,j - 1(t). So, by direct integration we obtain ^ t p_0,j(t) = ∫_0^t ^ u p_0,j - 1(u) u. The transition functions can be solved recursively starting from p_0,0(t). Let us determine the first few terms. For j = 1, we derive p_0,1(t) = ^- t∫_0^t ^ u p_0,0(u) u = ^- t∫_0^t ^ u^- uu = ( t) ^- t. The expression for p_0,1(t) is used to determine the second term: p_0,2(t) = ^- t∫_0^t ^ u ( u)^- uu = ( t)^2/2^- t. The third term is p_0,3(t) = ^- t∫_0^t ^ u( u)^2/2^- uu = ( t)^3/3!^- t. A pattern starts to show itself. Induction on j is used to show that the explicit expression p_0,j(t) = ( t)^j/j!^- t,j ∈, t ≥ 0 is correct. This verifies that the number of births in the interval [0,t] is indeed t. [M/M/∞ queue] We now set _i = and μ_i = i μ. This BD process models for example a population that grows exclusively through immigration with rateand all individuals die independently of each other with rate μ <cit.>; or packets arriving according to a Poisson process with ratethat are routed to their next destination after an exponential amount of time with rate μ. In the queueing context we refer to a birth as an arrival of a job and a death as a departure of a job.Suppose X(0) = 0 and we are interested in the transition functions p_0,j(t). For the event X(t) = j to occur, we require at least j arrivals. If k ≥ j jobs arrive, we furthermore require k - j departures. The probability that k jobs arrive in the time interval [0,t] follows from the Poisson distribution and is ^- t( t)^k/k!. Conditioning on the fact that there are k arrivals in the time interval [0,t], we know that the arrival instant of each job is independent of the arrival instants of other jobs and is moreover uniformly distributed in the interval [0,t]. So, the probability q(t) that a job is still in the system at time t follows by conditioning on the arrival time: q(t) = ∫_0^t ^-μ u1/tu = 1 - ^-μ t/μ t. The probability that j jobs remain at time t conditioned on k arriving in the interval [0,t] follows a Bernoulli distribution and leads to an explicit expression for p_0,j(t): p_0,j(t)= 0X(t) = j= ∑_k ≥ j0X(t) = j |k arrivals in [0,t]k arrivals in [0,t]= ∑_k ≥ jkj (1 - q(t))^k - j q(t)^j ^- t( t)^k/k!= ^- t q(t)( t q(t))^j/j! = ^- /μ (1 - ^-μ t)( /μ (1 - ^-μ t) )^j/j!. The explicit expression for p_0,j(t) allows for a simple determination of the transient mean as 0X(t) = ∑_j ≥ 0 j p_0,j(t) = /μ( 1 - ^-μ t). In conclusion, X(t) conditional on X(0) = 0 is a Poisson distribution at each time t with parameter (/μ) ( 1 - ^-μ t).For a similar discussion of the M/G/∞ queue, where the service time distribution is allowed to be any distribution (general, hence the G), see <cit.>. We now consider the first time at which the BD process { X(t) }_t ≥ 0 enters a state j, starting from a state i. We recall the definition of a hitting time random variable in (<ref>) as i,jinf{ t > 0 : lim_s ↑ t X(s) ≠ X(t) = j | X(0) = i }, We will make use of the LST i,j^- i,j,> 0. Recall that a LST uniquely characterizes the distribution of a random variable.[Regenerative structure]An irreducible BD process has a regenerative structure. Assume that at a particular time the BD process is in state 0. The process stays in state 0 for an exponential amount of time with parameter _0. After this time it transitions to state 1. Under the condition that the BD process is recurrent, it returns to state 0 after some time with probability 1. The time spent in state 0 is called an idle period and the time it takes to go from state 1 to state 0 is called the busy period. So, an irreducible BD process with recurrent states alternates between idle and busy periods, see <ref>. The terminology idle and busy period comes from the interpretation of a homogeneous BD process as the M/M/1 single server queue. In state 0 the server is idle and in all other states the server is busy serving jobs.The length of a busy period is the hitting time random variable 1,0 with LST 1,0. Let us assume that the BD process is homogeneous with _i = and μ_i = μ. Note that 1,0 is the sojourn time in state 1 plus the time it takes to reach state 0 from the state the process transitions to. We derive 1,0 using this observation, a one-step analysis and the strong Markov property: 1,0 = μ/ + μ_+ μ/ + μ + _H_1·1_^-· 0 + / + μ_+ μ/ + μ + 2,0. Due to the BD structure of the Markov process, we have 2,0 = 2,1 + 1,0, where 2,1 and 1,0 are independent random variables. More importantly, for homogeneous BD processes, the time it takes to go from state 2 to state 1 is exactly the same as the time it takes to go from state 1 to state 0 and in general the time it takes to go from state n ≥ 1 to state n - 1. So, 2,0 = 1,0^2 and we know that 1,0 is a solution to the polynomial x^2 - ( + μ + ) x + μ = 0. This equation has the two roots x_±() =+ μ + ±√(( + μ + )^2 - 4 μ)/2. A LST of a non-negative random variable has absolute value less than one for allwith > 0. Since 0 < |x_-()| < 1 < |x_+()| for > 0, 1,0 =+ μ +- √(( + μ + )^2 - 4 μ)/2. The expectation of the length of the busy period is determined from its LST 1,0 = - /1,0|_ = 0 = 1/μ - , < μ,∞, = μ, and we agree to write 1,0 = ∞ if ∫_0^∞ f_1,0(t) t < 1, which indicates that starting in state 1, there is a non-zero probability that state 0 will never be hit. This is the case if > μ; we do not prove this statement. Here we already see the relation with positive recurrence (< μ), null recurrence (= μ) and transience (> μ), that was observed in <ref>. If the system initially is in state 0 and the target state is n, we can write the hitting time 0,n as a sum of independent random variables: 0,n = 0,1 + 1,2 + ⋯ + n - 1,n. The independence property is crucial in the analysis that will follow. Clearly, 0,1 is an exponential random variable with parameter _0. More importantly, 0,n turns out to be a sum of n exponential random variables. Albeit true, this result is rather counterintuitive. Consider for example 0,2 = 0,1 + 1,2. Here 0,1 is still an exponential random variable with parameter _0, while 1,2 is definitely not an exponential random variable, yet their sum is. The crux lies in the fact that 0,2 is the sum of two exponential random variables where both parameters are different from _0. The hitting time 0,n is distributed as the sum of n exponential random variables:Keilson provides two analytical proofs of <ref> in <cit.> and <cit.>, but does not characterize the parameters of the exponential random variables. Fill <cit.> succeeds in proving the same result using probabilistic arguments and moreover characterizes the parameters of the exponential random variables. 0,n = X_0^(n) + X_1^(n) + ⋯ + X_n - 1^(n), with X_i^(n)∼θ_i^(n) and θ_i^(n) the n positive eigenvalues of -Q^(n), where Q^(n) is the transition rate matrix of the BD process on the states { 0,1,…,n } with n an absorbing state.In terms of the Laplace transform, we require to prove 0,n = ∏_i = 0^n - 1θ_i^(n)/θ_i^(n) + N^(n)/D^(n). A one-step analysis and the strong Markov property gives n,n + 1 = _n/_n + μ_n ++ μ_n/_n + μ_n + n - 1,n + 1,n ≥ 1. Using n - 1,n + 1 = n - 1,nn,n + 1 the above equation results in the recursion n,n + 1 = _n/_n + μ_n +- μ_n n - 1,n,n ≥ 1. Next, multiply (<ref>) by 0,n and use 0,n = 0,n - 1n - 1,n to obtain 0,n + 1 = _n 0,n/_n + μ_n +- μ_n 0,n/0,n - 1,n ≥ 1. We proceed by induction. The claim (<ref>) is true for n = 1, since 0,1 = _0/(_0 + ). Assume the claim is true for n, then (<ref>) reads 0,n + 1 = _n N^(n) N^(n - 1)/ (_n + μ_n + ) N^(n - 1) D^(n)- μ_nN^(n) D^(n - 1). The denominator of 0,n + 1 will be a polynomial of degree n + 1. Moreover, <cit.> establishes that 0,n + 1 has n + 1 negative real simple poles. Thus, 0,n + 1 also has the form (<ref>), proving the claim.Proving that the θ_i^(n) are the n positive eigenvalues of -Q^(n) is outside the scope of this book, an interested reader is pointed to <cit.>. Hitting times and transition functions are inherently linked. Let { Y(t) }_t ≥ 0 be a modified process which is identical to the BD process { X(t) }_t ≥ 0, except that the target state j is absorbing. Since state j in the process { Y(t) }_t ≥ 0 is absorbing, we know that if Y(t) reaches state j, it stays there forever. In other words, if Y(t) reaches state j at time t^* < t, it will still be in state j at time t. This leads to a relation between the hitting time i,j defined in (<ref>) and the transition functions of the modified BD process { Y(t) }_t ≥ 0:The intuition for (<ref>) appears in <cit.>, but also follows from the standard probabilistic reasoning outlined above the equation. i,j < t = iY(t) = j.§ EQUILIBRIUM DISTRIBUTION The BD processes that we study are irreducible. The irreducibility property implies that the BD process can go from any state i to any state j. For an irreducible Markov process the unique equilibrium distribution exists if it is positive recurrent. For BD processes we derive a necessary and sufficient condition for positive recurrence and examine the equilibrium distribution.In <ref> the concepts of global and local balance are introduced. In the context of a BD process, the global balance equations are constructed by equating the rate into and out of state i, yielding _0 p(0)=μ_1 p(1), (_i + μ_i) p(i)= _i - 1 p(i - 1) + μ_i + 1 p(i + 1),i ≥ 1. The latter equation is a second-order linear recurrence equation. Whenever possible, a relation between p(i) and p(i - 1) is far more convenient to work with and often leads to simple ways to determine explicit expressions for the equilibrium distribution = [ p(i) ]_i ≥ 0. Local balance equations give this relation between p(i) and p(i - 1). They are derived by equating the flow into and out of the set of states A_i - 1 = {0,1,…,i - 1}. Since this set of states has a single state through which the process can enter and leave, the local balance equations result in the simple expression μ_i p(i) = _i - 1 p(i - 1),i ≥ 1. Note that the local balance equations can also be obtained from the global balance equations by substitutions. The local balance equations may be solved in a recursive fashion, yielding p(i) = p(0) ∏_j = 1^i _j - 1/μ_j,i ≥ 0, where the empty product ∏_j = 1^0 = 1. All equilibrium probabilities p(i) are expressed in terms of p(0). Finally, the normalization condition allows for the determination of p(0) from 1 = ∑_i ≥ 0 p(i)= p(0) ∑_i ≥ 0∏_j = 1^i _j - 1/μ_j. The following theorem now summarizes when an irreducible Markov process is positive recurrent. A necessary and sufficient condition for an irreducibleprocess to be positive recurrent is ∑_i ≥ 0∏_j = 1^i _j - 1/μ_j < ∞, and ensures that all p(i) > 0.By <ref>, if an irreducible Markov process has a solution , = 1 to the balance equations, then the Markov process is positive recurrent. Condition (<ref>) is sometimes referred to as the stability condition. Note that this condition is trivially satisfied when the state space is finite, which is not surprising since we know that a finite irreducible Markov process possesses a unique equilibrium distribution.Returning to the homogeneous BD processes, we see that the stability condition (<ref>) reduces to ∑_i ≥ 0( /μ)^i < ∞. So, if /μ < 1 the BD process is positive recurrent and an equilibrium distribution exists. The condition /μ < 1 makes the intuition for the sample paths in <ref> rigorous.The local balance equations (<ref>) can be derived by censoring the parts of the sample path of the Markov process when it is not inside the set A_i = {0,1,…,i} with i ≥ 1. The rate at which the process leaves state i is p(i) (_i + μ_i). The rate at which we enter state i is p(i - 1) _i - 1 plus the rate at which the process transitions to states outside A_i that return to state i. There is only one transition from a state in A_i, state i, to a state outside A_i, state i + 1. Taking into account the above observations, the balance equations of the censored process are p(i) (_i + μ_i) = p(i - 1) _i - 1 + p(i) _i . In this case, = G_i + 1,i, which is the probability that, starting from state i + 1, the Markov process reaches state i in finite time. Assuming the Markov process is irreducible and positive recurrent, we know that the process always returns to A_i. More importantly, due to the BD structure the process always returns to A_i via state i. Combining these two properties we derive G_i + 1,i = 1. The balance equations (<ref>) for the censored process reduce to p(i) μ_i = p(i - 1) _i - 1, which is a local balance equation. The index i in A_i was arbitrary, so (<ref>) holds for all i ≥ 1.[M/M/∞ queue] We return to <ref> concerning the M/M/∞ queue. Regardless of the value ofand μ, as long as they are finite, this BD process is positive recurrent: ∑_i ≥ 0∏_j = 1^i _j - 1/μ_j = ∑_i ≥ 0∏_j = 1^i ρ/j = ∑_i ≥ 0ρ^i/i! = ^ρ < ∞, with ρ/μ. Since the BD process is positive recurrent, the limiting distribution exists and is found by taking t →∞ in the transition functions of <ref>, resulting in p(j) = lim_t →∞ p_0,j(t) = lim_t →∞^- ρ (1 - ^-μ t)( ρ (1 - ^-μ t) )^j/j! = ^-ρρ^j/j!. We showed in <ref> that for each time t, X(t) has a Poisson distribution and also in equilibrium it follows a Poisson distribution.If the transition functions are not available, the equilibrium distribution can be derived using the balance equations. Let us do that now. Each BD process satisfies the local balance equations. In this case they read p(j) j μ =p(j - 1) ⇒ p(j) = ρ/j p(j - 1) = ⋯ = ρ^j/j! p(0). We have an expression for p(j) in terms of p(0). The probability of being in state 0 follows from the normalization condition as follows 1 = ∑_j ≥ 0 p(j) = p(0) ∑_j ≥ 0ρ^j/j!⇒ p(0) = ^-ρ. Combining (<ref>) and (<ref>) shows that the equilibrium distribution is also given by (<ref>). [M/M/s/s queue] We examine the M/M/∞ queue but set _i = 0 for i ≥ s. The interpretation of this queueing system is that of the M/M/∞ queue, but if s servers are occupied, no arriving jobs are allowed into the system. These jobs may be considered blocked, or lost, and correspondingly this system is referred to as the Erlang loss or Erlang-B system. An alternative interpretation is that of a system with s servers that allows a maximum of s jobs to be in the system simultaneously. The birth rates are _i = , 0 ≤ i < s and the death rates are μ_i = i μ, 1 ≤ i ≤ s. Since it is an irreducible BD process with a finite state space, all states are positive recurrent. The local balance equations are, with ρ/μ, p(j) j μ = p(j - 1) ,1 ≤ j ≤ s ⇒ p(j) = ρ^j/j! p(0),1 ≤ j ≤ s. So, the equilibrium probabilities of the M/M/s/s model have the same structure as the ones of the M/M/∞ model seen in <ref>, expect for the normalization. The normalization condition in this case is 1 = ∑_j = 0^s p(j) = p(0) ∑_j = 0^s ρ^j/j!⇒ p(0) = ( ∑_j = 0^s ρ^j/j!)^-1 and the equilibrium distribution has been determined. If s →∞ then p(0) converges to ^-ρ and the equilibrium distribution coincides with the one from the M/M/∞ model.A quantity of special significance is the probability that an arriving job is lost or blocked, which, by PASTA (see <ref>), is given by B(s,ρ) = p(s) = ρ^s/s!( ∑_j = 0^s ρ^j/j!)^-1. This is often termed the Erlang-B formula. It is easily verified that the probability of blocking satisfies the recurrence relationB(s + 1,ρ) = ρ B(s,ρ)/ρ B(s,ρ) + s + 1, which is useful for numerical computation. [M/M/1 queue] We consider a homogeneous BD process with _i = and μ_i = μ. This is also called an M/M/1 queue in queueing terminology. From (<ref>) we require < μ for the states to be positive recurrent. The equilibrium distribution is derived from (<ref>)–(<ref>) and found to be p(i) = (1 - ρ) ρ^i,i ≥ 0, with ρ/μ. [M/M/s queue] Consider a queueing system consisting of s servers and a common queue. Jobs arrive according to a Poisson process with rateand enter service if a server is idle. Serving a job takes μ time. If all servers are occupied, the job joins the end of the waiting line in the common queue. When a server finishes serving a job, he takes the first job from the waiting line and starts serving that job. If there is no waiting job, the server becomes idle. This model is often referred to as the Erlang-C model. Both <ref> and <ref> are special cases of this model.The total number of jobs in the system at time t, labeled X(t), evolves according to a BD process with _i = and μ_i =i μ, 0 ≤ i ≤ s - 1, s μ, i ≥ s. Applying <ref>, the BD process is positive recurrent iff, with ρ/μ, ∑_i ≥ 0∏_j = 1^i _j - 1/μ_j = ∑_i = 0^s - 1∏_j = 1^i _j - 1/μ_j + ∑_i ≥ s∏_j = 1^i _j - 1/μ_j = ∑_i = 0^s - 1ρ^i/i! + ρ^s/s!∑_i ≥ 0ρ^i/s^i < ∞. So, ρ/s < 1 ensures that an equilibrium distribution exists. From the local balance equations (<ref>) the equilibrium distribution is p(i) =p(0) ρ^i/i!, 0 ≤ i ≤ s - 1, p(0) ρ^i/s!s^i - s, i ≥ s, with p(0) = ( ∑_i = 0^s - 1ρ^i/i! + ∑_i ≥ sρ^i/s!s^i - s)^-1 = ( ∑_i = 0^s - 1ρ^i/i! + ρ^s/s!1/1 - ρ/s)^-1 representing the probability that the system is empty.A quantity of great importance is the probability that an arriving job must wait, which is, by the PASTA property, C(s,ρ) = ∑_i ≥ s p(i) = ( 1 + s!/ρ^s (1 - ρ/s) ∑_i = 0^s - 1ρ^i/i!)^-1. This is often referred to as the Erlang-C formula. It is easily verified that the probability of waiting satisfies the recurrence relationC(s + 1,ρ) = (1 - ρ/s) C(s,ρ)/s + 1 - ρ - ρ/s C(s,ρ), which is convenient for numerical calculations.The waiting time W of a job is the time between his arrival and the time he is taken into service, assuming an equilibrium state for the queueing system. We analyze the waiting time distribution of an arbitrary job. Note that, given that an arriving job must wait, the number of waiting jobs in front of him is geometrically distributed with parameter ρ/s. So, the number of service completions the arriving job must wait for is G + 1, where G is a geometrically distributed random variable with parameter ρ/s. Also note that the times between successive service completions are independent and exponentially distributed random variables with parameter s μ. Now observe that the sum of G + 1 independent and exponentially distributed random variables with parameter ν, where G itself is a geometrically distributed random variable with parameter p, is again exponentially distributed with parameter ν (1 - p). So given that an arriving job must wait, its waiting time is exponentially distributed with parameter s μ (1 - ρ / s) = s μ -. Therefore the unconditional waiting-time distribution is given by W > t = C(s,ρ) ^- (s μ - ) t, since the probability that an arbitrary job needs to wait is C(s,ρ). Denote by R_i,j, j > i the expected time spent in state j multiplied by _i + μ_i between two successive visits to state i. Conditioning on the state visited after the first jump of the Markov process gives, for i ≥ 1, R_i,i + 1 = (_i + μ_i)i ∫_0^i,iX(t) = i + 1t= (_i + μ_i) ( _i/_i + μ_i i + 1 ∫_0^i + 1,iX(t) = i + 1t + μ_i/_i + μ_i i - 1 ∫_0^i - 1,iX(t) = i + 1t) = _ii + 1 ∫_0^i + 1,iX(t) = i + 1t_i M_i + 1,i. M_i + 1,i is interpreted as the expected time spent in state i + 1 before the process reaches state i, given that the process starts in state i + 1. This quantity is determined from a one-step analysis, M_i + 1,i = 1/_i + 1 + μ_i + 1_H_i + 1 + _i + 1/_i + 1 + μ_i + 1 G_i + 1,i M_i + 1,i. A positive recurrent Markov process has G_i + 1,i = 1 and therefore M_i + 1,i = 1/μ_i + 1⇒ R_i,i + 1 = _i/μ_i + 1, which holds for all i. It seems that p(i) = p(i - 1) R_i - 1,i = ⋯ = p(0) ∏_j = 0^i - 1 R_j,j + 1 = p(0) ∏_j = 0^i - 1_j - 1/μ_j,i ≥ 1, which can be proven to hold.A formal proof of the relation p(i) = p(0) ∏_j = 0^i - 1 R_j,j + 1 can be found in Latouche and Ramaswami <cit.>. Section 4.6 in the same reference derives more properties of the R_j,j + 1. Neuts also discusses this quantity in <cit.>. Plugging p(i) = p(0) ∏_j = 0^i - 1 R_j,j + 1 into the global balance equations (<ref>)–(<ref>) gives _0 p(0)= μ_1 p(0) R_0,1, (_i + μ_i) p(0) ∏_j = 0^i - 1 R_j,j + 1 = _i - 1 p(0) ∏_j = 0^i - 2 R_j,j + 1 + μ_i + 1 p(0) ∏_j = 0^i R_j,j + 1,i ≥ 1. R_0,1 is determined from (<ref>). Dividing (<ref>) by p(0) ∏_j = 0^i - 2 R_j,j + 1 shows that R_i,i + 1 satisfies μ_i + 1 R_i - 1,i R_i,i + 1 - (_i + μ_i) R_i - 1,i + _i - 1 = 0,i ≥ 1, or R_i,i + 1 = _i + μ_i/μ_i + 1 - _i - 1/μ_i + 11/R_i - 1,i,i ≥ 1. If the BD process is homogeneous with _i = and μ_i = μ, then from the definition of R_i,j we deduce that all R_i,i + 1 are identical and we denote it by R. This implies that R is the solution to the quadratic equation μ R^2 - ( + μ) R += 0,i ≥ 1. If the BD process is positive recurrent, then R is the minimal non-negative solution to (<ref>). We return to these equations for R_i,i + 1 and R in <ref>.§ TAKEAWAYS Many probability text books cover birth–and–death (BD) processes, ranging from short descriptions of the balance equations and equilibrium distribution, to extensive chapters including many special cases and time-dependent analysis <cit.>. In fact, we also decided to include some time-dependent analysis starting from the Kolmogorov forward equations that describe the relations between transition functions. The time-dependent analysis of all Markov processes, also the ones treated in this book, can depart from Kolmogorov equations, but only exceptional cases like BD processes lead to equations that are amenable to analysis, let alone result in compact solutions like in some of the examples. For a more extensive treatment of the time-dependent analysis of BD process, including some deep connections with orthogonal polynomials, we refer to the classic work of Karlin and McGregor <cit.>.BD processes give rise to Markov process with states that can be arranged on a half-line. This special structures makes that instead of global balance, it suffices to work with local balance, which considerably reduces the complexity of the system of equations. While we see more examples in this book where local balance suffices (<ref>), for BD processes the local balance equations are particularly neat, and solved by the product-form solution in (<ref>). This solution can be obtained by a recursive argument that starts in state zero and follows the half-line from one state to the other. We will exploit such recursive structures more often, for instance in developing the theory of matrix-geometric methods presented in <ref>.We saw that the equilibrium distribution of a BD process can also be found using the global balance equations, for instance using generating functions. For BD processes this is a naive method that forgets to exploit the unique state space structure, but still gives the product-form solutions. In this book we see more examples that can be approached by either global or (more) local views. In these more involved examples of <ref>, the global view leads nowhere, while the local view (not necessarily local balance, but at least a flow argument between a reduced number of states) provides a handle for finding a product-form solution. CHAPTER: QUEUEING NETWORKS This chapter deals with structured classes of Markov processes that arise from considering queueing networks, so systems of queues in which jobs or customers following routes to traverse multiple stations. The structure of these Markov processes shows strong dependencies between customers and stations, but nevertheless product-form solutions arise for some classes of networks.§ REVERSIBILITY For the purpose of introducing reversibility, or time-reversibility, we assume that the time index t belongs to , so that a Markov process is referred to as { X(t) }_t ∈. In this context, a stationary process has X(0) = x = p(x), where = [p(x)]_x ∈ is the stationary distribution. Consider a stationary Markov process { X(t) }_t ∈. Then the process { X(-t) }_t ∈ is a stationary Markov process with the same equilibrium distribution = [ p(x) ]_x ∈ and transition rates, for x ≠ y, x,yp(y)/p(x)y,x,x,y ∈, and x∑_y ≠ xx,y = ∑_y ≠ xx,yx.{ X(-t) }_t ∈ is a stationary process since X(-t) = x = p(x). Define Y(t)X(-t). Now, for x ≠ y, Y(t + h) = y | Y(t) = x = Y(t + h) = y, Y(t) = x/Y(t) = x= X(-t - h) = y, X(-t) = x/X(-t) = x= p(y) p_y,x(h)/p(x). Dividing both sides by h, letting h ↓ 0 and recalling (<ref>) gives the result.A similar proof of <ref> appears in Chen and Yao <cit.>.If a Markov process satisfies, for x ≠ y, p(x) x,y = p(y) y,x,x,y ∈ then the process is reversible. This definition implicates that all Markov processes that have a solution to the local balance equations are reversible Markov processes. In particular, all BD processes are reversible.[M/M/s/s queue] Recall the Erlang-B model, which is a BD process with i,i + 1 = and i + 1,i = (i + 1) μ for 0 ≤ i ≤ s - 1. The equilibrium distribution was derived in <ref> and is p(i) = p(0) (/μ)^i/i!,0 ≤ i ≤ s with p(0) given in <ref>. Using <ref>, for 0 ≤ i ≤ s - 1, p(i) i,i + 1 = p(0) (/μ)^i/i! = p(0) (/μ)^i + 1/(i + 1)! (i + 1)μ = p(i + 1) i + 1,i, verifying that the Markov process associated with the M/M/s/s queue is reversible. The following theorem plays a key part in the analysis of stochastic networks that are reversible. A reversible Markov process with state spaceand equilibrium distribution = [p(x)]_x ∈ that is truncated to A⊂ is again a reversible Markov process with equilibrium distribution p(x) = p(x)/∑_y ∈A p(y),x ∈A.Note that p(x)x,y = p(y)y,x by reversibility of the original process, so detailed balance is satisfied.See also Kelly and Yudovina <cit.> for some examples of the uses of truncating a reversible Markov process. [M/M/s/s queue] Employing <ref>, the equilibrium distribution of the Markov process associated with the M/M/s/s queue is the same as both the equilibrium distribution of the reversible Markov processes of the M/M/s queue and of the M/M/∞ queue truncated to the set A = { 0,1,…,s }. From <ref>, we know that the equilibrium probabilities are p(i)^(M/M/s) =p(0)^(M/M/s)ρ^i/i!, 0 ≤ i ≤ s - 1, p(0)^(M/M/s)ρ^i/s!s^i - s, i ≥ s, and p(i)^(M/M/∞) = p(0)^(M/M/∞)ρ^i/i!, with ρ/μ. Plugging both equilibrium probabilities into the right-hand side of (<ref>) produces the equilibrium distribution of the Markov process associated with the M/M/s/s queue. The queueing systems that we consider in this book have Poisson arrival processes. For many of these systems, the departure process is also a Poisson process where the departure rate is equal to the arrival rate, which we show in the following theorem. In queueing networks, the departure process of one queue can be the arrival process of another queue. Knowing that this arrival process is again a Poisson process makes the analysis of the network a lot easier. Consider a queue where jobs arrive according to a Poisson process with rateand leave at rate μ_i when i jobs are in the system. In equilibrium, the inter-departure times of jobs are exponentially distributed with mean 1/ and are independent of the number of jobs in the system.Denote by X(t) the number of jobs in the system at time t. The system is in equilibrium, which is equivalent to X(0) being distributed according to the equilibrium distribution . Let T be the time at which the first departure occurs and recall that X(T) is the number of jobs left behind by the first departure. Define the conditional joint transform R_i(,) ^- T^X(T)| X(0) = i,i ≥ 0. For i ≥ 1, either the first event is an arrival with probability /( + μ_i) or a departure with probability μ/( + μ_i). So, by the strong Markov property, for i ≥ 1, R_i(,)=+ μ_i/ + μ_i + ( / + μ_i R_i + 1(,) + μ_i/ + μ_i^i - 1) =/ + μ_i +R_i + 1(,) + μ_i/ + μ_i + ^i - 1 and R_0(,) = / +R_1(,). This gives the functional equations ( + ) R_0(,)=R_1(,), ( + μ_i + ) R_i(,)=R_i + 1(,) + μ_i ^i - 1,i ≥ 1. Define the PGF X∑_i ≥ 0 p(i) ^i and consider ^- T^X(T) = ∑_i ≥ 0 p(i) R_i(,). Multiply the i-th equation of (<ref>) by p(i) and sum over all i ≥ 1 to obtain ( + ) ∑_i ≥ 1 p(i) R_i(,) + ∑_i ≥ 1μ_i p(i) R_i(,) = ∑_i ≥ 1 p(i) R_i + 1(,) + ∑_i ≥ 1μ_i p(i) ^i - 1. Adding and subtracting p(0) R_0(,) on the left-hand side, using the local balance equations p(i - 1) = μ_i p(i), i ≥ 1 and (<ref>), results in ^- T^X(T) = / + X. So, the inter-departure time is exponentially distributed with parameterand is moreover independent of the number of jobs that are left behind by the departing job. § LOSS NETWORKS A loss network is a stochastic network consisting of nodes with links between nodes and jobs travelling over routes in the network. Jobs for each route arrive according to a Poisson process. A route is described by a number of links and for each link the number of capacity unit that is required to serve the job. A job holds the capacities in each link of its route simultaneously for an exponential amount of time, leaving the system afterwards. The capacity on each link is finite, however. So, an arriving job does not enter the network if it finds that a link on its route does not have enough free capacity. Such a job is lost, and therefore the network is called a loss network. Besides the equilibrium distribution, a key quantity in these networks is the probability that a job is lost.[A loss network]Consider a network of six nodes and links with capacities as shown in <ref>(a). There are three different routes in this network, see <ref>(b)-(d). Jobs for route 1 use the links 1, 3 and 5, arrive according to a Poisson process with rate _1 and, if admitted, hold simultaneously one unit of capacity on all three links in its route for an exponential amount of time with parameter μ_1. So, an arriving route-1 job is lost if there is no capacity available on links 1, 3 or 5. Route-2 jobs share link 5 with route-1 jobs and route-3 jobs share link 3 with route-1 jobs, but route-2 and route-3 jobs do not share a link. Nonetheless, there is still a large influence of route-2 jobs on the performance of route-3 jobs and vice versa. For example, if the arrival rate of route-3 jobs is large, all the capacity units of link 3 will be occupied. This means that almost all route-1 jobs will be lost and as a result, almost all route-2 jobs are admitted. In the following subsection we treat an example in greater detail. §.§ Multi-class Erlang-B model Consider a pool of c identical servers offered traffic from M job classes and denote the set of classes as M{1,2,…,M}. Class-m jobs arrive according to a Poisson process with rate _m and require an exponentially distributed service time with parameter μ_m. Denote by ρ_m _m/μ_m the offered traffic from class-m jobs. A class-m job requires the simultaneous use of b_m servers for the duration of its service. Arriving jobs for which there are not sufficiently many servers available leave the system immediately.The state of the system at time t may be described by a vector X(t)(X_1(t),X_2(t),…,X_M(t)) with X_m(t) representing the number of class-m jobs in the system at time t. Define {x∈_0^M : ∑_m = 1^M b_m x_m ≤ c } as the set of all feasible states. The process { X(t) }_t ≥ 0 is an irreducible Markov process with state space . Since its state space is finite, the equilibrium probabilities, now denoted as p(x), exist.Let m be a vector of dimension M with a 1 at position m, where indexing starts at 1. The equilibrium distribution satisfies the global balance equations ( ∑_m = 1^M _m x + m∈ + ∑_m = 1^M x_m μ_m ) p(x) = ∑_m = 1^M _m x_m > 0 p(x - m)+ ∑_m = 1^M (x_m + 1) μ_m x + m∈ p(x + m), for all states x∈, together with the normalization condition ∑_x∈ p(x) = 1. Let us try to solve for p(x) using an educated guess. If there would be infinite number of servers, then jobs of all classes are allowed to enter the system, removing dependencies between classes and we would expect a product-form solution. So, let us see if a product-form solution works here as well. In particular, use the form p(x) = 1/G(c,M)∏_m = 1^M y_m^x_m/x_m!, where y_m still needs to be determined and G(c,M) is a normalization constant. Assume all indicator functions in (<ref>) evaluate to 1. This indicates that we are in the interior of the state space . Plugging (<ref>) into (<ref>) and multiplying both sides by G(c,M) gives ∑_m = 1^M _m ∏_n = 1^M y_n^x_n/x_n! + ∑_m = 1^M x_m μ_m ∏_n = 1^M y_n^x_n/x_n!= ∑_m = 1^M _m y_m^x_m - 1/(x_m - 1)!∏_n ∈M∖{ m }y_n^x_n/x_n! + ∑_m = 1^M (x_m + 1) μ_m y_m^x_m + 1/(x_m + 1)!∏_n ∈M∖{ m }y_n^x_n/x_n!. If we now choose y_m = ρ_m, then the first summation on the left-hand side is equal to the second summation on the right-hand side and the second summation on the left-hand side is equal to the first summation on the right-hand side. We conclude that p(x) = 1/G(c,M)∏_m = 1^M ρ_m^x_m/x_m!, x∈ satisfies (<ref>) if all indicator functions evaluate to 1, but can also be shown to satisfy (<ref>) if this assumption is dropped. The normalization constant follows from the normalization condition (<ref>) and is G(c,M) = ∑_x∈∏_m = 1^M ρ_m^x_m/x_m!.We now consider the system occupancy in terms of the number of busy servers. Denote by p(i) the probability that i servers are busy for i = 0,1,…,c. Define _i {x∈ : ∑_m = 1^M b_m x_m = i } as the set of all states with exactly i servers busy. The probabilities p(i) may then be formally expressed in terms of the probabilities p(x) as p(i) = ∑_x∈_i p(x). The probability that a class-m job is blocked can directly be obtained from the probabilities p(i) as B_m = ∑_i = c - b_m + 1^c p(i). Of course, the blocking probability B_m may also be directly expressed in terms of the probabilities p(x) as B_m = ∑_x∈ : x + m∉ p(x) = 1 - ∑_x + m∈ p(x). This last summation can be rewritten as ∑_x + m∈ p(x) = 1/G(c,M)∑_x + m∈∏_n = 1^M ρ_n^x_n/x_n! = G(c - b_m,M)/G(c,M). Summarizing, the blocking probabilities B_m can be obtained from the ratio of the normalization constants for two systems with a different number of servers. In fact, the equilibrium distribution given above holds for any service time distribution with mean 1/μ_m (without proof). This means that the stationary distribution only depends on the service time distribution through its mean, and not on any higher moments. This is called an insensitivity property that is also encountered in the ordinary Erlang-B model, but also the Erlang-C model. Despite the elegant form, the expression (<ref>) is typically impractical for computing the probabilities p(i) and B_m. The number of feasible states in the above model and therefore also the number of terms in the normalization constant, grows rapidly with c and M. This makes the numerical evaluation of the normalization constant directly through brute-force summation prohibitively demanding for even moderately large values of c and M.We now discuss an alternative procedure for calculating the probabilities p(i) and the blocking probabilities first described in Kaufman <cit.> and Roberts <cit.>. The probabilities p(i) satisfy the recurrence relation i p(i) = ∑_m = 1^M ρ_m b_m i ≥ b_m p_i - b_m,i = 0,1,…,c.Define R_m(i) ∑_x∈_i x_m p(x). First observe that, from the definition of p(i) and _i, i p(i)= ∑_x∈_i i p(x) = ∑_x∈_i∑_m = 1^M b_m x_m p(x) = ∑_m = 1^M b_m ∑_x∈_i x_m p(x) = ∑_m = 1^M b_m R_m(i). From (<ref>), we have x_n p(x)= x_n/G(c,M)∏_m = 1^M ρ_m^x_m/x_m!= ρ_n/G(c,M)ρ_n^x_n - 1/(x_n - 1)!∏_m ∈M∖{ n }ρ_m^x_m/x_m! = ρ_n p(x - n). Substituting (<ref>) into (<ref>) yields R_m(i) = ρ_m ∑_x∈_i p(x - m) = ρ_m ∑_x∈_i - b_m p(x) = ρ_m i ≥ b_m p_i - b_m. Plugging (<ref>) into (<ref>) proves the claim.§.§ Equilibrium distributions for loss networks The multi-class Erlang-B model described in <ref> may be interpreted as a single `link' or transmission resource with c `circuits' or `trunks' (represented by the servers) offered `calls' or `connections' (represented by the jobs) from M classes. The single-link model may be generalized to networks of multiple links, where the various classes correspond to jobs that may traverse different routes (subsets of links), require different numbers of circuits, or a combination of these two features. Specifically, consider a network consisting of L links indexed by the set L{ 1,2,…,L }, offered traffic from M distinct job classes. Denote by c_l the capacity of, or, number of circuits in, link l. Class-m jobs arrive as a Poisson process with rate λ_m, and have exponentially distributed holding times with parameter μ_m. Denote by ρ_m _m/μ_m the offered traffic from class-m jobs. Class-m jobs require the simultaneous use of b_m,l circuits on link l for the duration of their holding time. Arriving jobs for which there are not sufficiently many circuits available leave the system immediately. The set of links R_m {l ∈L : b_m,l > 0 } may be interpreted as the route of class-k jobs. The route sets R_m need to satisfy certain `logical' constraints in order for routes to be contiguous paths in some underlying physical network topology. However, the subsequent analysis applies for completely arbitrary values of b_m,l.It is easily verified that the analysis in <ref> for the single-link model, in particular the equilibrium distribution given in (<ref>), readily extends to the above network scenario, with state space now replaced by {x∈_0^M : ∑_m = 1^M b_m,l x_m ≤ c_ll ∈L}. The equilibrium distribution of a loss network is given by p(x) = 1/G∏_m = 1^M ρ_m^x_m/x_m!, with normalization constant G = ∑_x∈∏_m = 1^M ρ_m^x_m/x_m!.We now give a proof using the concept of reversibility. Consider the case with infinite capacity c_1 = c_2 = ⋯ = c_L = ∞ (abbreviated as ic). In this case all jobs are accepted to the system and jobs in different classes are independent of each other. By this independence we have a product-form solution originating from the M/M/∞ queue: p_(x) = ∏_m = 1^M ^-ρ_mρ_m^x_m/x_m!. Truncating the state space from _0^M toand using <ref> gives the result. Clearly, the evaluation of the normalization constant G will be even more computationally demanding than in the single-link model. In general, there is no efficient numerical equivalent of the Kaufman-Roberts recursion presented in <ref>. In the important special case where b_m,l∈{ 0, 1 } for all m ∈M and l ∈L, the blocking probabilities for the various classes may be approximated using the so-called Erlang fixed-point approximation.Denote the blocking probability on link l as B_l'. Then the probability of a class-m job being blocked is expressed in terms of these link blocking probabilities as 1 - B_m = ∏_l ∈L :b_m,l = 1 (1 - B_l'), since each link in the class-m route needs to have at least one unit of capacity available. Now, assume that the blocking probabilities B_l' of blocking on link l are independent from link to link (which in a real network they are not!). In that case, the traffic offered to link l would be Poisson with rate σ_l = ∑_m = 1^M b_m,lρ_m ∏_k ∈L∖{ l } :b_m,k = 1 (1 - B_k'). By the Erlang-B formula, see also <ref>, the link blocking probabilities satisfy B_l' = B(c_l,σ_l) = σ_l^c_l/c_l!( ∑_j = 0^c_lσ_l^j/j!)^-1,l ∈L. A unique solution to these equations exists and therefore we are able to obtain the blocking probabilities for each class of jobs.The Brouwer fixed-point theorem states that a continuous map from a compact, convex set to itself has at least one fixed point. In our case (<ref>) defines a continuous map F : [0,1]^L → [0,1]^L and [0,1]^L is compact and convex, so at least one solution to (<ref>) exists, see also <cit.>. The uniqueness of this solution is established in <cit.>.[Blocking probabilities in a simple loss network] Consider again the loss network of <ref> and <ref>, where all three classes require one unit of capacity at each link in their route. Set c_l = 3, l ∈L and ρ_m = 1, m ∈M. Under the assumption of independent blocking probabilities, the Poisson traffic offered to each link is σ_1 = (1 - B_3')(1 - B_5'), σ_2 = (1 - B_3'), σ_3 = (1 - B_1')(1 - B_5') + (1 - B_2'),σ_4 = (1 - B_5'), σ_5 = (1 - B_1')(1 - B_3') + (1 - B_4'). We wish to determine these link blocking probabilities through the Erlang fixed-point equations (<ref>). A possible method of obtaining the solution is through straightforward successive substitutions. This method, however, does not guarantee convergence to the solution, but usually works in practice. Let us take this approach and use as an initial guess B_l' = 0, l ∈L, see <ref>. From this approximation we find that link 3 and 5 are blocked most often and there are two pairs of links that have the same blocking probabilities. The last observation can be explained by the fact that both pairs of links are on a route consisting of three links and a route of two links, and furthermore, the load offered by each class is the same. The blocking probabilities for a class (or route) are calculated from (<ref>): B_1 = 33.67%, B_2 = B_3 = 20.87%. § JACKSON NETWORKS In this section we consider the class of so-called Jackson networks, named after the queueing theorist J.R. Jackson. A Jackson network consists of M queues (or stations) with possibly state-dependent service rates. Specifically, when there are a total of x_m jobs at queue m, the service rate is v_m(x_m), with v_m(0) = 0, m = 1,2,…,M. Note that for example v_m(x_m) = min(x_m,s_m) models a situation where queue m has s_m identical servers. The service times at queue m are independent and exponentially distributed with parameter μ_m. At each of the queues, the jobs are served in order of arrival. Upon service completion at queue m, jobs either proceed to queue n with probability r_m,n or leave the system with probability r_m,0 = 1 - ∑_n = 1^M p_m,n, where `0' refers to outside the network. The probabilities r_m,n are commonly called routing probabilities, and the M × M matrix R = [r_m,n]_m,n = 1,2,…,M the routing matrix. Jobs can arrive from outside the network to any of the queues in the network.Let us first treat two examples of Jackson networks. §.§ Tandem queues Consider a system of M queues in series with s_m servers at the m-th queue. Jobs arrive to the first queue according to a Poisson process with rateand require independent and exponentially distributed service times with parameter μ_m at the m-th queue. Upon service completion at the m-th queue, jobs proceed to the (m + 1)-th queue, m = 1,2,…,M - 1 and a service completion at the final queue leads to the job departing the system. Define ρ_m /μ_m as the offered load at the m-th queue. For stability, assume ρ_m < s_m for all m = 1,2,…,M.here should a homogeneous structure in terms of the transitions. That is, the transition structure and the rate at which these transitions occur should be the same for all states in the interior; for all states on the vertical boundary; and for all states on the horizontal boundary.The state of the system at time t may be described by a vector X(t) = (X_1(t),X_2(t),…,X_M(t)) with X_m(t) representing the number of jobs at the m-th queue at time t. It is easily verified that the process { X(t) }_t ≥ 0 is a Markov process with state space = _0^M. Denote by p(x) the equilibrium probability of being in state x∈. The equilibrium probabilities satisfy the global balance equations (+ ∑_m = 1^M min( x_m, s_m ) μ_m ) p(x) = x_1 > 0 p(x - 1)+ ∑_m = 1^M - 1min( x_m + 1, s_m ) μ_m x_m + 1 > 0 p(x + m - m + 1)+ min( x_M + 1, s_M ) μ_M p(x + M), for all states x∈, and the normalization condition ∑_x∈ p(x) = 1. It is easily verified through substitution, as we did in <ref>, that the equilibrium distribution is a product-form solution p(x) = ∏_m = 1^M p_m(x_m), with p_m(i) =p_m(0) ρ_m^i/i!, 0 ≤ i ≤ s_m - 1, p_m(0) ρ_m^i/s_m! s_m^i - s_m, i ≥ s_m, and p_m(0) = ( ∑_i = 0^s_m - 1ρ_m^i/i! + ρ_m^s_m/s_m!1/1 - ρ_m/s_m)^-1. This indicates that the number of jobs at the various stations are independent, and the number of jobs at the m-th station is distributed as the number of jobs in an isolated queue with s_m servers, Poisson arrival at rateand exponentially distributed service times with parameter μ_m. Looking back at <ref>, the departure process of the first queue, which is the arrival process of the second queue, is Poisson with rateand moreover independent of the number of jobs in the first queue. So, we could have expected the product-form equilibrium distribution. §.§ Closed tandem queues Suppose that the previous tandem queue is modified as follows. Instead of a Poisson arrival process, we assume that there is a finite population of K jobs circulating through the tandem queues. Upon service completion at the M-th queue, jobs return to the first queue. To avoid trivialities, K > min(s_1,s_2,…,s_M), because otherwise there is no interaction between jobs, and each of the simply cycle through the M queues, independently of all others. We no longer need to assume that ρ_m < s_m since this system is always stable. This system is called a closed system because no outside arrivals are allowed into the system. The previous tandem queue is then aptly named open.As before, let X(t) be the vector of the number of jobs at time t at each queue. The process { X(t) }_t ≥ 0 is a Markov process on the state space = {x∈_0^M : ∑_m = 1^M x_m = K }, with equilibrium probabilities p(x). These probabilities satisfy the global balance equations ∑_m = 1^M min( x_m, s_m ) μ_m p(x) = ∑_m = 1^M - 1min( x_m + 1, s_m ) μ_m x_m + 1 > 0 p(x + m - m + 1)+ min( x_M + 1, s_M ) μ_M x_1 > 0 p(x + M - 1), for all states x∈, and the normalization condition ∑_x∈ p(x) = 1. For the closed tandem queuing network, the equilibrium distribution is given by p(x) = 1/G∏_m = 1^M p_m(x_m), with p_m(i) = ρ_m^i/i!, 0 ≤ i ≤ s_m - 1,ρ_m^i/s_m! s_m^i - s_m, i ≥ s_m, with the normalization constant G = ∑_x∈∏_m = 1^M p_m(x_m), and ρ_m γ/μ_m for some arbitrary constant γ > 0. This constant is arbitrary since it appears only as γ^K in both the numerator and denominator of p(x).Obviously, the number of jobs at the various stations are no longer independent, but the equilibrium distribution retains a remarkably simple structure. It looks as if we applied <ref> to the equilibrium distribution of the open tandem queue to obtain the equilibrium distribution of the closed tandem queue. However, <ref> requires the Markov process to be reversible, but that is not the case here. Consider M = 3 and examine the transition rate from state y = (1,0,0) to state z = (0,1,0), which is μ_1. For the Markov process to be reversible, we require p(y) q_y,z = p(z) q_z,y, but q_z,y = 0. So, the Markov process associated with the open tandem queue is in general not reversible and <ref> cannot be applied. §.§ Open Jackson networks The tandem queue of <ref> belongs to the class of open Jackson networks. As it turns out, open Jackson networks have a similar product-form solution for the equilibrium distribution. Here we treat open Jackson networks in full.In open Jackson networks jobs arrive from the external environment, and eventually leave the system. Specifically, jobs are assumed to arrive at queue m as a Poisson process with rate _m, m = 1,2,…,M. We will assume that r_m,0 > 0 for at least one value of m, because otherwise it would be impossible for jobs to leave, and the system would definitely be unstable.Denote by _m the total arrival rate at queue m, including both external arrivals and transitions from other queues or queue m itself. In case the system is stable, _m must equal the total departure rate at queue m, including both external departures and transitions to other queues or queue m itself, and will also be called the throughput of queue m. In case the system is stable, the throughputs satisfy the following set of linear equations describing the flow of jobs through the system, the so-called traffic equations, _m = _m + ∑_n = 1^M _n r_n,m,m = 1,2,…,M, which may be written in vector-matrix notation as =+R, or equivalently ( - R) =, withthe identity matrix, = [_m]_m = 1,2,…,M the throughput vector, and = [_m]_m = 1,2,…,M the vector of exogenous arrival rates. The assumption that r_m,0 > 0 for at least one value of m implies that the matrix R has spectral radius strictly less than unity, and ensures that the matrix - R has a positive inverse, so that the throughput vector may be expressed as =( - R)^-1. Note that the service rates v_m(·) and parameters μ_m do not occur in the traffic equations, but of course they do determine whether or note the system is stable, and in turn determine when the traffic equations actually apply. Without proof, we state that the system is stable if _m < μ_m v_m^* for all m = 1,2,…,M, with v_m^* lim inf_x →∞ v_m(x). In particular, when r_m(x) = min(x,s_m), the system is stable when _m < μ_m s_m.The state of the system at time t may be described by a vector X(t)(X_1(t),X_2(t),…,X_M(t)), with X_m(t) the total number of jobs present at queue m at time t. It is easily verified that the process { X(t) }_t ≥ 0 is a Markov process with state space = _0^M. Assuming the stability condition to be fulfilled, denote by p(x) the equilibrium probability that the system is in state x. These probabilities satisfy the global balance equations ∑_m = 1^M ( _m + μ_m r_m(x_m) ) p(x) = ∑_m = 1^M _m x_m > 0 p(x - m)+ ∑_m = 1^M μ_m r_m(x_m + 1) r_m,0 p(x + m)+ ∑_m = 1^M ∑_n = 1^M μ_m r_m(x_m + 1) r_m,nx_n > 0 p(x + m - n) for all states x∈, along with the normalization condition ∑_x∈ p(x) = 1. The equilibrium distribution has the product form p(x) = 1/G∏_m = 1^M g_m(x_m), with ρ_m _m/μ_m, g_m(x) ρ_m^x / (∏_n = 1^x v_m(n)), and G = ∑_x∈∏_m = 1^M g_m(x_m) = ∏_m = 1^M ∑_x_m ≥ 0 g_m(x_m) ∏_m = 1^M G_m. Since lim inf_x_m →∞ v_m(x_m) = v_m^* and ρ_m < v_m^* we know that there exists an ϵ > 0 and N < ∞ such that v_m(x_m) > v_m^* - ϵ > ρ_m for x_m > M. Now, G_m= ∑_k ≥ 0ρ_m^k/∏_l = 1^k v_m(l)= ∑_k = 0^N ρ_m^k/∏_l = 1^k v_m(l) + ∑_k ≥ N + 1ρ_m^k/( ∏_l = 1^N v_m(l) ) ( ∏_n = N + 1^k v_m(n) )= ∑_k = 0^N ρ_m^k/∏_l = 1^k v_m(l) + ρ_m^N/∏_l = 1^N v_m(l)∑_k ≥ 1ρ_m^k/∏_n = N + 1^N + k v_m(n)< ∑_k = 0^N ρ_m^k/∏_l = 1^k v_m(l) + ρ_m^N/∏_l = 1^N v_m(l)∑_k ≥ 1( ρ_m/v_m^* - ϵ)^k < ∞. So, the assumption _m < μ_m v_m^* ⇔ρ_m < v_m^* ensures that G_m < ∞. Note that p(x) may be written as p(x) = ∏_m = 1^M ψ_m(x_m), where ψ_m(x)g_m(x)/G_m. Further observe that ψ_m(·) is the equilibrium distribution of the number of jobs at an isolated queue with a Poisson arrival process with rate _m, exponentially distributed service times with parameter μ_m and state-dependent service rate v_m(·).In case queue m has just a single unit-rate (v_m(x) = 1) server, the factor ρ_m represent the utilization of the server at queue m, and G_m = 1/(1 - ρ_m), so that its equilibrium distribution (indexed by a subscript m) is given by p_m(x_m) = (1 - ρ_m) ρ_m^x_m. In case there are infinitely many servers at queue m (v_m(x) = x), we obtain that G_m = ^ρ_m, which means that p_m(x_m) = ^-ρ_mρ_m^x_m/x_m!.Now let p_m(x_m) be the equilibrium probability that there are a total of x_m jobs present at queue m. Since ψ_m(·) is a probability distribution, it follows that p_m(x_m)= ∑_y∈ :y_m = x_m p(y) = ∑_y∈ :y_m = x_m∏_n = 1^M ψ_n(y_n) = ψ_m(x_m) ∏_n ≠ m∑_y_n ≥ 0ψ_n(y_n) = ψ_m(x_m), and therefore p(x) = ∏_m = 1^M p_m(x_m).This implies two important properties of open Jackson networks. First of all, the total number of jobs present at queue m has the same equilibrium distribution as that of an isolated queue with a Poisson arrival process of rate _m, exponentially distributed service times with parameter μ_m, and state-dependent service rate v_m(·). Second, the numbers of jobs present at the various queues are independent in equilibrium. These are two crucial properties that however need to be applied and interpreted with caution. For example, the first property might suggest that the aggregate arrival process at queue m, including both external arrivals and transitions from other queues, is Poisson with rate _m. This is indeed the case in some particular Jackson networks such as the open tandem queues considered in this chapter. However, in general this is not the case. Also, the second property is rather striking in view of the strong interaction due to the transitions among the various queues. The interaction in fact implies that the state of one queue can influence the state of other queues at future time instants, which might seem to contradict the stated independence. In order to resolve the paradoxical situation, it is critical to note that the independence only holds for the joint number of jobs at each queue at the same time epoch in equilibrium, and not for the states of different queues at different instants in time or in transient regimes. §.§ Closed Jackson networks In closed Jackson networks there are no external arrivals or departures. Instead, there is a fixed population of K jobs which circulate through the system. In contrast to the case of open networks, we now have ∑_m = 1^M r_m,n = 1 for all m = 1,2,…,M; the routing matrix R = [r_m,n]_m,n = 1,2,…,M is stochastic. In order to ensure that the equilibrium distribution does not depend on the initial state, we assume that the matrix R is irreducible, which means that the matrix - R has rank M - 1.Like in the case of open networks, denote by _m the total arrival rate at queue m, now however consisting exclusively of transitions from other queues or queue m itself. Without any further assumptions, _m will be equal to the total departure rate or throughput at queue m, again now consisting however exclusively of transitions to other queues or queue m itself. The throughputs satisfy the following set of linear equations, the so-called traffic equations, _m = ∑_n = 1^M _n r_n,m,m = 1,2,…,M, which may be written in vector-matrix notation as =R, or equivalently ( - R) =, with = [_m]_m = 1,2,…,M the throughput vector. In contrast to the case of open networks, the traffic equations no longer have a unique solution. Note that scaling a solution, that is, multiplying all throughputs with a common scalar value, will again yield a solution since the traffic equations are first-order homogeneous. Because the matrix - R has rank M - 1, the traffic equations do however uniquely determine the relative values of the throughputs: they determine the throughputs up to a common scaling factor.As in the case of open networks, the state of the system at time t may be described by X(t)(X_1(t),X_2(t),…,X_M(t)), with X_m(t) representing the total number of jobs present at queue m at time t. It is easily verified that the process { X(t) }_t ≥ 0 is a Markov process with state space {x∈_0^M : ∑_m = 1^M x_m = K }. Denote by p(x) the equilibrium probability that the system is in state x∈. These probabilities satisfy the global balance equations ∑_m = 1^M μ_m v_m(x_m) p(x) = ∑_m = 1^M ∑_n = 1^M μ_m v_m(x_m + 1) r_m,nx_n > 0 p(x + m - n) for all states x∈, along with the normalization condition ∑_x∈ p(x) = 1. Through substitution it can be verified that the equilibrium distribution is p(x) = 1/G∏_m = 1^M g_m(x_m), with ρ_m κ_m/μ_m, g_m(x) ρ_m^x / (∏_n = 1^x v_m(n)), and G = ∑_x∈∏_m = 1^M g_m(x_m). Here _m is the relative throughput at queue m, so thatis the solution to the traffic equations satisfying = 1. The scaling factor κ may be chosen arbitrarily, for example so as to obtain `convenient' ρ_m values. In order to see that κ may be chosen arbitrarily, observe that the numerator and denominator of p(x) both have the term κ^K and therefore cancels.Note that the equilibrium distribution has a product form, just like in the case of open networks. While the various terms in the product look similar, they are no longer distributions, and hence the two important properties that we observed for open Jackson networks no longer hold. Some reflection indeed shows that it is not possible for the number of jobs present at queue m to have the same equilibrium distribution as that in an isolated queue with a Poisson arrival process, for the simple reason that the number of jobs at queue m is at most K, whereas the number of jobs in the latter situation is unbounded. Likewise, it follows that it is not possible for the various numbers of jobs at the queues to be independent, for the simple reason that if there are K jobs at one of the queues for example, all the other queues are known to be empty. §.§ Normalization constant in closed Jackson networks Although (<ref>) provides a simple expression for the normalization constant G in closed networks, brute-force numerical evaluation is prohibitively demanding for all but the smallest networks. The number of terms in the summation is M + K - 1M - 1 which rapidly grows with the values of M and K.We now describe a more efficient numerical procedure for calculating the normalization constant. For convenience, we assume that the various queues either have a single server (v_m(x) = 1) or infinitely many servers (v_m(x) = x), and are labeled such that queues 1,2,…,J are infinite-server queues while queues J + 1,J + 2,…,M are single-server queues. Infinite-server queues are not really `queues', in the sense that jobs never need to wait but immediately enter service upon arrival. However, they provide a useful paradigm for modeling various kinds of delays, such as think times of users, availability periods of machines, or transit times among queues. The normalization constant may then be expressed as G = ∑_x∈( ∏_m = 1^J ρ_m^x_m/x_m!) ∏_m = J + 1^M ρ_m^x_m = ∑_x∈( ∏_m = 1^J 1/x_m!) ∏_m = 1^M ρ_m^x_m. Now define _j,k{x∈_0^j : ∑_m = 1^j x_m = k } and let G(j,k) ∑_x∈_j,k∏_m = 1^j ρ_m^x_m/x_m!,j = 0,1,…,J, and G(j,k)∑_x∈_j,k( ∏_m = 1^J ρ_m^x_m/x_m!) ∏_m = J + 1^j ρ_m^x_m= ∑_x∈_j,k( ∏_m = 1^J 1/x_m!) ∏_m = 1^j ρ_m^x_m,j = J + 1,J + 2,…,M. Note that G = G(M,K). Observe that for j = 1,2,…,J, G(j,k)= ∑_x∈_0^j : x = k( ∏_m = 1^j 1/x_m!) ∏_m = 1^j ρ_m^x_m= 1/k!∑_x∈_0^j : x = kkx_1,…,x_j∏_m = 1^j ρ_m^x_m = 1/k!( ∑_m = 1^j ρ_m )^k, where we used the multinomial theorem. For j = J + 1,J + 2,…,M we obtain a recursion instead of an explicit expression: G(j,k)= ∑_x∈_0^j : x = k( ∏_m = 1^J 1/x_m!) ∏_m = 1^j ρ_m^x_m= ∑_x∈_0^j : x = k, x_j = 0( ∏_m = 1^J 1/x_m!) ∏_m = 1^j ρ_m^x_m + ∑_x∈_0^j : x = k, x_j > 0( ∏_m = 1^J 1/x_m!) ∏_m = 1^j ρ_m^x_m= ∑_x∈_0^j - 1 : x = k( ∏_m = 1^J 1/x_m!) ∏_m = 1^j - 1ρ_m^x_m + ρ_j ∑_x∈_0^j : x = k - 1( ∏_m = 1^J 1/x_m!) ∏_m = 1^j ρ_m^x_m= G(j - 1,k) + ρ_j G(j,k - 1). Using the above recursive relationship, G(M,K) can be efficiently computed starting from G(J,k) = 1/k! ( ∑_m = 1^j ρ_m )^k and G(j,0) = 1, j = J + 1,J + 2,…,M. §.§ Mean-value analysis in closed Jackson networks Previously, we described an efficient numerical procedure for calculating the normalization constant associated with the equilibrium distribution in closed Jackson networks with only single-server and infinite-server queues. In case we are not interested in the entire equilibrium distribution, but only in mean number of jobs at each queue or the mean sojourn times (time spent by a job in a station), there exists an even more efficient recursive procedure, called mean-value analysis (MVA).The mean-value analysis algorithm was developed by Reiser and Lavenberg <cit.> and relies heavily on the arrival theorem. The MVA algorithm as presented in <cit.> is more general than what is presented here. For example, different service mechanism are allowed and the authors consider networks of multiple closed systems that share stations (queues), yet still retain the product-form equilibrium distribution. Just like in the previous section, we assume that queues 1,2,…,J are infinite-server queues, while queues J + 1,J + 2,…,M are single-server queues.Mean-value analysis is based on the following property, often referred to as `arrival theorem', which we state without proof. Consider an arbitrary arrival instant at queue m, that is, a time epoch where a job makes a transition to queue m (possibly coming from queue m itself after a service completion). Then the joint equilibrium distribution at that time instant, not counting the arriving job, is the same as the joint equilibrium distribution of the same system, but with K - 1 rather than K jobs. In other words, when the job arrives at queue m, it sees the system as if it had never been present.The arrival theorem originated from Lavenberg and Reiser <cit.>, see also Sevcik and Mitrani <cit.>. In open Jackson networks, the distribution of the number of jobs at each queue is identical at arrival instants, departure instants and random points in time.Since we are interested in results in stationarity, we abuse notation and remove the time index t from the state variables. In order to formally state and use the above property, it is convenient to add a superscript a to indicate state variables at arrival instants (excluding the arriving job itself), and further explicitly indicate the population size in brackets. Then the above property may be written as X^(K)= x = X(K - 1) = x, for all x∈, or equivalently, X^(K + 1) = x = X(K) = x, which implies for example X_m^(K + 1) = X_m(K). In case of a infinite-server queue, the mean sojourn time S_m of a job at queue m is simply the mean service time S_m(K) = 1/μ_m,m = 1,2,…,J. In case of a single-server queue, the mean sojourn time of a job at queue m can be easily related to the number of jobs found upon arrival: S_m(K) = (X_m^(K) + 1)/μ_m = (X_m(K - 1) + 1)/μ_m,m = J + 1,J + 2,…,M. In turn, the mean sojourn time is related to the mean queue length via Little's law: _m(K) S_m(K) = X_m(K) with _m(K) the throughput at queue m given that there are K jobs in total in the system. The throughputs may be determined from the traffic equations (<ref>), up to a common scaling factor κ(K), namely (K) = κ(K) , whererepresents the vector of relative throughput with = 1, which can be uniquely determined from the traffic equations, and κ(K) is a common scaling factor depending on the total number of jobs in the system. Now, the number of jobs in the system is constant, so by summing over all m = 1,2,…,M on both sides of (<ref>) we obtain ∑_m = 1^M _m(K) S_m(K) = ∑_m = 1^M X_m(K) = K which gives an expression for κ(K): κ(K) = K ( ∑_m = 1^M _m S_m(K))^-1. Together, the above relationships provide a recursive procedure for calculating S_m(K) and X_m(K) for any desired value of K, starting from X_m(0) = 0, m = 1,2,…,M. We summarize the mean-value analysis in <ref>.[A trucking company] A large international trucking company has to move its spare parts from warehouse A to warehouse B. Upon arriving to warehouse A, the trucks wait to be served by a crew that loads the spare parts into the truck. The crew takes an exponential amount of time with mean 1 to load a single truck. The loaded truck then drives to warehouse B where another crew unloads the truck, taking an exponential amount of time with mean 6/5 per truck. The time it takes to drive from warehouse A to warehouse B (or back) takes is exponential distributed with mean 4. The trucking company does not comply with the regulations and laws and allows the truck drivers to make as many trips in a row as they want to earn some extra money. After unloading at warehouse B one third of the truck drivers decides to drive back to warehouse A and make another trip. The remaining fraction of the drivers goes to a motel nearby warehouse B and starts the drive to warehouse A after an exponential amount of time with mean 12. The trucking company is interested in the impact of the number of trucks (and drivers) on the number of trucks per time unit that unload at warehouse B.The moving operation can be modeled as a closed Jackson network. We identify two single-server queues (warehouses A and B) and three infinite-server queues (drive from A to B, drive from B to A and stay at the motel). See <ref> for the numbering of the stations. The trucking company is interested in _5(K) for various values of K.The routing matrix is R = [ 0 0 0 0 1; 0 0 0 1 0; 0 1 0 0 0; 1 0 0 0 0; 0 1/3 2/3 0 0 ]. The relative throughputsare determined from the traffic equations (<ref>) with = 1 and we find = 1/14 [ 3 3 2 3 3 ]. So, _1(K) = _2(K) = _4(K) = _5(K) and S_1(K) = S_2(K) = 4 and S_3(K) = 12 for all K. So, we only report S_4(K), S_5(K), _3(K), _5(K) and X_m(K), m = 1,2,…,5. Applying <ref> produces the results in <ref>.There are many scenarios to consider that could improve on this situation. The trucking company can train the unloading crew to become faster, if the mean unloading time reduces to 5/6, the throughput for K = 20 trucks increases by 13.7% to _5(20) = 0.8738. If the trucking company were to increase the money earned per trip, a fraction 1/2 does another trip. In that case, the throughput for K = 20 trucks only slightly increases by 2.5% to _5(20) = 0.7876. The first option seems better, but it does increase the mean sojourn time for the loading station by approximately 1 time unit. § TAKEAWAYS Starting from a basic birth–and–death process, and the concept of reversibility, we were able to find in an elegant manner the equilibrium distribution for the rich class of loss networks. While loss networks give rise to multi-dimensional Markov processes, their state space allows for local balance arguments with balance equations that are readily solved, leading to the beautiful product-form solution in <ref>. As pointed out, the catch here is the normalization constant, whose computation requires the enumeration of all states in the state space and needs tailor-made algorithms.Markov processes intimately related to loss networks are also studied in statistical mechanics, in the form of interacting particle systems. While the terminology is different (Markovian assumptions become Glauber dynamics, product-form solution is called Gibbs measure and the normalization constant is known as the partition function), the Markov process description and analytic methods are largely the same. For thorough treatments of such interacting particle systems we refer to Liggett <cit.>.We then proceeded to queueing networks, again giving rise to multi-dimensional Markov processes. But for these processes, local balance fails, and the global balance equations are then the unavoidable point of departure. Nevertheless, structure was there to be discovered, the first glimpse captured by Burke’s Theorem, telling us that the output process of one queue with Poisson arrivals is again Poisson. This property then naturally leads to the guess that networks of queues with external Poisson arrivals can be decomposed into isolated queues with arrival rates that match in expectation the arrival rates in the networks. Mathematically, such an educated guess translates into substituting a product of product forms into the global balance equations, and showing that indeed this is the unique solution and hence the unique equilibrium distribution. Although elegant and sound, this educated guess approach is somewhat unsatisfying, because it is non-constructive. In analysis, however, solving a difference or differential equations by clever substitutions is one of the key techniques. We shall continue to work with educated guesses for finding product-form solution whenever this is necessary, e.g., for more advanced Markov processes in <ref>.The network models in this chapter make it possible to consider real-world networked systems with a host of applications. Loss networks were used for instance to describe the topology and performance of the internet <cit.> and queueing models can describe complex manufacturing processes <cit.>.Loss networks and queueing networks are examples of stochastic networks, one of the richest topics in the field of applied probability. Text books with prominent roles for such network are Buzacott and Shantikumar <cit.>, Chen and Yao <cit.>, Kelly and Yudovina <cit.> and Whittle <cit.>. CHAPTER: QUASI-BIRTH–AND–DEATH PROCESSES Quasi-birth–and–death (QBD) processes are the natural two-dimensional generalization of the birth–and–death process. QBDs live on a countable state space that consists of one infinite dimension and one finite dimension. The finite dimension is added on top of what would otherwise be a BD process. Before we develop the general theory of a QBD process, let us treat some examples that show the extension of a BD process to a QBD process.§ VARIATIONS OF BIRTH–AND–DEATH PROCESSES§.§ Machine with setup times Let us consider a machine processing jobs in order of arrival. Jobs arrive according to a Poisson process with rateand the processing times are exponential with mean 1/μ. For stability we assume that ρ/μ < 1. The machine is turned off when the system is empty and it is turned on again when a new job arrives. The setup time is exponentially distributed with mean 1/θ. Turning off the machine takes no time. We are interested in the effect of the setup time on the sojourn time of a job.The state of the system may be described by X(t)(X_1(t),X_2(t)) with X_1(t) representing the number of jobs in the system at time t and X_2(t) describes if the machine is turned off (0) or on (1) at time t. The process { X(t) }_t ≥ 0 is a Markov process with state space { (i,j) ∈_0 ×{ 0,1 }}. The transition rate diagram is displayed in <ref>. It looks similar to the one for the BD process, in for example <ref>, except that each state i has been replaced by a set of states { (i,0),(i,1) }. This set of states is called level i. In BD processes transitions are restricted to neighbouring states, while in QBD processes the transitions are restricted to neighbouring levels. The horizontally aligned set of states { (0,j),(1,j),…} is often referred to as phase j.For the current model, define i = { (i,0),(i,1) }, i ≥ 0 as the set of states with i jobs in the system, that is i is level i. We can then write the state space as 0∪1∪2∪⋯Let p(i,j) denote the equilibrium probability of state (i,j) ∈. Clearly, p(0,1) = 0 since state (0,1) is transient. State (0,1) is included in the state space for notational convenience: all levels consists of two states. From the transition rate diagram we obtain by equating the flow out of a state and the flow into that state the following set of global balance equations, p(0,0) = μ p(1,1), ( + θ) p(i,0)=p(i - 1,0),i ≥ 1, ( + μ) p(i,1)=p(i - 1,0) + θ p(i,0) + μ p(i + 1,1),i ≥ 1. The structure of the equations (<ref>)–(<ref>) is closely related to balance equations of the M/M/1 queueing model, see for example <ref>. This becomes more striking by introducing vectors of equilibrium probabilities _i = [ p(i,0) p(i,1) ] and writing (<ref>)–(<ref>) in vector-matrix notation: _0 _0^(0) + _1 _-1^(1) = ,_i - 1_1 + _i _0 + _i + 1_-1 = ,i ≥ 1, where _-1 = [ 0 0; 0 μ ], _0 = [ -( + θ) θ; 0 -( + μ) ], _1 = [ 0; 0 ],_0^(0) = -_1, _-1^(1) = [ 0 0; μ 0 ]. Obviously, if we can determine the equilibrium probabilities p(i,j), then we also compute the mean number of jobs in the system, and by Little's law, the mean sojourn time. We now present three methods to determine the equilibrium probabilities. The first one is known as the matrix-geometric method, the second is referred to as the spectral expansion method, and the third one employs partial generating functions. Let us start with the matrix-geometric approach. We will introduce the first two methods in greater detail in the later sections of this chapter. The last method will appear in various places of this book, but is more well-known overall.We first simplify the balance equations (<ref>) by eliminating the vector _i + 1. By equating the flow from level i to level i + 1 to the flow from level i + 1 to level i we obtain (p(i,0) + p(i,1)) = p(i + 1,1)μ, or, in vector-matrix notation, _i ^* = _i + 1_-1 where ^* = [ 0; 0 ]. Substituting this relation into (<ref>) produces _i - 1_1 + _i ( _0 + ^* ) = ,i ≥ 1, which allows us to express _i in terms of _i - 1: _i = - _i - 1_1 ( _0 + ^* )^-1 = _i - 1 R, where R- _1 ( _0 + ^* )^-1 = [ / + θ/μ; 0/μ ]. Iterating (<ref>) leads to the matrix-geometric solution _i = _0 R^i,i ≥ 0. Notice that this is very similar to the solution for the M/M/1 model, which is p(i) = p(0) ρ^i, i ≥ 0. Finally, _0 follows from the equations (<ref>) and the normalization condition ∑_(i,j) ∈ p(i,j) = _0(- R )^-1 = 1. From (<ref>) we obtain the mean number of jobs in the system as X_1 = ∑_i ≥ 1 i _i= ∑_i ≥ 1 i _0 R^i= _0 R (I - R)^-2, and the mean sojourn time is S = X_1/.The matrix R is critical in the matrix-geometric approach. It is called the rate matrix and has an interesting and useful probabilistic interpretation. Element (j,k) of R is the expected time spent in state (i + 1,k) multiplied by element (j,j) of -_0 before the first transition to a state in level i, given the initial state (i,j). This immediately means that zero rows in _1 lead to zero rows in R. Recall the hitting-time random variables of <ref>, which we now use with a slight modification. For any set A⊂, Ainf{ t > 0 : lim_s ↑ t X(s) ≠ X(t) ∈A}. Note that we suppress the dependence on the initial state, since that will be clear from the expectation that we are determining. Using the hitting-time random variable, we can write (R)_j,k as (R)_j,k = (-_0)_j,j (i,j) ∫_0^iX(t) = (i + 1,k)t . Let us derive element (0,0) of R. Using a one-step analysis and the strong Markov property at the sojourn time in state (i,0) we have (R)_0,0 = ( + θ)(i,0) ∫_0^iX(t) = (i + 1,0)t = ( + θ) / + θ (i + 1,0) ∫_0^iX(t) = (i + 1,0)t =(i + 1,0) ∫_0^H_(i + 1,0) 1 t = / + θ, where H_x was defined as the sojourn time in state x. Possibly more interesting is the derivation of element (1,1) of R. Using a similar analysis as above we get (R)_1,1 = ( + μ)(i,1) ∫_0^iX(t) = (i + 1,1)t =(i + 1,1) ∫_0^iX(t) = (i + 1,1)t . We continue by conditioning on the number of times the process visits state (i + 1,1) before reaching level i. The probability q(n) that state (i + 1,1) is visited n times (where the initial visit is counted) before reaching level i is q(n) = μ/ + μ( / + μ)^n - 1,n ≥ 1, since if the process transitions to state (i + 2,1), it returns to state (i + 1,1) with probability 1 by positive recurrence due to ρ < 1. If the process visits state (i + 1,1) a total of n times before reaching level i, then it spends in expectation n/( + μ) time in state (i + 1,1). Combining these observations, we find (R)_1,1 = ∑_n ≥ 1n/ + μ·μ/ + μ( / + μ)^n - 1= μ/ + μ∑_n ≥ 1 n ( / + μ)^n = /μ. Element (R)_0,1 is equal to (R)_1,1, because if the process transitions from state (i,0) to (i + 1,0), it reaches state (i + 1,1) before level i with probability 1, allowing for the exact same analysis and result.We now demonstrate the spectral expansion method. This method first seeks solutions of the equations (<ref>) of the simple form _i = y x^i,i ≥ 0, where y = [ y(0) y(1) ] is a non-zero vector and |x| < 1. The latter is required, since we want to be able to normalize the solution afterwards. Substitution of this form into (<ref>) and dividing by common powers of x gives y( _1 + x _0 + x^2 _-1) = . So, the desires values of x are the roots inside the unit circle of the determinant equation ( _1 + x _0 + x^2 _-1 ) = 0. In this case we have ( _1 + x _0 + x^2 _-1 ) = ( - ( + θ)x)(μ x - )(x - 1) = 0. We can read of the roots with |x| < 1, which are x_1 = / + θ,x_2 = /μ. For i = 1,2, let y_i be the non-zero solution of y_i ( _1 + x_i _0 + x_i^2 _-1) = . Solving this linear system of equations gives the solutions y_1 = [ 1 -θ x_1/ - ( + μ)x_1 + μ x_1^2 ], y_2 = [ 0 1 ]. Note that, since the balance equations are linear, any linear combination of the two simple solutions satisfies (<ref>). Now the final step of the spectral expansion method is to determine a linear combination that also satisfies the boundary equations (<ref>). So we set _i = ξ_1 y_1 x_1^i + ξ_2 y_2 x_2^i,i ≥ 0, where the coefficients ξ_1 and ξ_2 follow from the boundary equations (<ref>) and the normalization condition ∑_(i,j) ∈ p(i,j) = ∑_i ≥ 0( ξ_1 y_1 x_1^i + ξ_2 y_2 x_2^i )= ξ_1 y_1 /1 - x_1 + ξ_2 y_2 /1 - x_2 = 1. Since the balance equations are dependent, we may omit one of the equations of (<ref>), and, for example, only use 0 = p(0,1) = ξ_1 y_1(1) + ξ_2 y_2(1), together with the normalization condition to determine the (unique) coefficients ξ_1 and ξ_2.Using representation (<ref>) we obtain X_1 = ∑_i ≥ 1 i _i= ξ_1 y_1 x_1 /(1 - x_1)^2 + ξ_2 y_2 x_2 /(1 - x_2)^2.The two methods presented above are closely related: x_1 and x_2 are the eigenvalues of the rate matrix R and y_1 and y_2 are the corresponding left eigenvectors.The third and final method uses generating functions. Introduce the partial generating functions j∑_i ≥ 0 p(i,j) ^i,j = 0,1, defined for all || ≤ 1. Multiplying (<ref>) and (<ref>) by ^i and summing over all i ≥ 1 yields ( + μ)(0 - p(0,0))= 0 , ( + μ)(1 - p(0,1))= 1 + θ (0 - p(0,0))+ μ/ (1 - p(0,1) - p(1,1) ). Using p(0,1) = 0 and (<ref>), we get 0 = p(0,0)/1 - / + θ, and 1 = θ0 - ( + θ) p(0,0)/( - 1)(μ/ - ) = ρ p(0,0)/(1 - / + θ)(1 - ρ)= p(0,0)ρ/ρ - / + θ( 1/1 - ρ - 1/1 - / + θ). The probability p(0,0) follows from the normalization condition 01 + 11 = 1, which results in p(0,0) = (1 - ρ) θ/ + θ. From (<ref>) and (<ref>) and 1/(1 - x) = ∑_i ≥ 0 x^i, |x| < 1 we conclude that for i ≥ 0, p(i,0)= p(0,0) ( / + θ)^i, p(i,1)= p(0,0)ρ/ρ - / + θ( ρ^i - ( / + θ)^i ), which agrees with the form (<ref>). The mean number of jobs in the system X_1 and the mean sojourn time S can also be determined by combining the PASTA property and Little's law. Based on PASTA we know that the average number of jobs in the system seen by an arriving job equals X_1, and each of them (also the one being processed) has a (residual) processing time with mean 1/μ. With probability 1 - ρ the machine is not in operation on arrival, so that the job also has to wait for the setup phase with mean 1/θ. Further, the job has to wait for its own processing time. Combining these observations, we have S = (1 - ρ) 1/θ + X_11/μ + 1/μ, and together with Little's law X_1 = S, we find S = 1/μ/1 - ρ + 1/θ. The first term at the right-hand side is the mean sojourn time in the system without setup times (the machine is always on). The second term is the mean setup time. Clearly, the mean setup time is exactly the extra mean delay caused by turning off the machine when there is no work. In fact, it can be shown (by using, for example, a sample path argument) that the extra delay is an exponential time with mean 1/θ.§.§ Erlang services We consider a single-server queue. Jobs arrive according to a Poisson process with rateand they are served in order of arrival. The service times are Erlang-r distributed with mean r/μ. For stability we require that the occupation rate ρr/μ is less than one. This system can be described by a QBD process { X(t) }_t ≥ 0 with levels 0 = {(0,0),(0,1),…,(0,r) } and i = { (i,1),(i,2),…,(i,r) }, with i ≥ 1, where level i indicates the number of jobs waiting in the queue and phase j is the remaining number of service phases of the job in service. The state space is denoted by 0∪1∪⋯. The transition rate diagram is shown in <ref>. Note that by setting r = 1 we get a homogeneous BD process on the states _0, modeling the M/M/1 queue.Let p(i,j) denote the equilibrium probability of state (i,j) ∈. From the transition rate diagram we get the following balance equations for the states (i,j) with i ≥ 1, ( + μ) p(i,j)=p(i - 1,j) + μ p(i,j + 1),j = 1,2,…,r - 1, ( + μ) p(i,r)=p(i - 1,r) + μ p(i + 1,1), or, in vector-matrix notation, _i - 1_1 + _i _0 + _i + 1_-1 = ,i ≥ 1, where _i = [ p(i,1) p(i,2)⋯ p(i,r) ], _-1 = [ ⋯ 0 μ; 0; ⋮; ], _0 = [ 0; μ 0; ⋱ ⋱; μ 0 ] - ( + μ), and _1 =, where all unlabeled entries are 0. We first determine the probabilities p(i,j) using the matrix-analytic method. Define an excursion as a sample path of the process starting in level i, moving to levels higher than i and ending at first return to level i. From the transition rate diagram we see that the number of excursions per time unit that end in state (i,r) is p(i + 1,1) μ. Note that this is the only state in which an excursion can end. On the other hand, the number of excursions per time unit that start in state (i,k), immediately go to state (i + 1,k) and ultimately end the excursion in state (i,r) is p(i,k). The number of excursions per time unit that end in state (i,r) is found by summing over all possible starting states, so we get ∑_k = 1^r p(i,k). In vector-matrix form this leads to _i + 1_-1 = _i _1 G, where the matrix G is called the auxiliary matrix of the matrix-analytic method and element (j,k) of G is interpreted as the probability that, starting in state (i,j), i ≥ 1, the first passage to level i - 1 happens in state (i - 1,k). This immediately means that zero columns in _-1 lead to zero columns in G. For the model at hand G = [ 0 ⋯ 0 1; ⋮ ⋮ ⋮; 0 ⋯ 0 1 ]. We can substitute the relation (<ref>) into (<ref>) to obtain _i = - _i - 1_1 ( _0 + _1 G)^-1, where we note that the inverse exists. By iterating this equation we get _i = _0 ( - _1 ( _0 + _1 G)^-1)^i,i ≥ 0. Finally the probabilities p(0,0) and _0 follow from the balance equations for the states in 0 and the normalization condition. The above relation also shows that the matrix-geometric and matrix-analytic methods for QBD processes are closely related: the rate matrix R = - _1 ( _0 + _1 G)^-1, since a relation like (<ref>) of the previous example also holds for the current model.We again apply the spectral expansion method to find the equilibrium distribution. We substitute the simple form p(i,j) = y(j) x^i,i ≥ 0, j = 1,2,…,r, into the balance equations (<ref>)–(<ref>) and divide by common powers of x to find ( + μ) y(j) x=y(j) + μ y(j + 1) x,j = 1,2,…,r - 1, ( + μ) y(r) x=y(r) + μ y(1) x^2, From (<ref>) we deduce y(j + 1)/y(j) = ( + μ) x - /μ x = , so we can set y(j) = ^j, j = 1,2,…,r. Substituting this back into (<ref>)–(<ref>) gives ( + μ) x=+ μ x, ( + μ) x=+ μ x^2/^r - 1. This set of equations is equivalent to x= β^r, 0= ( + μ) β^r - ( + μβ^r + 1). We will apply Rouché's theorem, see <ref>, to establish that (<ref>) has r roots inside the unit disk. Define f(β)( + μ) β^r and g(β)- ( + μβ^r + 1). Since both functions are polynomials, they are analytic functions for all β∈. Clearly, f(β) has r roots in the complex unit disk and we wish to establish |f(β)| > |g(β)| for |β| = 1 so that by Rouché's theorem, we have that (<ref>) has r roots in the complex unit disk. Observe that |f(β)| = f(|β|) and |g(β)| ≤ - g(|β|). Therefore we only require to that f(|β|) > - g(|β|) for |β| = 1, but, for |β| = 1, f(|β|) = f(1) =+ μ = - g(1) = - g(|β|). To resolve this issue, we essentially evaluate f(|β|) and g(|β|) along the circle |β| = 1 - ϵ. We use the Taylor expansion at |β| = 1 to get f(1 - ϵ) = f(1) - ϵ f'(1) + o(ϵ) and similarly for g(1 - ϵ). So, we require to show that f(1 - ϵ) > g(1 - ϵ) for ϵ small. However, since f(1) = - g(1) the only thing we need is f'(1) < - g'(1), with f'(1) = r( + μ) and -g'(1) = (r + 1)μ, which is indeed the case by the stability condition r< μ. Finally, we have established that (<ref>) has r roots in the complex unit disk. In <cit.> the authors establish for a more general model that these roots are unique.Label the r roots inside the unit disk of (<ref>) as β_1,β_2,…,β_r with corresponding x_k = β_k^r, k = 1,2,…,r. We have r basis solutions of the form p(i,j) = β_k^j x_k^i,k = 1,2,…,r. The next step is to take a linear combination of these basis solutions p(i,j) = ∑_k = 1^r _k β_k^j x_k^i,i ≥ 0, j = 1,2,…,r and determine the coefficients _1,_2,…,_r and p(0,0) from the balance equations of level 0 and the normalization condition. §.§ Tandem queue with blocking The final example is related to both loss networks and open Jackson networks, but is not an example of either of the two. The network consists of two stations. Jobs arrive to the first station according to a Poisson process with rate . The first station is an single-server queue where jobs are served in order of arrival and service takes an exponential amount of time with mean 1/μ_1. Jobs leaving the first station are routed to the second station. The second station is an Erlang-B model with r servers. A service at station 2 is exponentially distributed with mean 1/μ_2. After receiving service at station 2, the job leaves the system. A departing job from station 1 that finds all servers occupied in station 2 is blocked and leaves the system as well. The first station is an M/M/1 queue and is therefore stable if < μ; the second station is always stable.The state of the system may be described by X(t)(X_1(t),X_2(t)) with X_i(t) the number of jobs at station i at time t. The state space of this QBD process is 0∪1∪⋯ with levels i = { (i,0),(i,1),…,(i,r) }, i ≥ 0. The transition rate diagram is shown in <ref>.Let p(i,j) denote the equilibrium probability of state (i,j) ∈. From the transition rate diagram we obtain the global balance equations for level 0, with 0 < j < r, p(0,0)= μ_2 p(0,1), ( + j μ_2) p(0,j)= μ_1 p(1,j - 1) + (j + 1) μ_2 p(0,j + 1), ( + r μ_2) p(0,r)= μ_1 ( p(1,r - 1) + p(1,r) ),and for level i ≥ 1, with 0 < j < r, ( + μ_1) p(i,0)=p(i - 1,0) + μ_2 p(i,1), ( + μ_1 + j μ_2) p(i,j)=p(i - 1,j) + μ_1 p(i + 1,j - 1)+ (j + 1) μ_2 p(i,j + 1), ( + μ_1 +r μ_2) p(i,r)=p(i - 1,r)+ μ_1 ( p(i + 1,r - 1) + p(i + 1,r) ),We have learned from <ref> that the output process of the first station is a Poisson process with rate . So it is not at all unreasonable to think that both stations operate independently and therefore the equilibrium distribution is a product of the equilibrium distributions of the first and second station. Both equilibrium distributions were already derived in <ref>, see <ref>. Define ρ_1 /μ_1 and ρ_2 /μ_2, and let us validate if p(i,j) = 1 - ρ_1/∑_k = 0^rρ_2^k/k!ρ_1^i ρ_2^j/j! is the equilibrium distribution of the tandem queue. Note that the normalization condition ∑_i ≥ 0∑_j = 0^r p(i,j) = 1 is satisfied. It can be easily verified that (<ref>) is a solution to the global balance equations by substituting (<ref>) into (<ref>) and (<ref>). In conclusion, even though this tandem queue network has state-dependent routing, it still retains the explicit product-form equilibrium distribution that was encountered in the open Jackson networks of <ref>.The approach of making an educated guess for the equilibrium distribution and verifying its correctness through the global balance equations and the normalization condition is a powerful approach that can quickly lead to the solution. However, it is crucial that the problem is well understood so that intuition can lead to a correct guess for the form of the equilibrium distribution. Alternatively, the spectral expansion method leads to the same result, but more computations should be done to get there. For this method, we substitute the simple form p(i,j) = y(j) x^i into the global balance equations and try to determine both parameters. This takes considerably more work than immediately guessing the correct expression for p(i,j) as we have done for the current model.§ GENERAL QUASI-BIRTH–AND–DEATH PROCESSES From the previous three examples we have seen that a QBD process consists of one infinite dimension and one finite dimension. The state space of a QBD processes can be partitioned in levels, where level 0 sometimes has a different number of states. This structure holds for the QBD processes that we are interested in. In particular, 0{ (0,0),(0,1),…,(0,b) }, i{ (i,0),(i,1),…,(i,r) }, i ≥ 1, with b and r non-negative finite integers, so that the state space is given by 0∪1∪2∪⋯ We denote the state of the QBD process at time t as X(t)(X_1(t),X_2(t)) where X_1(t) describes the level and X_2(t) describes the phase at time t.Throughout this chapter we focus on homogeneous QBD processes, which means that transition rates are level-independent, possibly except for the transition rates from and to level 0.QBD processes that are not homogeneous are called level-dependent QBD processes. These processes admit similar solutions to those presented for the three examples in <ref>, but now require level-dependent R or G matrices. For a detailed description of the methods involved see Bright and Taylor <cit.> or Kharoufeh <cit.>. As stated in <ref>, the analogy with a BD process follows from the fact that transitions from a state within level i ≥ 1 can only go to a state within level i - 1, i, or i + 1. The transition rate matrix of a homogeneous QBD process has block-tridiagonal structure Q = [_0^(0)_1^(0); _-1^(1)_0^(1)_1; _-1_0_1; _-1_0_1; ⋱ ⋱ ⋱; ], when the states are ordered according to their level and in increasing order within a level. The subscript n of _n denotes the change in levels for a transition. Element (j,k) of _n is the transition rate from state (i,j) to state (i + n,k) with i ≥ 2. Note that elements (j,j), 0 ≤ j ≤ r of _0 are the exception to this rule; element (j,j) is negative, but it's absolute value is exactly the rate at which the process leaves (i,j), i ≥ 2. This makes the row sums of Q zero. The additional superscript l in _n^(l) indicates the dependence on the level l.The matrix _0^(0) is of dimension (b + 1) × (b + 1); _1^(0) is of dimension (b + 1) × (r + 1); _-1^(1) is of dimension (r + 1) × (b + 1); and _0^(1), _-1, _0 and _1 are square matrices of dimension r + 1. Note that _-1 + _0 + _1 is a transition rate matrix that describes the behavior of the QBD process in the vertical direction only. The matrixhas negative entries on the main diagonal and non-negative entries elsewhere with row sums equal to zero.§ MODELING QBD PROCESSES In this section we present some examples in various application fields of Markov processes that are QBD processes. We focus mainly on the modeling aspect: the translation of a problem description to a QBD process with a state space and transition matrices.[An insurance company] Claims arrive to an insurance company according to a Poisson process with rate . A claim is important with probability p. To achieve low waiting times for important claims, the insurance company is allowed to hold at most r of these important claims at the same time; if new important claims arrive, they are diverted to a different insurance company. Standard claims are resolved one-by-one independently of the important claims and take μ_1 time each. Important claims are also resolved one-by-one and take μ_2 time each.Denote by X_1(t) and X_2(t) the number of standard and important claims at time t and by X(t)(X_1(t),X_2(t)) the state of the system. Then, { X(t) }_t ≥ 0 is a QBD process with b = r, _-1 = μ_1, _1 = (1 - p), and _0 = [-μ_1 p; μ_2-μ p; μ_2-μ p; ⋱ ⋱ ⋱; μ_2-μ p; μ_2 -μ +p ] - , where μμ_1 + μ_2,is the identity matrix and unlabeled elements of _0 are zero. [Experiments that require setup] A scientist is performing experiments. Requests for an additional experiment arrive according to a Poisson process with rate . An experiment requires two phases of setup; both take θ time. Once setup is completed, experiments can be performed one after the other, where an experiment takes μ time. However, when a request for an additional experiment arrives, the scientist gets distracted and the current experiment and the setup process have to be redone. Knowing this, the scientist does no setup when there are no experiments to be done.Denote by X_1(t) the number of experiments that still need to be done at time t, let X_2(t) be the number of setup phases completed and let the state of the system be described by X(t)(X_1(t),X_2(t)). Then, { X(t) }_t ≥ 0 is a QBD process with b = 0, r = 2, _-1 = [ 0 0 0; 0 0 0; 0 0 μ ], _0 = [ -θθ0;0 -θθ;00 -μ ] - , _1 = [ 0 0; 0 0; 0 0 ]. [A single-server queue in a random environment] A single server with an infinite capacity queue is serving jobs one at a time. An exogenous process (the random environment) changes the parameters of the system, where this process can be in any of r + 1 phases. If the random environment is in phase n, then jobs arrive according to a Poisson process with rate _n and are served with exponential rate μ_n. The only restriction required on the exogenous process is that transitions occur after some exponential time and that all states within a level can be reached.Let the state of the system be denoted by X(t)(X_1(t),X_2(t), where X_1(t) is the total number of jobs in the system at time t and X_2(t) is the phase of the random environment at time t. Then, { X(t) }_t ≥ 0 is a QBD process with b = r, _-1 = μ_0,μ_1,…,μ_r, _1 = _0,_1,…,_r, and _0 = E - _-1 - _1, where E is the generator of the exogenous process and x is a square matrix with the vector x on the main diagonal. [Make to order and make to stock ] Standard products and customer-specific prototypes are produced by the same high-tech company. Demand for standard products arrives according to a Poisson process with rate _1 and demand for prototypes according to a Poisson process with rate _2. If the company has no outstanding orders, it makes standard products to stock. The company is willing to have at most r standard products on stock to avoid high holding costs. A demand for a standard product is immediately satisfied whenever stock is available, otherwise the standard product is produced to order. Prototypes are customer specific and are therefore made to order. Since prototypes yield higher monetary returns, producing these products has preemptive priority over producing standard products. Producing either product takes μ time.If we denote by X_1(t) the total number of outstanding orders (standard plus prototypes) at time t, by X_2(t) the number of standard products on stock at time t and by X(t)(X_1(t),X_2(t)) the state of the system, then { X(t) }_t ≥ 0 is a QBD process with b = r, _-1 = μ, _1 = _1 + _2,_2,…,_2, and _0 = [0; _10;_10;⋱⋱;_10 ] -( + μ), _0^(0) = _0 + [ 0 μ; 0 μ; ⋱ ⋱; 0 μ; 0 ], where _1 + _2. [An encryption server with inspection] A computing facility has a single server that encrypts files. Tasks arrive according to a Poisson process with rateand wait in an infinite queue if the server is busy. Before encrypting a file, the server inspects the contents of the file and decides on a certain technology to use. Inspecting a file takes θ time. The server uses three different encryption types: Data Encryption Standard (type 1), Advanced Encryption Standard (type 2), and RC4 (type 3). A file requires type n encryption with probability p_n and p_1 + p_2 + p_3 = 1. Type n encryption takes μ_n time.Let X_1(t) be the number of files that still need to be encrypted at time t and let X_2(t) be the encryption type of the file being encrypted at time t, where X_2(t) = 0 indicates that the server is still in the process of inspecting the file (or idle, if X_1(t) = 0 as well). The state of the system is denoted by X(t)(X_1(t),X_2(t)). The Markov process { X(t) }_t ≥ 0 is a QBD process with b = 0, r = 3, _1 =, _-1 = [ 0 0 0 0; μ_1 0 0 0; μ_2 0 0 0; μ_3 0 0 0 ], _0 = [-θ θ p_1 θ p_2 θ p_3; 0-μ_1 0 0; 0 0-μ_2 0; 0 0 0-μ_3 ] - . § STABILITY CONDITION From here on we will assume that the QBD process { X(t) }_t ≥ 0 is irreducible and that the transition rate matrixhas exactly one communicating class. The condition for this Markov process to be positive recurrent is an intuitive one and is easily extended from the homogeneous BD processes case. In the homogenous BD case the process is positive recurrent (or stable) if the birth rate is smaller than the death rate. This implies that the process does not drift off to infinity, because the net drift (birth rate minus death rate) is negative. For the QBD case we establish a similar condition for positive recurrence based on the net drift.The QBD process adds a finite number of phases to the BD process and transition rates to the left and right can vary from phase to phase. Just like the BD case, the QBD process should be stable if the mean drift to the left is larger than the mean drift to the right. This way, the process does not drift off to higher and higher levels. Now, the mean drift to the left or right depends on the transition rates at each phase, and more importantly, depends on the fraction of time the process spends in each of its phases. The last quantity is determined from the transition rate matrixof the vertical direction. Let x be the equilibrium distribution of the vertical direction: x = , x = 1. Element j of x is interpreted as the fraction of time that the QBD process is in phase j when it is far away from level 0 (so that boundary effects do not play a role). With this in mind, the mean drift from level i to level i - 1 is x_-1 and the mean drift from level i to level i + 1 is x_1. The net mean drift is then x_1- x_-1 and the process is positive recurrent—also called stable—if the net mean drift is negative. This condition is known as Neuts' mean drift condition <cit.> or the stability condition and we present it here as a theorem. Theprocess { X(t) }_t ≥ 0 is positive recurrent if and only if x_1< x_-1 with x = [ x(0) x(1)⋯ x(r) ]the equilibrium distribution of the Markov process with transition rate matrix _-1 + _0 + _1 x = , x = 1. [An encryption server with inspection] We derive the stability condition of the QBD process of <ref>. The transition rate matrix of the vertical direction is = [-θ θ p_1 θ p_2 θ p_3; μ_1-μ_1 0 0; μ_2 0-μ_2 0; μ_3 0 0-μ_3 ]. The dependent system of equations (<ref>) can easily be solved by replacing the left-most column in the generator (<ref>) by ones (we briefly refer to this modified generator as ^*) and solve the system x^* = [ 1 0 0 0 ] to obtain x = 1/1 + ∑_n = 1^3 θ p_n/μ_n[ 1 θ p_1/μ_1 θ p_2/μ_2 θ p_3/μ_3 ]. So, for this process, the stability condition (<ref>) reads < θ/1 + ∑_n = 1^3 θ p_n/μ_n = 1/1/θ + ∑_n = 1^3 p_n 1/μ_n. The mean drift to the right is clear: from every phase an arrival can occur with rate . The mean drift to the left is the inverse of the mean service time. The service time consists of the setup phase (exponential with mean 1/θ) plus the encryption, where type-n encryption occurs with probability p_n and is exponential with mean 1/μ_n. § MATRIX-GEOMETRIC METHOD The aim of the matrix-geometric methodThe matrix-geometric method was pioneered by Evans <cit.> and Wallace <cit.>, fully developed by Neuts <cit.>, and discussed at length in the classical work of Latouche and Ramaswami <cit.>. is to characterize the equilibrium probabilities p(i,j) lim_t →∞X_1(t) = i, X_2(t) = j,(i,j) ∈, as a matrix-geometric distribution in terms of the levels. In the examples of <ref> we have seen that the rate matrix R plays a key role. This is also true for the general class of QBD processes. Recall from <ref> that element (R)_j,k is the expected time spent in state (i + 1,k) multiplied by -(_0)_j,j before the first return to level i, given the initial state (i,j) with i ≥ 1. Note that -(_0)_j,j is the rate at which the process leaves state (i,j), i ≥ 1. From the interpretation of R we directly conclude that zero rows of _1 correspond to zero rows in R.We denote the equilibrium probability vectors as _0[ p(0,0) p(0,1)⋯ p(0,b) ],_i[ p(i,0) p(i,1)⋯ p(i,r) ],i ≥ 1, and = [ _0 _1⋯ ]. The balance equations for the QBD process with transition rate matrix Q partitioned by levels are given by _0 _0^(0) + _1 _-1^(1) = ,_0 _1^(0) + _1 _0^(1) + _2 _-1 = ,_i - 1_1 + _i _0 + _i + 1_-1 = ,i ≥ 2. The next result, appearing in <cit.>, describes the matrix-geometric structure of the equilibrium probability vectors. Notice that (<ref>) is obtained by substituting (<ref>) into (<ref>). Provided theprocess { X(t) }_t ≥ 0 is irreducible and positive recurrent, the equilibrium probability vector , satisfying Q = , = 1, is given by _i + 1 = _i R = _1 R^i,i ≥ 1, where R, called the , is the minimal non-negative solution of the matrix-quadratic equation R^2 _-1 + R _0 + _1 = 0. The equilibrium probability vectors _0 and _1 follow from the system of equations _0 _0^(0) + _1 _-1^(1) = ,_0 _1^(0) + _1 ( _0^(1) + R _-1)= , and the normalization condition _0+ _1 ( - R)^-1 = 1. For the computation of the rate matrix R we may rewrite (<ref>) in the form R = - ( _1 + R^2 _-1) _0^-1. The matrix _0 is indeed invertible, since it is a transient generator, which means that it is the transition rate matrix of a transient Markov process. More precisely, for a transient (substochastic) transition probability matrix P associated with a Markov chain, we know that P^n → 0 as n → 0. This convergence is geometric, the decay parameter of which is the largest eigenvalue of P, which is less than 1. So, the series ∑_n ≥ 0 P^n = ( - P)^-1 converges and therefore the inverse of I - P exists. Now, in the continuous time setting we can construct the transient transition probability matrix P as P =+ Δ Q, where 0 < Δ < max_i - (Q)_i,i. Since P is a transient transition probability matrix, the series ∑_n ≥ 0 P^n = ( - P)^-1 = (-Δ Q)^-1, and therefore Q is invertible.The above fixed point equation (<ref>) may be solved by straightforward successive substitutions, so R_n + 1 = - ( _1 + R_n^2 _-1) _0^-1,n ≥ 0, starting with R_0 = 0 and R_n ↑ R as n →∞.Neuts shows that R_n ↑ R as n →∞, see <cit.> for the proof in case of a discrete-time QBD chain. More sophisticated and efficient numerical schemes for determining R have been developed, in particular, cyclic reduction <cit.> and logarithmic reduction <cit.>. For computational purposes a stopping criterion is required: we show one choice for a stopping criterion in <ref>. Note that for a matrix A the max norm is A _max_i,j |(A)_i,j|. §.§ Explicit solutions for the rate matrix We have seen in <ref> that the rate matrix R can be determined explicitly. Also the example of <ref> admits an explicit solution for R. This is not always the case, however. We now review two cases in which the rate matrix R can be determined explicitly.Ramaswami <cit.> treats the two cases in which the R matrix can be obtained explicitly.The first case assumes that the transition rate matrix _-1 with transitions to the left is of the form _-1 = , whereis a column vector andis a stochastic row vector, both of dimension r + 1: = [ _0; _1;⋮; _r ],= [ _0 _1⋯ _r ],= 1, , > where x > indicates that all elements of x are non-negative and at least one element is positive. This means that all rows of _-1 are the same up to some scaling: from all states (i + 1,j), j = 0,1,…,r the probability of jumping to state (i,k), k = 0,1,…,r is independent of the starting state in level i + 1. We investigate the consequences for the rate matrix R.Substitution of (<ref>) into the balance equations for level i > 1 yields _i - 1_1 + _i _0 + _i + 1 = . To eliminate _i + 1 from this equation we derive a relation between _i and _i + 1 by equating the flow between level i and level i + 1: _i _1= _i + 1_-1 = _i + 1 = _i + 1. Substituting (<ref>) into (<ref>), we obtain _i - 1_1 + _i _0 + _i _1= , which can be rewritten as _i = _i - 1 R,i > 1, with the explicit formulation R = - _1 ( _0 + _1 )^-1, where _0 + _1 is invertible, since it is a transient generator.The second case for which R can be solved explicitly is when _1 is of the form _1 = . Similarly to the first case, this means that all rows of _1 are the same up to some scaling.From the recursive scheme (<ref>) we obtain R_0 = 0,R_1 = -_1 _0^-1 = - _0^-1 = _1, with row vector _1 = - _0^-1. Repeating the iteration shows that all R_n's are of the form R_n = _n, where _n > is a row vector of dimension r + 1. Since R_n ↑ R as n →∞, we conclude that _n ↑ and R = , for some vector >. So, R is a matrix of rank 1 and has a single non-zero eigenvalue which is equal to R. This implies that R^i = ()^i - 1 R = η^i - 1 R,i ≥ 1 where η = R and equal to the spectral radius R defined as Amax{ |η_0|,|η_1|,…,|η_r| }, where η_0,η_1,…,η_r are the eigenvalues of a matrix A of dimension r + 1.Observing (<ref>), the matrix-geometric form in <ref> reduces to _i + 1 = _1 R^i = η^i - 1_1 R = η^i - 1_2,i ≥ 1. What remains is to determine η. The spectral radius η of R for the case _1 = can be characterized as the unique root in (0,1) of the determinant equation ( _1 + η_0 + η^2 _-1) = 0.Using (<ref>) in (<ref>) results in R = - _1 ( _0 + η_-1)^-1. The eigenvalue η then follows from 0= ( R - η) = ( - _1 ( _0 + η_-1)^-1 - η) = ( ( _1 + η ( _0 + η_-1) ) ( - _0 - η_-1)^-1) = ( _1 + η_0 + η^2 _-1) ( ( - _0 - η_-1)^-1). Since η < 1, - _0 - η_-1 is nonsingular. So, η satisfies 0 = ( _1 + η_0 + η^2 _-1). Establishing that there is a single η∈ (0,1) follows from <cit.>.§.§ Exact solutions for the rate matrix If the transition matrices _-1, _0, and _1 are all upper or all lower triangular, then the interpretation of the elements of R shows us that the rate matrix is also upper or lower triangular. Determining R in this case is made easier by exploiting its structure.This class of QBD processes (or slight variations of it) is briefly mentioned in <cit.> and has hence been studied extensively in the literature and has lead to many variations of exact or explicit solutions for R. Some examples include exploiting structural properties of (<ref>) in <cit.>; lattice path counting solutions <cit.>; and renewal-reward based approaches in <cit.>.We will highlight the application of the methodology in <cit.> to QBD processes with upper triangular transition matrices. Since R is upper triangular, R^2 has the same upper triangular structure and has elements (R^2)_i,j = ∑_k = i^j (R)_i,k (R)_k,j, 0 ≤ i ≤ j ≤ r. From (<ref>) we deduce that the diagonal elements are the minimal non-negative solution of (R)_i,i^2 (_-1)_i,i + (R)_i,i (_0)_i,i + (_1)_i,i = 0,0 ≤ i ≤ r. So, (R)_i,i =-(_0)_i,i - √(( (_0)_i,i)^2 - 4 (_-1)_i,i (_1)_i,i)/2(_-1)_i,i. The elements on the superdiagonal of R are determined recursively and also follow from (<ref>), for 0 ≤ i < j ≤ r: ∑_k = i^j ∑_l = i^k (R)_i,l (R)_l,k (_-1)_k,j + ∑_k = i^j (R)_i,k (_0)_i,j + (_1)_i,j = 0. Solving the above equation for (R)_i,j yields (R)_i,j = - R_(i,j) + (_1)_i,j/( (R)_i,i + (R)_j,j) (_-1)_j,j + (_0)_j,j,0 ≤ i < j ≤ r, where R_(i,j)= ∑_k = i^j - 1( ∑_l = i^k (R)_i,l (R)_l,k (_-1)_k,j + (R)_i,k(_0)_k,j)+ ∑_k = i + 1^j - 1 (R)_i,k (R)_k,j (_-1)_j,j, with the convention ∑_n = n_0^n_1 f(n) = 0 if n_0 > n_1. The above equation is a recursion along the superdiagonals of R, which should be solved starting at the superdiagonal closest to the main diagonal and moving to the top right corner of the matrix.We finally note that the inverse of an upper triangular matrix is again upper triangular; the same applies for lower triangular matrices. For the determination of _0 and _1 the inverse (I - R)^-1 is required. Provided the diagonal elements of an upper triangular matrix A of dimension r + 1 are non-zero, the inverse can be determined as (A^-1)_i,i = 1/(A)_i,i,0 ≤ i ≤ r, (A^-1)_i,j = - (A^-1)_i,i∑_k = i + 1^j (A)_i,k (A^-1)_k,j,0 ≤ i < j ≤ r, where the recursion should be solved along the superdiagonals, starting at the main diagonal, followed by the superdiagonal closest to the main diagonal, and so forth, exactly as for R.§ MATRIX-ANALYTIC METHOD The main object of study in the matrix-analytic methodThe matrix-analytic method was pioneered by Neuts <cit.> in the 1980's and developed further in <cit.>. As we will see in later chapters, the matrix-geometric and matrix-analytic methods have their own area of application, but these areas overlap in case of QBD processes. A review of the matrix-analytic method and recent tutorial on the application of this method can be found in <cit.> and <cit.>, respectively. is the auxiliary matrix G. Element (G)_j,k is a first passage probability, defined as the probability that, starting at level i ≥ 2 in state (i,j), the first passage to levels i - 1 and below happens in state (i - 1,k). Note that indeed, if i ≥ 2, the first passage probabilities do not depend on i due to the homogeneous transition structure. Moreover, if the QBD process is positive recurrent and irreducible, then G is a right stochastic matrix.Similar to the derivation of G in <ref>, define an excursion as a sample path of the process starting at level i, moving to levels higher than i and ending at first return to level i. Clearly, the number of excursions per time unit that end in state (i,j) is equal to ∑_k = 0^r p(i + 1,k) (_-1)_k,j. Next, the number of excursions per time unit that start from state (i,k), immediately go to state (i + 1,l), and ultimately end the excursion in state (i,j) is p(i,k) (_1)_k,l (G)_l,j, where we exploit the interpretation of the elements of G. Summing over all possible starting states in level i and the state first visited in level i + 1 also gives us the number of excursions per time unit that end in state (i,j), namely ∑_k = 0^r p(i,k) ∑_l = 0^r (_1)_k,l (G)_l,j. Equating both expressions for the number of excursions per time unit that end in state (i,j), 0 ≤ j ≤ r and writing it in vector-matrix form yields _i + 1_-1 = _i _1 G. Substituting this relation in (<ref>) produces _i - 1_1 + _i ( _0 + _1 G) = . Based on this probabilistic derivation we have the following equivalent result to <ref> for the matrix-analytic method.The formal proof of <ref> follows from <ref> in combination with <cit.>. Provided theprocess { X(t) }_t ≥ 0 is irreducible and positive recurrent, the stationary probability vector , satisfying Q = , = 1, is given by _i + 1 = _i _1(- _0 - _1 G)^-1 = _1 ( _1(- _0 - _1 G)^-1)^i,i ≥ 1, where G is the right stochastic solution of the matrix-quadratic equation _-1 + _0 G + _1 G^2 = 0. The equilibrium probability vectors _0 and _1 follow from the system of equations _0 _0,0 + _1 _1,-1 = ,_0 _0,1 + _1 ( _1,0 + _1 G )= , and the normalization condition _0+ _1 ( - R)^-1 = 1. We can immediately conclude from <ref> and (or <cit.>) that there exist multiple relations between the rate matrix R and the auxiliary matrix G. We have _1 G = R _-1, R = _1(- _0 - _1 G)^-1 and G = (-_0 - R _-1)^-1_-1.In a similar fashion as for the rate matrix R, the auxiliary matrix G can be determined by successive substitutionFaster and more efficient algorithms than successive substitution exist to determine G; a good example is cyclic reduction <cit.>., see <ref>. Explicit and exact results exist also for the auxiliary matrix G, derived in an analogous manner to <ref>.§ SPECTRAL EXPANSION METHOD A third approach to determining the equilibrium probabilities does not make use of matrices but rather eigenvectors and eigenvalues. This approach is called the spectral expansion method.The spectral expansion method was developed by Mitrani <cit.>. The basic idea of this method is to first try and find basis solutions of the form _i = y x^i - 1,i ≥ 1, where y = [ y(0) y(1)⋯ y(r) ]≠ and |x| < 1, satisfying the balance equations (<ref>) for i ≥ 2. We require that |x| < 1, since we want to be able to normalize the equilibrium distribution. Substitution of (<ref>) in (<ref>) and dividing by common powers of x yields y( _-1 + x _0 + x^2 _1 ) = . These equations have a non-zero solution for y if ( _-1 + x _0 + x^2 _1 ) = 0. The desired values of x are the roots x with |x| < 1 of the determinant equation (<ref>). Equation (<ref>) is a polynomial equation of degree 2(r + 1). Suppose that r̃ + 1 roots x satisfy |x| < 1 and for now let us assume that these roots are different. Let y_k, k = 0,1,…,r̃ be the corresponding non-zero solutions of (<ref>) for x = x_k. Each solution _i = y_k x_k^i, k = 0,1,…,r̃ satisfies the global balance equations (<ref>). These solutions are moreover linearly independent. We can linearly combine the r̃ + 1 solutions to obtain a solution that satisfies the global balance equations (<ref>): _i = ∑_k = 0^r̃ξ_k y_k x_k^i,i ≥ 1, where ξ_k, k = 0,1,…,r̃ are arbitrary constants. So far, we have obtained expressions for _1,_2,…, which still contains r̃ + 1 unknowns ξ_k. Now, to determine these unknowns and _0, we turn to the global balance equations for levels 0 and 1. Equations (<ref>) and (<ref>) are a set of b + 1 + r + 1 linear equations involving the b + 1 unknown probabilities of level 0 and the r̃ + 1 unknowns constants ξ_k. The set of equations (<ref>) and (<ref>) only has b + 1 + r linearly independent equations, but an additional independent equation is provided by the normalization condition. In conclusion, the set of equations (<ref>) and (<ref>) only has a unique solution if the number of equations and unknowns match, which means that r̃ is required to be equal to r. Since an irreducible and positive recurrent (by <ref>) QBD process has a unique solution to the global balance equations, we have the following theorem. An irreducible and positive recurrentprocess has r + 1 solutions x with |x| < 1 of (<ref>). Assume these roots are different and label them x_0,x_1,…,x_r. Let y_k, k = 0,1,…,r be the non-zero solution of (<ref>) for x = x_k. The linear combination of basis solutions _i = ∑_k = 0^r ξ_k y_k x_k^i,i ≥ 1, is the unique equilibrium distribution. The coefficients ξ_0,ξ_1,…,ξ_r and _0 are the unique solution to the global balance equations of levels 0 and 1, _0 _0,0 + ∑_k = 0^r ξ_k y_k _1,-1 = ,_0 _0,1 + ∑_k = 0^r ξ_k y_k _1,0 + ∑_k = 0^r ξ_k y_k x_k _-1 = , and the normalization condition _0+ ∑_k = 0^r ξ_k y_k /1 - x_k = 1. The roots x_0,x_1,…,x_r do not have to be different. If we assume that, when a root x occurs k times, it is possible to find k linearly independent solutions of (<ref>), then the analysis proceeds in exactly the same way. In case there are less than k independent solutions, we would also have to consider more complicated basis solutions of the form i y x^i - 1 (or even higher powers of i).The relation between the matrix-geometric representation (<ref>) and the spectral expansion (<ref>) is clear: the roots x_0,x_1,…,x_r are the eigenvalues of R with corresponding left eigenvectors y_0,y_1,…,y_r.§ TAKEAWAYS Birth–and–death (BD) processes live on the positive half-line and move either to the left or right after exponential times. These basic features of BD processes, laid out in <ref>, proved essential for the theory developed in this part of the book. In <ref> we exploited the BD structure to construct multi-dimensional versions, first for loss networks and then for queueing networks.In this chapter we added a finite number of states to each state of that BD half-line, to construct quasi-birth–and–death (QBD) processes that live on a semi-infinite strip of states. The basic recursion method for BD processes was then lifted to the more general QBD setting to obtain the equilibrium distribution. Where BD processes result in geometric equilibrium distributions, QBD processes obey similar geometric forms, but with scalars replaced by matrices. This also explains why the main analytic technique introduced in this chapter is called the matrix-geometric method.The matrix-geometric method exploits the fact that the QBD process has a highly structured state space, which allows for describing the balance equations in terms of transitions in the horizontal direction only. All transitions in the vertical directions are described in terms of finite matrices that appear in the balance equations. The matrix-geometric method is than the analytic methods for solving the system of matrix equations. The central step is to prove the existence of the unique rate matrix R in <ref>. We have also presented efficient algorithms to determine R numerically. Taken together, this provides a powerful computational framework for QBD processes.Besides the matrix-geometric method, we have also demonstrated the matrix-analytic method with its auxiliary matrix G and the spectral expansion method. The matrix-analytic method is similar in scope to the matrix-geometric method, but its focus is on the transitions to the left with as a result the first passage probabilities in the G matrix. The spectral expansion method decomposes the R matrix into its eigenvectors and eigenvalues and linearly combines them to construct the product-form solution. At first sight the extension from BD processes to QBD processes might seem less spectacular than the extension to the network models in <ref>. To fully appreciate the wide scope of QBD processes, the key insight is that the computational complexity of the matrix-geometric method is determined by the finite dimension, i.e., the dimension of the rate matrix R. This is remarkable, because without exploiting the special QBD structure, we would face a Markov process living on an infinite state space, and simply trying to solve the global balance equations would in many cases be prohibitively difficult. So if a Markov process can be brought into a QBD form, this brings enormous computational advantages.Take as an example a single-server queue with generally distributed i.i.d. inter-arrival times and generally distributed i.i.d. service times. Approximate the inter-arrival and service times by phase-type distributions and use the finite dimension to keep track of these phases. Only arrivals and service completions then result in horizontal transitions, while all other events trigger transitions in the vertical direction. The fairly intractable general queueing systems is then converted into a QBD process, and performance analysis of the systems becomes straightforward.This example shows that effort should be put in constructing the QBD process, and then one can reap the benefits of reduced complexity. More generally, the additional finite dimension in QBDs can keep track of enormous amounts of information and this partly explains why so many real-world systems can be modeled as QBD processes <cit.>. In the next chapter we discuss the extension of QBD processes to Markov process of a similar structure, but with the possibility to take large steps in the horizontal direction. CHAPTER: QUASI-SKIP-FREE PROCESSES Quasi-skip-free (QSF) processes are the generalization to two dimensions of the Markov processes associated with the M/G/1 system and the G/M/1 system. The QSF process has the same state space as the QBD process, but its transition structure is different. Whereas the QBD process is skip-free in both directions, the QSF process allows for transitions of larger size in one of the two directions. A distinction is made for processes that are QSF to the right and to the left, since each process requires a different solution approach for the equilibrium distribution.§ VARIATIONS OF SKIP-FREE PROCESSES In this section we analyze two QSF processes that both are constructed from a QBD process. The resulting QSF processes are skip-free in different directions and therefore require different solution methods to obtain their equilibrium distributions. §.§ Machine with setup times and batch arrivals Let us consider an adaptation of the machine with setup times as mentioned before in <ref>. The machine processes jobs in order of arrival. Jobs arrive in batches to the system: batches of size 1 and 2 arrive according to Poisson processes with rates _1 and _2. The processing time of a job is exponentially distributed with mean 1/μ. For stability we assume that ρ (_1 + 2)/μ < 1. The machine is turned off when the system is empty and it is turned on again when a new batch of jobs arrives. The setup time is exponentially distributed with mean 1/θ. Turning off the machine takes an exponential amount of time with mean 1/γ.The state of the system may be described by X(t)(X_1(t),X_2(t)) with X_1(t) representing the number of jobs in the system at time t and X_2(t) indicates whether the machine is turned off (0) or on (1) at time t. The process { X(t) }_t ≥ 0 is a Markov process with state space { (i,j) ∈_0 ×{ 0,1 }}. The transition rate diagram is displayed in <ref> and resembles the QBD variant in <ref>, depicting the transition rate diagram of the system where jobs arrive one by one and turning off the machine takes no time.For the current model, define i = { (i,0),(i,1) }, i ≥ 0 as the set of states with i jobs in the system, that is, i is level i. We can then partition the state space as 0∪1∪2∪⋯The Markov process { X(t) }_t ≥ 0 is QSF to the left: the process cannot skip any levels when transitioning to the left, whereas it can skip a level when transitioning to the right due to a batch arrival of size 2. We demonstrate how the matrix-analytic method can be applied to determine the equilibrium distribution of this QSF process.Let p(i,j) denote the equilibrium probability of state (i,j) ∈. From the transition rate diagram we can obtain the balance equations by equating the flow out of a state and the flow into that state. For the boundary states we have, with _1 + _2, p(0,0) = γ p(0,1), ( + γ) p(0,1) = μ p(1,1), ( + θ) p(1,0) = _1 p(0,0), ( + μ) p(1,1) = _1 p(0,1) + θ p(1,0) + μ p(2,1), and for i ≥ 2, ( + θ) p(i,0)= _1 p(i - 1,0) + _2 p(i - 2,0), ( + μ) p(i,1)= _1 p(i - 1,0) + _2 p(i - 2,0) + θ p(i,0) + μ p(i + 1,1). Let us introduce the vectors of equilibrium probabilities _i = [ p(i,0) p(i,1) ] and write (<ref>)–(<ref>) in vector-matrix notation: _0 _0^(0) + _1 _-1 = ,_0 _1 + _1 _0 + _2 _-1 = ,_i - 2_2 + _i - 1_1 + _i _0 + _i + 1_-1 = ,i ≥ 2, where _-1 = [ 0 0; 0 μ ], _0 = [ -( + θ) θ; 0 -( + μ) ],_k = [ _k0;0 _k ], k = 1,2, _0^(0) = [-0;-( + ) ]. The balance equations (<ref>)–(<ref>)are referred to as the boundary equations. We show how the matrix-analytic method for QBD processes can be applied to processes that are QSF to the left.The auxiliary matrix G plays a key role in the matrix-analytic method. Element (j,k) of G is interpreted as the probability that, starting in state (i,j), i ≥ 2, the first passage to level i - 1 happens in state (i - 1,k). More generally, element (j,k) of G^n is interpreted as the probability that, starting in state (i,j), i ≥ n + 1, the first passage to level i - n happens in state (i - n,k). This immediately implies that zero columns in _-1 lead to zero columns in G. For the model at hand G = [ 0 1; 0 1; ]. This matrix appears when we censor the Markov process to particular sets of states. Define the union of levels 0,1,…,i as i0∪1∪⋯∪i. Censoring the Markov process to i means that we only observe the Markov process when it resides in a state in i and transitions to states outside this set are redirected appropriately to states inside the set.Suppose we censor the Markov process to i, i ≥ 2 and write down the balance equations for level i. From i - 2 the process transitions to i with rate _i - 2_2. From i - 1 the process transitions to i with rate _i - 1_1, but also with rate _i - 1_2 G, since in that case the process transitions to i + 1 and returns to i according to the probabilities described in G. Similarly, the contribution of i to the balance equations is _i (_0 + _1 G + _2 G^2). So, for the censored process the balance equations for i are _i - 2_2 + _i - 1( _1 + _2 G ) + _i ( _0 + _1 G + _2 G^2 ) = ,i ≥ 2. We can derive similar balance equations for 1 when censoring the process to 1 and for 0 when censoring the process to 0 = 0: _0( _1 + _2 G ) + _1 ( _0 + _1 G + _2 G^2 )= ,_0 ( _0^(0) + _1 G + _2 G^2 )= . Equation (<ref>) is a homogeneous system of equations which does not have a unique solution, butwe can at least conclude that _0 is proportional to [ ]. We supplement this system of equations with the normalization condition to be able to uniquely determine _0.We examine the series ∑_i ≥ 0_i with the goal of finding another expression involving _0. Abbreviate _0 = _0 + _1 G + _2 G^2, _1 = _1 + _2 G, _2 = _2. Use (<ref>)–(<ref>) to rewrite the summation ∑_i ≥ 0_i: ∑_i ≥ 0_i= _0 + _1 + ∑_i ≥ 2_i = _0 - _0 _1 _0^-1 - ∑_i ≥ 2( _i - 2_2 + _i - 1_1 ) _0^-1= _0 - _0 _1 _0^-1 - ∑_i ≥ 0_i - 2_2 _0^-1 - ∑_i ≥ 1_i - 1_1 _0^-1= _0 - ∑_i ≥ 0_i ( _1 + _2 ) _0^-1. Hence, ∑_i ≥ 0_i (+ ( _1 + _2 ) _0^-1) = _0, which gives ∑_i ≥ 0_i = _0 _0 ( _0 + _1 + _2 )^-1, where the inverse is given by ( _0 + _1 + _2 )^-1 = [ -1/θ 1/θ( 1 + θ + μ/_1 + 2_2 - μ);0 1/_1 + 2_2 - μ ]. So, the normalization condition is 1 = ∑_i ≥ 0_i= _0 _0 ( _0 + _1 + _2 )^-1. Substituting the normalization condition (<ref>) for any of the equations in (<ref>) allows us to calculate _0. Armed with _0 we are able to solve (<ref>) for _1, after which we can iteratively calculate _i, i ≥ 2 from (<ref>), stopping when the accumulated probability mass is close to 1. §.§ A batch machine subject to breakdowns Consider a batch machine that is subject to breakdowns. Depending on the details of the jobs, the batch machine can sometimes serve two jobs at the same time, but sometimes only a single job. For simplicity we assume that with probability 1/2 the batch machine serves a single job and with the same probability two jobs. The service time is independent of the number of jobs in service and is exponentially distributed with rate 2μ. Jobs arrive to the system according to a Poisson process with rate . The machine breaks down after an exponential amount of time with rate(irrespective of whether it is serving a job or not) and repair takes an exponential amount of time with rate θ. Every time the machine breaks down, the repair is started immediately. If there is only a single job in the system, the machine serves this single job with probability 1.Notice that the machine works a fraction θ/( + θ) of the time and is in repair a fraction /( + θ) of the time. So, the rate at which the server can serve jobs is θ/( + θ) · 2μ (1 · 1/2 + 2 · 1/2) and therefore the stability condition is ρ/θ/ + θ (μ + 2μ) < 1.The state of the system may be described by X(t)(X_1(t),X_2(t)) with X_1(t) representing the number of jobs in the system at time t and X_2(t) describes if the machine is working (1) or not (0) at time t. The process { X(t) }_t ≥ 0 is a Markov process with state space { (i,j) ∈_0 ×{ 0,1 }}. The transition rate diagram is displayed in <ref>. We use the same levels as in the previous example, i.e., i = { (i,0),(i,1) }, i ≥ 0.The Markov process { X(t) }_t ≥ 0 is QSF to the right: the process cannot skip any levels when transitioning to the right, whereas it can skip a level when transitioning to the left with a batch service of size 2. We demonstrate how the matrix-geometric method and spectral expansion method can be adapted to determine the equilibrium distribution of this QSF process.Let p(i,j) denote the equilibrium probability of state (i,j) ∈. From the transition rate diagram we can obtain the balance equations by equating the flow out of a state and the flow into that state. For the boundary states we have, ( + θ) p(0,0) =p(0,1), ( + ) p(0,1) = 2μ p(1,1) + μ p(2,1) + θ p(0,0), ( + θ) p(1,0) =p(0,0) +p(1,1), ( + 2μ + ) p(1,1) =p(0,1) + θ p(1,0) + μ (p(2,1) + p(3,1)), and for i ≥ 2, ( + θ) p(i,0)=p(i - 1,0) +p(i,1), ( + 2μ + ) p(i,1)=p(i - 1,0) + θ p(i,0) + μ (p(i + 1,1) + p(i + 2,1)). Let us introduce the vectors of equilibrium probabilities _i = [ p(i,0) p(i,1) ] and write (<ref>)–(<ref>) in vector-matrix notation: _0 _0^(0) + _1 _-1^(1) + _2 _-2 = ,_i - 1_1 + _i _0 + _i + 1_-1 + _i + 2_-2 = ,i ≥ 1, where _-2 = _-1 = [ 0 0; 0 μ ], _0 = [ -( + θ) θ; -( + 2μ + ) ],_1 = [ 0; 0 ], _0^(0) = [ -( + θ) θ;-( + ) ], _-1^(1) = [00;0 2μ ]. The balance equations (<ref>) are referred to as the boundary equations. We will first show how the matrix-geometric method for QBD processes can be applied to processes that are QSF to the right.Recall from the QBD processes that the matrix-geometric method expresses the equilibrium probability vectors as _i = _0 R^i,i ≥ 0. Substituting (<ref>) into (<ref>) gives _i - 1( _1 + R _0 + R^2 _-1 + R^3 _-2) = ,i ≥ 1, which holds if the rate matrix R is the solution to the matrix equation _1 + R _0 + R^2 _-1 + R^3 _-2 = 0. It can be shown that R is the unique minimal non-negative solution of (<ref>) and has spectral radius less than 1, which shows that ( - R)^-1 exists. Equation (<ref>) can be solved via successive substitutions in a similar fashion as <ref>.The boundary equilibrium probability vector _0 can be determined by substituting (<ref>) in the balance equations(<ref>): _0 ( _0^(0) + R _-1^(1) + R^2 _-2) = . This homogeneous system of equations does not have a unique solution. However, if we substitute any of its equations by the normalization condition 1 = ∑_i ≥ 0_i= _0+ ∑_i ≥ 1_1 R^i - 1 = _0+ _1 (- R )^-1, we get a non-homogeneous system of equations with unique solution _0, and through (<ref>) we find all _i.We now demonstrate the spectral expansion method. Recall that this method tries to find basis solutions of the form _i = y x^i,i ≥ 0, where y = [ y(0) y(1) ]≠ and |x| < 1, satisfying the balance equations (<ref>). We require that |x| < 1, since we want to be able to normalize the _i. Substitution of (<ref>) in (<ref>) and dividing by common powers of x yields y( _1 + x _0 + x^2 _-1 + x^3 _-2) = . These equations have a non-zero solution for y if ( _1 + x _0 + x^2 _-1 + x^3 _-2) = 0. The desired values of x are the roots x with |x| < 1 of the determinant equation (<ref>). In this case (<ref>) is a polynomial of degree four in x. One of the solutions of (<ref>)is x = 1, since _1 + _0 + _-1 + _-2 is the transition rate matrix of the Markov process that describes the phase transitions. Now, (<ref>) reads (x - 1)( x^3( + θ)μ + x^2 ( + 2θ)μ - x ( + 2μ + θ +) + ^2 ) = 0, which has two roots x_1 and x_2 inside the open unit disk.These rootshave an explicit expression, but are more difficult to write down because they originate from a cubic equation. For k = 1,2, let y_k be the non-zero solution of y_k ( _1 + x_k _0 + x_k^2 _-1 + x_k^3 _-2) = . Note that, since the balance equations are linear, any linear combination of the two solutions satisfies (<ref>). Now the final step of the spectral expansion method is to determine a linear combination that also satisfies the boundary equations (<ref>). So we set _i = ξ_1 y_1 x_1^i + ξ_2 y_2 x_2^i,i ≥ 0. We can determine the coefficients ξ_1 and ξ_2 by substituting (<ref>) into (<ref>), which gives ∑_k = 1^2 ξ_k y_k ( _0^(0) + x_k _-1^(1) + x_k^2 _-2) = . We can substitute the normalization condition for one of the above homogeneous equations to uniquely determine the coefficients ξ_1 and ξ_2. The normalization condition states that 1 = ∑_i ≥ 0_i = ξ_1 y_1/1 - x_1 + ξ_2 y_2/1 - x_2. Determination of the coefficients is now a straightforward task.§ GENERAL QUASI-SKIP-FREE PROCESSES From the previous examples we have seen that processes that are QSF to the left or right share the same state space as the QBD process that we have encountered in <ref>. We can therefore use the similar level definitions as before: 0{ (0,0),(0,1),…,(0,b) }, i{ (i,0),(i,1),…,(i,r) }, i ≥ 1, with b and r non-negative finite integers and partition the state space as 0∪1∪2∪⋯ We denote the state of the QSF process at time t as X(t)(X_1(t),X_2(t)) where X_1(t) describes the level and X_2(t) describes the phase at time t.Throughout this chapter we focus on homogeneous QSF processes, which means that transition rates are level-independent, possibly except for the transition rates from and to level 0. We can now identify the two types of QSF processes. Using the level-independent _n and level-dependent _n^(m) transition sub-matrices, we have that the transition rate matrix Q of a process that is QSF to the left is of the form Q = [_0^(0)_1^(0)_2^(0)_3^(0)_4^(0) ⋯; _-1^(1)_0_1_2_3 ⋯; _-1_0_1_2 ⋯; _-1_0_1 ⋯; ⋱ ⋱ ⋱ ]. The transition rate matrix Q of a process that is QSF to the right is given by Q = [_0^(0)_1^(0); _-1^(1)_0_1; _-2^(2) _-1_0_1; _-3^(3) _-2 _-1_0_1; ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ].By comparing these transition rate matrices (<ref>)–(<ref>) with the transition rate matrix of the QBD process in (<ref>), we see that the QBD process is a process that is QSF to both the left and the right.The balance equations Q = in case of a process that is QSF to the left are given by _0 _0^(0) + _1 _-1^(1) = ,_0 _i^(0) + ∑_k = 1^i + 1_k _i - k = ,i ≥ 1, and for a process that is QSF to the right we have ∑_k ≥ 0_k _-k^(k) = ,_0 _1^(0) + ∑_k ≥ 1_k _1 - k = ,∑_k ≥ i - 1_k _i - k = ,i ≥ 2.Each of the two examples in <ref> are analyzed by using different methods. The success of a method depends on the skip-free direction of the QSF process. That is, the matrix-geometric method or the spectral expansion method do not work for processes that are QSF to the left and one cannot use the matrix-analytic method to analyze processes that are QSF to the right. The application of these methods to QSF processes is similar to their application to QBD processes, so the treatment of these methods in the next sections will be brief.§ STABILITY CONDITION There is a natural extension of the stability condition for QBD processes seen in <ref> to the stability condition for processes that are QSF to the left or right. We summarize the results for both types in a single theorem.Recall that the transition rate matrixdescribes the transition behavior of the phases. We unify both types of QSF processes by setting ∑_i = -∞^∞_i. From here on we will assume that the QSF process { X(t) }_t ≥ 0 is irreducible and that the transition rate matrixhas exactly one communicating class. Let x be the equilibrium distribution of the Markov process with transition rate matrix : x = , x = 1. Using distribution x we can formulate a mean drift condition that generalizes the one for QBD processes. It asserts that the mean drift to the right is smaller than the mean drift to the left. Theprocess { X(t) }_t ≥ 0 is positive recurrent if and only if x∑_i ≥ 1_i< x∑_i ≤ -1_i with x = [ x(0) x(1)⋯ x(r) ] the equilibrium distribution of the Markov process with transition rate matrixx = , x = 1. § MATRIX-GEOMETRIC METHOD The matrix-geometric method is applicable to processes that are QSF to the right and have as their transition rate matrix the one shown in (<ref>). Instrumental for the approach is the rate matrix R that gives rise to the matrix-geometric relation _i = _1 R^i - 1,i ≥ 1. In the QBD case, the matrix R is the minimal non-negative solution of the matrix-quadratic equation (<ref>). In the QSF case this equation is no longer quadratic. In fact, R is the minimal non-negative solution of ∑_i ≥ 0 R^i _1 - i = 0. The largest eigenvalue (in terms of absolute value) of the matrix R is less than one, which ensures that - R is invertible. Of course, given that R satisfies (<ref>), it readily follows that the matrix-geometric representation (<ref>) satisfies the balance equations (<ref>): substitution of (<ref>) into (<ref>) gives _i - 1∑_k ≥ 0 R^k _1 - k = 0,i ≥ 2, which is valid because of (<ref>). Equation (<ref>) can be rewritten as R = - ( _1 + ∑_i ≥ 2 R^i _1 - i) _0^-1. To numerically solve this equation we first have to truncate the infinite sum at K say, and then compute an approximation for R by successive substitutions as done in <ref>. The larger K, the better the resulting approximation for R, but also the higher the computational effort to compute this approximation. We finally mention that the rate matrix R has the same probabilistic interpretation as in a QBD process.Once we have determined R, we can solve for the remaining equilibrium probability vectors _0 and _1. Substituting (<ref>) into the balance equations (<ref>)–(<ref>) for levels 0 and 1 gives _0 _0^(0) + _1 ∑_k ≥ 1 R^k - 1_-k^(k) = ,_0 _1^(0) + _1 ∑_k ≥ 1 R^k - 1_1 - k = . Replacing one of the boundary equations with the normalization condition 1 = ∑_i ≥ 0_i= _0+ _1 ∑_i ≥ 1 R^i - 1 = _0+ _1 ( - R)^-1 allow us to uniquely determine _0 and _1. We do, however, need to truncate the infinite series in (<ref>)–(<ref>) to be able to numerically determine _0 and _1.§ MATRIX-ANALYTIC METHOD Processes that are QSF to the left do not have a matrix-geometric representation of the equilibrium probability vectors. In <ref> we have developed a recursive scheme involving the auxiliary matrix G to determine the equilibrium probability vectors. In general, for processes that are QSF to the left, G is the minimal non-negative solution of ∑_i ≥ 0_-1 + i G^i = 0. Similarly as for the calculation of R, we are required to truncate the infinite sum at K say, and then approximate G using successive substitutions, which leads to <ref>. The matrix G has the same probabilistic interpretation as in a QBD process.In <ref> we censored the QSF process to i for all i and wrote down the balance equations for i. We can use the same principal in the present case to develop a recursive scheme for _i. We will use the following notation to develop that scheme: Γ_i^(0)∑_k ≥ 0_i + k^(0) G^k, i ≥ 0, Γ_i ∑_k ≥ 0_i + k G^k, i ≥ 0. The matrix Γ_i^(0) describes the rates at which the QSF process enters i from 0 in a single transition when the process is censored to i. The matrix Γ_i describes the rates at which the QSF process enters i + j from j, j ≥ 1 in a single transition when the process is censored to i + j.Using these definitions, we can censor the QSF process to 0 to develop the relation _0 Γ_0^(0) =. Censoring the QSF process to i, i ≥ 1, provides the recursive relationThe recursive scheme in the matrix-analytic method for levels i ≥ 1 is called Ramaswami's formula and was developed by Ramaswami in <cit.>. This formula is numerically stable because it involves only additions and multiplications of non-negative matrices and vectors; subtractions could lead to numerical instability due to the loss of significant figures. _0 Γ_i^(0) + ∑_j = 1^i _j Γ_i - j = ,i ≥ 1. Now, if we are able to determine _0, then we can use (<ref>) to determine any _i. The homogeneous system of equations (<ref>) does not have a unique solution, so we aim to supplement this system with the normalization condition. Let us add (<ref>) over all i ≥ 1, which gives _0 ∑_i ≥ 1Γ_i^(0) + ∑_i ≥ 1∑_j = 1^i _j Γ_i - j = . Interchanging the double summation yields _0 ∑_i ≥ 1Γ_i^(0) + ∑_j ≥ 1_j ∑_i ≥ jΓ_i - j = , or _0 ∑_i ≥ 1Γ_i^(0) + ∑_j ≥ 1_j ∑_i ≥ 0Γ_i = . Post-multiplying by the inverse of ∑_i ≥ 0Γ_i (assuming it exists) and then post-multiplying bygives _0 ∑_i ≥ 1Γ_i^(0)( ∑_i ≥ 0Γ_i )^-1 + ∑_j ≥ 1_j= 0, and by the normalization condition ∑_i ≥ 0_i= 1 this leads to _0 ∑_i ≥ 1Γ_i^(0)( ∑_i ≥ 0Γ_i )^-1 + 1 - _0 = 0, and therefore _0 (- ∑_i ≥ 1Γ_i^(0)( ∑_i ≥ 0Γ_i )^-1) = 1. By substituting (<ref>) for any of the equations in (<ref>) the value of _0 can be uniquely determined.Many of the equations required to determine the equilibrium probability vectors involve infinite sums that need to be truncated for actual computations. Furthermore, there needs to be a criterion for when the computations can be stopped. A natural stopping criterion is by examining the accumulated probability mass. In <ref> we demonstrate the implementation of the matrix-analytic approach.§ SPECTRAL EXPANSION METHOD The spectral expansion method only works for processes that are QSF to the right. This follows naturally from the fact that the spectral expansion method uses the eigenvalues and left eigenvectors of the rate matrix R of the matrix-geometric method, which is also only applicable to processes that are QSF to the right. Recall that the aim of the spectral expansion method is to linearly combine basis solutions of the form _i = y x^i - 1,i ≥ 1. By substituting (<ref>) into (<ref>) and dividing by common powers of x we obtain y( _1 + x _0 + x^2 _-1 + x^3 _-2 + ⋯) = . As we have argued before in the QBD case, these equations have a non-zero solution for y if ( _1 + x _0 + x^2 _-1 + x^3 _-2 + ⋯) = 0. Even though this determinant equation involves unbounded powers of x, it still provides us with exactly r + 1 solutions for x that lie inside the open unit disk. For a detailed discussion as to why this is the case, see <ref>. For numerical calculation purposes the determinant equation needs to be truncated. A rule of thumb could be to discard any terms with powers higher 3r, for example.Label the roots x of (<ref>) inside the closed unit disk as x_0,x_1,…,x_r and associate with the roots the corresponding non-zero eigenvectors y_0,y_1,…,y_r found from (<ref>). We assume that the eigenvectors are linearly independent, which is the case if roots x_k are different, but independence can also be the case even if some of the roots are identical. Now, each solution _i = y_k x_k^i, k = 0,1,…,r satisfies the global balance equations (<ref>). We can linearly combine these solutions as _i = ∑_k = 0^r ξ_k y_k x_k^i - 1,i ≥ 1, where ξ_k, k = 0,1,…,r are constants that we still need to determine. Substituting (<ref>) into the balance equations for levels 0 and 1 gives _0 _0^(0) + ∑_l = 0^r ξ_l y_l ∑_k ≥ 1 x_l^k - 1_-k^(k) = ,_0 _1^(0) + ∑_l = 0^r ξ_l y_l ∑_k ≥ 1x_l^k - 1_1 - k = . We can simplify the infinite sum in (<ref>) by using (<ref>): _0 _1^(0) - ∑_l = 0^r ξ_l y_l 1/x_l_1 = . Replacing one of the boundary equations with the normalization condition 1 = ∑_i ≥ 0_i= _0+ ∑_i ≥ 1∑_k = 0^r ξ_k y_k x_k^i - 1 = _0+ _1 ∑_k = 0^r ξ_k y_k/1 - x_k allow us to uniquely determine _0 and _1. We do, however, need to truncate the infinite series in (<ref>) to be able to numerically determine _0 and _1. § TAKEAWAYS This chapter exploited two structural properties we have encountered in earlier chapters. In <ref> we saw how constructing embedded Markov processes could help analyze processes with larger jumps to the left or to the right. In <ref> we saw how the skip-free structure of birth–and–death processes could be extended to two-dimensional quasi-birth–and–death processes that remained skip-free in one dimension, but could skip in the other. This chapter combined both features in quasi-skip-free (QSF) processes that can skip states in both dimensions.This additional flexibility comes with mathematical challenges, but the main methods used in earlier chapters again work, albeit in a more advanced form. The matrix-geometric and matrix-analytic method involved solutions of matrix equations with unbounded powers of R and G. The spectral expansion method worked after considering the determinant equation over infinitely many powers. For pratical purposes, these techniques require truncation of the infinite sums to be amenable for numerical calculations. The three techniques together provide a good handle, both analytically and algorithmically, on analyzing the rich class of QSF processes.PART:Advanced processes CHAPTER: PRIORITY SYSTEMS In this chapter we consider a priority systemPriority queueing systems have been studied for a long time. Cobham <cit.>, Davis <cit.> and Jaiswal <cit.> are among the first to formulate and analyze priority models. The earliest studies derive expressions for the marginal distribution of each class, whereas we are interested in the joint distribution of the underlying Markov. More recently, attention has been given to systems with an arbitrary number of priority classes in Sleptchenko et al<cit.> or to multi-server priority systems in Wang, Baron, Scheller-Wolf <cit.> and Selen and Fralix <cit.>.with a single exponential server that serves jobs of high and low priority arriving according to Poisson streams. High-priority jobs are served before low-priority jobs, low-priority jobs are only served when there are no high-priority jobs in the system. Whenever a high-priority job enters the system and a low-priority job is in service, the low-priority job is removed from service and placed at the head of the line while the high-priority job is immediately taken into service. This priority rule is referred to as preemptive priority: the high-priority job preempts the service of the low-priority job.To determine the equilibrium distribution of the two-dimensional Markov process of the number of high- and low-priority jobs associated with this single-server priority system we take three approaches. The first approaches recursively solves the balance equations by using second-order difference equations. The second approach translates the balance equations into a quadratic (functional) equation to find the bivariate PGF of the joint equilibrium distribution of high- and low-priority jobs. The third approach casts the balance equations into a QBD matrix structure, and uses the matrix-geometric and matrix-analytic methods to find a product-form solution for the equilibrium distribution. Due to the structure of the transition rate diagram the elements of the infinite-dimensional rate matrix R and auxiliary matrix G are easily determined.§ MODEL DESCRIPTION AND BALANCE EQUATIONS We distinguish the two job classes by numbering them: class-1 jobs have preemptive priority over class-2 jobs. The arrival process of class-n jobs is a Poisson process with rate _n. Each class-n job requires an exponentially distributed service time with rate μ_n. Since the service requirements are exponentially distributed and thus memoryless, the residual service time of a class-2 job that was removed from service again has an exponential distribution with the same rate μ_2. Denote by ρ_n _n/μ_n the amount of work brought into the system per time unit by class-n jobs.Let X_n(t) be the number of class-n jobs in the system at time t and denote the state of the system by X(t)(X_1(t),X_2(t)). Then { X(t) }_t ≥ 0 is a Markov process on the state space _0^2. It is apparent from the transition rate diagram in <ref> that the state space is irreducible. The system is stable if the total amount of work brought into the system per time unit is strictly less than one. We therefore assume ρρ_1 + ρ_2 < 1, to guarantee positive recurrence and the existence of the equilibrium distribution. Denote the equilibrium probability of being in state (i,j) as p(i,j).The balance equations for the interior of the state space are given for i,j ≥ 1 by ( + μ_1) p(i,j)= _1 p(i - 1,j) + μ_1 p(i + 1,j) + _2 p(i,j - 1) with _1 + _2. For the horizontal and vertical axis we have ( + μ_1) p(i,0)= _1 p(i - 1,0) + μ_1 p(i + 1,0),i ≥ 1, ( + μ_2) p(0,j)= _2 p(0,j - 1) + μ_2 p(0,j + 1) + μ_1 p(1,j),j ≥ 1. Finally, for the origin the balance equation is p(0,0) = μ_1 p(1,0)+ μ_2 p(0,1). Since the server always works at unit rate whenever there is work to do, p(0,0) = 1 - ρ.§ DIFFERENCE EQUATIONS APPROACH We exploit the upward structure of the transition rate diagram by first solving the balance equations for j = 0 and working our way up by increasing j one step at a time. For j = 0, (<ref>) is a homogeneous difference equation of order two: μ_1 p(i + 1,0) - ( + μ_1) p(i,0) + _1 p(i - 1,0) = 0,i ≥ 1. We have the general solution p(i,0) = c_0,0 x_1^i + c_0,1 x_2^i,i ≥ 0 where x_1 and x_2 are the roots of the quadratic equation μ_1 x^2 - ( + μ_1) x + _1 = 0 satisfying 0 < x_1 < 1 < x_2. Since the solution of (<ref>) needs to be normalized, and ∑_i ≥ 1 x_2^i = ∞, we set c_0,1 = 0. This gives with xx_1, p(i,0) = c_0,0 x^i,i ≥ 0, where we leave c_0,0 undetermined for now.For j = 1, (<ref>) is a nonhomogeneous difference equation of order two: μ_1 p(i + 1,1) - ( + μ_1) p(i,1) + _1 p(i - 1,1) = - _2 p(i,0),i ≥ 1 and its solution will be a combination of the general solution of the homogeneous equation and a particular solution of the nonhomogeneous equation. Clearly, the solution of the homogeneous equation is c_1,0 x^i, where c_1,0 is a constant that we determine later. For the solution to the nonhomogeneous equation we guess that it is of the form p(i,1) = c_1,1i + 11 x^i. Substituting this guess into (<ref>) and dividing by x^i - 1 gives μ_1 c_1,1i + 21 x^2 - ( + μ_1) c_1,1i + 11 x + _1 i1 = - _2 c_0,0 x. Since x satisfies (<ref>), we obtain c_1,1 = _2 c_0,0/ + μ_1 - 2 μ_1 x and p(i,1) = c_1,0i + 10 x^i + c_1,1i + 11 x^i.For j = 2, (<ref>) gives μ_1 p(i + 1,2) - ( + μ_1) p(i,2) + _1 p(i - 1,2) = - _2 p(i,1),i ≥ 1. The solution of the homogeneous version of (<ref>) is c_2,0 x^i, where c_2,0 is a constant that we determine later. Substituting the guess p(i,2) = c_2,1i + 21 x^i + c_2,2i + 22 x^i into (<ref>) and dividing by x^i - 1 gives μ_1 x^2 ( c_2,1i + 31 + c_2,2i + 32) - ( + μ_1) x ( c_2,1i + 21 + c_2,2i + 22)+ _1 ( c_2,1i + 11 + c_2,2i + 12) = -_2 x ( c_1,0i + 10 + c_1,1i + 11). We use i + 31 = ( 1 + 2/i + 1) i + 11, i + 32 = ( 1 + 4/i + 2/i(i + 1)) i + 12,i + 21 = ( 1 + 1/i + 1) i + 11, i + 22 = (1 + 2/i) i + 12, and the fact that x satisfies (<ref>) to simplify (<ref>) to c_2,11/i + 1i + 11 (2 μ_1 x - ( + μ_1)) + c_2,2i + 12( ( 4/i + 2/i(i + 1)) μ_1 x - 2/i ( + μ_1) ) = -_2 ( c_1,0i + 10 + c_1,1i + 11). Grouping terms in front of the binomial coefficients gives i + 10( c_2,1 ( 2 μ_1 x - ( + μ_1) ) + c_2,2μ_1 x ) + i + 11 c_2,2( 2 μ_1 x - ( + μ_1) ) = - i + 10 c_1,0_1 - i + 11 c_1,1_1. Matching the coefficients of the binomial coefficients finally shows that c_2,2 = _2 c_1,1/ + μ_1 - 2 μ_1 x,c_2,1 = _2 c_1,0 + μ_1 x c_2,2/ + μ_1 - 2 μ_1 x.Repeating this procedure leads to the general expression p(i,j) = ∑_k = 0^j c_j,ki + jk x^i,i,j ≥ 0, where the coefficients satisfy the recursion, for j ≥ 1, c_j,j = _2 c_j - 1,j - 1/ + μ_1 - 2 μ_1 x, c_j,k = _2 c_j - 1,k - 1 + μ_1 x c_j,k + 1/ + μ_1 - 2 μ_1 x,1 ≤ k ≤ j - 1. The coefficients c_j,0, j ≥ 0 still need to be determined. Since p(0,0) = 1 - ρ we have c_0,0 = 1 - ρ. Substituting (<ref>) into the balance equation (<ref>) gives c_1,0 = ( - μ_1 x)/μ_2 c_0,0 - c_1,1. From (<ref>) we obtain, for j ≥ 1, c_j + 1,0 = 1/μ_2∑_k = 0^j c_j,k( μ_1 x j + 1k - ( + μ_2) jk)+ ρ_2 ∑_k = 0^j - 1 c_j - 1,kj - 1k - ∑_k = 1^j + 1 c_j + 1,kj + 1k. We outline the computation of the coefficients in <ref>.§ GENERATING FUNCTION APPROACH Define the bivariate PGF x,y∑_i ≥ 0∑_j ≥ 0 p(i,j) x^i y^j,|x| ≤ 1, |y| ≤ 1. Note that x,0 = ∑_i ≥ 0 p(i,0) x^i and 0,y = ∑_j ≥ 0 p(0,j) y^j are the probability generating functions of the equilibrium probabilities of the states on the horizontal and vertical axis. Furthermore, x,1 and 1,y are the probability generating functions of the number of class-1 and class-2 jobs in the system, respectively.We shall now perform a series of operations on the balance equations to obtain an expression for x,y. Multiply both sides of (<ref>) by x^i y^j and sum over all i,j ≥ 1 to obtain (<ref>). Multiply both sides of (<ref>) by x^i and sum over all i ≥ 1 to obtain (<ref>). Finally, multiply both sides of (<ref>) by y^j and sum over all j ≥ 1 to obtain (<ref>). ( + μ_1) ∑_i ≥ 1∑_j ≥ 1 p(i,j) x^i y^j= _1 ∑_i ≥ 1∑_j ≥ 1 p(i - 1,j) x^i y^j+ μ_1 ∑_i ≥ 1∑_j ≥ 1 p(i + 1,j) x^i y^j+ _2 ∑_i ≥ 1∑_j ≥ 1 p(i,j - 1) x^i y^j, ( + μ_1) ∑_i ≥ 1 p(i,0) x^i= _1 ∑_i ≥ 1 p(i - 1,0) x^i+ μ_1 ∑_i ≥ 1 p(i + 1,0) x^i, ( + μ_2) ∑_j ≥ 1 p(0,j) y^j= _2 ∑_j ≥ 1 p(0,j - 1) y^j + μ_2 ∑_j ≥ 1 p(0,j + 1) y^j+ μ_1 ∑_j ≥ 1 p(1,j) y^j. Summing (<ref>)–(<ref>) and (<ref>) and using simplifications based on the definition of x,y such as ∑_i ≥ 1∑_j ≥ 1 p(i,j) x^i y^j= ∑_i ≥ 0∑_j ≥ 0 p(i,j) x^i y^j - ∑_i ≥ 0 p(i,0) x^i - ∑_j ≥ 0 p(0,j) y^j + p(0,0) = x,y - x,0 - y,0 + 0,0, shows that x,y satisfies the functional equation h_1(x,y) x,y = h_2(x,y) 0,y + h_3(x,y) 0,0 with h_1(x,y)_1 x y (1 - x) + _2 x y (1 - y) - μ_1 y (1 - x), h_2(x,y) - μ_1 y (1 - x) + μ_2 x (1 - y), h_3(x,y) - μ_2 x (1 - y). The question is whether we can solve functional equation (<ref>). Recall that 0,0 = p(0,0) = 1 - ρ. Setting y = 1 in (<ref>) gives x,1 = 0,1/1 - ρ_1 x, where 0,1 is the probability that there are no class-1 jobs in the system. Clearly, 0,1 = 1 - ρ_1 and therefore x,1 = 1 - ρ_1/1 - ρ_1 x = ∑_i ≥ 0 (1 - ρ_1) ρ_1^i x^i, which shows that the number of class-1 jobs follows a geometric distribution with parameter ρ_1. Due to the preemptive priority, class-1 jobs do not have to wait for class-2 jobs and therefore experience the system as if it were a standard M/M/1 queue. Now choose x so that the left-hand side of (<ref>) vanishes to obtain an expression for 0,y. For a fixed y with 0 < |y| ≤ 1, h_1(x,y) is a second degree polynomial in x, and hence 0 = h_1(x,y) ⇔ 0 = ρ_1 x^2 - (1 + ρ_1 + _2/μ_1(1 - y) ) x + 1. For a fixed y with 0 < |y| ≤ 1, (<ref>) has a unique solution x = (y) with |x| ≤ 1.For now, fix a y with 0 < |y| < 1. We use Rouché's theorem, see <ref>, to prove that (<ref>) has a unique solution within the closed unit disk. Denote the closed unit disk byand the unit circle by . Define the functions f_1(x,y)- (1 + ρ_1 + _2/μ_1(1 - y) ) x,g(x) ρ_1 x^2 + 1. Clearly, f_1(x,y) has only one root x = 0 in . We aim to show that |f_1(x,y)| > |g(x)|,x ∈, so that it follows from Rouché's theorem that f_1(x,y) + g(x) also has one root in .Then, |f_1(x,y)|= | 1 + ρ_1 + _2/μ_1 - _2/μ_1 y | |x| ≥( 1 + ρ_1 + _2/μ_1 - _2/μ_1 |y| ) |x|f_2(|x|,|y|) and |g(x)| = | ρ_1 x^2 + 1 | ≤ρ_1 |x|^2 + 1 = g(|x|). It suffices to show that f_2(|x|,|y|) > g(|x|) for |x| = 1 and 0 < |y| < 1, which is clearly the case.However, when |x| = |y| = 1 we have f_2(1,1) = g(1). In order to use Rouché's theorem for that particular case, we essentially evaluate f_2(|x|,1) and g(|x|) on the circle |x| = 1 + ϵ with ϵ small and positive. To accomplish this, we use the Taylor expansion f_2(1,1 + ϵ) = f_2(1,1) + ϵ f_2'(1,1) + o(ϵ) and verify that f_2(1,1 + ϵ) > g(1 + ϵ). Since f_2(1,1) = g(1) we are left to show that f_2'(1,1) > g'(1). Now, f_2'(1,1) = / |x| f_2(|x|,1) |_|x| = 1 = 1 + ρ_1 and g'(1) = / |x| g(|x|) |_|x| = 1 = 2ρ_1, which proves f_2'(1,1) > g'(1) since ρ_1 < 1. So, for sufficiently small ϵ > 0 we have that f_2(|x|,1) > g(|x|) for |x| ∈ (1,1 + ϵ], which proves the claim. The unique solution x = (y) within the closed unit disk can easily be computed from the second-degree polynomial (<ref>): (y) = _1 + μ_1 + _2(1 - y) - √((_1 + μ_1 + _2(1 - y))^2 - 4 _1 μ_1)/2_1. We proceed by plugging x = (y) and 0,0 = 1 - ρ into (<ref>) to obtain 0,y = -h_3((y),y) 0,0/h_2((y),y) = μ_2 (y)(1 - y)(1 - ρ)/-μ_1 y (1 - (y)) + μ_2 (y)(1 - y) so that (<ref>) gives the expression x,y = 1 - ρ/h_1(x,y)( h_2(x,y) μ_2 (y)(1 - y)/-μ_1 y (1 - (y)) + μ_2 (y)(1 - y) + h_3(x,y) ).We have converted the balance equations (<ref>)–(<ref>) into the functional equation (<ref>) and found a solution for x,y in (<ref>). So we took the direct, explicit relations between the equilibrium probabilities to the transform domain to find an indirect description of the equilibrium probabilities in terms of a complex-valued function x,y. Given this bivariate PGF of the joint equilibrium distribution, we can use <ref> to numerically invert x,y to obtain any p(i,j). In <ref> we demonstrate how the algorithm parameters j_1 and j_2 influence the accuracy of the solution and the computation time required to obtain this solution.Setting x = 1 in (<ref>) gives 1,y = 1 - ρ/ρ_2μ_1 (1 - (y))/-μ_1 y (1 - (y)) + μ_2 (y) (1 - y). Denote the equilibrium distribution of the number of class-2 jobs by p_2(·) so that 1,y = ∑_k ≥ 0 p_2(k) y^k. In <ref> we will see that the root (y) is a PGF: (y) = ∑_k ≥ 0 g_k y^k with { g_k }_k ≥ 0 the elements of the auxiliary matrix G for which we have an exact expression, see <ref> and <ref>. We derive a recursion for the probabilities p_2(·) by matching coefficients of the generating functions on both sides of (<ref>). Substituting in (<ref>) the series expression (y) = ∑_k ≥ 0 g_k y^k and 1,y = ∑_k ≥ 0 p_2(k) y^k gives ( μ_2 g_0 - μ_1 y + ∑_l ≥ 1( μ_2 g_l + (μ_1 - μ_2) g_l - 1) y^l ) ∑_k ≥ 0 p_2(k) y^k = 1 - ρ/ρ_2μ_1 ( 1 - ∑_l ≥ 0 g_l y^l ). For convenience, define a_l1 - ρ/ρ_2μ_1 (1 - g_0), l = 0, - 1 - ρ/ρ_2μ_1 g_l, l ≥ 1, b_lμ_2 g_0, l = 0,μ_2 g_1 + (μ_1 - μ_2) g_0 - μ_1, l = 1,μ_2 g_l + (μ_1 - μ_2) g_l - 1, l ≥ 2, so that (<ref>) becomes ∑_k ≥ 0∑_l ≥ 0 b_l p_2(k) y^k + l = ∑_m ≥ 0∑_n = 0^m b_m - n p_2(n) y^m = ∑_m ≥ 0 a_m y^m which, by coefficient matching, leads to the recursion b_0 p_2(0)= a_0, b_0 p_2(m)= a_m - ∑_n = 0^m - 1 b_m - n p_2(n),m ≥ 1.The numerical inversion algorithm shown in <ref> can also be used to determine the equilibrium probabilities. We show the equilibrium distribution for an example in <ref>. From the ratio p_2(j + 1)/p_2(j) in <ref> it is clear that the distribution of the number of class-2 jobs is not geometric. § QBD APPROACHES The two-dimensional Markov process { X(t) }_t ≥ 0 is a QBD process with levels i = { (i,0),(i,1),…}, i ≥ 0 and an infinite number of phases per level. The infinite-dimensional transition matrix Q can be partitioned into levels as Q = [ _0^(0) _1;_-1 _0 _1; _-1 _0 _1;_-1 _0 _1; ⋱⋱⋱;], where _-1 = μ_1, _1 = _1, withthe infinite-dimensional identity matrix, and _0 = - ( + μ_1)+ [0 _2; 0 _2;⋱⋱ ] and _0^(0) = - + [0 _2;μ_2 -μ_2 _2; μ_2 -μ_2 _2;⋱⋱⋱ ].We first use the matrix-geometric methodMiller <cit.> was the first to model the single-server priority system with two classes as a QBD process. He exploits the structure of the transition rate diagram to determine the joint equilibrium distribution. to determine the equilibrium distribution. Define the vectors _i [ p(i,0) p(i,1)⋯ ]. The rate matrix R satisfies the matrix-quadratic equation R^2 _-1 + R_0 + _1 = 0 and the equilibrium probabilities follow from _i + 1 = _i R,i ≥ 0, where the boundary probabilities are computed as _0 _0^(0) + _1 _-1 = ⇔_0 ( _0^(0) + R _-1) = and the normalization condition is _0 ( - R)^-1 = 1, whereis a vector of ones.At this point we can already obtain highly accurate approximations of the equilibrium distribution by truncating all matrices in (<ref>) to size n × n with n large and use successive substitutions to determine R (see <ref>). However, we can do better than that by exploiting the specific structure of the transition rate diagram.Since the transitions within levels i ≥ 1 are strictly upward in the vertical direction and the transition rate diagram is homogeneous, we already know from the probabilistic interpretation of the elements of the rate matrix (see <ref>) that R = [ r_0 r_1 r_2 ⋯; r_0 r_1 ⋯; r_0 ⋯; ⋱ ]. It is precisely this structure that makes it possible to solve for the elements of R using a recursive procedure. That is, component-wise the equations (<ref>) read μ_1 r_0^2 - ( + μ_1)r_0 + _1= 0,μ_1 ∑_l = 0^k r_k - l r_l - ( + μ_1) r_k + _2 r_k - 1 = 0,k ≥ 1. Since R is the minimal non-negative solution of (<ref>) we know that the solution of (<ref>) is given by r_0 = _1 + μ_1 + _2 - √((_1 + μ_1 + _2)^2 - 4 _1 μ_1)/2 μ_1. Substituting the solution for r_0 into (<ref>) gives the recursion r_k = _2 r_k - 1 + μ_1 ∑_l = 1^k - 1 r_k - l r_l/√(( + μ_1)^2 - 4 _1 μ_1),k ≥ 0, where the empty sum ∑_j = 1^0 is zero.The boundary probabilities can also be determined recursively due to the specific form of R and p(0,0) = 1 - ρ. In particular, μ_2 p(0,1)= ( - μ_1 r_0) p(0,0) = ( - μ_1 r_0) (1 - ρ),μ_2 p(0,j + 1)= ( ( + μ_2)p(0,j) - _2 p(0,j - 1)- μ_1 ∑_k = 0^j p(0,j - k) r_k ),j ≥ 1. A similar simplification as for p(0,j) is obtained for the equilibrium probabilities p(i,j): p(i + 1,j) = ∑_k = 0^j p(i,j - k) r_k.The numerical implementation of the matrix-geometric method is explained in <ref>. The positive integers i_ and j_ determine the subset of the state space ' { (i,j) ∈ : 0 ≤ i ≤ i_, 0 ≤ j ≤ j_} for which the equilibrium probabilities are determined exactly.For a QBD process, the matrix-geometric and matrix-analytic method are nearly identical. For that reason we do not describe how to determine the auxiliary matrix G of the matrix-analytic method. Instead, we focus on the probabilistic interpretation of the elements of the G (see <ref>) to immediately derive a recursion for the equilibrium probabilities.Sleptchenko et al <cit.> developed the method that exploits the transition structure to determine the elements { g_k }_k ≥ 0 and use these elements to write the excursions in two ways as we have done in <ref> for the matrix-analytic method.The auxiliary matrix G is given by G = [ g_0 g_1 g_2 ⋯; g_0 g_1 ⋯; g_0 ⋯; ⋱ ]. The element g_k can be interpreted as a first passage probability: it is the probability that, starting at level i ≥ 1 in state (i,j), the first passage to level i - 1 occurs in state (i - 1,j + k). The first passage probabilities do not depend on the starting state due to the homogeneous transition behavior in the interior of the state space. The { g_k }_k ≥ 0 are obtained from a recursion relation similar to the one for { r_k }_k ≥ 0 and given by g_0= μ_1/ + μ_1 + _1/ + μ_1 g_0^2, g_k= _2/ + μ_1 g_k - 1 + _1/ + μ_1∑_l = 0^k g_k - l g_l,k ≥ 1.We now use the first passage probabilities to derive an expression for the equilibrium probabilities in the interior of the state space. Let an excursion refer to a sample path of the Markov process that starts in level i, reaches levels higher than i and ends on first passage to level i. The number of excursions per time unit that ends in state (i,j) is p(i + 1,j) μ_1. Alternatively, this rate is also given by the number of excursions starting from level i per time unit that end in state (i,j). The number of excursions per time unit that starts in state (i,j - k) is p(i,j - k) _1, a fraction g_k of which ends its excursion in (i,j). Since these two rates are equal, we obtain the recursion μ_1 p(i + 1,j) = _1 ∑_k = 0^j p(i,j - k) g_k,i,j ≥ 0. It remains to determine the equilibrium probabilities on the vertical boundary. To that end, we censor the Markov process to 0. This leads to the transition rate diagram in <ref>. For the censored Markov process we can simply equate the number of transitions per time unit that enter and leave the set { (0,0),(0,1),…,(0,j) }, which yields μ_2 p(0,j + 1) = _2 p(0,j) + _1 ∑_k = 0^j p(0,j - k) ( 1 - ∑_l = 0^k g_l ). Starting from p(0,0) = 1 - ρ, all equilibrium probabilities can be obtained through (<ref>)–(<ref>). An alternative choice for a level is the vertically aligned set of states j = { (0,j),(1,j),…}, j ≥ 0. We write variables with a bar to reflect that they belong to the alternative choice for the levels.The transition matrix Q̅ can be partitioned into these levels as Q̅ = [ _0^(0) _1;_-1 _0 _1; _-1 _0 _1;_-1 _0 _1; ⋱⋱⋱;]. The matrix _-1 has zeroes everywhere except (_-1)_0,0 = μ_2. The Markov process can only go from level j + 1 to level j by using the transition from (0,j + 1) to (0,j). Due to the probabilistic interpretation of the elements of the auxiliary matrix G̅ we can immediately write that G̅ = [ 1 0 ⋯; 1 0 ⋯; ⋮ ⋮ ⋱ ].Denote the vectors _j [ p(0,j) p(1,j)⋯ ]. By censoring the Markov process to the set of states 0∪1∪⋯∪j we can write the balance equations for level j as _j - 1_1 + _j ( _0 + _1 G̅) = ,j ≥ 1, Finally, censoring the Markov process to level 0 shows that _0( _0^(0) + _1 G̅) = . These balance equations can be solved recursively in essentially the same way as the difference equations approach of <ref> solves the balance equations.§ BUSY PERIOD TRANSFORMS Key elements of both the generating function approach and the QBD approach are related to busy periods in a single-server system. We explain what a busy period is, derive its LST and mention where busy periods play a role in both approaches.A busy period in a single-server system is a length of time that starts when a first job arrives to an empty system and ends when a departing job leaves the system empty again. Let B_,μ denote the length of a busy period in an M/M/1 queue with arrival rateand service rate μ. In <ref> we have proven that the LST of B_,μ—now denoted by ,μ—is given by ,μ =+ μ +- √(( + μ + )^2 - 4μ)/2,> 0. We now mention a few relations between the busy period B_,μ and its transform ,μ and key elements of the approaches used in this chapter to determine the equilibrium distribution.The unique root (y) of the generating function approach can be expressed in terms of the busy period transform as (y) = _1,μ_1_2(1 - y). The root has an interpretation in terms of the PGF of the number of class-2 jobs A that arrives during a busy period of class-1 jobs. Condition on the length S_1 of the service of the first class-1 job to obtain y^A = ∫_0^∞y^A | S_1 = t f_S_1(t) t. A number K_2 of class-2 jobs joins the queue during this first service, but the busy period of class-1 jobs might not have ended yet. In particular, during this first service a number K_1 of class-1 jobs has joined the queue and each class-1 job induces a busy period of class-1 jobs that generates a number of class-2 arrivals, which is statistically identical to A. Note that K_1 ∼_1 t and K_2 ∼_2 t. By conditioning on the number of class-1 and class-2 arrivals in the interval [0,t], we see that y^A = ∫_0^∞∑_i ≥ 0∑_j ≥ 0y^A | S_1 = t, K_1 = i, K_2 = j· f_S_1(t) K_1 = iK_2 = jt = ∫_0^∞∑_i ≥ 0∑_j ≥ 0y^j + A^(1) + ⋯ + A^(i)· f_S_1(t) K_1 = iK_2 = jt, where A^(k) denotes the number of class-2 arrivals during the class-1 busy period started by the k-th class-1 job during the service of the first class-1 job. The random variables A^(k) are i.i.d. We can now substitute the probability density and mass functions of S_1, K_1 and K_2 to obtain y^A = ∫_0^∞∑_i ≥ 0∑_j ≥ 0y^A^i y^j μ_1 ^-μ_1 t(_1 t)^i/i!^-_1 t(_2 t)^j/j!^-_2 tt = μ_1 ∫_0^∞^-t(_1(1 - y^A) + _2(1 - y) + μ_1)t = μ_1/_1(1 - y^A) + _2(1 - y) + μ_1. From (<ref>), we see that (y) satisfies the exact same equation as (<ref>) and since |y^A| < 1 we conclude that (y) = y^A.The first passage probabilities g_k can be given in terms of a busy period B__1,μ_1 and a Poisson process { N__2(t) }_t ≥ 0 with rate _2. By examining the transition rate diagram in <ref>, we see that g_k =N__2(B__1,μ_1) = k, which is the probability that during a class-1 busy period, exactly k class-2 jobs arrive. The PGF of { g_k }_k ≥ 0 is given by ∑_k ≥ 0 g_k y^k= ∑_k ≥ 0 N__2(B__1,μ_1) = k y^k = ∑_k ≥ 0N__2(B__1,μ_1) = k y^k =y^N__2(B__1,μ_1). Conditioning on the length of the busy period and using the PGF of a Poisson distribution with parameter _2 t yields ∑_k ≥ 0 g_k y^k= ∫_0^∞ y^N__2(B__1,μ_1)| B__1,μ_1 = t f_B__1,μ_1(t) t = ∫_0^∞^-_2(1 - y)t f_B__1,μ_1(t) t = _1,μ_1_2(1 - y), which shows that ∑_k ≥ 0 g_k y^k = y^A = (y). The first passage probabilities are explicitly given by g_0 = _1,μ_1_2 and for k ≥ 1, g_k = γ_1^k _1,μ_1_2∑_l = 0^k - 1 C_l k - 1 + lk - 1 - lγ_2^l, where C_l1/(l + 1) 2ll are the Catalan numbers and γ_1= _2/_1(1 - 2 _1,μ_1_2) + μ_1 + _2,γ_2= _1 _1,μ_1_2/_1(1 - 2 _1,μ_1_2) + μ_1 + _2.We prove the claim by verifying that ∑_k ≥ 0 g_k y^k = _1,μ_1_2(1 - y). For now, abbreviate _1,μ_1_2 as L.We have ∑_k ≥ 0 g_k y^k= ( 1 + ∑_k ≥ 1 (γ_1 y)^k ∑_l = 0^k - 1 C_l k - 1 + lk - 1 - lγ_2^l ) = ( 1 + γ_1 y ∑_k ≥ 0 (γ_1 y)^k ∑_l = 0^k C_l k + lk - lγ_2^l ). Interchanging the two summations gives ∑_k ≥ 0 (γ_1 y)^k ∑_l = 0^k C_l k + lk - lγ_2^l= ∑_l ≥ 0 C_l γ_2^l ∑_k ≥ lk + lk - l (γ_1 y)^k = ∑_l ≥ 0 C_l (γ_1 γ_2 y)^l ∑_k ≥ 0k + 2lk (γ_1 y)^k. From the negative binomial distribution we know that the generating function of the binomial coefficient is ∑_k ≥ 0k + Kk^k = 1/(1 - )^K + 1, so that ∑_l ≥ 0 C_l (γ_1 γ_2 y)^l ∑_k ≥ 0k + 2lk (γ_1 y)^l = 1/1 - γ_1 y∑_l ≥ 0 C_l ( γ_1 γ_2 y/(1 - γ_1 y)^2)^l. Now use the generating function of the Catalan numbers ∑_k ≥ 0 C_k ^k = 1 - √(1 - 4)/2. to get 1/1 - γ_1 y∑_l ≥ 0 C_l ( γ_1 γ_2 y/(1 - γ_1 y)^2)^l = 1 - √(1 - 4γ_1 γ_2 y/(1 - γ_1 y)^2)/2γ_1 γ_2 y/1 - γ_1 y Substituting this back into (<ref>) yields ∑_k ≥ 0 g_k y^k= ( 1 + 1 - √(1 - 4γ_1 γ_2 y/(1 - γ_1 y)^2)/2γ_2/1 - γ_1 y) = 2 γ_2/1 - γ_1 y + 1 - √(1 - 4γ_1 γ_2 y/(1 - γ_1 y)^2)/2γ_2/1 - γ_1 y. Substituting γ_1 γ_2 y/(1 - γ_1 y)^2 = _1 _2y/(_1(1 - 2 ) + μ_1 + _2(1 - y))^2,γ_2/1 - γ_1 y = _1 /_1(1 - 2 ) + μ_1 + _2(1 - y), into (<ref>), multiplying the numerator and denominator of (<ref>) by _1(1 - 2 ) + μ_1 + _2(1 - y) and recognizing thatsatisfies _1 ^2 - ( + μ_1)+ μ_1 = 0 finally proves the claim. § TAKEAWAYS The Markov process associated with the single-server priority system has no downward transitions in the interior of the state space. This structure allowed for a simple solution using the generating function approach and while modeling the Markov process as a QBD process. A Markov process with a structure in which there are no upward transitions is amenable to the same solution approaches. The single-server priority system is one of many models that possesses this structure; a few others can be found in <cit.>.Due to the upward structure the balance equations could be solved recursively by treating them as second-order difference equations. The balance equations for i > 0 and j = 0 are homogeneous difference equations and were easily solved by substituting a product-form solution. The balance equations for i,j > 0 are nonhomogeneous difference equations, where the constant term is the rate at which the process enters the state from the state directly below. The nonhomogeneous difference equations could also be solved and the final expressions for the equilibrium probabilities involve coefficients that needed to be calculated recursively.For the generating function approach the upward structure meant that the functional equation for x,y did not involve x,0 associated with the equilibrium probabilities of the states on the horizontal axis. This allowed for a direct determination of 0,y as a function of the root (y) and ultimately led to an explicit expression for x,y.In terms of the QBD approach the upward structure ensured that the infinite-dimensional rate matrix R and auxiliary matrix G were upper triangular. Both R and G satisfied a matrix-quadratic equation, that, due to the upper triangular structure, could be solved recursively. The upward structure was used once more to derive recursions for the equilibrium probabilities on the vertical boundary and in the interior of the state space.The transition behavior in the interior of the state space has some additional structure: on top of being strictly upward in the vertical direction, the transition behavior in the horizontal direction mimics the transition behavior of an M/M/1 queue with arrival rate _1 and service rate μ_1. The relations between busy periods of an M/M/1 queue and key elements of the generating function approach and the QBD approach came as no surprise.The approaches we saw in this chapter are not restricted to Markov process with no downward transitions in the interior of the state space. Specifically, the approaches work whenever there are no downward, upward, leftward or rightward transitions, see <ref> and <cit.>. For example, when there are no upward transitions—the second case in <ref>—x,0 appears in the functional equation (<ref>) instead of 0,y and the matrices R and G are lower triangular instead of upper triangular. By swapping the two coordinates the third case with no rightward transitions in <ref> reduces to the first case with no upward transitions and the fourth case with the leftward transitions reduces to the second case with no downward transitions. CHAPTER: GATED SYSTEMS In this chapter we consider an exponential single-server queueing system where access to the system is regulated by a gate. Jobs arrive according to a Poisson process and first have to wait behind this gate. Whenever there are no jobs left in the system, the gate opens and all waiting jobs are transferred to the system without further delay. The gate closes immediately after the transfer and the server starts service. If there are no jobs in the system nor behind the gate, then the gate remains open until a job arrives. An arriving job is then immediately transferred to the system, the gate closes and service starts. Notice that the system cannot be empty unless there are no jobs behind the gate. We are interested in the joint distribution of the number of jobs behind the gate and in the system.The system can be described as a two-dimensional Markov process with as dimensions the number of jobs behind the gate and in the system. We shall determine the equilibrium distribution of this Markov process using three different approaches. The first approach casts the balance equations into the generating function domain and determines the generating function of the joint equilibrium distribution using an iterative approach.Rietman and Resing <cit.> study the gated system with general service times using the generating function approach. In the exponential case that we consider in this chapter, we are able to develop an explicit expression for the equilibrium distribution in terms of an infinite sum of product forms. The second approach uses the matrix-geometric method and exploits the downward structure in the interior of the state space to explicitly determine the elements of the rate matrix R. However, the second approach strands here and numerical approximations are required to determine the equilibrium probabilities. The third approach is called the compensation approachResing and Rietman <cit.> introduced the gated single-server system and determined the equilibrium distribution using the compensation approach of Adan, Wessels and Zijm <cit.>. and exploits the fact that a product-form solution satisfies the balance equations for the states in the interior of the state space. These product-form solutions are linearly combined to also satisfy the remaining balance equations. The first and third approach both lead to infinite sum expressions for the equilibrium probabilities.§ MODEL DESCRIPTION AND BALANCE EQUATIONS Jobs arrive according to a Poisson process with rate . Each job requires an exponentially distributed service time with rate μ. Denote by ρ/μ the amount of work brought into the system per time unit.Let X_1(t) be the number of jobs behind the gate at time t and let X_2(t) be the number of jobs in the system at time t. Further, denote the state of the system by X(t)(X_1(t),X_2(t)). Then { X(t) }_t ≥ 0 is a Markov process with state space { (i,j) : i ∈_0, j ∈}∪{ (0,0) }. It is apparent from the transition rate diagram in <ref> that the state space is irreducible. To guarantee positive recurrence and the existence of the equilibrium distribution we assume that ρ < 1. Let p(i,j) denote the equilibrium probability of being in state (i,j).The balance equations for the interior of the state space are given by ( + μ) p(i,j) =p(i - 1,j) + μ p(i,j + 1),i,j ≥ 1. For the vertical axis we have ( + μ) p(0,1)=p(0,0) + μ p(0,2) + μ p(1,1), ( + μ) p(0,j)= μ p(0,j + 1) + μ p(j,1),j ≥ 2, and the balance equation at the origin is p(0,0) = μ p(0,1). Combining (<ref>) with (<ref>) gives p(0,1) = μ p(0,2) + μ p(1,1).Observe that { X_1(t) + X_2(t) }_t ≥ 0 is also a Markov process. More specifically, it is the Markov process associated with an M/M/1 queue with arrival rateand service rate μ and equilibrium probabilities p(k), where p(k) = ∑_i + j = k p(i,j) = (1 - ρ) ρ^k. Since an empty system can only occur when there are no jobs behind the gate we clearly have p(0,0) = 1 - ρ and therefore ∑_i ≥ 0∑_j ≥ 1 p(i,j) = ρ.§ GENERATING FUNCTION APPROACH Define the bivariate generating function x,y∑_i ≥ 0∑_j ≥ 1 p(i,j) x^i y^j - 1,|x| ≤ 1, |y| ≤ 1. Since 1,1 = ρ < 1, x,y is not a probability generating function. From (<ref>) or (<ref>) we derive that 0,0 = p(0,1) = (1 - ρ) ρ.We shall now perform a series of operations on the balance equations to obtain an expression for x,y. Multiply both sides of (<ref>) by x^i y^i - 1 and sum over all i,j ≥ 1 to obtain ( + μ) ∑_i ≥ 1∑_j ≥ 1 p(i,j) x^i y^j - 1= ∑_i ≥ 1∑_j ≥ 1 p(i - 1,j) x^i y^j - 1 + μ∑_i ≥ 1∑_j ≥ 1 p(i,j + 1) x^i y^j - 1.Multiply both sides of (<ref>) by y^j - 1, sum over all j ≥ 2, and add (<ref>) to obtain ∑_j ≥ 1 p(0,j) y^j - 1 + μ∑_j ≥ 2 p(0,j) y^j - 1= μ∑_j ≥ 1 p(0,j + 1) y^j - 1 + μ∑_j ≥ 1 p(j,1) y^j - 1. Summing (<ref>)–(<ref>), multiplying both sides by y/μ and using simplifications based on the definition of x,y such as ∑_i ≥ 1∑_j ≥ 1 p(i,j + 1) x^i y^j - 1 = ∑_i ≥ 1∑_j ≥ 0 p(i,j + 1) x^i y^j - 1 - 1/y∑_i ≥ 1 p(i,1) x^i = 1/y( x,y - 0,y - x,0 + 0,0), shows that x,y satisfies the functional equation h(x,y) x,y = y,0 - x,0 + (y - 1) 0,0 with h(x,y)y ( 1 + ρ(1 - x) ) - 1.It is easily seen that for a fixed x we have that h(x,y) = 0 for y = (x) with (x) = 1/1 + ρ(1 - x). Notice that v(x) has a simple pole at x = 1 + 1/ρ > 1. Since |(x)| ≤ 1 if |x| ≤ 1 we have by substituting (x) into (<ref>) that x,0 = (x),0 + ((x) - 1) 0,0. We will iterate (<ref>) to obtain an expression for x,0. First, define ^∘ k(x)( ∘∘⋯∘_)(x),n ≥ 1, where the operator ∘ denotes a composition: (f ∘ g)(x) = f(g(x)). The composition ^∘ k(x) is explicitly given by ^∘ k(x) = 1 - ρ^k - x ρ (1 - ρ^k - 1)/1 - ρ^k + 1 - x ρ (1 - ρ^k),k ≥ 1.For k = 1 the claim is true. Assume that (<ref>) holds for k. We show that it also holds for k + 1. We have ^∘ (k + 1)(x) = 1/1 + ρ(1 - ^∘ k(x)) and by substituting in the expression for ^∘ k(x) and multiplying the denominator and numerator by 1 - ρ^k + 1 - x ρ (1 - ρ^k) we get that ^∘ (k + 1)(x) is given by (<ref>). Hence, by induction, we conclude that the claim holds for all k ≥ 1. Iterating (<ref>) gives x,0 = lim_k →∞^∘ k(x),0 + 0,0∑_k ≥ 1 (^∘ k(x) - 1). Since ^∘ k(x) - 1 = (ρ^k) as k tends to ∞, the infinite series in (<ref>) is convergent. It is easily seen from <ref> that lim_k →∞^∘ k(x) = 1 independent of x and therefore x,0 = 1,0 + 0,0∑_k ≥ 1 (^∘ k(x) - 1). We determine 1,0 by setting x = 0 in (<ref>). Since ^∘ k(0) - 1 = - 1 - ρ/ρρ^k + 1/1 - ρ^k + 1, we get 0,0 = 1,0 - 0,01 - ρ/ρ∑_k ≥ 1ρ^k + 1/1 - ρ^k + 1, which, by 0,0 = (1 - ρ)ρ, indicates that 1,0 = (1 - ρ)^2 ∑_k ≥ 1ρ^k/1 - ρ^k. Substituting the expressions for 1,0 and 0,0 into (<ref>) yields x,0 = (1 - ρ) ( (1 - ρ) ∑_k ≥ 1ρ^k/1 - ρ^k + ρ∑_k ≥ 1 (^∘ k(x) - 1) ). Define (x) ∑_k ≥ 1 ( ^∘ k(x) - 1 ) and substitute x,0, y,0 and 0,0 into (<ref>) to find that x,y satisfies x,y = (1 - ρ) ρ/h(x,y)( (y) - (x) + y - 1 ) = (1 - ρ)ρ(x)/(x) - y( 1 - y + (x) - (y) ). Since (x) = ((x)) - 1 + (x) we can write the expression for x,y as x,y = (1 - ρ)ρ(x) ( 1 + ((x)) - (y)/(x) - y). The fraction is ((x)) - (y)/(x) - y = ∑_k ≥ 1^∘ (k + 1)(x) - ^∘ k(y)/(x) - y, where the k-th summand is equal to (1 - ρ)^2 ρ^k 1/(x)/(1 - ρ^k + 2)(1 - ρ^k + 1)1/1 - x ρ1 - ρ^k + 1/1 - ρ^k + 21/1 - y ρ1 - ρ^k/1 - ρ^k + 1. If we substitute k = 0 into (<ref>) we get 1. So, we find from (<ref>) that x,y = ∑_k ≥ 0(1 - ρ)^3 ρ^k + 1/(1 - ρ^k + 2)(1 - ρ^k + 1)1/1 - x ρ1 - ρ^k + 1/1 - ρ^k + 21/1 - y ρ1 - ρ^k/1 - ρ^k + 1. Expanding the terms (1 - x ρ (1 - ρ^k + 1)/(1 - ρ^k + 2))^-1 and (1 - y ρ (1 - ρ^k)/(1 - ρ^k + 1))^-1 as geometric series shows that x,y is given by, for |x|,|y| ≤ 1, ∑_i ≥ 0∑_j ≥ 1∑_k ≥ 0(1 - ρ)^3 ρ^k + 1/(1 - ρ^k + 2)(1 - ρ^k + 1)( ρ1 - ρ^k + 1/1 - ρ^k + 2)^i ( ρ1 - ρ^k/1 - ρ^k + 1)^j - 1 x^i y^j - 1. Comparing (<ref>) with the definition of x,y in (<ref>) shows that the equilibrium probabilities are explicitly given by p(i,j) = ∑_k ≥ 0 c_k _k^i _k^j - 1,i ≥ 0, j ≥ 1 with _k = ρ1 - ρ^k + 1/1 - ρ^k + 2, _k = ρ1 - ρ^k/1 - ρ^k + 1, c_k = (1 - ρ)^3 ρ^k + 1/(1 - ρ^k + 2)(1 - ρ^k + 1).§ MATRIX-GEOMETRIC METHOD The two-dimensional Markov process { X(t) }_t ≥ 0 is a Markov process that is QSF to the right (also called a G/M/1-type Markov process) with levels i = { (i,1),(i,2),…}, i ≥ 0. We ignore state (0,0) since it does not appear in the balance equations (<ref>), (<ref>) and (<ref>). Consistent with the indexing of levels, in this section the indexing of vectors and matrices starts at 1.The infinite-dimensional transition matrix Q can be partitioned into levels as Q = [ _0^(0) _1;_-1 _0 _1;_-2_0 _1;_-3 _0 _1;⋮ ⋱⋱;], where _-k has zeroes everywhere, except (_-k)_1,k = μ, _1 =, withthe infinite-dimensional identity matrix, _0 = -( + μ) + [ 0; μ 0; μ 0; ⋱ ⋱ ] and _0^(0) = _0 + [ μ 0 ⋯; 0 0; ⋮ ⋱ ].Define the vectors _i [ p(i,1) p(i,2)⋯ ]. The rate matrix R satisfies the matrix equation R _0 + _1 = 0 and the equilibrium probabilities follow from _i + 1 = _i R,i ≥ 0, where the boundary probabilities are computed as _0 _0^(0) + ∑_k ≥ 1_k _-k = ⇔_0 ( _0^(0) + ∑_k ≥ 1 R^k _-k) = and the normalization condition is _0 ( - R)^-1 = ρ since p(0,0) = 1 - ρ, whereis a vector of ones.A highly accurate approximation of the equilibrium distribution can be obtained by truncating all matrices in (<ref>) to size K × K with K large and using successive substitutions to determine R (see <ref>). To determine _0 from (<ref>) both the matrices and the infinite sum must be truncated. However, we can do better than that by exploiting the specific structure of the transition rate diagram.Since the transitions between the levels are not upward, we know from the probabilistic interpretation of the elements of the rate matrix (see <ref>) that R = [ r_0; r_1 r_0; r_2 r_1 r_0; ⋮ ⋮ ⋮ ⋱ ]. The elements of R can be determined explicitly. Component-wise the equations (<ref>) read -( + μ) r_0 += 0, -( + μ) r_k + μ r_k - 1 = 0,k ≥ 1, so we obtain r_k = ( μ/ + μ)^k / + μ,k ≥ 0.Determining _0 exactly is difficult. The balance equations (<ref>) for level 0 involve infinite-dimensional matrices and an infinite sum. We propose the following approximation scheme for _0: truncate the vector and all matrices in (<ref>) to have dimension K; truncate the infinite sum to K; replace one equation with the normalization condition _0 ( - R)^-1 = ρ; and numerically solve for _0. The inverse of - R can be calculated exactly, see <ref>.The vectors _i, i ≥ 1 follow from (<ref>), which reads as p(i + 1,j) = ∑_k ≥ 0 p(i,j + k) r_k. Unfortunately, also this expression involves an infinite sum. Truncating the sum once more to K finally gives an approximation for the equilibrium probabilities.<ref> shows how to derive the approximate equilibrium distribution using the matrix-geometric method. The parameter K determines the accuracy of the obtained approximate equilibrium probabilities: the dimension of all matrices and infinite sums are truncated to K. So, increasing K increases the accuracy of the results, but also requires more computation time.Recall that the total number of jobs behaves like an M/M/1 queue and therefore we have the exact equilibrium probabilities in (<ref>). Clearly, for k ≥ 1, p(k) = ∑_i = 0^k - 1 p(i,k - i). In <ref> we compare p(k) obtained using <ref> with the exact values of (<ref>).§ COMPENSATION APPROACH We make the educated guess that p(i,j) is of the form ^i ^j - 1. Substitute this guess into the balance equations (<ref>) and divide by common powers to obtain (ρ + 1)= ρ + ⇒ = ρ/ρ + 1 -f(). Any pair (,) that satisfies (<ref>), satisfies the balance equations (<ref>). Moreover, any linear combination of product-form solutions that each, by itself, satisfies (<ref>) also satisfies (<ref>), which is a crucial property that we shall exploit. Since the equilibrium distribution must be normalized, only solution pairs (,) with ||,|| < 1 are of interest.We will construct a linear combination of solutions that satisfy the balance equations for the states in the interior to also satisfy the balance equations (<ref>) on the vertical axis. If both (<ref>) and (<ref>) are satisfied, then the remaining balance equation (<ref>) is automatically satisfied, since the balance equations are dependent.Rearrange (<ref>) to (ρ + 1)p(0,j) - p(0,j + 1) = p(j,1),j ≥ 2. Let us take as initial term p(i,j) = c_0 _0^i _0^j - 1 with _0 = 0, _0 = f(_0) = ρ / (ρ + 1) and c_0 > 0 some constant. The choice _0 = 0 is essential; we argue why in <ref>. Since the pair (_0,_0) satisfies (<ref>), the initial term p(i,j) satisfies the balance equations of the interior, but does it also satisfy (<ref>)? Substitute p(i,j) = c_0 _0^i _0^j - 1 into (<ref>) to get 0 = c_0 _0^j. It is clear that the above equality does not hold, however, in this section we will abuse notation and write `=' anyway. Clearly, p(i,j) does not satisfy (<ref>). Let us therefore add another product-form term to compensate for the error c_0 _0^j. Set p(i,j) = c_0 _0^i _0^j - 1 + c_1 _1^i _1^j - 1 and substitute this into (<ref>) to get c_1(ρ + 1) _1^j - 1 - c_1 _1^j = c_0 _0^j + c_1 _1^j. Since we want to compensate for the error introduced by the initial term, we chose c_1 and _1 such that c_1(ρ + 1) _1^j - 1 - c_1 _1^j = c_0 _0^j. Equation (<ref>) must hold for all j ≥ 1 and it is therefore immediate that we must choose _1 = _0. We want the pair (_1,_1) to satisfy (<ref>) and therefore conclude that _1 = _0, _1 = f(_1),c_1 = c_0 _0/ρ + 1 - _0. By compensating once and choosing c_1, _1 and _1 as in (<ref>) we have introduced a new error on the right-hand side of (<ref>), namely c_1 _1^j. We compensate a second time: add a product-form term to the solution to get p(i,j) = c_0 _0^i _0^j - 1 + c_1 _1^i _1^j - 1 + c_2 _2^i _2^j - 1 and compensate for the error term c_1 _1^j introduced by the previous compensation step. Similarly as for the previous compensation step, we set _2 = _1, _2 = f(_2),c_2 = c_1 _1/ρ + 1 - _1. Substituting this three-term solution p(i,j) into (<ref>) gives zero on the left-hand side, but an error term c_2 _2^j on the right-hand side.The procedure is clear: compensation step k adds a term c_k _k^i _k^j - 1 to the current solution to compensate for the error term c_k - 1_k - 1^j introduced during compensation step k - 1. The terms are chosen according to _k = _k - 1, _k = f(_k),c_k = c_k - 1_k - 1/ρ + 1 - _k - 1.Now, if the error terms c_k _k^j tend to zero sufficiently fast as k →∞, then the linear combination of product-form solutions p(i,j) = ∑_k ≥ 0 c_k _k^i _k^j - 1,i ≥ 0, j ≥ 1, is finite and satisfies (<ref>) and (<ref>). From _k = f(_k - 1) and _k = _k - 1 it can be verified that _k= ρ1 - ρ^k + 1/1 - ρ^k + 2, _k = ρ1 - ρ^k/1 - ρ^k + 1, c_k= c_0 ∏_l = 0^k - 1_l/ρ + 1 - _l = c_0 ρ^k 1 - ρ^2/1 - ρ^k + 21 - ρ/1 - ρ^k + 1. From these explicit expressions it is clear that c_k _k^j → 0 for k →∞. In fact, the error terms c_k _k^j tend to zero geometrically fast (with rate ρ). Since 0 < _k,_k < 1 and c_k > 0, we know that (<ref>) is maximal if i = 0 and j = 1. Therefore, if (<ref>) is finite for i = 0 and j = 1, then it is finite for all i ≥ 0, j ≥ 1. We have p(0,1) = ∑_k ≥ 0 c_k = c_0 ∑_k ≥ 0ρ^k 1 - ρ^2/1 - ρ^k + 21 - ρ/1 - ρ^k + 1 < c_0 ∑_k ≥ 0ρ^k < ∞. The constant c_0 follows from the normalization condition (<ref>): ρ = ∑_i ≥ 0∑_j ≥ 1∑_k ≥ 0 c_k _k^i _k^j - 1 = ∑_k ≥ 0 c_k 1/1 - _k1/1 - _k = c_0 1 - ρ^2/(1 - ρ)^2. So, c_0 = ρ(1 - ρ)^2/1 - ρ^2 and therefore c_k = (1 - ρ)^3 ρ^k + 1/(1 - ρ^k + 2)(1 - ρ^k + 1). <ref> shows how the compensation parameters _k and _k are generated. From <ref> it is clear that if _0 > ρ then _k → 1 and _k → 1 for k →∞. This in turn means that by (<ref>) c_k →∞ as k →∞ and the error terms c_k _k^j →∞. Hence, it is clear that _0 must satisfy 0 ≤_0 < ρ. However, we have made the specific choice _0 = 0. We demonstrate why that choice is essential.Let us fix an alternative _0 with 0 < _0 < ρ. In that case, substituting the initial term p(i,j) = c_0 _0^i _0^j - 1 in (<ref>) results in two error terms: the term c_0 _0^j on the right-hand side and the term (ρ + 1) _0^j - 1 - _0^j on the left-hand side. So, we would need to add two terms to compensate for the two errors. <ref> shows that an infinite sequence of _k and _k is generated in two directions, where in one direction _k and _k tend to -∞, thus leading to a divergent infinite series expression. Continuing in this way leads to a divergent infinite series expression for the equilibrium probabilities.On the contrary, the choice _0 = 0 results in only one error term, which generates a convergent infinite series. As an edge case, choosing _0 = f^∘ k(0) for some k, the sequence that is generated in the left- and downward direction terminates when the coordinate (0,0) is hit (the correct initial value!). We have seen that the compensation approach solves the balance equations by inserting a linear combination of product-form solutions. The linear combination contains a countably infinite number of product-form solutions and therefore a procedure is required to select the right product-form terms. These product-form solutions all have one thing in common: they satisfy the balance equations (<ref>) of the states in the interior of the state space.For numerical purposes the infinite sum expression must be truncated. We outline a simple procedure to determine an approximation of any equilibrium probability in <ref>. Just as for the matrix-geometric approach, we compare the values obtained for p(k) from <ref> and the exact values in (<ref>). Comparing <ref> with <ref>, it seems that the compensation approach produces better approximations of the equilibrium probabilities while requiring less computation time.§ TAKEAWAYS The Markov process associated with the gated single-server system has no upward transitions in the interior of the state space. However, it does have transition from the states on the horizontal axis to states on the vertical axis, a property that makes the analysis of the gated single-server system challenging.For the generating function approach, the transitions from the horizontal axis to the vertical axis ensured that both x,0 and y,0 appear, while 0,y did not appear in the functional equation for x,y. Substituting the root (x) into the functional equation led to an expression of x,0 in terms of the same generating function evaluated in a different point, namely (x),0. By iteratively substituting (x),0,((x)),0,… an infinite sum expression was obtained for x,0. In our case the function (x) was easy to work with and allowed for an explicit determination of x,0. Finally, an explicit expression involving three infinite summations was obtained for x,y. The expression for x,y revealed that each equilibrium probability has an explicit expression in terms of an infinite sum of product-form terms.Even though the elements of the rate matrix R of the matrix-geometric method were determined explicitly, it seems that this method suffered the most from the complex balance equations of the states on the vertical axis. We had to resort to numerical approximations of the equilibrium probabilities by truncating relevant matrices, vectors and infinite summations.The compensation approach was well-suited for Markov processes with this structure in the transition rate diagram. The approach identified that a product-form solution satisfies the balance equations of the states in the interior of the state space. These product-form solutions were then linearly combined to also satisfy the balance equations of the states on the vertical axis. Finally, we showed that the error terms tend to zero and that the infinite sum expression is convergent so that the infinite sum expression indeed described the equilibrium probabilities.The generating function approach and the compensation approach both led to the same product-form solution. Whereas the generating function approach can be used to obtain the generating function of the equilibrium probabilities for a broad class of Markov processes, the compensation approach is more limited in scope. However, if the compensation approach can be applied, then it leads to an explicit expression for the equilibrium probabilities. We consider another model where the compensation approach can be applied in <ref>. CHAPTER: PRODUCTION SYSTEMS In this chapter we consider three production systems that give rise to two-dimensional Markov processes. The first system produces standard items to stock and non-standard items to demand. The second system produces items in two phases. When all demand for items is fulfilled, the system is allowed to complete the first phase of the production and place these half-finished items on stock.The first and second system are hybrid systems that combine two production disciplines: make-to-order and make-to-stock. The analysis of these two systems is presented in Adan and van der Wal <cit.>. Williams <cit.>, Van Donk <cit.> and Carr et al <cit.> treat hybrid systems and answer questions such as which item to stock and which item to produce to order and what capacity is required. The third system is a production line with two machines and three processing steps. The first and last step are both executed by machine one. Machine one works on step one items and immediately switches to items that require processing in the last step whenever they become available.The third model is called a re-entrant line model and is analyzed in Adan and Weiss <cit.> using the same techniques as in this book. The stability condition is derived in Weiss <cit.>. Similar re-entrant line models (without the infinite supply of work) can be found in Chen and Meyn <cit.> and Dai and Weiss <cit.>. For each system we present a tailor-made solution method to obtain the equilibrium distribution.§ STOCKING STANDARD ITEMS Consider a single-server system that produces both standard items to stock and non-standard items to demand. When there is no unfulfilled demand for either product, the server (machine or worker) produces standard items to stock in anticipation of future demand. We assume that at most J units of standard items can be placed on stock. Demand for standard items are delivered from stock. However, if there is no stock, then the server produces standard items to satisfy the demand. Non-standard items are never delivered from stock, but are produced to order. Demand for standard and non-standard items arrives according to Poisson processes with rates _1 and _2. We denote _1 + _2. The production times for both items are exponentially distributed with rate μ. Producing items to satisfy demand preempts the production of standard items to stock. The sample paths of this system alternate between the server producing as many standard items to stock as possible in its otherwise idle time and the server satisfying demand for both standard and non-standard items.Let X_1(t) be the total number of unfulfilled demand (both standard and non-standard items) at time t and let X_2(t) be the number of standard items on stock. Denote the state of the system by X(t)(X_1(t),X_2(t)). Then { X(t) }_t ≥ 0 is a Markov process with state space { (i,j) ∈_0^2 : 0 ≤ j ≤ J }. It is apparent from the transition rate diagram in <ref> that the state space is irreducible. To guarantee positive recurrence and the existence of the equilibrium distribution we require that ρ / μ < 1. Let p(i,j) denote the equilibrium probability of being in state (i,j).The Markov process { X(t) }_t ≥ 0 is a QBD process with levels i{ (i,0),(i,1),…,(i,J) },i ≥ 0. We use the matrix-geometric method to determine the equilibrium distribution. To that end, define the vectors _i [ p(i,0) p(i,1)⋯ p(i,J) ].We display the balance equations in vector-matrix notation. The balance equations for the interior levels i, i ≥ 1 are _i - 1_1 + _i _0 + _i + 1_-1 = , where _-1 = μ, _1 = _2+ [ _10⋯0;00;⋮ ⋱ ;00 ] and _0 = -( + μ)+ [0 ; _10;_10;⋱⋱;_10 ]. For the boundary level 0 we have the balance equation _0 _0^(0) + _1 _-1 = , where _0^(0) = _0 + [ 0 μ; 0 μ; ⋱ ⋱; 0 μ; 0 ] + [ 0 0; ⋱ ⋮; 0 0; 0 ⋯ 0 μ ].The rate matrix R satisfies the matrix-quadratic equation R^2 _-1 + R _0 + _1 = 0, and the equilibrium probabilities follow from _i + 1 = _i R,i ≥ 0. The boundary probabilities _0 are computed from (<ref>) by inserting _1 = _0 R: _0 ( _0^(0) + R _-1) = and from either the normalization condition _0 ( - R)^-1 = 1 or using p(0,J) = 1 - ρ. Clearly, we can use successive substitutions, see <ref>, to determine the rate matrix R from (<ref>). However, we can do better and obtain exact expressions by exploiting the structure of the transition rate diagram.In all levels except 0 the process cannot move upwards. So, from the probabilistic interpretation (see <ref>) we know that the rate matrix R is a lower triangular matrix. We have seen this before in <ref> and <ref>. Moreover, many of its elements are identical due to the homogeneous transition structure for phases 1 until J. In particular, we can write R = [ b_0; b_1 r_0; b_2 r_1 r_0; b_3 r_2 r_1 r_0; ⋮ ⋱; b_J r_J - 1 ⋯ r_0 ]. With this representation in mind, the system of equations (<ref>) can be written component-wise as μ b_0^2 - ( + μ) b_0 += 0,μ r_0^2 - ( + μ) r_0 + _2= 0, and μ( b_i b_0 + ∑_k = 1^i r_i - k b_k ) + _1 r_i - 1 - ( + μ) b_i= 0,1 ≤ i ≤ J,μ∑_k = j^i r_i - k r_k - j + _1 r_i - (j + 1) - ( + μ) r_i - j = 0,1 ≤ j < i ≤ J. Since R is the minimal non-negative solution, we get that b_0 = ρ and r_0 =+ μ - √(( + μ)^2 - 4_2 μ)/2μ. Many of the equations (<ref>) are identical. We introduce d = i - j in (<ref>) and find μ∑_k = 0^d r_d - k r_k + _1 r_d - 1 - ( + μ) r_d = 0,1 ≤ d ≤ J - 1. Starting from the initial values b_0 and r_0 we can solve for r_d for 1 ≤ d ≤ J - 1 using (<ref>) and then solve for b_i for 1 ≤ i ≤ J using (<ref>). Finally, we construct the matrix R according to (<ref>).We still need to solve for the boundary probabilities _0. The balance equations (<ref>) can be solved iteratively. Component-wise these equations read, for 1 ≤ j ≤ J - 1, 0= μ p(0,J - 1) -p(0,J) + μ p(0,J) r_0, 0= μ p(0,j - 1) - ( + μ) p(0,j) + _1 p(0,j + 1)+ μ∑_k = 0^J - j p(0,j + k) r_k, 0= -( + μ) p(0,0) + _1 p(0,1) + μ∑_k = 0^J p(0,k) b_k. Using p(0,J) = 1 - ρ we can solve for all boundary probabilities by starting with the equation of phase J and working our way down. Since we have the additional equation p(0,J) = 1 - ρ, equation (<ref>) is redundant, since we can determine p(0,0) from (<ref>) for j = 1.<ref> summarizes the matrix-geometric method for the model that combines production of standard items to stock and non-standard items to demand.The number of unfulfilled demand in equilibrium is denoted by X_1. Using <ref> we can determine key performance indicators such as the expected number off unfulfilled demand X_1 and the probability X_1 ≥ 2 that two or more unfulfilled orders are in the system. We show both performance indicators in <ref> as a function of the maximum stock level J. Clearly, increasing J when J is relatively small has a larger positive impact on these indicators than when J is already relatively large.§ STOCKING HALF-FINISHED ITEMS The next production system that we consider is one that produces items in two phases. The first and second phases take an exponential amount of time with parameters μ_1 and μ_2. There is a single server (machine or worker) that produces the items. The first phase is identical for all items. Therefore, some half-finished items (items for which only the first phase is completed) can be placed on stock in anticipation of future demand. We assume that at most J units of half-finished items can be placed on stock. Demand for a single item arrives according to a Poisson process with rate . When demand arrives, the server immediately takes a half-finished item from stock and finishes its second phase, or, if there is no stock, starts immediately with the first phase. The behavior of the production system is cyclical: the server produces as much stock as possible in its otherwise idle time and then satisfies demand as it comes in until all demand is satisfied and the server returns to producing stock.Let X_1(t) be the number of unfulfilled demand at time t and let X_2(t) be the number of half-finished items in the system at time t. Denote the state of the system by X(t)(X_1(t),X_2(t)). Then { X(t) }_t ≥ 0 is a Markov process with state space { (i,j) ∈_0^2 : 0 ≤ j ≤ J }. <ref> shows the transition rate diagram. The state space is irreducible because from each state all other states can be reached. The Markov process is positive recurrent if ρ (1/μ_1 + 1/μ_2) < 1 and then the equilibrium distribution exists. Let p(i,j) denote the equilibrium probability of being in state (i,j).The Markov process { X(t) }_t ≥ 0 is a QBD process with levels as in (<ref>). We use generating functions to determine the equilibrium distribution. Since it is a QBD process, also other approaches such as the matrix-geometric or matrix-analytical methods are applicable, but we do not demonstrate them.The balance equations for the interior levels i, i ≥ 1 are ( + μ_2) p(i,J)=p(i - 1,J), ( + μ_2) p(i,j)=p(i - 1,j) + μ_2 p(i + 1,j + 1),2 ≤ j ≤ J - 1, ( + μ_2) p(i,1)=p(i - 1,1) + μ_2 p(i + 1,2) + μ_1 p(i,0), ( + μ_1) p(i,0)=p(i - 1,0) + μ_2 p(i + 1,1). For 0 we have the balance equations p(0,J)= μ_1 p(0,J - 1), ( + μ_1) p(0,j)= μ_1 p(0,j - 1) + μ_2 p(1,j + 1),1 ≤ j ≤ J - 1, ( + μ_1) p(0,0)= μ_2 p(1,1).Define the generating functions j∑_i ≥ 0 p(i,j) ^i,|| < 1, j = 0,1,…,J associated with the equilibrium probabilities of phase j. We derive expressions for these generating functions, starting with phase J and working our way down.Multiplying (<ref>) by ^i and summing over all i ≥ 1 yields ( + μ_2) ∑_i ≥ 1 p(i,J) ^i = ∑_i ≥ 1 p(i - 1,J) ^i. Adding and subtracting ( + μ_2)J0 on the left-hand side and bringing all J terms to one side allows us to write J = J0 + μ_2/ + μ_2 -= J01/1 - / + μ_2. Since J0 = p(0,J) = 1 - ρ, we obtain an explicit expression for J. In (<ref>) we recognize the geometric series p(i,J) = (1 - ρ) ( / + μ_2)^i.Next, we multiply (<ref>) by ^i + 1 and sum over all i ≥ 1 to obtain, for 2 ≤ j ≤ J - 1, ( + μ_2) ∑_i ≥ 1 p(i,j) ^i = ^2 ∑_i ≥ 1 p(i - 1,j) ^i - 1 + μ_2 ∑_i ≥ 1 p(i + 1,j + 1) ^i + 1. Using definition (<ref>) we can write this as ( + μ_2)( j - j0 ) = ^2 j+ μ_2 ( j + 1 - j + 10 - / yj + 1y|_y = 0). Equation (<ref>) for j involves j0, which is unknown at this point. We derive an additional equation to eliminate j0 from (<ref>). Define the set of states in phase j as A_j = { (0,j),(1,j),…} and the union of the sets in the first j phases as A_≤ j = ∑_k = 0^j A_k. Since the Markov process is in equilibrium, the rate at which the process enters and leaves the set of states A_≤ j is equal. For 1 ≤ j ≤ J - 1, this balance equation reads μ_1 p(0,j) = μ_2 ∑_k ≥ 1 p(k,j + 1), or, in terms of the generating functions, μ_1 j0 = μ_2 (j + 11 - j + 10). Using (<ref>) to eliminate j0 from (<ref>) yields (+ μ_2 - ) j = μ_2 ( j + 1 - j + 10 - / yj + 1y|_y = 0)+ (+ μ_2 ) μ_2/μ_1 (j + 11 - j + 10). As a result, the generating function j is expressed in terms of j + 1 evaluated at some points.We continue by examining (<ref>). Multiply both sides by ^i + 1 and sum over all i ≥ 1 to obtain ( + μ_2) ∑_i ≥ 1 p(i,1) ^i= ^2 ∑_i ≥ 1 p(i - 1,1) ^i - 1 + μ_2 ∑_i ≥ 1 p(i + 1,2) ^i + 1 + μ_1 ∑_i ≥ 1 p(i,0) ^i. Adding (<ref>) for j = 1 and simplifying using the definition (<ref>) yields ( + μ_2 - ) 1 = (μ_2 - μ_1) 10 + μ_2 ( 2 - 20 )+ μ_1 0. Eliminate 10 using (<ref>) to derive ( + μ_2 - ) 1 = (μ_2 - μ_1) μ_2/μ_1 (21 - 20)+ μ_2 ( 2 - 20 ) + μ_1 0.We derive a second expression for 1 and 0. Multiply both sides of (<ref>) by ^i + 1, sum over all i ≥ 1 and add (<ref>) to obtain ( + μ_1) ∑_i ≥ 0 p(i,0) ^i= ^2 ∑_i ≥ 1 p(i - 1,0) ^i - 1 + μ_2 ∑_i ≥ 0 p(i + 1,1) ^i + 1. Simplify this expression by using definition (<ref>) and balance equation (<ref>): ( + μ_1 - ) 0 = μ_2 1 - μ_2^2/μ_1 (21 - 20).Substituting (<ref>) into (<ref>) then completes the system of equations for j, 0 ≤ j ≤ J. Starting from the explicit expression of J in (<ref>) we iteratively solve j for j = J - 1, J - 2, …, 0 from (<ref>), (<ref>) and (<ref>).It must be noted here that it seems that the terms J - j, 0 ≤ j ≤ J - 2 can be written as a polynomial of degree j + 1 in 1/(1 - /( + μ_2) ). However, an explicit expression of the coefficients in each polynomial is difficult to obtain, so we do not that discuss this here.The equilibrium probabilities can be determined by taking derivatives, or by using a standard inversion algorithm for univariate generating function such as the one we presented in <ref>.Let X_2 be the number of half-finished items on stock in equilibrium. As a performance indicator of the system, we can compute, for J even, X_2 ≤ J/2 = ∑_j = 0^J/2j1, which is the probability to be low on stock. No inversion algorithm is required to determine X_2 ≤ J/2, since j1 can easily be computed from the solution j. In <ref> we show this probability for various values of J. By increasing J, X_2 ≤ J/2 decreases, which indicates that for a larger fraction of orders, only the second processing phase remains at the arrival instant.§ RE-ENTRANT LINE The third system is a re-entrant line consisting of two machines that produce items. Each item undergoes three processing steps. In the first step it is processed by the first machine, in the second step by the second machine and it finally returns to the first machine for its third processing step. As is typical in a manufacturing environment, there are always items that can be processed in the first step. We therefore assume that there is an infinite number of items awaiting the first processing step. Service times in each step are exponentially distributed with rates μ_1, μ_2 and μ_3. <ref> shows the re-entrant line.Since the first machine processes items for the first step and for the third step, we need a policy that dictates which item the first machine should serve whenever there are items in both queues. The policy we study here prioritizes processing the items in the third queue. More precisely, we assume that this priority is preemptive: whenever an item arrives in the third queue, machine 1 will stop processing an item from queue 1 and start processing the item from queue 3, only to resume processing items in queue 1 when queue 3 is empty. So, machine 1 will undergo cycles of work on items in queue 1, which are called push periods (pushing items into the system), and work on items in queue 3, which are called pull periods (pulling items from the system). The re-entrant line is therefore also sometimes called a push-pull system <cit.>.Let X_2(t) and X_3(t) be the number of items at the second and third queue at time t. Denote the state of the system by X(t)(X_2(t),X_3(t)). Then { X(t) }_t ≥ 0 is a Markov process with state space _0^2. The transition rate diagram in <ref> shows that the state space is irreducible. The states are positive recurrent if 1/μ_1 + 1/μ_3 > 1/μ_2. The intuition behind this condition is that if it does not hold, then the arrival rate to machine 2 will be 1/(1/μ_1 + 1/μ_3), which exceeds its service rate μ_2. The proof of (<ref>) is shown in <cit.>. For now we assume that the condition holds and we prove that it is a sufficient condition later. Let p(i,j) denote the equilibrium probability of being in state (i,j).The balance equations for the interior states are (μ_2 + μ_3) p(i,j) = μ_3 p(i,j + 1) + μ_2 p(i + 1,j - 1),i,j ≥ 1. On the vertical axis we have the balance equations μ_3 p(0,j) = μ_2 p(1,j - 1) + μ_3 p(0,j + 1),j ≥ 1. The balance equations on the horizontal axis are (μ_1 + μ_2) p(i,0) = μ_1 p(i - 1,0) + μ_3 p(i,1),i ≥ 1, and at the origin we have μ_1 p(0,0) = μ_3 p(0,1).We solve for the equilibrium probabilities by directly working with the balance equations. We attempt to solve the balance equations by inserting a product-form solution ^i ^j with 0 < ||,|| < 1 to ensure that we can normalize the solution. Substituting this product form in both (<ref>) and (<ref>) and dividing by common powers results in the following system of equations: (μ_2 + μ_3)= μ_3 ^2 + μ_2 , (μ_1 + μ_2)= μ_1 + μ_3 . We determineandby solving this system of equations. From (<ref>) we have that = μ_1/μ_1 + μ_2 - μ_3 . Substituting (<ref>) in (<ref>) and multiplying both sides by μ_1 + μ_2 - μ_3 yields a cubic equation in : f() μ_3^2 ^3 - μ_3 (μ_1 + 2μ_2 + μ_3) ^2 + (μ_1 + μ_2) (μ_2 + μ_3)- μ_1 μ_2 = 0. One of the roots of this equation is = μ_2/μ_3. We can therefore factorize (<ref>) as f() = (μ_3- μ_2) ( μ_3 ^2 - (μ_1 + μ_2 + μ_3)+ μ_1 ) = 0. Let us study the function f() in more detail, see also <ref>. We know that f(0) = - μ_1 μ_2 < 0 and lim_→∞ f() = ∞. At = μ_2/μ_3 we compute the derivative f'(): f'( μ_2/μ_3) = μ_1 μ_3 - μ_1 μ_2 - μ_2 μ_3 = μ_1 μ_2 μ_3 ( 1/μ_2 - 1/μ_3 - 1/μ_1) < 0, due to the stability condition (<ref>). Because of these properties and the fact that f() = 0 is a cubic equation, we know that f() = 0 has three positive roots: one at μ_2/μ_3, and one smaller and one larger than μ_2/μ_3. We now show that the smallest root is in (0,1).Define g()- (μ_1 + μ_2 + μ_3) ,h() μ_3 ^2 + μ_1. Clearly, g() has a single root in || < 1. Now, for || = 1, we have |g()| = (μ_1 + μ_2 + μ_3) || = μ_1 + μ_2 + μ_3 and |h()| = |μ_3 ^2 + μ_1| ≤μ_3 |^2| + μ_1 = μ_1 + μ_3. So, for || = 1, we know that |g()| > |h()| and according to Rouché's theorem, see <ref>, g() + h() has a single root in || < 1. This root is given by = μ_1 + μ_2 + μ_3 - √((μ_1 + μ_2 + μ_3)^2 - 4μ_1 μ_3)/2μ_3. Substituting the rootpresented in (<ref>) into (<ref>) yields after some manipulations = μ_1/μ_2 (1 - ). Since the rootsatisfies both 0 << 1 and < μ_2/μ_3, we know from the latter condition and (<ref>) that also 0 << 1.At this point we have a solution for the balance equations of the states (i,j) with i ≥ 1 and j ≥ 0. We substitute this solution in (<ref>) to get μ_3 p(0,j) = μ_2 ^j - 1 + μ_3 p(0,j + 1),j ≥ 1. Since this equation holds for all j ≥ 1, we must have p(0,j) = c ^j, j ≥ 1, where c follows from substituting this solution in (<ref>): c = μ_2 /μ_3(1 - ) = μ_1/μ_3 .The remaining (not yet normalized) equilibrium probability p(0,0) is determined from (<ref>) and can be seen to equal 1.At this point we return to the stability condition (<ref>). The solution of the balance equations is non-zero and since it is geometric, it is immediately seen to be absolutely convergent. So, as a result of <ref>, the Markov process is positive recurrent. Since we assumed (<ref>) to hold, we know that it is a sufficient condition for positive recurrence.The solutions that we have obtained are not yet normalized. By multiplying them by the normalization constant C, we obtain p(i,j) =C, i = j = 0, C μ_1/μ_3^j - 1, i = 0, j ≥ 1, C ^i ^j, i ≥ 1, j ≥ 0. The normalization condition reads 1 = ∑_i ≥ 0∑_j ≥ 0 p(i,j) = C ( 1 + μ_1/μ_31/1 -+ /1 - 1/1 - ). Using (<ref>) we write 1= C ( 1 + μ_1/μ_21/1 -+ μ_1/μ_31/1 - ) = C 1/μ_21/μ_31/1 - 1/1 - ·( μ_2 μ_3 (1 - )(1 - ) + μ_1 μ_3 (1 - ) + μ_1 μ_2 (1 - ) ). We focus on the term in parentheses. Eliminateusing (<ref>) to obtain μ_2 μ_3 (1 - μ_1/μ_2(1 - ))(1 - ) + μ_1 μ_3(1 - ) + μ_1 μ_2 (1 - μ_1/μ_2(1 - )) = μ_2 μ_3 (1 - ) + μ_1 μ_3 ( - ^2) + μ_1 μ_2 - μ_1^2 (1 - ) = μ_2 μ_3 (1 - ) + μ_1 ( - μ_3 ^2 + (μ_1 + μ_3 )- μ_1 + μ_2). Sincesatisfies (<ref>), we can simplify (<ref>) and finally obtain C = μ_3/μ_1 + μ_3 (1 - ).In conclusion, provided (<ref>) holds, the Markov process associated with the re-entrant line has the equilibrium probabilities p(i,j) = μ_3/μ_1 + μ_3 (1 - ), i = j = 0,μ_1/μ_1 + μ_3 (1 - ) ^j - 1, i = 0, j ≥ 1,μ_3/μ_1 + μ_3 (1 - ) ^i ^j, i ≥ 1, j ≥ 0, whereandare given in (<ref>) and (<ref>).From the explicit expression (<ref>) we can easily determine other key performance indicators. For example, the marginal distribution p_2(·) of the number of items at machine 2 is given by p_2(i) = ∑_j ≥ 0 p(i,j) = μ_3/μ_1 + μ_3 (1 - ) (1+ 1/1 - ), i = 0,μ_3/μ_1 + μ_3 (1 - ) ^i/1 - , i ≥ 1.§ TAKEAWAYS The three production systems in this chapter shared the common property that they produce items whenever they would otherwise be idle. For the first and second model this was clear: if the server would otherwise be idle, then in the first case standard items are produced to stock and in the second case the first phase of the production process is completed. In the third production system, machine one produces items from queue one whenever there are no items awaiting their third processing step.The Markov processes associated with the first and second production system both led to two-dimensional Markov processes with one finite dimension and only nearest-neighbor transitions, which made them QBD processes. The transition rate diagrams of both systems had no upward transitions in all levels except for level 0. This structure was exploited to obtain exact expressions for the equilibrium distribution. In the first model we used the matrix-geometric method to determine the equilibrium distribution. But, instead of using the successive substitutions algorithm to determine R, we noticed that R must be lower triangular and that many of its elements must be identical. We have seen this before in <ref>. However, in this case the transition structure in phase 0 was different from the transition structure in all other phases, which made that the boundary elements of R are different from the other elements.For the second model we again exploited the downward transition structure. By introducing a generating function for each phase, we could recursively determine all generating functions, starting from the known generating function for phase J. This approach allowed for an easy determination of the probability that there are j half-finished items on stock, since this is equal to j1.Both the first and the second model could be analyzed in multiple ways. The matrix-geometric (and matrix-analytic) method of the first model could be used to determine the equilibrium distribution of the second model. In <cit.>, the two production systems are analyzed using the difference equations approach outlined in <ref>. Instead of starting in phase 0 as in the priority systems of <ref>, we started with phase J and worked our way down to phase 0. This is in line with the two approaches that we have seen in this chapter.The third system has two infinite dimensions. To determine the equilibrium distribution of this model, we showed that the balance equations in the interior and on the horizontal axis were satisfied by a product-form solution. This product-form solution could be extended to also hold on the vertical axis and the origin by suitably multiplying it by a constant, see also <cit.>.CHAPTER: JOIN THE SHORTEST QUEUE In this chapter we consider a system consisting of two exponential single-server queues in parallel. Jobs arrive to the system according to a Poisson process and join the shortest of the two queues. If the queue lengths are equal, then the job joins either queue with equal probability. Once a job has joined one of the two queues, it stays there until it has completed service. We are interested in the joint distribution of the number of jobs in both queues.This join the shortest queue systemThe join the shortest queue system is a classical topic in queueing theory and can be analyzed in various ways. Haight <cit.> originally introduced the problem. Kingman <cit.> uses generating functions and complex analysis to determine the equilibrium distribution. Cohen <cit.> and Cohen and Boxma <cit.> reduce the analysis of the equilibrium distribution to finding a solution of a boundary value problem. Hooghiemstra, Keane and Van De Ree <cit.> introduce a power-series method to calculate the equilibrium distribution for a more general class of queueing systems. Blanc <cit.> applies this numerical algorithm to obtain results for the join the shortest queue model. Halfin <cit.> obtains bounds for the equilibrium distribution by using linear programming techniques. gives rise to a Markov process in two dimensions describing the joint queue-length distribution. However, this state description leads to a transition rate diagram that is inhomogeneous in the interior of the state space and thus, is difficult to analyze. We therefore transform the state space and create a Markov process on the positive half-plane, where we can eliminate one of the two quadrants due to symmetry. This leaves us to analyze a Markov process on the positive quadrant with a homogeneous transition rate diagram in the interior of this quadrant. We then use the compensation approach to determine the equilibrium distribution of this process in the form of an infinite series of product forms.§ MODEL DESCRIPTION AND BALANCE EQUATIONS Jobs arrive according to a Poisson process with rate 2ρ to two parallel queues. Each job requires an exponentially distributed service time with rate 1. Due to symmetry, ρ is the average amount of work brought into each queue per time unit.Let X_1(t) and X_2(t) be the number of jobs at the first and second queue at time t. Denote the state of the system of the system by X(t)(X_1(t),X_2(t)). Then { X(t) }_t ≥ 0 is a Markov process with state space _0^2. It is apparent from the transition rate diagram in <ref> that the state space is irreducible. To guarantee positive recurrence and the existence of the equilibrium distribution we assume that ρ < 1.<ref> shows that the states on the diagonal (i,i), i ≥ 0 divide the state space into regions with different transition structures. In each state below the diagonal the first queue has more jobs and an arriving job joins the second queue. For the states above the diagonal the situation is reversed. This is why the Markov process has an inhomogeneous transition structure, which complicates the analysis of the equilibrium distribution. We make the analysis easier by moving to a different state description and using a symmetry argument. Define Y_1(t) min(X_1(t),X_2(t)) as the minimum queue length at time t and Y_2(t)X_2(t) - X_1(t) as the difference between the two queue lengths at time t. The state of the system is now Y(t)(Y_1(t),Y_2(t)) and the process { Y(t) }_t ≥ 0 is a Markov process on the state space ' _0 ×.Each element ofcorresponds to exactly one element of ' and vice versa. For example, the state (m,n) ∈' corresponds to the state (m,m + n) ∈ if n > 0 and to (m - n,m) if n ≤ 0. The state space ' is irreducible becauseis irreducible and { Y(t) }_t ≥ 0 is positive recurrent if ρ < 1. So, determining the equilibrium distribution of the Markov process { Y(t) }_t ≥ 0 gives us the equilibrium distribution of the Markov process { X(t) }_t ≥ 0. Let p(m,n) denote the equilibrium probability of { Y(t) }_t ≥ 0 being in state (m,n) ∈'.The join the shortest queue policy does not favor any of the two servers in particular and the servers are identical, which makes the queue index interchangeable. As a result, the equilibrium probability that there are i jobs in the first queue and j jobs in the second queue is equal to the equilibrium probability that there are j jobs in the first queue and i jobs in the second queue. Hence, p(m,n) = p(m,-n), n > 0 by symmetry. If we can calculate p(m,n) for m,n ≥ 0, then we know the complete equilibrium distribution.The transition rate diagram of { Y(t) }_t ≥ 0 is shown in <ref>. Notice that the transition structure in each quadrant is homogeneous. Since determining p(m,n), m,n ≥ 0 is enough to obtain the complete equilibrium distribution, we only present the balance equations for the states in the positive quadrant ((m,n) with m,n ≥ 0). To that end, we exploit the symmetry property p(m,n) = p(m,-n), n > 0 and consider balance equations that only involve the equilibrium probabilities p(m,n), m,n ≥ 0. For m ≥ 1, n ≥ 2, 2(1 + ρ) p(m,n) = 2ρ p(m - 1,n + 1) + p(m,n + 1) + p(m + 1,n - 1), and for m ≥ 1, 2(1 + ρ) p(m,1) = 2ρ p(m - 1,2) + p(m,2) + ρ p(m,0) + p(m + 1,0). For the vertical axis we have, for n ≥ 2, (1 + 2ρ) p(0,n) = p(0,n + 1) + p(1,n - 1), and (1 + 2ρ) p(0,1) = p(0,2) + p(1,0) + ρ p(0,0). The balance equations for the horizontal axis are, for m ≥ 1, (1 + ρ) p(m,0) = 2ρ p(m - 1,1) + p(m,1), and at the origin ρ p(0,0) = p(0,1). Substituting (<ref>) and (<ref>) into (<ref>) and (<ref>) gives, for m ≥ 1, 2(1 + ρ) p(m,1)= 2ρ p(m - 1,2) + p(m,2)+ ρ/1 + ρ( 2 ρ p(m - 1,1) + p(m,1) )+ 1/1 + ρ( 2 ρ p(m,1) + p(m + 1,1) ), (1 + 2ρ) p(0,1)= p(0,2) + 1/1 + ρ( 2 ρ p(0,1) + p(1,1) ) + p(0,1).The equations (<ref>), (<ref>), (<ref>) and (<ref>) together form the balance equations of the states (m,n) with m ≥ 0, n ≥ 1. These equations only involve the equilibrium probabilities p(m,n), m ≥ 0, n ≥ 1. If we can determine these equilibrium probabilities as a solution to (<ref>), (<ref>), (<ref>) and (<ref>), then, through (<ref>) and (<ref>) and the symmetry p(m,n) = p(m,-n), n ≥ 0, we obtain p(m,n) for all (m,n). We adopt the following terminology for balance equations in three subsets of the state space { (m,n) : m ≥ 0, n ≥ 1 }. We refer to the balance equations (<ref>) as the balance equations of the interior; to (<ref>) as the balance equations of the vertical boundary; and to (<ref>) as the balance equations of the horizontal boundary. Define the bivariate PGF x,y∑_m ≥ 0∑_n ≥ 0 p(m,n) x^m y^n,|x| ≤ 1, |y| ≤ 1. We can obtain an expression for x,y by manipulating the balance equations (<ref>)–(<ref>). Multiplying the balance equation of state (m,n), m,n ≥ 0 by x^m y^n and summing over all m,n ≥ 0 produces the functional equation h_1(x,y) x,y = h_2(x,y) x,0 + h_3(x,y) 0,y with h_1(x,y) (1 + 2ρ x)x - 2(1 + ρ)xy + y^2, h_2(x,y) (1 + 2ρ x)x - (1 + ρ)xy - ρ x y^2, h_3(x,y) (y - x)y.We will not use the functional equation to determine x,y, but instead work directly with the balance equations to determine p(m,n), m,n ≥ 0 using the compensation approach. The functional equation will appear to be useful later on to determine the normalization constant, see also Kingman <cit.>.§ COMPENSATION APPROACH We have already seen the compensation approachAdan, Wessels and Zijm <cit.> develop the compensation approach which can be used to analyze the join the shortest queue model, but also many related models with state-dependent routing <cit.>. in <ref>. Recall that the compensation approach linearly combines product-form solutions ^m ^n. Each product-form solution is chosen such that it satisfies the balance equations (<ref>) of the interior. In each compensation step a single product-form solution is added.In a vertical compensation step a product-form solution is added such that the resulting linear combination of product-form solutions satisfies the balance equations of both the states in the interior (<ref>) and on the vertical boundary (<ref>). However, in doing so, the resulting linear combination does not satisfy the balance equations (<ref>) on the horizontal boundary.Each vertical compensation step is followed by a horizontal compensation step. In this step, a product-form solution is added such that the resulting linear combination of product-form solutions satisfies the balance equations of both the states in the interior (<ref>) and on the horizontal boundary (<ref>). Similarly to the vertical compensation step, the horizontal compensation step results in a linear combination that does not satisfy the balance equations (<ref>) on the vertical boundary.The procedure is repeated and each horizontal compensation step is followed by a vertical compensation step. We ultimately obtain two countably infinite linear combinations of product-form solution (one series each for the horizontal and vertical compensation steps). If the two series converge absolutely, then the error terms on each boundary converge sufficiently fast to zero. Finally, if the sum of the equilibrium probabilities is absolutely convergent, then by <ref>, the solution can be normalized to obtain the equilibrium distribution. §.§ Constructing the equilibrium distribution We make the educated guess that p(m,n) in the interior is of the form ^m ^n. Substitute this guess into the balance equations (<ref>) and divide by common powers to obtain 0 = ^2 + 2 ρ^2 + ^2 - 2(1 + ρ) . We have the following result regarding roots of (<ref>).* For every fixedwith || ∈ (0,1), equation (<ref>) has exactly one rootinside the open circle of radius ||.* For every fixedwith || ∈ (0,1), equation (<ref>) has exactly one rootinside the open circle of radius ||.(i) Divide (<ref>) by ^2 and set z = / to obtain the second-degree polynomial 0 = (2ρ + ) z^2 - 2(1 + ρ) z + 1. Define f(z)-2(1 + ρ) z, g(z)(2ρ + ) z^2 + 1 and the regionas the unit disk with the unit circle as the boundary . Clearly, f(z) has a single root in . Now, for z ∈, or equivalently |z| = 1, |f(z)|= 2(1 + ρ)|z| = 2 + 2ρ, |g(z)|= | (2ρ + ) z^2 + 1| ≤ (2ρ + ||) |z|^2 + 1 = 2ρ + || + 1. Since || < 1 we conclude that |f(z)| > |g(z)| for z ∈. Then, by Rouché's theorem, see <ref>, f(z) + g(z) has a single root inside the unit circle. This proves that (<ref>) has a single rootinside the circle with radius ||.(ii) Divide (<ref>) by ^2 and set z = / to obtain the second-degree polynomial 0 = z^2 + ( - 2(1 + ρ))z + 2ρ. Define f(z)( - 2(1 + ρ))z, g(z)z^2 + 2ρ and the sameandas in (i). Clearly, f(z) has a single root in . Now, for z ∈, or equivalently |z| = 1, |f(z)|= |- 2(1 + ρ) | |z|^2 ≥ | || - 2(1 + ρ) | > 1 + 2ρ, |g(z)|= |z^2 + 2ρ| ≤ |z|^2 + 2ρ = 1 + 2ρ, where the last inequality for f(z) follows from || ∈ (0,1). So, |f(z)| > |g(z)| for z ∈ and Rouché's theorem proves the claim. Let us, for now, further assume that the equilibrium probabilities along the horizontal and vertical boundary are also satisfied by a product-form solution p(m,n) = ^m ^n. We can substitute this solution in the balance equations for the horizontal boundary (<ref>): 0 = ^2 + ( (1 + ρ) + 3ρ - 2(1 + ρ)^2 ) + 2ρ ((1 + ρ) + ρ) and for the vertical boundary (<ref>): 0 = ^2 - (1 + 2ρ) + . In <ref> we show the curves (,) satisfying (<ref>), (<ref>) and (<ref>), respectively. Wherever two curves intersect, we know that that pair (,) satisfies those balance equations simultaneously. We find four of such pairs. Three of them are not useful since they do not satisfy 0 < ||,|| < 1. The remaining fourth pair satisfies simultaneously the balance equations of the interior (<ref>) and the horizontal boundary (<ref>). In general we can state that there is no pair (,) with 0 < ||, || < 1 that satisfies simultaneously the balance equations of the interior and the vertical boundary, but there is a single pair (,) that satisfies simultaneously the balance equations of the interior and the horizontal boundary. It is easy to derive this pair from the system of equations (<ref>) and (<ref>): (,) = (ρ^2,ρ^2/(2 + ρ)). In <ref> we numerically verify that this pair dictates the tail behavior of the equilibrium probabilities for m and n large.We see that if m and n are large, then p(m,n) ≈ h_0 _0^m _0^n for some coefficient h_0, and parameters _0 and _0. We have simulated a join the shortest queue model with ρ = 0.8 to determine _0 and _0 from the ratios p(m + 1,n)/p(m,n) and p(m,n + 1)/p(m,n), see <ref>. The simulation confirms that _0 = ρ^2 and _0 = ρ^2/(2 + ρ) describe the tail behavior for large m and n. A rigorous derivation of _0 and _0 is given in, e.g., <cit.>, but we do not show it here. Since normalization follows at the end of the compensation procedure, we can now set h_0 = 1.The pair (_0,_0) = (ρ^2,ρ^2/(2 + ρ)) is the only pair that satisfies simultaneously the balance equations of the interior and the horizontal boundary. In fact, this property characterizes the initial product-form solution h_0 _0^m _0^n. Since the initial solution does not satisfy the balance equations (<ref>) on the vertical boundary—as we have already concluded from <ref>—we need to compensate for the error introduced on the vertical boundary. It is important that in each compensation step—vertical or horizontal—the correction term that is added should be small compared to h_0 _0^m _0^n in order to not disturb the asymptotic behavior for large m or n.In the vertical compensation step we add a single product-form term to the initial solution and construct h_0 _0^m _0^n + v ^m ^n. We refer to v ^m ^n as the compensation term. We will choose v,andsuch that this linear combination satisfies both the balance equations of the interior (<ref>) and the vertical boundary (<ref>). Inserting it into (<ref>) gives for all n ≥ 2, (1 + 2ρ) ( h_0 _0^n + v ^n ) = h_0 _0^n + 1 + v ^n + 1 + h_0 _0 _0^n - 1 + v ^n - 1. Since this equation holds for all n ≥ 2, we must have that = _0. We further want the pair (,_0) to satisfy the balance equations of the interior, so we pick = _1 as the root of (<ref>) for fixed = _0 satisfying |_1| < |_0|. There also exists the root _0 of (<ref>) for fixed = _0 satisfying |_0| < |_0|, but that would turn the compensation term into the initial term, which makes that root not useful. By choosing = _1 and = _0, we know that the linear combination h_0 _0^m _0^n + v _1^m _0^n satisfies the balance equations of the interior. What remains is to choose v = v_0 in such a way that the linear combination h_0 _0^m _0^n + v_0 _1^m _0^n satisfies (<ref>). We now describe the method of choosing this coefficient in a general setting. Consider the product form h ^m ^n with 0 < || < || < 1 and some coefficient h, that satisfies the balance equations (<ref>) of the interior and stems from a solution that satisfies the balance equations of the interior and the horizontal boundary. For this fixed , letbe the root that satisfies (<ref>) with || < ||. Then there exists a coefficient v such that p(m,n) = h ^m ^n + v ^m ^n satisfies (<ref>) and (<ref>). The coefficient v is given by v = -- / -h.Notice that both (,) and (,) satisfy (<ref>). So, the linear combination h ^m ^n + v ^m ^n satisfies (<ref>) for any h and v.Inserting the linear combination into (<ref>) and dividing by common powers yields (h + v) ( 2(1 + ρ) - ^2 - ) = h+ v . Since (,) and (,) both satisfy (<ref>) we know that += 2(1 + ρ) - ^2. Substituting this relation into (<ref>) proves the claim. We apply <ref> to find that we must choose v_0 = - _1 - _0/_0 - _0 h_0. With these choices for the coefficient and the parameters of the compensation term, the linear combination h_0 _0^m _0^n + v_0 _1^m _0^n satisfies (<ref>) and (<ref>). However, adding the term v_0 _1^m _0^n introduces an error on the horizontal boundary for which we need to compensate.In a horizontal compensation step we add a compensation term to compensate for the error introduced during the vertical compensation step. So, we form the linear combination h_0 _0^m _0^n + v_0 _1^m _0^n + h ^m ^n. We will choose h,andsuch that this linear combination satisfies both the balance equations of the interior (<ref>) and on the horizontal boundary (<ref>). We know that h_0 _0^m _0^n already satisfies (<ref>) and (<ref>), so we do not need to take this term into account. Substituting the sum of the remaining two terms into (<ref>) gives for m ≥ 1, 2(1 + ρ) ( v_0 _1^m _0 + h ^m ) = 2ρ( v_0 _1^m - 1_0^2 + h ^m - 1^2 ) + v_0 _1^m _0^2 + h ^m ^2+ ρ/1 + ρ( 2 ρ( v_0 _1^m - 1_0 + h ^m - 1) + v_0 _1^m _0 + h ^m )+ 1/1 + ρ( 2 ρ( v_0 _1^m _0 + h ^m ) + v_0 _1^m + 1_0 + h ^m + 1). Since this equation holds for all m ≥ 1, we must have that = _1. We want the pair (_1,) to satisfy the balance equations of the interior, so we pick = _1 as the root of (<ref>) for fixed = _1 satisfying |_1| < |_1|. Just as in the vertical compensation step, we can discard the other root of (<ref>). So, by choosing = _1 and = _1, we know that the linear combination h_0 _0^m _0^n + v_0 _1^m _0^n + h _1^m _1^n satisfies the balance equations of the interior. What remains is to choose h = h_1 in such a way that the linear combination h_0 _0^m _0^n + v_0 _1^m _0^n + h_1 _1^m _1^n satisfies (<ref>). We now describe the method of choosing this coefficient in a general setting. Consider the product form v ^m ^n with 0 < || < || < 1 and some coefficient v, that satisfies the balance equations (<ref>) of the interior and stems from a solution that satisfies the balance equations of the interior and the vertical boundary. For this fixed , letbe the root that satisfies (<ref>) with || < ||. Then there exists a coefficient h such that p(m,n) = v ^m ^n + h ^m ^n satisfies (<ref>) and (<ref>). The coefficient h is given by h = - (ρ + )/ - (1 + ρ)/(ρ + )/ - (1 + ρ) v.Notice that both (,) and (,) satisfy (<ref>). So, the linear combination v ^m ^n + h ^m ^n satisfies (<ref>) for any v and h.Inserting the linear combination into (<ref>) and dividing by common powers yields v ( 2(1 + ρ)- 2ρ^2 - ^2 ) + h ( 2(1 + ρ)- 2ρ^2 - ^2 ) = ρ/1 + ρ( 2ρ (v+ h ) + v+ h )+ 1/1 + ρ( 2ρ (v+ h ) + v ^2+ h ^2 ). Since (,) and (,) satisfy (<ref>) we can simplify the coefficients of v and h on the left-hand side to obtain v ^2 (1 + ρ) + h ^2 (1 + ρ)= ρ( 2ρ (v+ h ) + v+ h )+ ( 2ρ (v+ h ) + v ^2+ h ^2 ). So, h = -(2ρ + )(ρ + ) - ^2(1 + ρ)/ (2ρ + )(ρ + ) - ^2(1 + ρ) v. Sinceandare roots of (<ref>) we have the relation (2 ρ + ) = ^2. Using this relation proves the claim. Applying <ref> shows that we must choose h_1 = - (ρ + _1)/_1 - (1 + ρ)/(ρ + _1)/_0 - (1 + ρ) v_0 to ensure that the linear combination h_0 _0^m _0^n + v_0 _1^m _0^n + h_1 _1^m _1^n satisfies (<ref>) and (<ref>). Adding the compensation term h_1 _1^m _1^n, however, introduces an error on the vertical boundary for which another vertical compensation step needs to be performed.It is clear how the compensation procedure works: after an initial product-form solution is constructed, it alternates between horizontal and vertical compensation steps to compensate for the error introduced on the vertical or horizontal boundary in the previous compensation step. In every vertical compensation step we just need to compensate for the error introduced by the compensation term of the previous horizontal compensation step; the linear combination of product-form solutions at the time of the previous vertical compensation step namely already satisfies the balance equations of the interior and on the vertical boundary! Obviously, the same statement can be made for the horizontal compensation step.<ref> shows the indexing of the terms of the compensation procedure. <ref> can be used to generate a finite number of compensation parameters and <ref> shows how the compensation parameters _i and _i are generated.The compensation procedure ultimately leads to a series expression for the equilibrium probabilities: p(m,n) = ∑_i ≥ 0 h_i _i^m _i^n + ∑_i ≥ 0 v_i _i + 1^m _i^n,m ≥ 0, n ≥ 1. If the errors terms converge sufficiently fast to zero, then the series converges. Moreover, if the sum of p(m,n) over all states is absolutely convergent, then it can be normalized to produce the equilibrium distribution and the balance equation (<ref>) in state (0,1) is also satisfied by this series expression, because we can sum over all other balance equations—which are already satisfied—to produce the balance equation in state (0,1). Hence, what remains to be done is (i) to show that the two series in (<ref>) converge absolutely and that ∑_m ≥ 0∑_n ≥ 1 |p(m,n)| < ∞; and (ii) to determine the normalization constant. §.§ Proving convergence of the series We will study the absolute convergence of the two series in (<ref>) by determining, for m ≥ 0, n ≥ 1, R_1(m,n) lim_i →∞| h_i + 1_i + 1^m _i + 1^n/h_i _i^m _i^n|, R_2(m,n) lim_i →∞| v_i + 1_i + 2^m _i + 1^n/v_i _i + 1^m _i^n|. The coefficients h_i and v_i and the roots _i and _i are non-zero for all i, which allows us to divide by these quantities in (<ref>). The coefficients cannot be zero, since this would indicate that there exists a product-form solution that satisfies the balance equations of the interior, horizontal boundary and the vertical boundary. From <ref> we know that such solution does not exist. By inspecting (<ref>) we know that all roots _i and _i are non-zero.If the limits (<ref>) exist and are strictly less than one, then we have proven that the two series in (<ref>) converge absolutely. We can rewrite (<ref>) as R_1(m,n) = lim_i →∞| h_i + 1/v_i_i + 1^m/_i + 1^m_i + 1^m + n/_i + 1^m + n/h_i/v_i_i^m/_i^m_i^m + n/_i + 1^m + n|, R_2(m,n) = lim_i →∞| v_i + 1/h_i + 1_i + 2^m/_i + 1^m_i + 1^m + n/_i + 1^m + n/v_i/h_i + 1_i + 1^m/_i^m_i^m + n/_i + 1^m + n|. If we can determine the limits of the fractions present in (<ref>) as i →∞, then we can also determine R_1(m,n) and R_2(m,n).First, let us study the sequence of 's and 's in greater detail. Each _i generates a _i through (<ref>) that satisfies |_i| < |_i| and each _i generates an _i + 1 through (<ref>) that satisfies |_i + 1| < |_i|. So, we have the ordering |_0| > |_0| > |_1| > |_1| > ⋯ This indicates that _i and _i + 1 are the two roots of (<ref>) for a fixed = _i with |_i + 1| < |_i| < |_i| and _i and _i + 1 are the two roots of (<ref>) for a fixed = _i + 1 with |_i + 1| < |_i + 1| < |_i|. We therefore have that _i and _i + 1 satisfy _i _i + 1 = 2 ρ_i^2, _i + _i + 1 = 2(1 + ρ) _i - _i^2 and _i and _i + 1 satisfy _i _i + 1 = _i + 1^2/2ρ + _i + 1, _i + _i + 1 = 2(1 + ρ)/2ρ + _i + 1_i + 1. Since _0,_0 > 0 it follows from (<ref>) and (<ref>) by induction that all _i and _i are positive. More importantly, the parameters _i and _i decrease geometrically fast, which we establish now. There exists c ∈ (0,1) such that 0 < _i, _i < c^i, i ≥ 0.For a fixed , letbe the root of (<ref>) satisfying <. Define t() /. In <ref> we show that lim_↓ 0 t() exists and is less than 1, so that t() < 1 for ∈ [0,ρ^2] by <ref>. Since the interval [0,ρ^2] is closed and bounded, we have that c_1 max_∈ [0,ρ^2] t() < 1. Perform the same procedure for a fixedto obtain a second bound c_2. So, _i < _i c_1 and _i + 1 < _i c_2. Set cc_1 c_2 to prove the claim. A consequence of <ref> is that _i → 0 and _i → 0 as i →∞.The following results on the asymptotic behavior of _i/_i and _i + 1/_i will be used to evaluate (<ref>).* For a fixed _i, let _i be the root of (<ref>) with _i < _i. Then, as i →∞ the ratio _i/_i →_ with _ < 1 the smaller root of 0 = 2 ρ^2 - 2(1 + ρ)+ 1, where the roots are _ = 1 + ρ±√(1 + ρ^2)/2 ρ.* For a fixed _i, let _i + 1 be the root of (<ref>) with _i + 1 < _i. Then, as i →∞ the ratio _i + 1/_i → 1/_ with _ > 1 the larger root of (<ref>).(i) In (<ref>), set = _i and = _i, divide by _i^2, set = _i/_i and let i →∞ to obtain (<ref>). It is easy to see that _ > 1/(2ρ) for 0 < ρ < 1 and since __ = 1/(2ρ) we conclude that _ < 1.(ii) In (<ref>), set = _i + 1 and = _i, divide by _i^2, set ζ = _i + 1/_i and let i →∞ to obtain 0 = ζ^2 - 2(1 + ρ) ζ + 2 ρ. We are interested in the root of (<ref>) smaller than one, which is 1/_, since ζ satisfies the same equation as 1/. We can also determine v_i/h_i and h_i + 1/v_i as i →∞. This is the final ingredient in the evaluation of (<ref>).* Consider the setting of . Then, as i →∞, v_i/h_i→1/(2ρ) - _/_ - 1/(2ρ).* Consider the setting of . Then, as i →∞, h_i + 1/v_i→ - _/_.(i) Using the indexing of the compensation parameters, (<ref>) becomes v_i = - _i + 1 - _i/_i - _i h_i. Divide both sides of (<ref>) by h_i and multiply by _i/_i to obtain v_i/h_i = 1 - _i + 1/_i/_i/_i - 1. For i →∞, we have by <ref> that _i + 1/_i → 1/_ and _i/_i → 1/_. So, for i →∞, v_i/h_i→1 - 1/_/1/_ - 1 = __ - _/_ - __, and then __ = 1/(2ρ) proves the claim.(ii) Using the indexing of the compensation parameters, (<ref>) becomes h_i + 1 = - (ρ + _i + 1)/_i + 1 - (1 + ρ)/(ρ + _i + 1)/_i - (1 + ρ)v_i. Divide both sides of (<ref>) by v_i and multiply by _i/_i to obtain h_i + 1/v_i = - (ρ + _i + 1)_i/_i + 1 - (1 + ρ)_i/(ρ + _i + 1) - (1 + ρ)_i. For i →∞ we have that _i + 1→ 0, _i → 0 and _i/_i + 1 = _i/_i + 1·_i + 1/_i + 1→_ / _, which proves the claim. We can now determine the limits (<ref>). Applying <ref> produces R_1(m,n) = R_2(m,n) = 1/(2ρ) - _/_ - 1/(2ρ)( _/_)^m + n - 1. If we define θ_ 2ρ_ = 1 + ρ±√(1 + ρ^2), then it is easy to see that θ_ < 1 and θ_ > 1 for 0 < ρ < 1. More importantly, for m ≥ 0, n ≥ 1, R_1(m,n) = R_2(m,n) = 1 - θ_/θ_ - 1( θ_/θ_)^m + n - 1 < 1, because, for 0 < ρ < 1, 1 - θ_/θ_ - 1 = 1 + 2ρ( ρ - √(1 + ρ^2)) < 1. Since R_1(m,n) and R_2(m,n) are both less than one, we know that the two series in (<ref>) converge absolutely. For a series to converge, its summands must tend to zero. So, for m ≥ 0, n ≥ 1, lim_i →∞ h_i _i^m _i^n = 0, lim_i →∞ v_i _i + 1^m _i^n = 0. This shows that the error terms introduced in each vertical and horizontal compensation step indeed tend to zero.The continuous-time analog of a result from Foster <cit.>, shown in <ref>, states that if the solution p(m,n) satisfies all balance equations, is non-zero, and ∑_m ≥ 0∑_n ≥ 1 |p(m,n)| ≤∑_m ≥ 0∑_n ≥ 1( ∑_i ≥ 0 | h_i _i^m _i^n | + ∑_i ≥ 0 | v_i _i + 1^m _i^n | ) < ∞, then the solution can be normalized to produce the equilibrium distribution. The solution is non-zero because p(m,n) = _0^m _0^n + ( _1^m _0^n ),m ≥ 0, n ≥ 1, and for m large p(m,n) is positive. We prove that (<ref>) holds. Since the summands in (<ref>) are positive, we can interchange the order of the summations to obtain ∑_m ≥ 0∑_n ≥ 1( ∑_i ≥ 0 | h_i _i^m _i^n | + ∑_i ≥ 0 | v_i _i + 1^m _i^n | ) = ∑_i ≥ 0|h_i|/1 - |_i||_i|/1 - |_i| + ∑_i ≥ 0|v_i|/1 - |_i + 1||_i|/1 - |_i|. We show that the two series converge. To that end, define R_3 lim_i →∞| |h_i + 1|/1 - |_i + 1||_i + 1|/1 - |_i + 1|/|h_i|/1 - |_i||_i|/1 - |_i||, R_4 lim_i →∞| |v_i + 1|/1 - |_i + 2||_i + 1|/1 - |_i + 1|/|v_i|/1 - |_i + 1||_i|/1 - |_i||, which can be written as R_3= lim_i →∞| |h_i + 1|/|v_i|1/1 - |_i + 1|1/1 - |_i + 1||_i + 1|/|_i + 1|/|h_i|/|v_i|1/1 - |_i|1/1 - |_i||_i|/|_i + 1||, R_4= lim_i →∞| |v_i + 1|/|h_i + 1|1/1 - |_i + 2|1/1 - |_i + 1||_i + 1|/|_i + 1|/|v_i|/|h_i + 1|1/1 - |_i + 1|1/1 - |_i||_i|/|_i + 1||. By applying the results of <ref> and the fact that _i → 0 and _i → 0 as i →∞, we find R_3 = R_4 = 1 - θ_/θ_ - 1 < 1, so that (<ref>) holds.In conclusion, due to <ref>, the series in (<ref>) is the unique (up to a multiplicative constant) solution to the balance equations (<ref>), (<ref>), (<ref>) and (<ref>) and can be normalized to produce the equilibrium distribution. Divide (<ref>) by the normalization constant C and merge the two series to obtain p(m,n) = C^-1∑_i ≥ 0 (h_i _i^m + v_i _i + 1^m) _i^n,m ≥ 0, n ≥ 1. §.§ Normalization constant We use the PGF x,y to determine the normalization constant C. First, eliminate the p(m,0), m ≥ 0 in the definition of x,y using (<ref>) and (<ref>) to get x,y = p(0,0) + ∑_m ≥ 1 p(m,0) x^m + ∑_m ≥ 0∑_n ≥ 1 p(m,n) x^m y^n = 1/ρ p(0,1) + 1/1 + ρ∑_m ≥ 1( 2ρ p(m - 1,1) + p(m,1) ) x^m+ ∑_m ≥ 0∑_n ≥ 1 p(m,n) x^m y^n. Second, substituting the series expression (<ref>) into (<ref>) gives x,y = C^-1[1/ρ∑_i ≥ 0 (h_i + v_i) _i+ 1/1 + ρ∑_m ≥ 1∑_i ≥ 0( h_i (2ρ + _i) _i^m - 1 + v_i (2ρ + _i + 1) _i + 1^m - 1) _i x^m+ ∑_m ≥ 0∑_n ≥ 1∑_i ≥ 0 (h_i _i^m + v_i _i + 1^m) _i^n x^m y^n ]. Third, changing the order of the summations and simplifying the geometric series finally gives x,y = C^-1[1/ρ∑_i ≥ 0 (h_i + v_i) _i+ 1/1 + ρ∑_i ≥ 0( h_i (2ρ + _i) x/1 - _i x + v_i (2ρ + _i + 1) x/1 - _i + 1 x) _i+ ∑_i ≥ 0( h_i 1/1 - _i x + v_i 1/1 - _i + 1 x) _i y/1 - _i y]. Notice that the PGF x,y is valid for |x| < 1/_0 and |y| < 1/_0. The expression (<ref>) is called a partial fraction decomposition of the PGF x,y. This decomposition shows that x = 1/_i and y = 1/_i are the simple poles of x,y, which implies that the function x,y approaches infinity as x approaches 1/_i or y approaches 1/_i.We determine the normalization constant by deriving two expressions for the leading term in the asymptotic expansion of x,0 as x ↑ 1/_0. To that end, we set y = 0 in (<ref>) to obtain x,0 = C^-1[1/ρ∑_i ≥ 0 (h_i + v_i) _i+ 1/1 + ρ∑_i ≥ 0( h_i (2ρ + _i) x/1 - _i x + v_i (2ρ + _i + 1) x/1 - _i + 1 x) _i ]. Now, as x ↑ 1/_0 = 1/ρ^2, x,0 = C^-11/1 + ρ h_0 (2ρ + _0)1/ρ^2/1 - _0 x_0 + (1) = 1/C ρ (1 + ρ) (1/ρ^2 - x) + (1), where we used that h_0 = 1 and _0 = ρ^2/(2 + ρ) and property (<ref>).For a second expression for the leading term, we investigate the functional equation (<ref>). If we pick the pair (x,y) such that h_1(x,y) = 0 and |x| < 1/_0, |y| < 1/_0, then we find that x,0 and 0,y are related according to 0 = h_2(x,y) x,0 + h_3(x,y) 0,y. Apply relation (<ref>) to three pairs (x,y) in the following order: (1/(2ρ),1), (1/(2ρ),1/ρ) and (1/ρ^2,1/ρ). All three pairs satisfy h_1(x,y) = 0. For the first pair (x,y) = (1/(2ρ),1) we have 0 = h_2(1/2ρ,1) 1/2ρ,0 + h_3(1/2ρ,1) 0,1. Notice that 0,1 is the fraction of time the first server is idle. The offered load to the system is 2ρ per unit time, so that by symmetry we know that 0,1 = 1 - ρ. So, from (<ref>) we obtain that 1/(2ρ),0 = 1 - ρ. For the second pair (x,y) = (1/(2ρ),1/ρ) we have 0 = h_2(1/2ρ,1/ρ) 1/2ρ,0 + h_3(1/2ρ,1/ρ) 0,1/ρ and find 0,1/ρ = (1 - ρ)(2 - ρ). Now, for the third pair (x,y) = (1/ρ^2,1/ρ), we let x ↑ 1/ρ^2 and y → 1/ρ. To that end, we need the solution of h_1(x,y) = 0 for a fixed x. This solution is given by y = υ(x) with υ(x) = (1 + ρ) x - √(x(x(1 + ρ^2) - 1)). Observe that if x ↑ 1/ρ^2, then υ(x) → 1/ρ. Substituting the pair (x,y) = (x,υ(x)) into (<ref>) gives the relation x,0 = - h_3(x,υ(x))/h_2(x,υ(x))0,υ(x). Then, as x ↑ 1/ρ^2 we find that 0,υ(x)→0,1/ρ = (1 - ρ)(2 - ρ), h_3(x,υ(x)) → h_3(1/ρ^2,1/ρ) = (1 - 1/ρ) / ρ^2, and h_2(x,υ(x)) = - (1 - ρ)(2 + ρ)/2ρ( x - 1/ρ^2) + (x - 1/ρ^2). By combining these asymptotic results, we obtain from (<ref>) a second expression for the leading term in the asymptotic expansion of x,0. For x ↑ 1/ρ^2, x,0 = 2(1 - 1/ρ)(2 - ρ)/ρ (2 + ρ)(x - 1/ρ^2) + (1). Finally, combining (<ref>) and (<ref>) gives, as x ↑ 1/ρ^2, 1/C ρ (1 + ρ) (1/ρ^2 - x) = 2(1 - 1/ρ)(2 - ρ)/ρ (2 + ρ)(x - 1/ρ^2). Solving this relation for C gives the explicit expression C = ρ(2 + ρ)/2(1 - ρ^2)(2 - ρ).§ COMPARISON WITH RANDOM ROUTING The compensation procedure allows us to easily calculate the equilibrium distribution using <ref>. From the equilibrium distribution we can determine performance measures such as the expected number of jobs in the system. Let X denote the total number of jobs in the system in equilibrium. Then, X = 0 = p(0,0),X = x = ∑_m = 0^x p(m,x - m) + ∑_m = 0^x - 1 p(m,m - x)= p(x,0) + 2 ∑_m = 0^x - 1 p(m,x - m),x ≥ 1, where we used p(m,x - m) = p(m,m - x) by symmetry, and therefore X = ∑_x ≥ 0 x ( p(x,0) + 2 ∑_m = 0^x - 1 p(m,x - m) ).For numerical purposes the number of compensation steps needs to be finite and the infinite summation in (<ref>) should be truncated. We first present a simple method to perform an appropriate number of compensation steps, see <ref>. Essentially, <ref> is the same as <ref>, but now selects the number K according to some preset target level: when the relative change in the equilibrium probability p(m,n) goes below a certain threshold ϵ, the compensation procedure is terminated.One way to choose the truncation level of the infinite series (<ref>) is described in <ref>. We base the truncation level on the criterion that almost all probability mass is captured in the distribution of X.<ref> allow us to determine X to any prescribed accuracy. We can compare these results with a naive random routing policy and demonstrate that the join the shortest queue routing policy is superior.Random routing means that each job joins either queue with equal probability, irrespective of the number of jobs at each server. Due to the Poisson splitting, random routing ensures that each queue operates as an M/M/1 queue with arrival rate ρ and equilibrium probabilities (1 - ρ)ρ^i. We denote by X_ the total number of jobs in the system with random routing and derive X_ = x = ∑_k = 0^x (1 - ρ) ρ^x - k (1 - ρ) ρ^k = (x + 1) (1 - ρ)^2 ρ^x. Then, we get that X_ = ∑_x ≥ 0 x X_ = x = (1 - ρ)^2 ∑_x ≥ 0 x (x + 1) ρ^x = 2ρ/1 - ρ. This result is also easily derived from the fact that under random routing both servers have independent Poisson input and the expected total number of jobs is the sum of the expected number of jobs in each queue (ρ/(1 - ρ)).<ref> compares join the shortest queue routing to random routing for various values of ρ. In terms of the expected number of jobs in the system, join the shortest queue routing is superior to random routing. For small ρ, an arriving job usually finds an empty system. In that case, both routing policies operate equally well. For larger ρ, join the shortest queue routing outperforms random routing. This routing policy balances the number of jobs at each server, and therefore utilizes the servers more efficiently than the random routing policy. Moreover, as ρ↑ 1 the join the shortest queue system behaves as a pooled system, which means that it behaves as if there is a single queue served by two servers instead of two separate queues with one server each.§ TAKEAWAYS The straightforward choice of taking the number of jobs at each queue as the dimensions of the Markov process led to an inhomogeneous transition rate structure. By performing a simple coordinate transformation and using the symmetry of the two servers and the join the shortest queue routing we were able to formulate a Markov process that did have a homogeneous transition rate structure in the interior. Due to this symmetry, we only needed to determine the equilibrium probabilities for the states (m,n) with m ≥ 0 and n ≥ 1.The compensation approach worked by linearly combining product-form solutions to satisfy all balance equations. These product-form solutions all satisfied the balance equations of the interior. In each step of the compensation procedure, a single product-form solution was added to the linear combination so that the resulting linear combination satisfied the balance equations on one of the two boundaries. In the next step, a single product-form solution was added to satisfy the balance equations on the other boundary. This process was repeated and finally led to an infinite sum of product-form solutions. Then, showing that this infinite sum converged, established that it was the unique equilibrium distribution.For the gated single-server system in <ref>, compensation was only necessary on a single boundary. For the join the shortest queue model, however, we had to compensate on two boundaries. This creates two different, alternating compensation steps. The compensation approach applied to the gated single-server system is therefore inherently `simpler', which was demonstrated by the fact that the parameters _i and _i can be obtained explicitly, whereas this was not possible for the join the shortest queue system. Furthermore, for the gated single-server system _i,_i did not tend to zero, while the coefficients c_i did, and for the join the shortest queue system this is reversed: _i,_i tended to zero, while the coefficients h_i,v_i did not.The compensation approach is not limited to the join the shortest queue system. It applies to a more general class of models, which we now briefly describe. For a Markov process in the positive quadrant, the compensation approach can be applied when it obeys the following conditions: (i) there should be only transitions to neighboring states; (ii) in the interior of the state space, there should be no transitions to the North, North-East, and East; and (iii) there should a homogeneous structure in terms of the transitions, i.e., the transition structure and the rate at which these transitions occur should be the same for all states in the interior, for all states on the vertical boundary, and for all states on the horizontal boundary. It can be shown that these conditions imply that _i,_i → 0, which, as we saw in <ref>, is not necessary for convergence of the series expression for p(m,n). 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Aside from the named number sets, all sets are denoted by calligraphic letters such as A. p.18 p.73 defined asequal in distribution vector of zeros of appropriate dimension A indicator function of the event Avector of ones of appropriate dimension (A)_i,j element (i,j) of matrix A A^-1 or (A)^-1 inverse of a matrix A A^c complement of a set A set of complex numbers (A) determinant of a matrix A i vector of zeros of appropriate dimension with a 1 at position i X expectation of a random variable X xf(X) expectation of a functional of a process { X(t) }_t ≥ 0 given X(0) = x n Erlang-n distribution with parameter exponential distribution with parameterf_X(·) probability density function of a random variable X F_X(·) cumulative distribution function of a random variable Xp geometric distribution with failure probability p and support _0 (or sometimes ) imaginary part of ∈complex unit _n transition rate submatrices in a QBD or QSF process from level i to level i + n, independent of i _n^(i) transition rate submatrices in a QBD or QSF process from level i to level i + n X Laplace-Stieltjes transform of the random variable X evaluated at the point, _0 = { 1,2,3,…}, _0 = { 0 }∪ X probability generating function of the random variable X evaluated at the pointPoisson distribution with parameterA probability of event A A | B conditional probability xf(X) probability of a functional of a process { X(t) }_t ≥ 0 given X(0) = x Q transition rate matrix of a Markov process set of real numbersreal part of ∈state space of a Markov process X standard deviation of a random variable X ,closed unit disc and unit circle v^ transpose of a vector v X variance of a random variable X X ∼μ the random variable X has distribution μset of integer numbersCHAPTER: ABBREVIATION INDEXtocchapterAbbreviation index p.15p.75BD birth–and–death iff if and only if i.i.d. independent and identically distributed LST Laplace-Stieltjes transform LT Laplace transform PASTA Poisson arrivals see time-averages PGF probability generating function QBD quasi-birth–and–death | http://arxiv.org/abs/1709.09060v1 | {
"authors": [
"Ivo Adan",
"Johan van Leeuwaarden",
"Jori Selen"
],
"categories": [
"math.PR"
],
"primary_category": "math.PR",
"published": "20170926143855",
"title": "Analysis of structured Markov processes"
} |
[][email protected] of Physics, Indian Institute of Technology, Delhi[][email protected] Department of Physics, Indian Institute of Technology, DelhiCooperation between micro-organisms give rise to novel phenomena like clustering, swarming in suspension. We study the collective behavior of the artificial swimmer called Taylor line at low Reynolds number using multi-particle collision dynamics method. In this paper we have modeled bi-motility mixtures of multiple swimmers in 2 dimension, which differ from each other by the velocity with which they swim. We observe that the swimmers can segregate into slower and faster ones depending on the relative difference in velocity of the 2 type of swimmers. We also observe thatcontribution of slower swimmers towards clustering, on an average, is much larger compared to faster ones, although we employ a homogeneous mixture. When the difference in velocity is large between the swimmers, the faster ones move away from the slower ones towards the boundary. On the other hand, when the relative difference in velocity is very small, the slower and faster swimmer mix together to form big clusters. At later time even for small difference in velocity the swimmers segregate into fast and slow swimmer clusters.Mixing and de-mixing of model microswimmers in bi-motility mixtures Sujin B. Babu December 30, 2023 ===================================================================The collective motility of micro-organisms is quint-essential in a range of biological activities <cit.>. Large scale cooperative movement is seen in micro-organisms which propagate by virtue of deformations along the cell body. Such swimming strategies are commonly seen in spermatozoa, C. elegans and various flagellated microswimmers <cit.>. Micro-swimmers demonstrate various aggregating patterns such as swarming, clustering or band formation <cit.>. Aggregation is a consequence of homology in the system and is hydro-dynamically favorable as it reduces energy consumption in transport <cit.>. The real systems comprise of swimmers having a range of different motilities. In addition, a portion of swimmers may be unhealthy or may employ atypical swimming strategies and hence immensely differ in propagating strengths. To study how the cluster formation is favored in case when the species are not perfectly homologous, several simulations have been performed <cit.>. Consequently, a positive feedback between clustering and segregation has been reported <cit.>. Previous works comprising mixture of active and passive self-propelled particles or rods indicate spontaneous segregation in the system <cit.>. These processes occur even in absence of cell to cell signaling or chemotaxis <cit.>. In this Letter, we employ numerical simulations to investigate the collective dynamics of microswimmers which show propulsion via planar beating mechanisms in Newtonian fluid. We analyze the cooperation between the swimmers in a bi-motility mixture which results into the aggregation and segregation among their own types. Understanding this cooperation is prerequisite to deeper understanding of collective motion of microswimmers. In the present study, we consider only hydrodynamic and steric interactions between the swimmers. To model artificial swimmers, we use a two dimensional discretized model of Taylor’s sheets termed as Taylor Lines <cit.>. The Taylor Line hydrodynamically interacts with the fluid using a sinusoidal bending wave which moves along the body. To simulate fluid environment, Multi-Particle Collision Dynamics (MPC) is used which employs “coarse-grained” particles of mass m=1 <cit.>. As the Taylor line is continuously pumping energy into the system, we have used MPC with Anderson Thermostat and angular momentum conservation (MPC-AT+a) <cit.>. The method consists of consecutive collision and streaming steps. In ballistic streaming step, the coordinates of the fluid particles {r_i} having velocity {v_i} are updated with integration time Dt.For the collision step, the particles are segregated into collision cells of length l=1 and are imparted random velocities chosen from the Gaussian distribution of variance k_B T/m=1 such that the momentum of the cell is conserved, where k_B is Boltzmann constant and T is the temperature. The position of grid with respect to box is randomly changed in each step so as to incorporate Galilean invariance.We have chosen the density of fluid as 10 m/l^3. The unit of time is [t]=l√((m/k_B T)).Each Taylor line consists of a sequence of N beads each of mass 10m at an equilibrium distance of 1/2 l. These beads interact with the nearest neighbors by Hooke's spring potential and bending potential V_B=κ/2∑_i=1^N-1[t_i+1-R(α_i)t_i] <cit.> that keeps the consecutive beads aligned at an angle α_i with the bending rigidity κ=pk_B T where p is the persistent length. A sinusoidal wave of beating frequency ν is generated along the contour of the Taylor line by varying curvature c(i,t)=α_i/l, spontaneously, with time, t and bead position, i.c(i,t)=b·sin[ 2π( ν t + 2i/N) + ϕ]To get a directed motion we choose p=5×10^3 · l (N-1). The factor of 2 with i assures that there are two waves trains packed inside the Taylor line and ϕ is the the initial phase shift. In order to model the interaction among various swimmers, we use a truncated Leonard-Jones potential,V_I = 4ϵ[(r_o/r)^12 - (r_o/r)^6 ] , r^2<2^3 r_owhere r is the separation between the beads of different Taylor lines, r_o istaken to be equal to l and ϵ=13.75 <cit.> is the strength of the potential. Using this potential along with the intra-swimmer potentials, we calculate the acceleration on every bead and update their position with integration time step dt = 0.01Dt. To simulate the hydrodynamic interaction between the swimmer and the fluid, we make the beads to participate in every collision step.This ultimately incorporates the Taylor line into the fluid environment.We make use of both rigid and periodic boundary walls in this study. In case of rigid boundary, we use L× L square and a circle of radius R as the kinds of wall.To mimic fluid flow at the rigid walls of a confinement, we bounce back the fluid particles crossing the walls and employ “Ghost” particle method in order to satisfy no-slip (rigid) boundary condition <cit.>. We allow the Taylor lines to easily slide on the walls by implementing bounce forward rule on the beads. For the purpose of our simulations we choose Dt = 0.01, N to be 100, and b to be 0.15 which gives the equilibrium wavelength of 22.59 and amplitude 2.27.A single swimmer in periodic boundary condition, yields a velocity of 0.0025 - 0.0224 for the respective frequency range of 0.001 - 0.009 similar to the work ofand the Reynolds number in the range of 0.003 - 0.028.showed that if the velocity of each swimmer was chosen from a Gaussian distribution, the cooperation of swimmers was enhanced when the variance, σ<3%. In the present study we will analyze a simple system consisting of two types of swimmers which differ only in the swimming velocity. Since for Taylor line the velocity is directly proportional to the beating frequency ν <cit.>, we vary the frequency of actuation in our simulations. The beating frequency of the faster swimmer is given by ν_a and that of slower swimmer is given by ν_b where we define δν = |ν_a-ν_b|/⟨ν⟩. The range of δν in the present study is10^-2 - 1. We have performed simulations with a number density of the swimmers ranging from ρ= 1.5 × 10^-3. Initially, the swimmers are scattered inside MPC fluid with random center of mass coordinate, orientation, initial phase and direction of motion.In Fig.<ref> we have shown the snapshots as obtained from the simulation for three different boundary conditions, where red color signifies slow swimmer and green signifies fast swimmer. In Fig.<ref> we show the aggregates formed due to circular rigid boundary condition, where we observe that the slow swimmers are usually along the center. A visual inspection reveals that the swimmers have segregated into slow and fast swimmer clusters. In Fig.<ref> we have shown the aggregate formed due to rigid square boundary condition, where we observe that the faster swimmers are forming cluster at the corners of the square. Here too a segregation between fast and slow swimmers can be seen. Supplemental Material<cit.>, Fig.S1 shows the snapshots of the system at different time instants. While in Fig.<ref> we have used periodic boundary condition and observe that the segregation is accompanied by the formation of bands because boundaries do not exit in this case as reported by .... et.al. All the results we discuss from hereon are for rigid circular boundary condition only. To quantify the collective behavior, we calculate the cluster size of the swimmer as follows. We consider two swimmers to be part of a cluster if they simultaneously satisfy two conditions for one complete beating period of the faster swimmer. First, if the minimum distance between at least 10% of the beads of the two swimmers is less than the 2.27a, which is the amplitude of swimmer in our case. Second, if the angle between the end-to-end vectors <cit.>, is less than π/6. See Supplemental Material<cit.>, Fig.S2 for an illustration of clusters. In Fig<ref> we show the evolution of the average cluster size, ⟨ n ⟩= {∑ n·Π(n)}/∑Π(n), where Π(n) is the cluster size distribution. We observe that initially ⟨ n ⟩ increases, signifying that clusters are being formed. These clusters keeps on growing un-till ⟨ n ⟩ reaches a steady state and then it oscillates around 10 indicating there is constant aggregation and fragmentation of the clusters. To understand whether we have a segregated state or a mixed state we employ a dimensional number called the segregation index (D) <cit.>. D = 1/2∑_clusters| n_a/N_a-n_b/N_b| where, n_a,b is the population of a or b type swimmers in a particular cluster and N_a,b is the total number of a or b type swimmers in the system. The summation runs over all the clusters in the system, which means D=1 implies completely segregated and D=0 implies fully mixed system. Fig.<ref> indicates D initially increases and reaches 1 signifying a completely segregated state and at later time it oscillates between 0.8 and 1. Also, we observe that when ⟨ n ⟩ is at a maximum D is at a minimum and vice versa. In Fig.<ref> we have shown the snapshot of the system when we have a maximum in the system for ⟨ n ⟩ and a minimum in D and we observe a mixed state where a cluster contains both fast and slow swimmer. Taylor lines always go towards the wall and form clusters close to the wall, these clusters then start moving along the walls. Even when the system is segregated both the types of swimmers would eventually encounter each other as one is slower than the other and will form mixed state. At t=0.85 × 10^5 the mixed swimmers start to de-mix and segregate into clusters of fast and slow swimmers. At this time we observe that D=1 and ⟨ n ⟩ is a minimum, which means the swimmer have completely segregated as can be observed from the snapshot Fig.<ref>. Again at t=1.05 × 10^5 we observe that the swimmers again mix together and form larger cluster as the ⟨ n ⟩ is a maximum and D is a minimum, i.e when segregation is a minimum we always have large cluster on average in our system.The cluster size distribution Π(n) in case of circular confinement is plotted in Fig. <ref>. A power law decay, Π(n) ∝ n^β, is observed for smaller cluster followed by an exponential decay for the higher values of n. The power law exponent β is independent of δν and depends only on the density of swimmers. Fig. <ref> shows β is approximately -1.4 for high density (ρ=0.0048) and approximately -1.9 for low density (ρ=0.0025) similar to what has been reported before for self propelled rods, both in case of simulations <cit.> as well as experiments <cit.>. β also does not depend on the time up to which the average is done, confirming that it is an inherent property of system. The decrease in exponent with density shows the importance of interactions for cluster formation. In Fig <ref> we have plotted the probability of finding a slow or fast swimmer at a distance r/R from the center of the circle, when the system has attained a steady state. Here we notice that the fast swimmers reach the wall earlier and stay near the wall as the distribution is nearly zero towards the center while has a sharp peak close to the walls. In the case of the slow swimmers the distribution is very broad, with a very small peak close to the walls. We know that the Taylor lines prefer to be closer to the walls, but in the case of slower swimmers when they reaches the wall the faster swimmer swim through the cluster and fragments the slow moving cluster. The fragmented segment again move towards the center. See Supplemental Movie.1 <cit.>.The power laws suggest intermittent behavior in cluster dynamics <cit.> resulting from aggregation and fragmentation at steady state. To quantify the contribution of the fast and slow swimmer towards cluster formation we introduce a parameter η defined as η_a,b= ∑(n_a,b/n)·Π(n)/∑Π(n)which give the contribution of either a or b swimmer in a cluster of size n. The time dependence of η is plotted in Fig. <ref> for time period before the system reaches a steady state. We can observe that the contribution of slower swimmers to clusters is higher than that of the faster swimmers.In Fig. <ref> we have plotted the evolution of η(a) and η(b) for 3 different values of δν. It can be observed that for both large and small values of δν, the values of η_b is always above the 0.5 and η_a is below 0.5. For the large difference in δν the slower swimmers contribute more for the cluster formation, while faster swimmer prefer to form smaller cluster. As δν is reduced, i.e., δν=0.2, η_a and η_b fluctuate around 0.5, where by the contribution towards the cluster by both the swimmers are almost the same. When the δν is small, we again observe that the contribution of the slower swimmer are much more than the faster swimmers. If we further decrease the δν we will observe that the contribution of both the kinds is the same.In the present work thus we are able to observe 3 different regions based on the clustering of swimmers. Fig. <ref> shows a time average of ⟨η⟩ for the whole simulation period vs. δν. Each of the point is averaged over 6 different configurations with ρ≈0.005. In region I, the faster ones push through the slower ones to reach the walls in small clusters while the slower ones are at that time dispersed around the center of the system. As a result, segregation index, as apparent from inset of Fig. <ref> and Supplemental Material <cit.> Fig. S3, is greater than 0.8 and, ⟨η_b ⟩>0.5 whereby the suspended slower ones will easily form clusters at center of the circular confinement. With the decrease in difference in beating frequency of swimmers in region II, the fast swimmers are unable to push through and there is virtually a competition between both the kinds of swimmers to form clusters in the confinement. As a result, there is almost, on an average, equal contribution from both the kinds to cluster and thus ⟨η_b ⟩ tends to 0.5. If the δν is decreased further in region III, both kinds of swimmers easily form clusters with each other as ⟨ D ⟩ <0.8. ⟨η_b ⟩>0.5 suggests that the concentration of slower swimmers is higher in a cluster and also the slow ones exploit the thrust of fast swimmers to form clusters. Thus the faster swimmers are always leading inside a cluster while there is a high density of slower swimmers at the back of the cluster. See Supplemental Movie.2 <cit.>. Gradually, the faster swimmers start swimming out of the cluster, thereby increasing the segregation. Thus, the parameter δν plays an important role in controllingthe cluster dynamics of system. We have also simulated systems with δν = 0.004 to observe that as δν tends to zero, η and ⟨ D ⟩ tend to 0.5 and zero respectively as expected.In conclusion, through our numerical simulation of hydrodynamically interacting Taylor's Lines in a confinement, we have shown that the cooperation between swimmers in a bi-motility mixture comprises of recurring mixing and de-mixing, which result in aggregation and finally segregation of the swimmer into fast and slow swimmer. This tendency of segregation has been reported in experiments <cit.> in which such binary mixtures are developed artificially or in natural response to external stimuli and also in recent simulations of active and passive particles <cit.>. However, we have shown that the system shows different behaviors depending on the relative difference in velocity. The results can be exploited to understand the collective dynamics among microswimmers in real systems which are composed of a continuous distribution of motility. We can infer that a stable cluster of swimmers comprises of those with small difference in ν, in which the slower ones are at the back guided by small numbers of faster ones, which is also observed experimentally <cit.>. When the difference in ν between clusters is large, the faster ones move away from the center assisting efficient swarming which has also been reported in the study of mixtures of healthy and dying microorganisms <cit.>. Our simulations reveal the novel kinds of cooperation between different microswimmers which stimulate the collective motion in a suspension. We would like to acknowledge HPC cluster at IIT Delhi, as well as baadal, the IITD’s private cloud, for allowing us to use the facility for the purpose of running the simulations.apsrev4-1 | http://arxiv.org/abs/1709.09085v1 | {
"authors": [
"Adyant Agrawal",
"Sujin B. Babu"
],
"categories": [
"cond-mat.soft",
"physics.bio-ph"
],
"primary_category": "cond-mat.soft",
"published": "20170926151755",
"title": "Mixing and de-mixing of model microswimmers in bi-motility mixtures"
} |
Exfoliation of single layer BiTeI flakes [ December 30, 2023 ======================================== § INTRODUCTION The singularity theorems of Hawking and Penrose (see <cit.>) show that solutions of Einstein's equations are “non-continuable” under rather general conditions, but do not provide very specific information on the structure of singularities. This motivated several attempts to try and provide an analytical description of singularities of solutions of Einstein's equations. Our approach in this paper is to try and determine how to perturb known exact solutions and to decide whether or not the type of singularity they possess is representative of the behavior of more general solutions.There is a technique which provides precisely this type of information for rather general classes of partial differential equations: the Fuchsian algorithm. It consists in constructing singular solutions with a large number of arbitrary functions by considering the equation satisfied by a rescaled unknown, which represents in fact the `regular part' of the solution. This new unknown satisfies a Fuchsian PDE, i.e., a system of the formtu⃗ t + Au⃗ = f(t,x_1,…,x_n,u⃗,u⃗_x),where A is a square matrix and f vanishes like some power of t as t→ 0. A general introduction to this algorithm, with several applications can be found in <cit.>, and a brief presentation is given in section 2 below. We just note here that non-singular solutions can also be constructed by the Fuchsian algorithm. In fact, the Cauchy problem itself reduces to a very special case of the method.We prove in this paper that the Fuchsian algorithm applies to Einstein's vacuum equations for Gowdy spacetimes, and establishes the existence of a family of solutions depending on the maximal number of arbitrary functions, namely four, in the `low-velocity' case, whose definition is recalled below. When one of these functions is constant, the solution actually extends to the `high-velocity' case as well. We will refer to the former solutions as `generic' and to the latter as `non-generic.' Earlier exact solutions are obtained by specializing the arbitrary functions in the solutions of this paper.In both cases, the solutions are `asymptotically velocity-dominated' (AVD) in the sense of Eardley, Liang and Sachs <cit.>, and precise asymptotics at the singularity are given. The reduction to Fuchsian form actually provides a mechanism whereby inhomogeneous solutions can become AVD in the neighborhood of such a singularity. The results explain the paradoxical features of numerical computations described next. §.§ Earlier results T^3×R Gowdy spacetimes <cit.> have spacelike slices, homeomorphic to the three-torus, on which a U(1)×U(1) isometry group acts. It is convenient to take as time coordinate the area t of the orbits of this two-dimensional group; the spacetime corresponds to the region t>0. The metric then takes the formds^2=e^λ/2t^-1/2(-dt^2+dx^2)+t[e^-Z(dy+X dz)^2 +e^Zdz^2],where λ, X and Z are functions of t and x only, and are periodic of period 2π with respect to x. We also letD=t_t. Form of the equations. With the above conventions, the equations take the form:D^2 X - t^2 X_xx= 2(DX DZ-t^2 X_x Z_x); D^2 Z - t^2 Z_xx= -e^-2Z((DX)^2 - t^2 X_x^2) λ_x= 2(Z_xDZ+e^-2ZX_xDX);Dλ= (DZ)^2+t^2Z_x^2+e^-2Z((DX)^2+t^2X_x^2),where subscripts denote derivatives.The last two equations arise respectively from the momentum and Hamiltonian constraints. It suffices to solve the first two equations, and we therefore focus on them from now on. Of course, one should also ensure that the integral of λ_x from 0 to 2π vanishes.If X=Z=0, we recover a metric equivalent to the (2/3,2/3,-1/3) Kasner solution. Other Kasner solutions are recovered for X=0 and Z=k ln t; the corresponding Kasner exponents are (k^2-1)/(k^2+3), 2(1-k)/(k^2+3) and 2(1+k)/(k^2+3). The equations for X and Z are often interpreted as expressing that (X,Z) generates a harmonic-like' map from 1+1 Minkowski space with values in hyperbolic space with the metricdZ^2+e^-2ZdX^2.The usual Cartesian coordinates on the Poincaré model of hyperbolic space are X and Y=e^Z, so that the metric coincides with the familiar expression (dX^2+dY^2)/Y^2.It is occasionally useful to use polar coordinates (w,ϕ) on the hyperbolic space, so that the metric on the target space isdw^2+sinh^2w dϕ^2.The equations for w and ϕ then take the formD^2w - t^2_xxw =1 2sinh 2w [(Dw)^2-t^2w_x^2]; D^2ϕ - t^2_xxϕ= -2 w [DwDϕ-t^2 w_xϕ_x]. For fixed t, the solution represents a loop in hyperbolic space.For extensive references on Gowdy spacetimes, see <cit.>.Exact solutions. Both sets of equations can be solved exactly if we seek solutions independent of x. In terms of the (X,Z) variables for example, these solutions have leading behavior of the formZ∼ k ln t + O(1); X = O(1),where k is a positive constant, and represent solutions in which the loop degenerates to a point which follows a geodesic and tends to a point at infinity in hyperbolic space.Motivated by this, it was suggested that a general solution corresponds to a loop, each of whose points asymptotically follows a geodesic and tends to some point at infinity in hyperbolic space. This regime is called the `geodesic loop approximation.'This is borne out in the case of the `circular loop' which corresponds to ϕ(x)=nx, n=1, 2,…, and w independent of x.However, the only case in which this behavior could be established for solutions containing an arbitrary function of x was the `polarized' case, defined by the condition X≡ 0 (see <cit.> and its references). The equation for Z is then a linear Euler-Poisson-Darboux equation, the general solution of which can be represented explicitly in terms of Bessel functions; this fact does not necessarily make the investigation of singularities straightforward, see [3]. This provides a family of solutions involving two arbitrary functions. These solutions haveZ∼ k ln t + O(1),where k now depends on x and can be arbitrary.Numerical computations suggest more complicated behavior in the full nonlinear system for X and Z <cit.>. Indeed, if one monitors the `velocity' v(x,t)=√((DZ)^2+exp(-2Z)(DX)^2), which should tend to |k(x)|, one finds that it is not possible to find solutions which satisfy v>1 on any interval as t→ 0. Even if one starts out with v>1 and solves towards t=0, the parameter v dwindles to values less than 1, except for some sharp spikes located near places where X_x=0, and which eventually disappear at any fixed resolution. They may persist longer at higher resolutions. Solutions such that v<1 are called `low-velocity,' and others are called `high-velocity.' A formal asymptotic computation, proposed in <cit.>, also suggests that the low-velocity case allows asymptotics that would not be available in the high-velocity case. This expansion is obtained by introducing a parameter η in front of the spatial derivative terms in the equations, and expanding the solution in powers of η.Note that solutions with k positive and negative are qualitatively quite different, even though they would have the same value for v.Since the numerical computations we wish to account for were performed in the (X,Z) variables, we will focus on them. However, we will briefly mention what happens in the (w,ϕ) variables, since the circular loop is then more simply described.The problem can be summarized as follows: if the geodesic loop approximation is valid, v approaches |k|. We therefore need a mechanism which forces |k|<1—but if v must be smaller than 1, how do we account for the polarized solutions? Also, should we restrict ourselves to k>0, given that the numerics do not give information on the sign of k?Results. Our results account for the various types of behavior observed on numerical and special solutions by exhibiting a solution with the maximum number of `degrees of freedom,' and which, under specialization, reproduces the main features listed above. We describe these results first for k>0.When k is positive, we first define new unknowns u(x,t) and v(x,t) by the relationsZ(x,t) = k(x) ln t + (x) + t^ u(x,t);X(x,t) = X_0(x) + t^2k(x)(ψ(x) + v(x,t)),whereis a small positive constant to be chosen later. The objective is to construct solutions of the form (<ref>-<ref>), where u and v tend to zero as t tends to zero. If 0<k<1, the periodicity condition ∮λ_xdx=0 is equivalent to∫_0^2π k(_x+2 X_0xψ e^-2)dx = 0,which we assume from now on. If k>1, wewill require in addition that X_0x≡ 0, for reasons described later. In both cases, we find that λ = k^2 ln t + O(1) as t→ 0.We then prove that, upon substitution of (5)-(6) into (1)-(2), we obtain a Fuchsian equation for (u,v), in which the right-hand side may contain positive and negative powers of t, as well as logarithmic terms. If there are only positive powers of t, possibly multiplied by powers of ln t, we prove an existence theorem which ensures that there are actual solutions of this form in which u and v tend to zero. In fact, one can derive iteratively a full expansion of the solution near the singularity at t=0. We prove that there are only positive powers of t in two cases: * if k lies strictly between 0 and 1; this provides a `generic' solution involving four arbitrary functions of x, namely k, X_0,and ψ.* if k>0 and X_0 is independent of x; this provides a solution involving only three functions of x and one constant. This case includes both the x-independent solutions and the polarized solutions, and explains why these cases do not lead to a restriction on k.The fact that high-velocity is allowed when X_0 is constant is to be compared with the numerical results which show spikes when X_x=0.If k is negative, one can proceed in a similar manner, except that one should start withZ = k(x)ln t + (x) + t^ u(x,t);X = X_0(x) + t^ v(x,t),where k,and X_0 are arbitrary functions. In fact, one can generate solutions with negative k from solutions with positive k. Indeed, if (X,Z) is any solution of the Gowdy equations, so is (X̃, Z̃), whereX̃ = X X^2 + Y^2, Z̃ = lnY X^2 + Y^2,with Y=e^Z as before. This corresponds to an inversion in the Poincaré half-plane.Our existence results can actually be applied in two different ways to the problem. One is to assume the arbitrary functions to be analytic and 2π-periodic, and to produce solutions which are periodic in x. One can also use the results to produce solutions which are only defined near some value of x. This is useful for cases when the solution is not conveniently represented in the (X,Z) coordinates, in which one of the points at infinity in hyperbolic space plays a distinguished role. In such cases, one can patch local solutions obtained from several local charts in hyperbolic space. §.§ Organization of the paperSection 2 presents a brief introduction to Fuchsian techniques.Section 3 is devoted to the reduction of the basic equations to Fuchsian form, and shows how the distinction between low- and high-velocity arises naturally from the Fuchsian algorithm (theorems 1 and 2).Section 4 proves the existence result (theorem 3) which produces the above solutions. It also shows the impact of the rigorous results on formal asymptotics.§ INTRODUCTION TO FUCHSIAN TECHNIQUES We briefly review the main features of Fuchsian methods as they are relevant to our results. The main advantages of these techniques are: * Fuchsian reduction provides an asymptotic representation of singular solutions of fairly general partial differential equations. * The arbitrary functions in this representation generalize the Cauchy data, in the sense that their knowledge is equivalent to the knowledge of the full solution. The Cauchy problem is itself a special case of the Fuchsian algorithm. * The reduction of a PDE to Fuchsian form explains why solutions should become AVD, i.e., how the spatial derivative terms can become less important than the temporal derivatives near singularities, even though the solution is genuinely inhomogeneous. The starting-point is a reinterpretation of the solution of the Cauchy problem for, say, a second-order equationF[u] = 0.The geometric nature of the unknown is not important for the following discussion.Solving the Cauchy problem amounts to showing that the solution is determined by the first two terms of its Taylor expansion:u = u^(0) +t u^(1) + ….One can think of u^(0) and u^(1) as prescribed on the initial surface {t=0}. This statement does not require any information on the geometric meaning of the unknown u, which may be a scalar or a tensor for instance.However, this representation may fail if the solution presents singularities. The Fuchsian approach seeks an alternative representation near singularities, in a form such asu = t^ν(u^(0) +t u^(1) + …).There are several issues that need to be dealt with if one seeks such a solution: * How do we construct such a series formally to all orders? The question is far from trivial because any amount of inhomogeneity for example can force the appearance of logarithmic terms at arbitrarily high orders. Furthermore, the arbitrary terms in the series can occur at very high orders even if the equation is only of second order.* How do we know there is one solution corresponding to this expansion, rather than infinitely many solutions differing by exponentially small corrections?* How restrictive is it to start with power behavior: in particular, is logarithmic behavior allowed?Once this has been done, the formal series can be used much in the same way as an exact solution would.It turns out that all of these issues can be addressed simultaneously by reducing the given equation to a Fuchsian PDE by the following program:First, identify the leading terms.This requires being able to find an expression a(x^q) in the coordinates x^q such that, upon substitution of a into the equation, the most singular terms cancel each other.Second, define a renormalized regular part v by setting, typically,u = a + t^m v.If a is a formal solution up to order k, it is reasonable to set m=k+ε. If the structure of logarithmic terms is made explicit, one can also specify the dependence of v on logarithmic variables, as in <cit.>. There is a considerable variety in the choices of the renormalized part v, and the list of possible cases where these ideas apply seems to be growing.Third, obtain the equation for v. It is important to ensure, by introducing derivatives of v as additional variables if necessary, that one is left with a Fuchsian system, that is, one of the formt v t + A v = t^ε f(t,x,v,∂_α v),where A is a matrix, which could depend on spatial variables, but should be independent of t—otherwise we could incorporate the time dependence into f. ∂_α v stands for first order spatial derivatives; a second-order equation is converted to such a form by adding derivatives of the unknowns as additional unknowns. In general, f can be assumed to be analytic in all of its arguments except t, because a may contain logarithms or other more complicated expressions.Fuchsian PDE are a generalization of linear ordinary differential equations with a regular, or Fuchsian singularity at t=0, such as the Bessel or hypergeometric equations.Once this reduction has been accomplished, general results on Fuchsian equations give us the desired results, intuitively because the equation can be thought of as a perturbation of the case when f=0. The initial-value problem for such equations can be solved in the non-analytic as well as the analytic case <cit.>.The Fuchsian form has several advantages, in addition to being the one which allows one to construct and validate the expansions in the first place: * It makes AVD behavior natural, because the spatial derivative terms appear only in f, which is preceded by a positive power of t. We therefore expect spatial derivative terms to be switched off at leading order, but to contribute at higher order. By contrast, the term t∂_t v behaves like a term of order zero, because it transforms any power t^j into a multiple of itself (namely jt^j).* It is invariant under restricted changes of coordinates which preserve the set t=0: if we change (t,x^α) into (t',x^'α), it suffices to require that t'/t be bounded away from zero and independent of x^α near t=0. One can even allow non-smooth changes of coordinates such as t'=t^α. Further generalizations are possible.* It is invariant under `peel-off': for instance, if we write v=v^(0)+tw, and assume for simplicity that ε =1, we find that w solves a Fuchsian system with A replaced by A+1. A more general property of this kind can be found in <cit.>. This explains why the Fuchsian form is adapted to the construction of formal solutions as well as to their justification.* It can be used to generate the formal expansion systematically: assume the solution is known to some order k. Substitute into f, and call g the result; now solve the resulting equation t∂_t v + A v = t^ε g for v. It is easy to see that the result will contain corrections of order higher than k. This method is useful if the exact form of the solution is unknown, or if it is very complicated. Let us now turn to examples.1. The Cauchy problem. The Cauchy problem can always be thought of as an initial-value problem for a first-order systemu t = f(t,x,u,∂_x u),where x=(x^α) stands for several space variables, and f is analytic in all its arguments to fix ideas. For instance, in the case of Einstein's equations in harmonic coordinates, u represents the list of all the components of the metric as well as their first time derivatives.Let us now take as principal part a the initial condition u^(0), and writeu=u^(0) + tv.If we insert this into f, we find that all of the v-dependent terms must contain a positive power of t. In other words,f=f^(0)+tg(t,x,v,∂_x v),where f^(0)=f(0,x,u^(0),∂_x u^(0)). The equation for v is thereforet∂ v∂ t + v = f^(0)+tg,which is a Fuchsian equation for (v-f^(0)), with A=1. The existence of solutions of Fuchsian systems ensures in this case that one can solve the initial-value problem. To recover a solution of Einstein's equations, one needs to handle the propagation of the constraints separately, as usual.2. A nonlinear ODE. Consider the equationu_tt = u^2,where subscripts denote derivatives, and u=u(t) is a scalar.Let us try to find a leading part of the form u∼ a t^s with a≠ 0. The left-hand side is then ∼ as(s-1)t^s-2 and the right-hand side is ∼ a^2 t^2s. If s(s-1)=0, it means that we are dealing with a Cauchy problem: u≈ a+u_1 t+… if s=0, and u≈ a t+u_2 t^2… if s=1. We therefore assume s(s-1)≠ 0. It is then necessary for the two sides to balance each other as t→ 0, which means that we needs-2=2ss(s-1) = a.This means that s=-2 and a=6. The principal part is 6/t^2, and the first step is complete.For the second step, let us define the renormalized unknown v byu=t^-2(6 + v t). Finally, let us write the equation for v. We find(D-5)(D+2)v = tv^2,where D=td/dt.This is a Fuchsian equation of second order, which can be converted into a first-order Fuchsian system by introducing (v,Dv) as two-component unknown. This would lead to an equation where the eigenvalues of A are 2 and -5.The knowledge of the eigenvalue -5 combined with general properties of Fuchsian systems ensures that there is a complete formal solution for v where the coefficient of t^5 in the expansion of v is arbitrary. One can convince oneself of this fact by direct substitution, but this is often cumbersome, because of the need to compute a formal solution to sixth order in this case. In general, the expansion of v contains also powers of tln t, but they are not necessary for this simple example.The same method applies to any equation of the formu_tt = u^2 + c_1(t)u + c_0(t) + c_-1(t)u^-1 +…and yields a convergent series solutionu(t)=t^-2∑_j,k u_j,kt^j(tln t)^k,which is entirely determined once the coefficient u_6,0 is prescribed. The translates u(t-t_0) of this solution form a two-parameter family of solutions, parametrized by (u_6,0,t_0), which is stable under perturbations (i.e., `generic'). It is possible to show that the other eigenvalue of A, namely 2, is related to the variation of the parameter t_0, although we do not dwell on this point.Logarithmic terms are not due to logarithms in the equation itself. For instance, the equation u_tt=u^2+t^2 has no solution which is free of logarithms.3. The Euler-Poisson-Darboux equation. As an example of a linear Fuchsian PDE, let us consider the Euler-Poisson-Darboux (EPD) equationu_tt - λ -1 tu_t =u_xx+u_yy,in two space variables to fix ideas. This equation has a variety of uses, from the solution of the wave equation in Minkowski space to computer vision. In particular, the Einstein equations in the `polarized' Gowdy spacetime (i.e., when X=0) reduce to the above equation with only one space variable, and with λ=0.To reduce it into Fuchsian form, one may let introduce new unknowns: v=u, v_0=tu_t, v_1=tu_x and v_2=tu_y (numerical subscripts do not denote derivatives). One then finds the system{[ t_t v - v_0 = 0; t_t v_0 - λ v_0 = t_xv_1+t_yv_2; t_t v_1 =t_x(v+v_0); t_t v_2 = t_y(v+v_0). ].The general solution can in this case be computed explicitly using the Fourier transform (or Fourier series in a finite domain) in terms of Bessel functions. The solution has the form U+Vln t, where U and V are series in t and do not involve logarithms.Fuchsian reduction applies directly to non-linear perturbations of the EPD equation. However, the non-linearity causes the appearance of products of logarithms. The Fuchsian algorithm, by ensuring that the solutions are actually functions of t and tln t, guarantees that the singularity of the logarithm is always compensated by powers of t. Remark: There are cases when it is useful to make a change of time variable.Consider an example such as(t_t)^2u - tu_xx=0.If we let (v,v_0,v_1)=(u,tu_t,tu_x), we obtain the system{[ t_t v = v_0; t_t v_0 = _xv_1; t_t v_1 = v_1 + t_xv_0, ].in which the term _xv_1 does not have a factor of t. We can nevertheless obviate this problem by letting t=s^2. The original equation then becomes(s_s)^2u - 4s^2u_xx=0;expanding and dividing through by s^2, we recover the Euler-Poisson-Darboux equation, up to the harmless factor of 4.4. Leading logarithms. The first case to be treated by Fuchsian PDE methods actually required a logarithmic leading term. We merely state the result, as it is developed extensively elsewhere <cit.>. Consider the equationη^ab∂_abu = e^u,in Minkowski space. This equation admits a Fuchsian reduction with singularity on any space-like hypersurface t=ψ(x), which is obtained by applying the above ideas to the equation satisfied by e^u. This generates a family of stable singularities which do not propagate on characteristic surfaces, since the singularity locus is space-like. There is a complete expansion of the solution at the singularity, and it is free of logarithms if and only if the singularity surface has vanishing scalar curvature (i.e., ^(3)R=0).To summarize, the Fuchsian approach to singularity formation consists in three steps: (1) identification of the leading part; (2) identification of a convenient renormalized unknown; (3) solution of the Fuchsian system for the new unknown. This technique is now applied to the Gowdy problem. § REDUCTION TO FUCHSIAN FORM§.§ General results In this section, we first reduce the Gowdy equations to a second-order system for u and v, which is then converted to a first-order Fuchsian system. The subscripts 0, 1 and 2 in this section do not denote derivatives.The equations now become:(D+)^2u = t^2-[k_xxln t + _xx+t^ u_xx]-exp(-2-2t^ u) {t^2k-((D+2k)(v+ψ))^2-t^2-2k-[X_0x+t^2k(v_x+ψ_x+2k_x(v+ψ)ln t)]^2 }; D(D+2k) v= t^2-2k X_0xx + 2t^(D+)u(D+2k)(v+ψ) + t^2[(v+ψ)_xx+4k_x(v_x+ψ_x)ln t + (2k_xxln t + 4k_x^2(ln t)^2)(v+ψ)]- 2t^2-2k[X_0x+t^2k(v_x+ψ_x+k_x(v+ψ)ln t)]×[k_xln t+_x+t^ u_x]. This second-order system will now be reduced to a first-order system. To this end, let us introduce the new variablesu⃗ = (u_0,u_1,u_2,v_0,v_1,v_2) = (u,Du,tu_x,v,Dv,tv_x).We then find[Du_0 =u_1;;Du_1 =-2 u_1-^2u_0+t^2-(k_xxln t +_xx)+t_x u_2; -exp(-2-2t^ u_0){t^2k-(v_1+2kv_0+2kψ)^2-t^2-2k-X_0x^2; -2t^1-X_0x(v_2+tψ_x+k_x(v_0+ψ) tln t);- t^2k-(v_2+tψ_x+2k_x(v_0+ψ) tln t)^2 };;Du_2 = t_x(u_0+u_1);;Dv_0 =v_1;;Dv_1 =-2kv_1 + t^2-2kX_0xx + t_x (v_2+tψ_x) + 4k_x(v_2+tψ_x) tln t;+(v_0+ψ)[2k_xxt^2ln t + 4(k_xtln t)^2]; +2t^(v_1+2kv_0+2kψ)(u_1+ u_0); -2X_0xt^2-2k(k_xln t+_x+t^_x u_0); -2t (_x (v_0+ψ)+2k_x(v_0+ψ)ln t) (k_xtln t+t_x+t^ u_2);;Dv_2 = t_x(v_0+v_1); ] This system has therefore the form(D+A)u⃗ = g(t,x,u⃗,u⃗_x),where the right-hand side g involves various powers of t, possibly multiplied with logarithms. We will chooseso that all of these terms nevertheless tend to zero as t goes to zero. The low-velocity case is precisely the one in which it is possible to achieve this without making any assumptions on the arbitrary functions entering in the system, namely k, X_0,and ψ.In fact, the high- and low-velocity cases are now distinguished by the absence or presence of the terms involving t^2-2k (and t^2-2k-).As is clear from the above equations, these terms disappear precisely if X_0 is a constant (i.e., X_0x=0).For any positive number σ, we define the matrixσ^A = exp(Alnσ):=∑_r=0^∞(Alnσ)^r r!.One checks by inspection that the matrix A has eigenvalues , 0, and 2k, and that there is a constant C such that |σ^A|≤ C for any σ∈(0,1) if >0. This can be seen for instance by reducing A and explicitly computing the matrix exponential.Note that this system is of Cauchy-Kowalewska type for t>0, and that the solutions will in fact be analytic in all variables for t>0. The issue is to construct solutions with controlled behaviour as t→ 0.We are interested in solutions of (<ref>) which satisfy u⃗=0 for t=0. Let us check that these solutions have the property that u_0 and v_0 solve the original Gowdy system. Since the second and fifth equation of the system satisfied by u⃗ are obtained directly from the second-order system, it suffices to check that u_1=Du_0, v_1=Dv_0, u_2=tu_0x and v_2=tv_0x. The first two statements are identical with the first and fourth equations respectively. As for the last two, we note that the first and third equations implyD(u_2-t_x u_0)=t_x(u_0+u_1-Du_0-u_0)=0.Since u_2-t_x u_0 tends to zero as t→ 0, it must be identically zero for all time, as desired. The same argument applies to v.The computations for the case k<0 are entirely analogous, and are therefore omitted.We now study the low- and high-velocity cases separately. §.§ Low-velocity case The following theorem gives the existence of a solution depending on four arbitrary functions in the case when k lies between zero and one:Let k(x), X_0(x), ϕ(x) and ψ(x) be real analytic, and assume 0<k(x)<1 for 0≤ x≤ 2π. Then there exists a unique solution of the form (5)–(6), where u and v tend to zero as t→ 0. Proof: By inspection, the vector u⃗ satisfies a system of the form (<ref>), where g can be written t^α f, provided that we take α andto be small enough. Letting t=s^m, we obtain a new system of the same form, but with α replaced by mα. By taking α large enough, we may therefore assume that we have a system to which theorem <ref> below applies. The result follows. §.§ High-velocity case The following theorem gives the existence of a solution depending on three arbitrary functions in the case when k is only assumed to be positive, and may take values greater than one. If k is less than one, we recover the solutions obtained above, but with X_0x=0:Let k(x), ϕ(x) and ψ(x) be real analytic, and assume X_0x=0 and k(x)>0 for 0≤ x≤ 2π. Then there exists a unique solution of the form (5)–(6), where u and v tend to zero as t→ 0. Proof: Since X_0x is now zero, we find that u⃗ satisfies, if >0, a Fuchsian system of the form (<ref>), where g can be written t^α f, provided that we take α andto be small enough. Letting as before t=s^m, we obtain a new system of the same form, but with α replaced by mα. By taking m large enough, we may therefore assume that we have a system to which theorem <ref> applies. The result follows.§ EXISTENCE OF SOLUTIONS OF FUCHSIAN SYSTEMS Consider quite generally a Fuchsian system, for a `vector' unknown u(x,t), of the form(D+A)u = F[u] := tf(t,x,u,u_x).We are now dropping the arrow on u for convenience.In this equation, A is an analytic matrix near x=0, such that σ^A≤ C for 0<σ<1. The number of space dimensions is n (n=1 for the application to the Gowdy problem). It suffices that the nonlinearity f preserve analyticity in space and continuity in time, and depend in a locally Lipschitz manner on u and u_x, i.e., that its partial derivatives with respect to these arguments be bounded when these arguments are.To fix ideas, one may assume that f is a sum of products of analytic functions of x, u and u_x by powers of t, t^k(x) and ln t. In fact, all one needs is to ensure the estimate in step 2 below.In this section, the number of space variables is arbitrary. We are only interested in positive values of t. The system (<ref>) has exactly one solution which is defined near x=0 and t=0, and which is analytic in x, continuous in t, and tends to zero as t↓ 0. Remark 1: The solutions are constructed as the uniform limit of a sequence of continuous functions which are analytic in x. They are classical solutions as well, by construction. However, by the Cauchy-Kowalewska theorem, they are also analytic in t away from t=0.Remark 2: The solution provided by the theorem will be defined for x in a complex neighborhood of a subset Ω of R^n. This can be applied to the Gowdy problem in two different ways: one can take Ω to be an interval of length greater than 2π, and note that the solution will be 2π-periodic if the right-hand side is, since it is given as a limit of a sequence all of whose terms are periodic. It is this solution which shows that the `geodesic loop approximation' corresponds to a generic family of exact solutions in the low-velocity case, and a non-generic family otherwise. However, one could also take Ω to be a small interval of length less than 2π, and generate solutions which are defined only locally. This second application can itself be useful in two contexts: (a) for generalisations of Gowdy spacetimes where the space variable is unbounded, or compactified in a different fashion; (b) for the description of `circular loop' type solutions, which correspond to a solution which depends linearly in the angular coordinate in terms of polar coordinates on the Poincaré half-plane. Proof: Let us begin by defining an operator H which corresponds to the inversion of (D+A). The proof will consist in showing that the operator v↦ G[v]:=F[H[v]] is a contraction for a suitable norm. Its fixed point generates a solution u=H[v] to our problem.Before we jump into the details, let us first motivate the strategy by examining some of the possible difficulties. For more details on the history of existence theorems in the complex domain, see Chapter 2 in <cit.>.The basic difficulty in achieving a successful iteration is that it is not clear at all how to build a measure of the size of u (that is, a function space norm) which remains finite after even one step of the iteration. The problem is that in order to control H[v], we need to estimate the spatial derivative of v in terms of a norm which only involves v.This cannot be remedied by adding information on the derivatives of v in the definition of the norm: we would then need to estimate both H[v] and its derivative, in order to have a well-defined iteration.In fact, this is an essential problem because the result would be false if the right-hand side involved second as well as first derivatives of u. Majorant methods are not appropriate because the nonlinearity f does not have an expansion in powers of t—only in powers of x for fixed t.It is not possible to estimate the derivative of an analytic function by its values on the same domain: think of the function √(1-z) on the unit disk, which is bounded on (-1,1) even though its derivative is not. However, by going into the complex domain, it is possible to estimate the derivatives of an analytic function in terms of its values on the boundary of a slightly larger domain.[In the non-analytic case, this problem is gotten around thanks to the additional assumption of hyperbolicity, by showing that there are expressions which can be estimated as though the right-hand side did not involve derivatives of v at all, see <cit.>, ana well as Chapter 2 of <cit.> for a broader introduction.] This is given by Cauchy's theorem, which expresses the value of an analytic function ϕ at any point as a weighted average of its values on any curve γ circling that point once:ϕ(z) =1 2π i∮_γϕ(ζ)dζζ-z.Differentiating with respect to z and taking absolute values, we see that we have a means of estimating the derivatives of ϕ from its values on a larger domain. However, we must move into the complex domain to achieve this. The transition to several variables offers no difficulty, because an analytic function of several variables is separately analytic in each of its arguments, and it therefore suffices to apply the above to each variable separately to obtain some estimate of derivatives—which is all we need. For instance, the relevant Cauchy integral formula in two variables is simplyϕ(z_1,z_2) =1 (2π i)^2∮∮ϕ(ζ_1,ζ_2)dζ_1 dζ_2 (ζ_1-z_1)(ζ_2-z_2),where the integration extends over a product of circles:|ζ_1-z_1|=r_1|ζ_2-z_2|=r_2.The proof below differs from the existence result used in <cit.> by the fact that the nonlinearity is not analytic anymore with respect to time. It is therefore necessary to check carefully that the estimates on f can still be carried out.We now present the proof of the result.Throughout the proof, the meaning of the letter C will change from line to line: it denotes various constants, the specific value of which is not needed.We letH[v] = ∫_0^1 σ^A(x)-1v(σ t) dσ.It is easily checked that this provides the solution of(D+A)u=v,with u(0)=0, provided that v=O(t) near t=0.We are ultimately interested in real values of x in some open set Ω, so that we work in a small complex neighborhood of the real set Ω. The proof in fact does not depend on the nature or size of this set. We also define two norms which will be useful. The s-norm of a function of x isu_s = sup{ |u(x)| : x∈ d(x,Ω)<s }.The a-norm of a function of x and t is defined by|u|_a = sup{s_0-s tu(t)_s √(1-t a(s_0-s)): t<a(s_0-s) }Note that this norm allows functions to become unbounded when t=a(s_0-s). This can be thought of basically as the boundary of the domain of dependence of the solution. For the reasons indicated earlier, the iteration would not be well-defined if we had worked simply with the supremum of the s-norm over some time interval.The objective is to prove that the iteration u_0=0, u_n+1=G[u_n] is well-defined and converges to a fixed point of G, which gives us the desired solution. This will be acheived by exhibiting a set of functions which contains zero and on which G is a contraction in the a-norm. Since a contraction has a unique fixed point, we also obtain uniqueness.We choose R>0 and s_0 such that F[0](t)_s_0≤ Rt. This can always be achieved since we are allowed to take R very large. Step 1: Estimating H. Using the definition of |u|_a, we find, with the notation ρ=σ t/a(s_0-s),[H[u](t)_s≤|u|_as_0-s ∫_0^1 |σ^A|σtσ(1-σta(s_0-s))^-1/2 dσ; = C|u|_as_0-s ∫_0^t/a(s_0-s) a(s_0-s) dρ√(1-ρ); ≤ C_0 a |u|_a. ] Step 2: Estimating F. Using Cauchy's integral representation, and the fact that f contains a factor of t, we claim that there is a constant C_1 such thatF[p]-F[q] _s(t) ≤C_1t s'-sp-q_s'if s'>s and p_s and q_s are both less than R; this constraint will be ensured in Step 3 thanks to the argument of the previous step.Indeed, F[p] is the product of t by a linear expression in the gradient of p, with coefficients which are Lipschitz functions of p; it is in fact in the Gowdy case an analytic function of these variables, x, and positive powers of t multiplied by logarithms. The bound on the s-norm ensures that all the partial derivatives of F with respect to p and ∇_x p are bounded by some constant C.Therefore, we have| F[p]-F[q] | ≤ Ct(|p-q|+|∇_x p-∇_x q|).We want to estimate the supremum of this expression as x varies so as to satisfy (x,Ω)<s. The first term is clearly less than or equal to p-q_s, and a fortiori no bigger than p-q_s'. The second is estimated by Cauchy's inequality on each component. Thus, for the first component, we writep(x,t)-q(x,t)=1 2π i∫_|z-x_1|=s'-s(p(z,x_2,…,t)-q(z,x_2,…,t))dz z-x_1.Differentiating with respect to x_1, we find|_1(p-q)| =| 1 2π i∫_|z-x_1|=s'-s(p(z,x_2,…,t)-q(z,x_2,…,t))dz (z-x_1)^2| ≤ 1 2π∫_|z-x_1|=s'-s|p(z,x_2,…,t)-q(z,x_2,…,t)||dz| (s'-s)^2≤ 1 2πp-q_s'2π(s'-s) (s'-s)^2,which provides the desired estimate for the second term as well.Step 3: G is contractive. Let us assume in this section that |u|_a and |v|_a are both less than R/2C_0a. We prove that|G[u]-G[v]|_a≤ C_2 a|u-v|_a.One should think of u and v as two successive terms u_n and u_n+1 in the iterative procedure.To obtain this inequality, we first writeG[u]-G[v]=∑_j=1^n F[w_j]-F[w_j-1],wherew_j = ∫_0^j/nσ^A-1u(σ t) dσ+∫_j/n^1 σ^A-1v(σ t) dσ.By the arguments of Step 1, we have w_j_s< R/2 for t<a(s_0-s).We therefore have, using Step 2 with p=w_j and q=w_j-1,G[u]-G[v]_s(t)≤∑_j=1^n Ct s_j-sw_j-w_j-1_s_j. Let us choose a sequence of numbers, s_j=s(j/n), wheres(σ) = 1 2 (s+s_0-σ t a).We now find[w_j-w_j-1_s_j= ∫_(j-1)/n^j/nσ^A-1[u(σ t)-v(σ t)] dσ_s_j; ≤ ∫_(j-1)/n^j/n C u-v_s(σ)(σ t) dσ/σ; ≤ ∫_(j-1)/n^j/nCt s_0-s(σ) |u-v|_a dσ√(1-σ t/a(s_0-s(σ))). ]Letting n tend to infinity, we find the estimateG[u]-G[v]_s(t)≤∫_0^1 Ct^2 |u-v|_a(s(σ)-s)(s_0-s(σ))dσ√(1-σ t/a(s_0-s(σ)) ).We now make the change of variables ρ=σ t/a(s_0-s). Note that(s(σ)-s)(s_0-s(σ))=(s_0-s)^2 4(1-ρ^2); 1-σ t a(s_0-s(σ)) = 1-ρ 1+ρ.We therefore find[ G[u]-G[v]_s(t)≤ C a t|u-v|_a s_0-s∫_0^t/a(s_0-s)dρ (1-ρ)^3/2; ≤C a t|u-v|_a s_0-s(1-t a(s_0-s))^-1/2;]Using the definition of the a-norm, we see that we have obtained the desired estimate. Step 4: End of proof. Let u_0=0 and define inductively u_n by u_n+1=G[u_n]. We show that this sequence converges in the a-norm if a is small. Since u_1_s_0≤ Rt, we have |u_1|_a<R/4C_0a if a is small. We may assume C_2 a<1/2. It follows by induction that |u_n+1-u_n|_a≤ 2^-n|u_1|_a, and |u_n+1|_a < R/2C_0a, which implies in particular Hu_n_s < R/2. Therefore all the iterates are well-defined and lie in the domain in which G is contractive. As a result, the iteration converges, as desired.Impact on formal expansions.The expansion of <cit.> amounts to seeking X and Z as functions of (t, x), expanding in , and then letting =1. Its convergence can therefore be derived from the analyticity of the solutions in x. Note that the reference solution in that paper is slightly more restrictive than those considered here: they are geodesic loops travelling to the right' in the Poincaré half-plane.The Fuchsian algorithm provides a different way of generating formal solutions: by following the existence proof itself. Thus, starting with u=0, we can compute F[0], then solve (D+A)u_1=F[0], which is a linear ODE in t, compute F[u_1], etc. The higher-order corrections are automatically generated even if their order is not known in advance.Remarks on the nature of the singularity.One could check that AVD Gowdy spacetimes with 0<k<1 or k>1 do have a curvature singularity at t=0 by directly computing the Kretschmann scalar B:= R_ijklR^ijkl (for large classes of such spacetimes, see [2], who uses symbolic manipulation; see also a brief remark in this direction at the end of [6]). We give a simpler argument which reduces the issue to the corresponding problem for Kasner spacetimes, where the answer is classical.Indeed, consider the orthonormal coframe(e^λ/4t^-1/4dt,e^λ/4t^-1/4dx,t^1/2e^-Z/2(dy+Xdz),t^1/2e^Z/2dz),and the dual frame {_a=_^k ∂_k}. One finds, by direct computation, that the Ricci rotation coefficients of this frame all have the form:γ^a_bc = C^a_bc t^-3(k^2+1)/4 (1+o(1)),where the leading-order coefficients C^a_bc are t-independent quantities which involve only k:its derivatives, or the functions X_0, φ and ψ do not affect the value of these coefficients. A similar property holds for the coefficients b^c_ab defined by [_a,_b] = b^c_ab_c. It follows that the product terms in the expression of the frame components of the curvature tensor are at most O(t^-3(k^2+1)/2). As for the Pfaffian derivative terms, it turns out that they are not more singular, because they are coordinate derivatives multiplied by appropriate frame components. There are still no x-derivatives at leading order. It follows that the most singular term in B as t→ 0 is in fact the same as the one corresponding to X_0=φ=ψ=0, and k=const., which is the Kasner case.In extrinsic terms, we may express the result as follows: if h is the mean curvature of the slices t=const., then B/h^4 tends to a non-zero constant for 0<k<1 or k>1, which has the same expression as in the Kasner case. In particular, B blows up like t^-3(k^2+1), so that we have a curvature singularity, QED.Remark 1: It is easy to check that this singularity is reached in finite proper time by observers with x=const., so that this space is indeed (past) geodesically incomplete.Remark 2: There is no change in the leading power of B as k goes through 1: only the coefficient of the leading term in B vanishes. We thank V. Moncrief for numerous helpful discussions during the early stages of this work. S. K. would like to thank the Max-Planck-Institut in Potsdam for its warm hospitality during two weeks.99b-m B. Berger and V. Moncrief, Numerical investigation of cosmological singularities, D48(10) (1994) 4676–4687.c P. T. Chruściel, On uniqueness in the large of solutions of Einstein's equations (Strong Cosmic Censorship'), Proc. CMA 27, ANU (1991).c-i-m P. T. Chruściel, J. Isenberg and V. Moncrief, Strong cosmic censorship in polarized Gowdy spacetimes, , 7 (1990) 1671–1680.e-l-s D. Eardley, E. Liang and R. Sachs, Velocity-dominated singularities in irrotational dust cosmologies, 13 (1972) 99–106.g R. H. Gowdy, Vacuum spacetimes and compact invariant hypersurfaces: topologies and boundary conditions, 83 (1974) 203–241.g-m2 B. Grubišić and V. Moncrief, Asymptotic behavior of the T^3×R Gowdy space-times, D47(6) (1993)2371–2382.h-e S. W. Hawking and G. F. R. Ellis, The Large-Scale Structure of Spacetime, Cambridge U. Press, 1973.nlw S. Kichenassamy, Nonlinear Wave Equations, Marcel Dekker, Inc., New York, 1996.syd S. Kichenassamy, WTC expansions and non-integrable equations, Studies in Appl. Math., to appear. See also The blow-up problem for exponential nonlinearities.hs S. Kichenassamy, Fuchsian equations in Sobolev spaces and blow-up, Journal of Differential Equations, 125 (1996) 299–327.inversibilite S. Kichenassamy, The blow-up problem for exponential nonlinearities, Communications in PDE, 21 (1&2) (1996) 125–162.gks S. Kichenassamy and G. K. Srinivasan, The structure of WTC expansions and applications, J. Phys. A: Math. Gen., 28:7 (1995) 1977–2004. | http://arxiv.org/abs/1709.09710v2 | {
"authors": [
"Satyanad Kichenassamy",
"Alan D. Rendall"
],
"categories": [
"gr-qc",
"math.AP",
"83C75, 83C15, 83C20, 35Q75, 35A20, 83C05"
],
"primary_category": "gr-qc",
"published": "20170927193733",
"title": "Analytic description of singularities in Gowdy spacetimes"
} |
{} OMSzplmmnremark[1][Remark]#1=4calc mindmap,shapes,arrowsOn Time Optimization of Centroidal Momentum DynamicsBrahayam Ponton^1, Alexander Herzog^1, Andrea Del Prete^1, Stefan Schaal^1,2 and Ludovic Righetti^1,3 This research was supported by New York University, the Max-Planck Society and the European Union's Horizon 2020 research and innovation programme (grant agreement No 780684 and European Research Council's grant No 637935). We would like to thank the reviewers for valuable comments on the first version of this article. ^1Max Planck Institute for Intelligent Systems, Tuebingen, Germany ^2University of Southern California, Los Angeles, USA ^3New York University, New York, USADecember 30, 2023 ===================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================empty emptyRecently, the centroidal momentum dynamics has received substantial attention to plan dynamically consistent motions for robots with arms and legs in multi-contact scenarios. However, it is also non convex which renders any optimization approach difficult and timing is usually kept fixed in most trajectory optimization techniques to not introduce additional non convexities to the problem. But this can limit the versatility of the algorithms. In our previous work, we proposed a convex relaxation of the problem that allowed to efficiently compute momentum trajectories and contact forces. However, our approach could not minimize a desired angular momentum objective which seriously limited its applicability. Noticing that the non-convexity introduced by the time variables is of similar nature as the centroidal dynamics one, we propose two convex relaxations to the problem based on trust regions and soft constraints. The resulting approaches can compute time-optimized dynamically consistent trajectories sufficiently fast to make the approach realtime capable. The performance of the algorithm is demonstrated in several multi-contact scenarios for a humanoid robot. In particular, we show that the proposed convex relaxation of the original problem finds solutions that are consistent with the original non-convex problem and illustrate how timing optimization allows to find motion plans that would be difficult to plan with fixed timing [Implementation details and demos can be found in the source code available at https://git-amd.tuebingen.mpg.de/bponton/timeoptimization.]. § INTRODUCTION Motion optimization for robots with arms and legs such as humanoids is a challenging task for many reasons including very-high dimensionality, problem discontinuities due to intermittent contacts, non convex optimization landscapes prone to local minima, realtime constraints to find a solution, quality of the solution for execution on a real robot, etc.Yet, these challenges have inspired researchers for many years to develop optimization methods that show really impressive simulation results<cit.>. On the more practical side however, the most successful methods have not been the most complex and computationally expensive ones, but the ones that provide sufficient flexibility to perform a desired task while being well suited for model predictive control <cit.>. Indeed the ability to compute motions in a receding horizon fashion is very important to provide the necessary reactivity to the robot behavior in uncertain environments.In recent years, the centroidal momentum dynamics model <cit.> has become a popular model for multi-contact dynamic full-body motions <cit.>. Indeed, this model, under the assumption of enough torque authority, provides sufficient conditions for planning dynamically feasible motions <cit.>, and is simple enough such that the problem could be solved with close to realtime rates <cit.>. Notably, it was shown in <cit.> that such plans could be successfully used on a humanoid robot.However, the centroidal momentum dynamics is not convex, which renders the optimization problem difficult. Recent works have looked at the mathematical structure of the problem to find more efficient optimization algorithms. In <cit.>, a multiple-shooting method is used to efficiently optimize the centroidal dynamics. In <cit.> a convex bound on the rate of change in angular momentum is used to minimize a worst-case bound on the l_1 norm of the angular momentum.In <cit.>, it was shown that the non convex part could be decomposed as a difference of quadratic functions. This allows an efficient convex approximation of the problem that can be used in sequential quadratic programming approaches.The paper also proposed a method to efficiently compute dynamically consistent full-body motions by alternating centroidal dynamics optimization with full robot kinematics optimization. In our previous work <cit.>, we proposed a convex relaxation of the problem that allowed to find solutions at realtime rates. However, our approach was using a proxy function to minimize the angular momentum: it was minimizing the sum of the squares of thequadratic functions composing the non-convex part of the equations. Thus, it was not possible to include an explicit target momentum in the cost function, whichlimits the space of desirable solutions where momentum is effectively minimized; which made this approach not well suited to be used directly in the alternating full-body optimization method proposed in <cit.>. The new approach presents a more general approximation of the nonconvex constraints, allowing to also investigate an under-studied aspect namely the importance of timing for centroidal momentum optimization. Several works have realized the importance of including time as an optimization variable <cit.> and in doing so have shown great simulation and experimental results. However, including time optimization is usually computationally very costly due to the non convexity introduced in the discretized dynamics.In this paper, we propose two methods for convex relaxation of the centroidal momentum dynamics optimization problem that allows to include an explicit angular momentum objective.Moreover, noticing that the non-convexity introduced by the time variable is of similar nature as the torque cross product allows us to use the same relaxation approaches for time optimization. The resulting algorithm allows to compute dynamically consistent plans close to realtime. Experiments demonstrate the computational efficiency of our approach in multi-contact scenarios. In particular, we show that the numerical solutions found in our method are very close to the original dynamics (measured by the amount of constraint violation), suggesting that the convex relaxation is a good approximation of the original problem. Then we show that optimizing time allows to find solutions that could not be easily found with fixed time optimization. We also combine our approach with the alternating approach proposed in <cit.> to compute dynamically consistent full-body motions.The remainder of this paper is structured as follows. In Sec. <ref>, we present the problem formulation. Then, in Sec. <ref>, we show how to efficiently find a solution to the centroidal momentum dynamics problem. We show experimental results in Sec. <ref> and conclude the paper in Sec. <ref>. § PROBLEM FORMULATIONThe equations of motion that describe the dynamic evolution of a floating-base rigid body system are given by M(q)q̈ + N(q,q̇) = S^Tτ_j + J_e^Tλ , where the robot state is denoted by q=[ x^T q_j^T ]^T, and comprises the position and orientation of a floating base frame in the robot relative to an inertial frame x ∈ SE(3), and the joint configuration q_j∈ℝ^n_j. The inertia matrix is denoted by M(q) ∈ℝ^(n_j+6) × (n_j+6), the vector of nonlinear terms N(q,q̇) ∈ℝ^n_j+6includes Coriolis, centrifugal, gravity and friction forces, the selection matrix 𝐒=[ 0^n_j× 6 I^n_j× n_j ] represents the system under-actuation, namely that x ∈ SE(3) is not directly actuated by the vector of joint torques τ_j∈ℝ^n_j, but indirectly through a vector of generalized forces λ and the Jacobian of the contact constraints J_e.The system under-actuation leads to a dynamics decomposition into an actuated (subscript a) and un-actuated parts (subscript u) as follows:M_a(q)q̈ + N_a(q,q̇)= τ_j + J_e,a^Tλ M_u(q)q̈ + N_u(q,q̇)= J_e,u^TλEquation (<ref>), known as the Newton-Euler equations, tells us that the systems' change of momentum depends on external contact forces. Any combination of forces λ and accelerations q̈ can be realized, if they are consistent with the underactuated dynamics (<ref>), and there is enough torque authority (<ref>) <cit.>. This natural decomposition suggests that satisfaction of the momentum equation (<ref>) is sufficient to guarantee dynamic feasibility, and equation (<ref>) ensures kinematic feasibility and torque limits. §.§ Dynamics Model As a consequence of the last observation, under the assumption of enough torque authority and kinematic reachability, a necessary condition for planning physically consistent motions is that the total wrench generated by external and gravitational forces (<ref>) equals the rate of momentum computed from the robot joint angles and velocities (<ref>)<cit.>. On the one hand, the centroidal momentum, computed from external forces, expressed at the robot center of mass is = [ ; ; ;] =[1/m;m 𝐠 + ∑_e_e; ∑_e (𝐩_e+ 𝐑^x,y_e_e-) ×_e + 𝐑^z_eτ_e;] wheredenotes the center of mass position,the linear andthe angular momentum.is a shortcut vector comprising ,and . The total robot mass is m and 𝐠 the gravity vector. The position of the e end-effector is denoted 𝐩_e, _e∈ℝ^2 is the center of pressure (CoP) expressed in local end-effector coordinates. 𝐑^x,y_e∈ℝ^3 × 2 represents the first two columns of the rotation matrix 𝐑_e∈ℝ^3 × 3 that maps quantities from end-effector frame to the inertial coordinate frame. _e∈ℝ^3 and𝐑^z_eτ_e∈ℝ^3 are forces and torques acting at contact point 𝐩_e+ 𝐑^x,y_e_e, represented in inertial frame. τ_e∈ℝ is the torque around the z, upward pointing, axis expressed in end-effector frame. 𝐑^z_e maps τ_e to the inertial coordinate frame.On the other hand, the centroidal momentum, computed from the robot joint angles and velocities, is given by <cit.> [ ;] =𝐀(𝐪) 𝐪̇ where 𝐀(𝐪) ∈ℝ^6 × n_j+6 is the centroidal momentum matrix. we highlight that the rate of change of momentum, as given by (<ref>), only depends on dynamic quantities, while the momentum, as given by (<ref>), only depends on kinematic quantities. This separation, suggested in <cit.>, allows the use of an iterative procedure, where one can alternate between a kinematic and a dynamic optimization to solve the joint problem. Both optimization procedures need only agree on the center of mass trajectory, momentum and contact locations and such agreement is enforced by a cost function in each optimization algorithm.The benefit of this separation is that the optimization problems can be solved more easily separately than as a joint problem and the inherent structure of each problem can be exploited in dedicated solvers. §.§ Trajectory Optimization In this paper we focus on the centroidal dynamics optimization problem and only briefly comment on the alternating full-body optimization procedure as we will use it in the experiments section. First, we discretize the differential equations (<ref>)-(<ref>) into algebraic equations and then use the iterative proceduredescribed in <cit.> to alternate between a dynamics and a kinematics optimization.§.§.§ Kinematics Optimization Problem We will not focus on the kinematic optimization. However, we would like to mention that we use at each time step an inverse kinematics procedure, whose objective is to track a desired center of mass position, linear and angular momenta, regularize rates of momentum, track desired motions for unconstrained end-effectors, regularize joint posture towards a default posture and regularize joint velocities and accelerations. The constraints include the evolution of linear and angular momenta using the centroidal momentum dynamics (<ref>), evolution of center of mass according to linear momentum, evolution of endeffectors based on the end-effector jacobians, joint limits and constraints for active end-effectors.§.§.§ Dynamics Optimization Problem We are interested in the optimization of dynamic motions including momentum trajectories, contact forces and timings, under the non-convex and non-linear centroidal momentum dynamics, which could later be realized by a low-level controller such as an inverse dynamics one <cit.>. We assume a fixed set of contacts, however they can be easily included as optimization variables constrained to lie within the stepping stones <cit.>. Formally, the objective to be minimized in our optimization problem is: min__e, τ_e, _e, Δ_tϕ_N(_n-_n_des) + ∑_t=1^n-1ℓ_t( - _des, _e, _e, τ_e, Δ_t) More specifically, we would like to minimize a terminal cost ϕ_N(_n-_n_des) that penalizes the difference between the final state _n and the desired final state _n_des, and a running cost ℓ_t( - _des, _e, _e, τ_e, Δ_t), that penalizes the tracking performance of a desired linear and angular momentum trajectories - _des, and regularizes the available controls, namely, forces _e, torques τ_e and time discretizations Δ_t.Desired momentum trajectories could be as trivial as zeros or could for instance come from the kinematic optimization.The constraints of the optimization problem include a discrete form of the centroidal momentum dynamics (<ref>): = [ _t; _t; _t; _t; _t;] = [ _t-1 + 1/m_tΔ_t;_t-1 + _tΔ_t;_t-1 + _tΔ_t; m 𝐠 + ∑_e_e,t;∑_eκ_𝐞,𝐭; ] The variable κ_𝐞,𝐭 (end-effector contribution to angular momentum rate _t) has been defined as κ_𝐞,𝐭 = (𝐩_e,t + 𝐑^x,y_e,t_e,t - _t) ×_e,t + 𝐑^z_e,tτ_e,t= ℓ_e,t×_e,t + 𝐑^z_e,tτ_e,t= [r] 0 -ℓ^ z_e,t ℓ^ y_e,t ℓ^ z_e,t 0 -ℓ^ x_e,t -ℓ^ y_e,t ℓ^ x_e,t 0 [ ^ x_e,t; ^ y_e,t; ^ z_e,t; ] + 𝐑^z_e,tτ_e,t where for simplicity of notation, we have introduced the change of variable ℓ_e,t = (𝐩_e,t + 𝐑^x,y_e,t_e,t - _t). Physical constraints such as friction cone, CoP within region of support and torque limits are given by _L^ x_e,t+_L^ y_e,t_2≤μ_L^ z_e,t, _L^ z_e,t≥ 0, ^x,y_e,t∈[ ^x,y_min, ^x,y_max], τ_e,t∈[ τ_min, τ_max] Δ_t∈[ Δ_min, Δ_max] 𝐩_e,t - _t≤ℓ_e^max where _L_e,t =𝐑^T_e,t_e,t is the end-effector force in local frame. (<ref>) states that forces belong to a friction cone with friction coefficient μ. Friction cones could alternatively be approximated by the usual pyramids as a polyhedral approximation. (<ref>) expresses that the CoP should be within a conservative region with respect to the real physical available region. (<ref>) constrains the torque to a bounded region, <cit.> also provides precise closed-form formulas for it under polyhedral approximation of the friction cone. Equation (<ref>) constrains the time discretization variable to a bounded region. Finally, equation (<ref>) constrains the distance between the current position of the center of mass _t and the end-effector contact point 𝐩_e,t to be less than the maximum length of the end-effector ℓ_e^max. It includes a constant offset, when the end-effectors of interest are the arms of the robot. We would like to conclude this section by highlighting that the non-convexities of this problem are the bilinear terms of eq. (<ref>). For instance, terms where the time discretization variable Δ_t appears or the cross products of the angular momentum rate terms κ_𝐞,𝐭. In traditional momentum optimization <cit.>, the focus is on the cross products of the angular momentum, that introduce bilinear constraints to the optimization problem. It is important to note that including time discretizations as optimization variables introduces nonconvex constraints of the same bilinear nature. Therefore, the goal of the next section will be to devise methods that can approximate bilinear constraints, while still allowing us to efficiently find a solution to the optimization problem.§ APPROACHIn this section, we describe a tool to express bilinear constraints (<ref>)-(<ref>) as a difference of convex functions, and then we show how to approximate these still nonconvex constraints, using the knowledge about their positive curvature. §.§ Disciplined Convex-Concave Programming In this subsection, we will describe a tool <cit.> for handling bilinear constraints in optimization problems. The method decomposes a bilinear or nonconvex quadratic expression into a difference of convex functions; in other words, it decomposes the bilinear expression into a difference of two terms, each of which is a convex function. After the decomposition, the constraint will continue to be nonconvex; however, the terms composing it will have known curvature, which will let us perform an efficient approximation.This approach, previously used and detailed in <cit.>, analytically decomposes the nonconvex quadratic expressions of the angular momentum (<ref>) into a difference of convex (quadratic) functions.The set of difference of convex functions C^± is defined as: C^± = { C^+(𝐱) - C^-(𝐱)| 𝐱∈ℝ^n, ..C^+, C^-: ℝ^n→ℝ,are convex functions} Expressions such as scalar products x^Ty can be decomposed into a difference of convex functions, such as Q^+-Q^- Q^+ = 1/4x+y^2, andQ^- = 1/4x-y^2 where Q^+∈C^+ and Q^-∈C^- are convex functions. Using this idea, we can easily identify scalar products and transform quadratic expressions in cross products into scalar products, which can then be defined as elements of C^±. As an example, we present the decomposition of a cross product: ℓ× = [ [ -ℓ_zℓ_y ]_𝐚_cvx^T[ _y; _z ]^𝐝_cvx, [ℓ_z -ℓ_x ]_𝐛_cvx^T[ _x; _z ]^𝐞_cvx,[ -ℓ_yℓ_x ]_𝐜_cvx^T[ _x; _y ]^𝐟_cvx ]^T and then each scalar product is defined as an element of C^±. ℓ× =1/4[ 𝐚_cvx +𝐝_cvx^2_2 - 𝐚_cvx - 𝐝_cvx^2_2; 𝐛_cvx +𝐞_cvx^2_2 - 𝐛_cvx - 𝐞_cvx^2_2; 𝐜_cvx +𝐟_cvx^2_2 - 𝐜_cvx - 𝐟_cvx^2_2; ] For simplicity of presentation, we denote 𝐩=𝐚_cvx+𝐝_cvx and 𝐪=𝐚_cvx-𝐝_cvx (𝐩, 𝐪∈ℝ^2). With this notation the first component of the torque contribution of an end-effector to the angular momentum rate dynamics (<ref>) becomes κ^x = 1/4[ 𝐩^T𝐩 - 𝐪^T𝐪 ] + τ^x, which is equivalent to the following formulation κ^x = 1/4[ p̅ - q̅ ] + τ^xp̅ = 𝐩^T𝐩q̅ = 𝐪^T𝐪where we have introduced the scalar variables p̅, q̅∈ℝ_+. Under this formulation the original nonconvex quadratic constraint has been now separated into a linear constraint, and two non-convex constraints (<ref>). The difference is that, while in the original constraint, the hessian is an indefinite matrix, the hessian of the new constraints is positive semi-definite and therefore each term has known curvature. §.§ Approximations of Quadratic Equality Constraints In the last section, we have presented a method to analytically decompose a bilinear constraint into a linear constraint and two nonconvex quadratic equality constraints with known curvature, see (<ref>). In this section, we present alternatives for dealing with each of the two nonconvex quadratic equality constraints with known curvature, namely, approximation using a trust region and use of soft constraints.§.§.§ Approximation using a trust region A nonconvex quadratic equality constraint with known curvature can be thought of as the intersection of a convex and a nonconvex inequality constraints (Fig. <ref>). The approximation of the equality constraint using a trust region consists in keeping only the convex inequality constraint, but restricting its interior with a trust region. The effect of this, as can be seen in Fig. <ref> is that the search space is close to the boundary of the curve, and therefore the values of y are close to x^2, as desired to approximate the original equality constraint.Generalizing this example to the context of our problem, the nonconvex constraint (<ref>) can be equivalently represented as the intersection of a convex constraint p̅≽𝐩^T𝐩 and a nonconvex constraint 𝐩^T𝐩≽p̅. The main idea of the method is to first obtain an initial guess of the optimal vector by solving the optimization problem using only the convex part of the search space, namely p̅≽𝐩^T𝐩 (as in <cit.>), and then refine the solution by introducing the trust region. While the trust region could be introduced as a trivial box constraint with a threshold, that would constrain the value of p̅ to values near 𝐩^T𝐩, the best affine trust region that exploits the knowledge about the curvature of the function and the information contained in the current value of the optimal vector is a linear approximation such as 𝐩_val^T𝐩_val + 2 𝐩_val (𝐩-𝐩_val)≽p̅ - σ, where σ is a positive value representing a threshold, enough to provide a feasible interior to the intersection of the constraints, and 𝐩_val is any value taken by the variables 𝐩 coming from the solution of the relaxed problem. Notice that if the hessian of the quadratic equality constraint where an indefinite matrix, this trust region would not constrain the problem as desired and instead lead to unbounded regions. The advantage of building the trust region this way is that σ represents the maximum amount of desired constraint violation, which brings firstan automatic way to constrain the values of 𝐩 around 𝐩_val that satisfy this amount, and second a method to further refine the solution by reducing the value of σ (that increase approximation accuracy), as required by convergence tolerances. While this approximation is still a relaxation of the constraint, it allows us to approximate the terms p̅, q̅, and consequently κ^x. This formulation has the advantage that we do not trade-off cost expressiveness, allowing e.g. quadratic terms over κ^x without losing the convexity properties.§.§.§ Approximation using soft constraints This method is similar in spirit to the last one. It will first drop the nonconvex terms to find an initial guess for the optimal vector and then introduce cost heuristics, whose goal is to bias the solutions towards the boundaries. Unlike the previous method, this method does not restrict strictly the search space, but instead biases the solutions towards the boundaries by pulling the variables towards an underestimator of the function.As in the previous case, the best affine underestimator, that exploits the knowledge about the curvature of the function and the current value of the optimal vector is a linearization 𝐩_val^T𝐩_val + 2 𝐩_val (𝐩-𝐩_val). The heuristic is a quadratic term in the cost that penalizes the difference between the variable p̅ and the linearization. This rewards the optimization for selecting values of p̅ that are close to the boundaries of the constraint and are therefore feasible for the nonconvex constraint. As in the previous case, we do not trade-off expressiveness of the cost, allowing for example quadratic terms over κ^x without losing the convexity properties.§.§.§ Convergence criteriaThe amount of constraint violation can be determined by how close the approximation is to the nonconvex constraint. In our setting, the convergence criteria is that the average error on center of mass and momentum trajectories, computed comparing the values of the corresponding variables coming from the relaxed optimization problem and the values computed integrating optimal torques and forces are below a certain residual error.For problems involving time optimization, we have noticed that the speed of convergence of angular momentum (being the last to converge) to optimal values strongly depends on the convergence of first linear momentum and then center of mass. Based on this observation, we use in time optimization problems a first phase where the problem bilinear constraints are approximated as described in this section and let the algorithm discover a motion plan and its timing. In a second phase, we focus on the convergence of the approximated angular momentum, and do so by keeping the approximation of its cross products as described in this section and replacing the second order approximation of time related constraints with first order ones. This allows the optimizer to still make adjustments on the center of mass, linear momentum, and timings, while allowing us to speed up and increase accuracy of the convergence of angular momentum.§ EXPERIMENTSWe have tested the algorithm in several multi-contact scenarios, including walking on an uneven terrain (Fig. <ref>), walking under a bar using also hands (Fig. <ref>), and a walking motion with low friction coefficient (Fig. <ref>). The resulting motions, reproduced in time agreement with the optimized plans, are visible in the attached video https://youtu.be/ZGhSCILANDw. §.§ Walking on uneven terrainThe first motion (Fig. <ref>) has been built so that double support time after a single support on left foot is short, while double support time after single support on right foot is longer. It then allows to see the effect of double support duration on momentum optimization. Stairs are also close to each other, such that legs require momentum to be lifted up, and minimum jerk trajectories that guide this kinematic motion have not been well tuned such that the effect of it is more visible. We test three scenarios: fixed time momentum optimization (blue), time optimization with no constraints on the total duration of the motion (orange) and time as an optimization variable with fixed total duration (green)As visible in Fig. <ref> (Momentum Optimization, in blue), the angular momentum in the Y direction has peaks at short time double supports, while during longer double supports less momentum is distributed along a longer timespan. For the optimization of both time and momentum (orange), timestep discretizations are included in the optimization, without a fixed total duration. As can be seen in the bottom plot, timestep durations are increased during the initial short time double supports. The optimization thus automatically distributes the angular momentum in the Y direction without peaks. The last optimization (fixed time horizon and momentum optimization, in green), is also capable of adapting the timestep durations, however, to be comparable to the original motion, the time horizon is fixed to the same value as in the first experiment. Here, we can also see the tendency of increasing the timestep durations at the short time double supports, while reducing time spent at the beginning and end of the motion, and when walking on the straight line. This also allows to decrease angular momentum in the Y direction, which was the goal of the optimization.We note (not shown), that the momentum coming from the kinematic optimizer can be tracked by the dynamic optimization.These experiments show how time can be used by the optimizer to reduce the overall momentum of the system and result in potentially easier to execute plans. For example, in an extreme case, where the available motion time makes the peaks of only the momentum optimized plan (in blue) much higher, the endeffector wrenches required to realize such a motion would also be higher. A different perspective on this could be a scenario with limitations on the endeffector wrench (such as walking under low friction) that would require a slower execution of the motion to make it feasible. §.§ Walking under a bar using hands In this example, we compare both relaxation methods (trust region and soft constraint) using fixed time (Fig. <ref>). We see that both relaxations lead to very similar solutions. This suggests both methods are similarly applicable, however in our current implementation the soft constraint relaxation was always significantly faster than the trust region one. For this specific problem, we also notice that time optimization brings marginal changes to the motion, which highlights that having more endeffectors available to control the motion can compensate the lack of timing adaptation. §.§ Walking with low friction coefficient In this example (Fig. <ref>), time optimization was critical to find a dynamically feasible motion. In this example, the friction coefficient is 0.4, the original time horizon is around 10 sec with a timestep duration of 0.1 sec. Under these conditions, the motion cannot be realized without optimizing time. Time optimization allows to automatically increase the time horizon (to around 16 sec) and the time available during double supports (timestep durations hitting its limit at 0.25 sec), which generates a physically realizable motion. Another alternative to generate the motion could be, as in the previous case, using hand contacts if available. §.§ Discussion on time optimizationAs shown in the previous experiments, including time as an optimization variable is useful, and depending on the kind of motion it can really produce much lower cost solutions or even make them feasible which is an infinite improvement. However, it increases the dimensionality of the problem to be solved and consequently the time required to solve it.§.§.§ Limitations of the convex approximationsThe problem at hand is nonconvex and thus hard to solve. The proposed heuristics help us reduce the effort required to find a solution by searching in a convex space an approximate solution (because of the constraint violations); however, the approach has also limitations. For instance, the trust region method could start with a bad initial guess, and the trust region built around it could render the local solution non optimal or even worse, it might overly restrict the problem making it primal infeasible. In the case of soft-constraint approximations, the penalty affects competing objectives, namely the amount of constraint violation and conditioning of the problem, which leads to problems such as having low constraint violation but poor problem conditioning or high constraint violation with faster convergence, therefore a reasonable tradeoff needs to be found. Source code provides more detail on the implementation and heuristics used in our experiments to speed up convergence, to find a solution realiably and robustly, and on how we build and refine trust region constraints and soft-constraint penalties.§.§.§ Time complexityTable <ref> shows information about the optimal problems being solved and time required to find a solution. Among the parameters being shown are the time horizon, the number of timesteps, number of variables, linear equality and inequality constraints, second order cones, size and nonzeros of the KKT matrix, and finally the time required to solve the problem. M1_time is the motion under low friction coefficient with time adaptation, while M1 is the same motion but with normal friction coefficient because time is fixed. M2 is the motion for walking under a bar using hands without time adaptation and M2_time with time adaptation.We can observe that the timings required to solve a dynamics optimization problem are in the order of a second, which is double the time required in <cit.>, because in this method, we also refine the solution after the relaxation, which allows us to directly penalize and track momentum, which increases the time required to find a solution, however improves its quality. Our solving time is comparable to the one reported in <cit.>, which corresponds to 1.23 sec for a motion duration of 8 sec using the centroidal wrench and 3.89 sec for the same motion using contact forces as control input, as in our case. Note that the time taken by the optimization is always faster than the duration of the motion and that, with appropriate warm start of the optimizer, receding horizon control is attainable.The solve time for a motion that includes time optimization is larger because more iterative refinements are required to converge to a good threshold of constraint violation. However, it could be further sped up doing only a few iterations of time optimization and then fixing time discretizations; increasing the accepted approximation error tolerance used as convergence criteria; or warm-starting a time optimization with a fixed-time dynamic optimization. The computation time is still lower than the plan time horizon, what makes it possible to run the algorithm online (the next plan can be computed, while the current one is being executed).§.§.§ Constraint violationsTable <ref> compares the average error in the center of mass, linear and angular momentum, computed integrating forces and torques with the original model and the approximate values in our relaxed formulation. As shown, errors on center of mass and linear momentum are marginal, while error on the angular momentum is small. The values shown are in an absolute scale. If they are compared to values shown in Figs. <ref>-<ref>, they are also small, because values in the figures are normalized by the robot mass.§ CONCLUSION We have presented two convex relaxation methods for the optimization of the centroidal dynamics and motion timing of a legged robot. Our approach is efficiently solvable and could therefore be used in receding horizon control. Moreover, the convex relaxation deviates only marginally from the original dynamics. Our approach has not been yet tested on a real robot, but this is the step coming. ieeetr | http://arxiv.org/abs/1709.09265v3 | {
"authors": [
"Brahayam Ponton",
"Alexander Herzog",
"Andrea Del Prete",
"Stefan Schaal",
"Ludovic Righetti"
],
"categories": [
"cs.RO"
],
"primary_category": "cs.RO",
"published": "20170926212547",
"title": "On Time Optimization of Centroidal Momentum Dynamics"
} |
Randomized experiments to detect and estimatesocial influence in networks Randomized experimentsSean J. Taylor Facebook [email protected] Dean Eckles Sloan School of Management and Institute for Data, Systems & Society, Massachusetts Institute of [email protected]* Sean J. Taylor and Dean Eckles December 30, 2023 ==================================Estimation of social influence in networks can be substantially biased in observational studies due to homophily and network correlation in exposure to exogenous events.Randomized experiments, in which the researcher intervenes in the social system and uses randomization to determine how to do so, provide a methodology for credibly estimating of causal effects of social behaviors. In addition to addressing questions central to the social sciences, these estimates can form the basis for effective marketing and public policy. In this review, we discuss the design space of experiments to measure social influence through combinations of interventions and randomizations.We define an experiment as combination of (1) a target population of individuals connected by an observed interaction network, (2) a set of treatments whereby the researcher will intervene in the social system, (3) a randomization strategy which maps individuals or edges to treatments, and (4) a measurement of an outcome of interest after treatment has been assigned.We review experiments that demonstrate potential experimental designs and we evaluate their advantages and tradeoffs for answering different types of causal questions about social influence. We show how randomization also provides a basis for statistical inference when analyzing these experiments. Keywords Field experiments, causal inference, Fisherian randomization inference, social interactions, spillovers, social networks§ INTRODUCTION There is a long tradition in the social sciences of examining how individual level behaviors diffuse and aggregate, including influential work by <cit.> and <cit.>, among many others <cit.>.Stylized models from this tradition have been used to explain some of the most important human phenomena, from which innovations are likely to gain widespread usage to who people vote for in elections. The fundamental building blocks of diffusion models are assumptions about how people change their behaviors in response to the behaviors of people they observe or interact with.These assumptions can vary in their disciplinary origins and sophistication — from epidemiological models to game-theoretic models with multiple equilibria.Randomized experiments provide a useful tool for testing theories.The increasing digitization and connectedness of human behaviors has made digital field experiments cheaper and easier to apply to social behaviors via contemporary communication technologies.This methodological paradigm shift has created opportunities for researchers hoping to understand the underpinnings of large-scale social behaviors in order to improve theory, make predictions, and compare hypothetical policies.In this review, we hope to make randomized experimentation more accessible to researchers seeking to contribute to our understanding of mechanisms for social influence and diffusion in social systems.We first discuss how randomized experiments can rule out potential confounding factors (Section <ref>).Because experiments require that the researcher intervenes in the social system, we devote Section <ref> to discussing the ethical consideration associated with employing digital field experiments.Section <ref> outlines the four components of a randomized experiment to detect or estimate social influence. This facilitates discussing the many design choices experimenters have, including defining the relevant network, what treatments can be employed, and how those treatments may be randomly assigned to subjects. In Section <ref>, we turn to the analysis of experiments in networks, where we focus on Fisherian randomization inference.Section <ref> discusses how the analysis of experiments can be extended in various ways in order to increase the usefulness of the results.This review complements more general references on design and analysis of randomized experiments <cit.>. Design and analysis with disjoint groups, has received substantial attention in economics and epidemiology <cit.>. On the other hand, there are few other reviews of design and analysis of experiments in networks. Compared with extant reviews <cit.>, we aim to integrate all the methods reviewed into a single causal model and discuss some design choices and analysis methods in detail.What exactly counts as social influence? Different fields distinguish among various processes by which people affect each other. For example, economists distinguish between peer effects caused by constraint, preference, and expectation interactions <cit.>, while other fields may make different distinctions. Thus, for some prior work, “social influence” denotes something more specific. However, given our methodological focus here, we choose to remain agnostic about the mechanisms and define social influence to include all processes by which an individual's behaviors affect another's, either directly or indirectly. Thus, we could have instead referred to “peer effects”, “diffusion”, or “social contagion”. Further theory-specific distinctions may motivate additional design and analysis choices. §.§ What makes randomized experiments different?We privilege information gained through randomized experiments because they create a different kind of knowledge than observational studies. In particular, because we know exactly how units are assigned to treatments, a properly implemented experiment rules out all alternative explanations for an observed correlation besides the causal one, and allows for both unbiased estimation of the effect of our intervention and statistical inference that is exact in finite samples <cit.>.Any observational analysis intended to answer questions about social influence must confront several potential biases that make it difficult to trust its conclusions.First, social networks are known to exhibit strong homophily <cit.>, creating network correlation of attributes, opinions, and behaviors through people's preferences for whom they spend time with.For instance, people with similar political beliefs may be more likely to form friendships <cit.>, and therefore homophily may readily explain cases of apparent political persuasion.Second, people who are connected in social networks are subject to similar exogenous shocks to their behavior, as when neighbors are exposed to similar marketing messages on billboards.A substantial program of research has been devoted to proving that what economists call “identification problems” in social influence are likely to be insurmountable without randomized experiments <cit.>.The intuitive reason is that without intervening in the social system, there are usually reasonable alternative explanations for correlations that do not involve a social influence effect.Despite their clear advantages, the use of randomized experiments is not a panacea for social scientists.Experiments are usually more costly to design and implement than observational studies because the researcher must alter people's behaviors or interactions in social system in some way.Interventions require substantial upfront costs for planning and implementation, including: recruitment of subjects, cost of the interventions themselves (financial or logistical), evaluation and exposure of risks of harm to subjects <cit.>.Because field experiments require researchers to impact the social systems they study at potentially very large scale, they can be associated with larger or different ethical challenges from other methods, which we summarize in Section <ref>.Experiments can also be problematic because, although they reduce concerns about bias, the variance of estimation becomes a first-order concern and the possibility of type II errors (commonly known as issues with experimental power) dominate due to the cost of sample size or the impossibility of the researcher creating large effects <cit.>.On a more positive note, we will see in Section <ref> that some well-designed experiments can require more straightforward analysis than observational studies.In addition, there are no internal validity concerns with experiments – they provide unbiased estimates for the social influence effects they were designed to measure. The two main constraints of an experimental methodology are which estimates are possible and the precision of those estimates.The randomization the researcher employs and structure of the network together determine what causal quantities of interest can be credibly estimated <cit.>.These estimands address counterfactual questions about which individual-level behaviors would obtain under alternative interventions. In one simple case, we may be able answer the question of how much an individual's probability of a behavior is increased by having exactly one peer (rather than no peers) who engages in that behavior.A more complex causal estimand might be the distribution of that behavior in the total population after a series of targeted (e.g., marketing) interventions or a policy change (e.g., by a government).As we will see, no single experiment can answer all possible causal questions and the experiment should be designed with some estimands in mind.The second constraint from using experiments is the precision of the effect estimates.Experimental data is often more costly to collect than observational data because treatments are not free and because observational data is abundant.In general, the precision of the estimates of social influence are limited by the direct effects of the intervention (weaker ones provide less experimental power) and the available sample size <cit.>.§.§ Ethical considerations for digital field experimentsThe reduced cost and increased feasibility of digital field experiments (DFEs) has led to increased experimentation over the past decade.While DFEs may help researchers answer many important questions about social influence, they can present more ethical challenges than observational research and even pre-digital lab and field experiments.To ground the discussion, we will refer to the four ethical principles proposed in the Belmont Report <cit.> and the subsequent Menlo Report <cit.> which are meant to provide guidance on human subjects research.Those principles – Respect for Persons, Beneficence, Justice, and Respect for Law and Public Interest – are briefly summarized principles in Table <ref>.For an in-depth, thorough treatment of ethics in research in the digital age, we refer the reader to Chapter 6 of <cit.> and for a recent discussion of institutional review processes to mitigate risk please see <cit.>.Rather than review those materials exhaustively, we use this subsection to discuss five ethical considerations that we consider to be particularly salient for digital field experiments.First, DFEs are implemented in software and therefore have very low variable costs with respect to the size of the treated population. It is no longer unusual for experiments to deploy treatments to millions of people <cit.>, amplifying their potential harm compared to more modest sample sizes.Additionally, treatments with network effects can, by research intention or not, cause detrimental effects for people who were not in the original treated population.Researchers acting in accordance with the ethical principle of Beneficence may have a more difficult time evaluating the potential risks of DFEs in networks because their potential effects on social systems are not obvious, intuitive, or even measured.Second, it may be difficult to identify whether subjects in DFEs are members of a vulnerable or protected population.When designing a DFE, researchers might find it challenging to estimate risks of harm because there is uncertainty about how many subjects could be adversely affected.Researchers might also be unable to reason about whether the benefits and risks of the research are distributed equitably across the population, in accordance with the principle of Justice. On many online or mobile platforms, researchers may not know if users are a reasonable age for consent or are particularly vulnerable to risk from the planned experiment.Third, DFEs typically use automated, large-scale collection of potentially sensitive and/or identifying information, e.g. location information or exchange of personal communication.Indeed, these data can be integral to the ability of the experiment to answer the research question of interest.For instance, a log of email communications can be used to infer a social network <cit.>, which is a key component for social influence studies.Persistent records of sensitive or identifying information can potentially be used for unintended purposes, causing harm to experimental subjects <cit.>.Fourth, because of their large scale and integration with existing technologies, DFEs often pose unique challenges for receiving informed consent, which is sometimes an implication of the ethical principle of Respect for Persons.Informing subjects of the experiment and receiving their consent can be disruptive to their normal experiences using various platforms and products (particularly if experiments are frequent, as is becoming more common).Furthermore, requiring informed consent can limit or bias the experimental population or prime the subjects, undermining or altering the treatment effects.Although informed consent is important component of Respect for Persons, deception may be permissible if the experiment complies with all other ethical principles and the deception does not strongly violate the norms of that setting <cit.>. Some experiments with potentially important benefits require deception in order ensure the research question can be suitably answered.For instance, in the employment discrimination field experiments <cit.> discuss, one could not credibly measure discrimination after informing employers of the nature of the research.Fifth, it may be difficult for researchers to comply with all laws, contracts, terms of service, or social norms because DFEs may involve partnerships with companies, span countries or other legal boundaries, or include subjects from many cultures.Inconsistent, overlapping, and sometimes unclear rules and norms lead to challenges for researchers hoping to understand all potential stakeholders and their associated goals and risks.These five considerations are not meant to be exhaustive – there are certainly other ways in which DFEs can present new ethical challenges for researchers.But we hope that this subsection has made clear that while experimental research has become easier to conduct on some dimensions, it has become more fraught on others – in particular in evaluating and mitigating the risk of harm to subjects.§.§.§ Recommendations for ethical research Taking into account the challenges identified, more research is needed to address the ethical implications in DFEs and to develop mitigating and creative strategies. In the meantime, researchers should do the utmost to:* Ensure that the research is ethical and beneficial for subjects; that it does not expose them to risk or harm (this may require escalation and further deliberation with other teams within the company, along with the assessment of alternative research methods that could be used). * Carefully assess if the collection and processing of sensitive data is essential for the research being conducted.* Determine if an experiment is strictly necessary for the objectives of the research (or if the same results can be obtained through less risky research, e.g. a smaller sample size).* Whenever possible, ensure that such collection and processing is done with prior informed consent given by the data subjects.* Only keep that data for the minimum necessary period of time andensure the proper de-identification of that data according to most effective and updated industry standards. § COMPONENTS OF A RANDOMIZED EXPERIMENTThe randomized experiment methodology has four main components:* A target population of units (i.e. individuals, subjects, vertices, nodes) who are connected by some interaction network.(Section <ref>)* A treatment which can plausibly affect behaviors or interactions. (Section <ref>)* A randomization strategy mapping units to probabilities of treatments. (Section <ref>)* An outcome behavior or attitude of interest and measurement strategy for capturing it. (Section <ref>) To summarize the relationship of these components, the researcher applies a treatment (2) to a target population (1) using a randomization strategy (3) and then measures the outcome behavior (4).The following four sections describe these four components, characterize the space of possibilities for each one, and provide examples from existing research.We introduce notation along the way that we will use in Sections <ref> and <ref>.For convenience that notation is summarized in Table <ref>. Lowercase letters designate particular fixed values of interest. §.§ Target population and interaction networkThe target population is the set of people whose interactions and behaviors the researcher seeks to study.If we were studying whether peers affect which movies we choose to watch, the population of interest might be movie-goers.Before and during the experiment the target population generates some data, which we list here:* We observe N individuals from some target population, indexed by i. This might be a sample or it may be the entire finite population.* We observe pre-treatment covariates for the individuals: X_i.Commonly researchers collect demographic information such as gender, physical location, or age.Often it is also useful to measure pre-experimental behaviors that are similar to the outcome of interest.* We observe an interaction network between people in the population A_ij where i, j ∈ [1, …, N]; alternatively, this is a network G = (V, E).This interaction network determines an exposure model — which individuals we expect to potentially influence each other and with what intensity.* We observe when the population engages in some behavior of interest D_i.* We observe some outcome variable associated with each individual, Y_i.For instance the researcher might survey them to ask often they smoke.We will discuss outcome measurement in more depth in Section <ref>. Substantively, we care about the effect of the behavior D_j on the outcome Y_i in the population.The special case where D_i = Y_i can be termed in-kind peer effects and is frequently studied, but it is easy to envision cases where the peer behavior of interest is different from the outcome (e.g. my friend's studying habits, measured as D_j, affect my probability of applying to college, Y_i).Selecting the target population often involves tradeoffs between external validity and the ability to collect data about behaviors, outcomes of interest, and relevant social interactions — and intervene. Researchers have used the following three strategies to solve this recruitment problem.First, researchers have continued to recruit convenience samples. As with classic lab experiments in social influence <cit.>, these are often students from universities and colleges for studies. These samples can facilitate either construction of artificial networks or measuring the subjects' networks (with, e.g., surveys, asking them to log into Facebook, measuring co-location). The later strategy can be used to conduct “lab experiments in the field” as existing networks are combining with artificial choices and treatments <cit.>. The former strategy has been increasingly been used in combination with online labor markets (such as Amazon's Mechanical Turk), which has created an important new source of experimental subjects <cit.>. These individuals can be assigned to positions in networks by researchers or through economic games played by the subjects themselves <cit.>; of course, this may limit external validity.Second, the last decade has led to a dramatic increase in experiments that are conducted on online social networks or in collaboration with the companies that run online communication services.<cit.> constructed a Facebook application in order to gather social network information, introduce a treatment (presence of viral features), and measure the outcome of interest (adoption of the application).Other researchers have worked directly with Internet firms to conduct experiments.<cit.> and <cit.> conducted experiments by implementing them in partnership with Facebook (see Bond et al., this volume), while <cit.> partnered with a social news website to introduce an experimental change.Third, researchers in education, development, labor economics, and ecology have conducted ambitious field experiments in samples of schools or classrooms <cit.>, villages <cit.>, and herds of animals <cit.> for which networks can be measured.§.§.§ Measuring or constructing the interaction network We use the term “interaction network”, which is vague, because what is usually denoted by “social network” will often not be the causal network of interest.[Another related term is “exposure model” – a model that determines which subjects are exposed to which other subjects.]In most settings there is some specific type of interaction we hypothesize to transmit the behavior we care about.The most intuitive definition is that interaction is “i considers j to be her friend,” but, even when this can be operationalized, further consideration of a particular research question may lead to other reasonable choices:* i saw a story j posted on Facebook <cit.>* i is made aware that her friend j likes a product <cit.>* i lives with j in a dormitory for a year <cit.>* i lives in the same household as j <cit.>* i is in the same training class as j<cit.> The researcher hopes that the chosen network captures salient interactions for the influence process she expects.This definition can vary depending on the outcome behavior of interest. In the case of the <cit.>, who study educational outcomes, the interaction network is prolonged co-habitation, while in the case of <cit.>, who study clicks on ads, it is merely that a Facebook friend's name can appear next to an advertisement.A more prolonged, socially important interaction network can plausibly cause larger changes in subject behavior. In the former case, the the researchers can study changes in more important and ingrained behaviors like studying habits, while in the latter the researchers must study more proximate outcomes (clicks on ads).There are many different possibilities for measuring, eliciting, or directly constructing interaction networks.If the research setting is an articulated social network, (e.g. an online social network such as Facebook, Instagram, Twitter, or Pinterest) the researcher may use that network's definition (followers, friends, subscriptions).This approach is convenient but often not the precise interaction network of interest.Most people have online “friends” with whom they never interact in person, as well as “real life” friends who they have not articulated ties with online.Facebook, Instagram, and Twitter use algorithmic ranking to determine which content users see, meaning that a friend or follow relationship on those platforms may not necessarily imply content visibility.If the plausible mechanism of influence is offline, then using an online network might bias estimates of causal effects.A misspecification of the interaction network can, even with randomization, bias measurement of social effects.In digital settings, interaction networks may be constructed incrementally as people's interactions in the social system are logged (i.e. A_ij = 1 if i chatted with person j during the study). For instance in <cit.>, the interaction network is determined by Facebook users seeing advertisements during their browsing sessions.The salient interaction network is easily captured by logging which users see which ads.In addition, logging the interactions which have the potential for transmittingbehaviors can improve precision by omitting interactions with no potential to transmit influence. <cit.> could have used other definitions of the interaction network (e.g. Facebook friendship), but these would have yielded biased and/or higher variance estimates of effects.Like observational research [e.g., the US National Longitudinal Study of Adolescent Health (AddHealth) study <cit.>], much measurement of social networks for randomized experiments has involved asking subjects who their friends, kin, etc., are. The specific questions can be selected to elicit the possibly domain-specific network of interactions. For example, <cit.> asked heads of rural households to household heads to list five friends that they most frequently discuss farming and finance with, anticipating that this would be a relevant network for social influence in adoption of weather insurance and spillovers from their intervention. Such questions require being able to uniquely identify the named peers, which may be challenging in the presence of common names and/or limited literacy. <cit.> thus used a complete photographic census of the villages in which they planning to intervene. When the goal is to measure an objective fact about behavioral interactions, incentives for subjects to truthfully report their friends and tie-strength to researchers could be helpful. For example, <cit.> use a game in which individuals report how much time they spend with peers and paying them more money if this report matches the peer's report.Researchers can infer interaction networks from communication meta-data, especially when it covers enough time to precisely measure interaction rates and the communication medium (e.g. email) is likely to be the medium through which influence is transmitted. Influential observational research has measured networks by counting exchanges of emails <cit.> or instant messages <cit.>. Beyond allowing for constructing a binary network, directed behaviors between individuals predict self-reported tie strength <cit.>. In a a randomized experiment, these measures can then be used to estimate how spillovers <cit.> or social influence <cit.> vary by tie strength. Choices by researchers in inferring networks from communications data can be non-trivial and have a substantive impact on results <cit.>.Finally, studies can be designed to directly construct the interaction network for the subjects, a strategy which is enabled by running digital experiments even if they happen to be conducted synchronously in behavioral research labs <cit.>. For example, <cit.> randomly vary the networks on which Amazon Mechanical Turkers play a public goods game.Since creating the interaction network requires the researcher to intervene in the social system, we will discuss this strategy in more depth in Section <ref>.§.§.§ Extensions to this framework Thus far we have described a randomized experiment with a single time period of post-treatment observation and a single outcome of interest.The DAG in Figure <ref> does not allow for the subject's behavior to affect the peer's behavior, which in turn affects the subject's behavior. There are two simple extensions which may be useful and more realistic.First, we might study the outcome at different points in time (e.g. instead of D_i and Y_i we might observe D_i(t) and Y_i(t) where t denotes either discrete or continuous time.Time-dependent behavior is a challenging empirical setting because the researcher will often need to model how the interaction network varies across time, as well as how the individual behavior evolves over time <cit.>.The second extension is from a single peer behavior and outcome of interest to multiple behaviors and outcomes.We might observe a set of people make decisions about a collection of products, ads, content items, or behaviors, meaning we would measure D_ik and Y_ik, where k indexes the items.Multiple items present an important opportunity to observe social influence processes play out repeatedly in the same population of individuals across the same interaction network. Studies which measure effects across multiple behaviors might provide a more generalizable estimate of effects or allow the researcher to understand effect heterogeneity on other dimensions. As we discuss in Section <ref>, this may offer additional opportunities in analysis.§.§ Experimental treatmentsTreatments are the means by which the researcher intervenes in the social system.The space of treatments is often very limited based on cost and practical constraints, risks to subjects, and simply what changes a researcher can possibly apply in a social system.We will consider the researcher intervening by setting variables Z_j and W_ij, usually through some random assignment procedure. Note that we do not assume the researcher can directly change D_j and A_ij, as these variables are chosen by individuals and can often only be affected through the researcher-controlled instruments.The case where this is possible is the special case of perfect compliance, which is rare in field experiments.Instead, we posit a (potentially estimable) compliance model that produces D_j and A_ij and which may also include pre-treatment variables and random noise. This section focuses on defining these treatments; we defer their random assignment to Section <ref> below.§.§.§ Subject-level treatments A binary subject-level treatment is denoted by Z_j ∈{0, 1}, where Z_j = 0 by default, and where this treatment is expected to affect behavior such that D_j(z_j) = f_i(z_j, ϵ_j), with observed D_j = f_i(Z_j, ϵ_j). The direct effects of the treatment may sometimes be of interest (e.g., effects of a message on voter turnout), but the idea here is that Z_j functions as an encouragement or instrumental variable with respect to D_j, allowing interpretation of spillovers from treatment as social influence via D_j. Thus, researchers can create these treatments primarily for this purpose of detecting social influence. For the treatment to be effective as an instrument, we must believe that f_i is such that changing Z_j sometimes changes D_j; for example, perhaps D_j(z_j) = 1{α + β z_j + ϵ_j > 0} with β≠ 0, which can be tested. Many interventions (e.g., providing information, advertisements) cause only small changes changes in the behavior, making detecting downstream social influence difficult.The special case where D_j = Z_j is known as perfect compliance. Noncompliance may also be only one-sided, such that if Z_j = 1 then D_j = 1. Say we are interested in social influence in adoption of a paid upgrade of an music streaming service. We could, as do <cit.>, purchase the upgrade for a active users at random, thus producing one-sided, rather than two-sided, noncompliance (i.e. users could still purchase the upgrade on their own if we did not).[Of course, in such cases we may wonder whether D_j (i.e. having the upgrade) was really the behavior we were interested in. Perhaps so — if most of the effects of peers' upgrades on subjects would be via a single indicator on the peers' profiles that they had upgraded.] Two-sided noncompliance seems to be more common in the social sciences, particularly among the difficult-to-change behaviors which are often most interesting to study (e.g., health behaviors, costly product purchases).Experiments using subject-level treatments within groups (i.e., networks consisting of disconnected cliques) to detect and estimate social influence — sometimes called partial population experiments <cit.> — have been adopted in economics and political science <cit.>. These designs are based on the expectation that treating a fraction of subjects can induce detectable changes in the population of individuals connected to them. A smaller number of such experiments have been conducted in networks; these too have often relied on having a network multiple connected components (e.g., villages, schools) <cit.> with few exceptions <cit.>.Knowledge of the interaction network can be crucial for the success of subject-level treatments.If there is uncertainty about which peers may be affected by a subject's treatment, then detecting effects can become more burdensome from a statistical standpoint because omitting edges or including irrelevant ones adds additional random variation in estimation.§.§.§ Interaction-network treatments In an interaction-network treatment, the researcher intervenes by setting W_ij, which affects the interaction network of the subjects in the experiment; that is, A_ij(w_ij) = g_ij(w_ij, U_i, U_j, ν_ij), with observed A_ij = g_i(W_ij, U_i, U_j, ν_ij). As above, if A_ij is binary we may posit that A_ij(w_ij) = 1{γ + δ w_ij + ν_ij > 0} with δ≠ 0. Then particular edges may exist (δ > 0) or not (δ < 0) because of the treatment. In the edge-formation case of δ > 0, we have treatments such as suggesting that two people become friends or introducing them <cit.>. Not all suggested edges will form, but we expect that some will.Researchers sometimes define the interaction network such that there is perfect compliance.There are numerous examples of randomized group formation with ostensibly perfect compliance. <cit.> use a novel group randomization to understand how the constituents of groups affect ideation.<cit.> and <cit.> use random assignment of college roommates and squadrons in order to understand how these groups affect various learning and development outcomes.Note that the degree of “compliance” depends on how the network is defined.Although roommate assignment creates perfect compliance for the network of roommates, there is still two-sided noncompliance for the network of friendships.Encouraging edge removal, preventing formation, or attenuating interaction (δ < 0) can also be possible, if challenging in practice, and would rely on the researcher discouraging at least one type of interaction between individuals in the population.As an extreme example, researchers studying smoking cessation could ask subjects to delete phone contacts for any friend they believe might encourage them to continue smoking.In the context of online communication technologies, whether some binary treatment should be understood as encouraging or discouraging interaction is relative to an arbitrary and temporary status quo. For example, <cit.> analyze an intervention that modifies the display of i's posts to j, varying the salience of the user interface elements for commenting on the post. Perfect compliance, or at least one-sided noncompliance, can also occur when there is some exhaustive channel by which interaction occurs. For example, <cit.> randomize along which edges notification for their Facebook application are sent, thus defining an interaction network that is a random subset of the Facebook friendship network. Similarly, <cit.> randomize whether whether or not a friend appears as social context for an advertisement. We have elsewhere called these mechanism experimental designs since they randomize whether particular mechanisms for social influence are active <cit.>. §.§ Randomization strategyA randomization strategy ϕ specifies a probability distribution over treatment assignments; here π_ϕ(Z) or π_ϕ(W), where Z is the N-vector of subject-level treatments Z_i and W is the matrix of edge-level treatments W_ij. The marginal distribution is thus a function that maps a subject (j) or edge (ij) to probability of treatment.More advanced experiments might additionally allow this function to depend on pre-treatment covariates X_j or the existing interaction network A_ij. The specific form of the randomization determines what causal questions the experiment is capable of, or especially suitable for, answering.§.§.§ Implementing randomization In practice, researchers tend to implement randomization using deterministic cryptographic hash functions to generate pseudo-random variables with specified distributions <cit.>. PlanOut is a domain-specific language for specifying randomization strategies that is used at Facebook and several other companies.[The design of PlanOut is described in <cit.> and it is available from https://github.com/facebook/planouthttps://github.com/facebook/planout.]Using variable-specific crypographic salts, PlanOut provides functionality for independent random assignment for multiple experiments, multiple variables, and multiple types of units (e.g., users, clusters, items, edges). The determinism of the hash functions ensures that a random assignment is “persistent”, without requiring the assignments be stored; that is, the assignments can be computed online and statelessly, as subjects arrive. PlanOut code implementing the i.i.d. randomization we described in the previous paragraph as well as some more advanced randomizations are shown in Listing <ref>.[caption=Example PlanOut code for subject-level treatment assignment., label=lst:planout_iid]# i.i.d. random assignment smoking_program = uniformChoice(choices=[0,1], unit=subject_id);# block random assignment smoking_program = uniformChoice(choices=[0,1], unit=subject_group_id);# hierarchical block random assignment smoking_program_prob = randomFloat(min=0, max=1, unit=subject_group_id); smoking_program = bernoulliTrial(p=smoking_program_prob, unit=subject_id); §.§.§ Subject-level treatment randomizations Here we consider randomizations for subject-level treatments. Consider the simplest possible randomization for a subject-level treatment is independent and identically distributed (i.i.d) Bernoulli random variable: Z_j ∼(0.5).[Often the literature on randomized experiments <cit.> starts with a completely randomized design, in which some fixed number N_1 of the N subjects are assigned to treatment. However, in the case of large digital field experiments implemented as described in Section <ref>, this cannot easily be done in online (i.e. streaming) assignment without complications.] Say Z_j is assignment to a smoking prevention program. We hypothesize that the program will reduce how much people in the study smoke (i.e. D_j is lower in expectation when Z_j = 1), and are further interested in using this randomization to learn about social influence in smoking. In the context of disjoint groups (i.e. a network consisting of multiple disjoint cliques), we can think of this randomization as a partial population experiment <cit.>, in that some of the population is treated and we can study behavior of their peers. This design is analogous to marketing interventions which seek to exploit spillovers or network effects in demand by providing discounts or promotions to a small subset of consumers <cit.>.In order for our randomization to enable detecting and estimating social influence, we will generally need variation in the treatments of the peers of our subjects. While many measures of peer treatment can be used, we will illustrate the points in this section with the fraction of i's peers who are treated:T_i = ∑_j=1^N A̅_ij Z_j,where A̅_ij = A_ij / ∑_j=1^N A_ij an entry in the row-normalized adjacency matrix, with A̅_ij = 0 if ∑_j=1^N A_ij = 0.The i.i.d. subject-level assignment described above and shown in the third panel of Figure <ref> has a very important limitation: if a subject has a substantial number of peers, then there is a vanishingly small probability that they will all be assigned to treatment; for example, if subject i has 10 peers then (T_i = 1) = (∑_i=1^10 Z_j = 10) < .01.So we are unlikely to be able to use an experiment with this type of randomization to answer counterfactual questions about having all (or even a large percentage) of a person's friends participate in the program. For some asymptotic sequences with growing degree, this will mean the variance of sample means for units with, e.g., all treated peers diverges <cit.>.Thus, we will often want to consider other randomizations.At the opposite extreme, we could assign treatment at the level of groups or clusters. For instance if students are grouped by classrooms, we could do the smoking prevention assignment at the classroom-level.Let c(j) be the classroom for subject j. Then a group-level randomization would be to assign each group an i.i.d Bernoulli, P_c ∼(0.5) and assign each student her group's assignment, Z_j = P_c(j). If we think the classrooms are disjoint cliques, we might posit an interaction network that is a block-diagonal matrix, such that A_ij = 1{c(i) = c(j)}. Note that in the case of disjoint groups, such an “everyone or nobody” randomization abandons the partial population idea. This randomization can help answer questions about what will happen should we deploy the program to everyone, but it cannot answer questions about social influence and thus whether the program can be deployed more cost-effectively by treating a smaller proportion of students.We may be able to dramatically reduce smoking in a classroom by encouraging 25% of the students to not smoke.In this group randomization, we will never observe a classroom with any quantity other than 0% or 100% of the students treated; see the first panel of Figure <ref>.Intermediate designs between these two extremes use a hierarchical (or, in this case, two-stage) randomization to creates additional dispersion in the quantity of students per classroom assigned to the treatment, but also makes subject's own treatment and their peers not perfectly dependent. For example, we can first draw a random uniform variable per classroom, P_g ∼(0, 1), and then for each student, we draw a Bernoulli random variable with their group's probability, Z_j ∼(p_c(j)).[ With a small number of groups, we may want to use a completely randomized design, rather than independent draws of P_c <cit.> consider optimal two-stage randomizations in the context of disjoint groups, given the goal of estimating some particular direct or indirect effects. ] For some randomization ψ, we call it overdispersed because _ψ(T_i) > _iid(T_i); that is, it greater variability in the fraction of peers treated than from i.i.d. subject-level randomizations. An overdispersed randomization could be useful for selecting a number of students to treat per classroom, given some budget, that will minimize smoking because it can provide an estimate smoking behavior under different many levels of treatment.Block-diagonalnetworks (e.g., villages assumed to not interact) make overdispersed randomizations easy to implement. With more general networks, there are more design choices, and it can be difficult to generate arbitrary degrees of overdispersion in friend treatment assignment probabilities. We would like a randomization such that the distribution of T_i has certain properties; for example, one heuristic, is we should have that (T_i = k) > ε for all feasible fractions k given i's degree. Or we may aim to maximize (T_i = k) for k ∈{0, 1}. One recently popular way to do so is to partition the network into clusters using existing graph partitioning algorithms, and then proceed with the cluster-randomized design <cit.>. Given the structure of the network, there will still be edges between clusters (fourth panel in Figure <ref>). For example, say we use state-of-the art methods to partitioning the Facebook friendship network; with only 1,000 clusters, already over 40% of edges will be between clusters <cit.>. Not only is graph partitioning challenging in large networks, but standard min-cut objectives will often just be a heuristic: we would instead prefer to optimize bias or total error in estimation of particular quantities. To facilitate such optimization, one can further treat the clusters, or some other model fit to the network (e.g., a more general stochastic block model <cit.>), as an approximation to the observed network. Thus, <cit.> propose using optimal designs for approximations to the observed network. A final design possibility with subject-level treatment randomizations is that treatment assignment probabilities can depend on pre-treatment covariates X_i in order to increase precision. Blocking or pre-stratification exactly balances some covariates between treatments, rather than simply balancing them in expectation, thus reducing the variance in effect estimates is attributable to the random assignment of treatments causing covariate imbalance in small samples <cit.>.For instance, in a small sample it could make a large difference in estimates if a subject who is very active or who has many friends is assigned to treatment or not.State-of-the-art blocking methods allow improving balancing on high-dimensional covariates and lead to higher-precision estimates of treatment effects <cit.>. While usually large samples make blocking irrelevant because post-stratification or regression adjustment can provide similar precision gains <cit.>, use of graph cluster randomization again reduces the effective number of units being randomized, perhaps making blocking a relevant design consideration.Pre-treatment covariates can be used to target specific subjects who may have certain network positions or be likelier to cause social influence based on some hypothesis or prior analysis.If a researcher wanted to test a seeding strategy based on network position a reasonable design would be select a set of influential candidate subjects <cit.> and treat a random fraction of them while reserving some others as a control <cit.>.§.§.§ Interaction-level treatment randomizationTreatments defined at the level of individual edges allow for further choices in randomization. Because this design space is so large, we consider some notable examples.Historically, many examples of interaction-level treatments come from experiments in the formation of random groups. Here the interaction network is set in advance by the researcher or by some exogenous process.From a notational standpoint, these designs amount to setting W_ij = 1 for blocks of subjects to induce variation in A_ij and, in turn, the distribution of quantities such as the fraction of adopting peers, ∑_j=1^N D_j A̅_ij.An important aspect of this type of randomization is that the resulting groups must exhibit variance on D_j, the behavior of interest.For the same reason that i.i.d. assignment in subject-level treatments may not cause sufficient variation in peer exposures, large random groups are unlikely to be useful for identifying causal effects <cit.>. As with subject-level treatments above, it may be desirable to introduce overdispersion in group composition.The random group assignment designs generally leverage existing group formation policies. In the case of <cit.>, which exploits the fact that roommate assignments at Dartmouth college are conditionally randomly assigned (directly setting A_ij = 1 for the “is roommate” relation), we may even consider this a natural experiment.On the other hand, <cit.> introduce novel group formation policies for squadrons at the United States Air Force Academy; here squadrons are groups of roughly 30 that cadets are required to spend the majority of their time with.As a further refinement, random group formation can be performed dynamically to allow for repeated measurements of the same individuals as they change social contexts.<cit.> use such a randomization to measure how group interactions between entrepreneurs affects their ideation.Their approach allows them to not only measure how changing groups affects their outcome of interest, but allows for longitudinal measurements of individual outcomes as well.Without leveraging existing group formation policies, researchers may be limited to encouraging the formation edges that involve less prolonged contact. Several experiments have randomly assigned subjects to different graph structures whether in an artificial setting <cit.> or in the context of an online health-related service <cit.>. Here the experiment is generally conceptualized at the level of entire replications of a particular graph. Thus, the outcomes and analyses may be defined and conducted in aggregate rather than at the individual level.One can think of these designs as randomizing A directly and then observing some aggregate network outcome, which is slightly more complex than the framework we propose here.Other edge-level treatments are best understood as conditional on peer behaviors and a pre-treatment network. These include what we have called mechanism experimental designs, which work by randomizing whether a social signal is delivered via particular channel. Mechanism designs <cit.> are equipped to answer counterfactuals about how peer behavior would be affected in the amplification or attenuation of the influence channel of interest.[An additional refinement of the model we outline here is subjects may be connected via multiple overlapping networks, such as in-person vs online interactions, and an experiment may cause changes in some of those networks but not others.]For example, in <cit.>, the only peers eligible for the experiment are those who have already liked a page on Facebook (conditioning on D_j = 1) and the randomization (assigning W_ij as a Bernoulli random variable with perfect compliance for A_ij) determines whether this behavior will be displayed when the focal user sees an ad. <cit.> use another mechanism design in exploiting the fact that notifications in their Facebook application are delivered to a random set of the user's friends.If we believe that these notifications are the only mechanism through which a Facebook friend might adopt the application, this amounts to randomly amplifying values of A_ij for the friends who received the notifications, while leaving it un-amplified for the remaining Facebook friends that were collected when the user installed the application.Just because a treatment is defined at the level of edges not mean the randomization is i.i.d at the edge level. <cit.> select random subsets of edges involving the same subject. In the context of a treatment that encourages providing feedback (likes and comments on Facebook, in this case) to a specified directed edge, <cit.> compare different possible randomizations. One sender-clustered design would randomly assign vertices to an encouragement to give all of their peers more feedback. Another recipient-clustered design would randomly assign vertices to have all of their peers encouraged to give them feedback This latter design is used in <cit.>, as simulations suggest it will often have precision advantages. Finally, other designs could, like some of the designs we considered in the previous section, interpolate between i.i.d. assignment of edges and either of these clustered designs.§.§ Outcome measurementPerhaps an underrated requirement of randomized experiments is the ability to measure an outcome appropriate to the research question at hand.Sometimes researchers invest more time and expense in intervening with their treatment than in measuring the outcome.However, precise, valid, and complete measurement plays a large role in the success of randomized experiments. A simple example is that, if outcomes are measured with noise, the resulting estimates will be less precise. Even more problematic are cases where some outcomes are missing, either randomly or not.<cit.> show that matching ad exposures to conversions via cookies — where matching is random but plausibly independent of treatment status — results in a substantial loss of experimental power.Other experiments might rely on surveys or self-reports to measure outcomes, which yields either a biased measurement (e.g., social desirability) or a treatment effect estimate for only a biased sub-population (survey takers).<cit.>, who study social influence for comment quality in public discussions, can only measure comment quality improvements for the set of subjects who choose to write comments.<cit.> measure voter turnout by matching Facebook users to people in the state voter files, which is a noisy process (match rates were about 40%) that was limited to 13 states because of the expense of acquiring voter file data.Digital field experiments present some opportunities and also limitations for experimenters.Many important outcomes are potentially observable, such as clicks on ads <cit.>, sharing and production of user-generated content <cit.>, and adoption of apps (both free and paid) <cit.>.However, digital platforms create comprehensive logs of digital behaviors, which are perhaps not the only behaviors of theoretical interest.For instance, while <cit.> apply a reasonable text-analysis procedure to measure people's emotions at scale, it is debatable whether a change in emotion is adequately captured by the text they choose to share on Facebook <cit.>.The sheer volume of data produced on digital platforms is a signal of how trivial the actions they collect can be.Despite dramatic advances in observability of human behavior, it continues to be a central research challenge to measure important outcomes and join them to experimentally assigned treatments.§ ANALYZING RANDOMIZED EXPERIMENTSOne frequent consequence of having a well-designed randomized experiment is that the data analysis is then straightforward. While this is true to some degree in experiments about social influence in networks, estimation and inference can both be complicated by the network. Causal and statistical inference in networks remains an active research area, with contemporary contributions to basic problems such as laws of large numbers and asymptotic inference in networks <cit.>.In this section, we review methods for estimation and inference (e.g., hypothesis testing) for social influence in network experiments. The known randomization of subjects or edges to treatments provides a “reasoned basis” for inference <cit.> with minimal assumptions even when we only observe a network with a single giant component. We thus focus on Fisherian randomization inference, but briefly review other methods.As with the experimental design, the primary goal in analysis is learning about social influence. Ideally, this means learning about effects of D_j on Y_i or of A_ij on Y_i. It will often be more straightforward to simply detect any effects of Z_j or W_ij on Y_i. This is because (a) the experimenter sets these, but usually only affects D_j on Y_i indirectly and (b) in measuring D_j and A_ij, we may not capture all of the ways that our treatments can affect subjects. Thus, we can often take evidence about effects of Z_j or W_ij as evidence of social influence, without being about to denominate these effects in terms of peer behaviors. We start with this simpler case. §.§ Effects of randomized treatmentsIn this section, we consider how to conduct inference about effects of randomized treatments. We start by considering inference about spillovers in experiments where subjects are randomly assigned to subject-level treatments; that is, we are interested in questions about whether subjects' outcomes are affected by others' treatments. If we assume that others' treatment only affect an individual through others' behaviors (Figure <ref>), then these tests are also tests of social influence.§.§.§ Testing sharp null hypotheses about spillovers Consider the null model in which there is a direct effect of a subject's own treatment, but no effects of others' treatments, including those of peers. [No spillovers with constant direct effects]There exists some τ such that Y_i(z_i) = τ z_i + ξ_i for all z ∈^N and i ∈ V.Note that under this null hypothesis Y_i - τ Z_i does not vary under alternative treatment assignments. This null hypothesis is a composite of null hypotheses of the form: [No spillovers with constant direct effects, τ_0]Y_i(z_i) = τ_0 z_i + ξ_i for all z ∈^N and i ∈ V.Hypothesis <ref> is a sharp null hypothesis, which allows inferring all of a unit's potential outcomes from its single, observed potential outcome. We can thus use Fisherian randomization inference, in which we exploit our knowledge of the distribution of Z (which we or the experimenter chose), to test this null hypothesis. This is often implemented as a permutation test with a test statistic chosen to be sensitive to the kinds of deviations from the null that we expect. For example, consider a larger model that includes a linear effect of the fraction of treated peers:Y_i = τ Z_i + ρ∑_j = 1^N Z_j A̅_ij + ξ_iwhere A̅_ij is an entry in the row normalized adjacency matrix. A non-zero ρ would correspond to a particular violation of Hypothesis <ref>. The score statistic for ρ can be used as a test statistic <cit.>, as can many other test statistics.Algorithm <ref> tests Hypothesis <ref> using Fisherian randomization inference. To test Hypothesis <ref>, researchers would generally test many particular values of τ (e.g., in a grid, or through a search algorithm) and take the supremum.[Randomization inference for Hypothesis <ref>]Inputs: test statistic T(·, ·) : 𝕐^N ×{0, 1}^N that is a function of units’ residual outcomes and the treatment vector; posited direct effect τ_0. * Compute residual outcomes given τ_0, Ỹ := Y - τ_0 Z.* For every r ∈{1, ..., R} and some τ_0: * Draw a new treatment vector Z^* consistent with the original randomization.* Compute value of test statistic with observed outcomes and permuted treatment T_null, r := T(Ỹ, Z^*). * Compare observed and null test statistics, yieldingp-value(τ_0) = 1/R∑_r = 1^R 1{T(Ỹ, Z) > T_null, r}. We would then reject Hypothesis <ref> for small p-values, instead concluding subjects are affected by others' treatments. While randomization inference frequently makes use of permutation tests, the two are not identical. Fisherian randomization inference makes use of knowledge about the exact distribution of variables that were randomized to conduct exact causal inference for a finite population of units. Often (e.g. with a single completely randomized treatment vector) this can be approximated to arbitrary precision through permutation of the treatment vector, but need not be if the distribution over treatments is more complicated. Furthermore, permutation tests of social influence are often used without the justification they are afforded by randomization; that is, they are often used when other assumptions would be needed to make them exact in finite samples or even good asymptotic approximations. For example, <cit.> make additional, strong assumptions about non-influence processes to justify the use of a permutation test to detect influence in observational data.Even in the case of randomized experiments, particular permutation tests may not be readily justified by the randomization. Without explicitly considering the relevant sharp null hypothesis, it can be easy to make mistakes that make the resulting permutation test invalid. For example, <cit.> test for spillovers from a randomly assigned encouragement to vote in the 2010 U.S. elections. This was implemented as as a permutation test that implicitly assumed the absence of direct effects, even though <cit.> elsewhere rejected that null hypothesis. <cit.> show that such tests can have dramatically inflated Type I error rates (i.e. they too often reject the null hypothesis when it is true). §.§.§ Inference for the magnitude of spilloversSay we use Algorithm <ref> and reject Hypothesis <ref>. We may further wish to quantify the magnitude of these spillovers from treatment.These methods can also be used to construct acceptance regions for more complex positive hypotheses about the size of spillovers in the network. To do this, we can use a similar test but with Equation <ref> specifying a sharp null hypothesis given a choice of τ and ρ. We can, for example, use a test statistic that measures model fit (e.g., sum of squared residuals) <cit.> and determine a region of τ and ρ values that we do not reject (i.e. an acceptance region). With only these two parameters, grid search is often feasible, but other search algorithms can be used. For more on this topic, see <cit.> and <cit.>.The preceding methods require testing a sharp null hypothesis or a composite null consisting of a parametrically defined set of sharp nulls. In particular, we imposed the constant effects assumption that the direct effect of the treatment τ was common to all units. If direct effects are heterogeneous, these tests could reject the null even when there are no spillover effects of treatment. To partially address this concern, we could expand the null model to allow effects to be heterogeneous by observed subject covariates X_i; however, this would not allow for latent heterogeneity in direct effects. Outside the context of networks, we might be confident that, at least asymptotically, good choices of test statistics would result in tests that are not asymptotically sensitive to this heterogeneity <cit.>; however, we lack such asymptotic results for networks. In the next sections, we consider alternative methods that do not make use of these homogeneity assumptions. Nonetheless, the preceding methods may have some advantages in practice (e.g., greater power).§.§.§ Conditional randomization inference in networksHow can we use randomization inference to test for spillovers without specifying the form of direct effects? Consider a null hypothesis of no spillovers in the absence of assumptions about constant direct effects of treatment. [No spillovers]Y_i(z) = Y_i(z') for all i ∈ V, and all pairs of assignment vectors z, z′ ∈{0,1}^N such that z_i = z'_i.This hypothesis is not sharp because it does not specify how each subject would have behaved if its treatment were different. Rather, it posits levels sets of Y_i(·). It is possible to test such non-sharp null hypotheses by using conditional randomization inference — that is, by conditioning on functions of the treatment vector Z <cit.>.Here consider the basic case of testing Hypothesis <ref>. In particular, we can designate a subset of subjects as focal subjects for which we examine their outcomes and condition on their observed treatment assignment <cit.>. Note that, conditional on the focal subjects receiving the same treatment, Hypothesis <ref> is now sharp for those subjects. We can implement this test as follows.[Conditional randomization inference for Hypothesis 2]Inputs: set of focal units V_F ∈ V, test statistic T(·, ·) : 𝕐^|V_F|×{0, 1}^N that is a function of focal units’ outcomes and the treatment vector. * Draw permuted treatment vector Z^* such that all focal units get the same treatment as observed, Z^*_i = Z_i for all i ∈ V_F* Compute value of test statistic with observed outcomes and permuted treatment T(Y_V_F, Z^*)* Repeat 1 and 2 for R times, storing results as the R-vector T_null.* Compare observed and null test statistics, yieldingp-value = 1/R∑_r = 1^R 1{T(Y_V_F, Z) > T_null, r}.We would then reject Hypothesis <ref>, and thus the stronger Hypothesis <ref>, for small values of this p-value. This test has the correct Type I error rate without any assumptions about the model for direct effects.How should the focal subjects be selected? Any choice is valid (i.e. results in correct Type I error rates), but this choice can affect power. First, in some cases, this choice may be obvious because of the availability of outcome data. For example, when joining treatment and network data with a second data set with outcomes, a researcher may only observe outcomes for a small fraction of subjects, which could then be designated the focal subjects <cit.>. Second, theory or prior observations may suggest that some subject may not respond to social influence; it may be desirable to not include them as focal units. Finally, the network itself can be used to select focal subjects to improve power <cit.>.It is possible to apply similar approaches to testing for higher-order spillovers, testing for spillovers on a second network, and other hypotheses about spillovers. When the null hypothesis allows for, e.g., spillovers from immediate neighbors on a relatively dense network, these methods may lack sufficient power to be useful. We refer readers to <cit.> for details. §.§.§ Extension to edge-level treatmentsWe have focused on the case where subjects, rather than edges, are assigned to treatments; however, similar methods can be used when edges are assigned as long as either (a) a sharp null hypothesis can be posited or (b) a non-sharp null hypothesis implies level sets that can be conditioned on. §.§ Estimating effects of peer behaviorsThus far we have described inference about effects of other subjects' randomly assigned treatments, while often the substantive questions are about effects of other subjects' behaviors (i.e. social influence). As noted above, if we assume that a subject's outcome is only affected by a peers' treatments via their behaviors, then evidence for spillovers from treatment is evidence for social influence. However, we are often interesting in quantifying the size of this social influence by, e.g., estimating effects of D_j or A_ij on Y_i.We can proceed as before by considering the following sharp null hypothesis, which specifies how a subject's outcomes vary with its own treatment and peers' behaviors.[Constant direct effects and social influence, (τ_0, θ_0)] Y_i(z, d) = τ_0 z_i + θ_0 ∑_j = 1^N d_j A̅_ij + ξ_i,for all z ∈{0,1}^N, d ∈^N, and i ∈ v. According to Hypothesis <ref>, subjects are unaffected by others' treatments except as reflected in their neighbors behaviors D_j. This a complete mediation assumption or exclusion restriction and is encoded in Figure <ref>. Combined with Z having been randomized, this is sufficient for function of Z to be used as instrumental variables for social influence. Following <cit.>, we can then test Hypothesis <ref> by noting that it implies that Y_i(z, d) - τ_0 z_i - θ_0 ∑_j = 1^N d_j A̅_ij is invariant in z, and thus that Algorithm <ref> can be applied with this alternative residualization of the outcomes.§.§ Other methods of analysis There are some other methods available for statistical inference about spillovers and social influence with randomized experiments. Under monotonicity assumptions (i.e. that treating more subjects can only increase all subjects' potential outcomes) and with bounded outcomes, it is possible to construct confidence intervals for effects attributable to the observed treatment assignment <cit.>. Or under local interference assumptions (i.e. subjects are only affected by immediate peers' treatments) and bounded degree, it is possible to do conservative asymptotic inference <cit.>.In some cases the presence of replication is helpful by allowing for plausible independence assumptions. First, there may be observation of multiple plausibly independent behaviors on a single network. For example, <cit.> randomize a mechanism of social influence for many different subjects and brands. In their estimation and statistical inference, they assume that outcomes that do not have a common subject or brand are independent. They then use statistical methods that account for dependence of observations within brands and users <cit.>. Ignoring or not properly accounting for dependence in analyzing such experiments would increase the type I error rate.Second, some networks consist of multiple sizable connected components (e.g., villages, schools), rather than a single giant component. However, often the lack of edges between components is an artifact of how the network is measured. For example, <cit.> measure kinship and friendship relationship among rural villagers in Honduras, but edges between villages are not measured. On the other hand, <cit.> measure inter-village edges, but nonetheless only allow for within-village dependence when conducting statistical inference. Thus, independence remains a potentially strong assumption.§ INTERPRETATION AND ADDITIONAL ANALYSESThe simplest possible randomized experiment with a binary treatment (i.e. an “A/B test”) could be used to estimate as little as a single causal parameter of interest — the average treatment effect.In many cases researchers have found that this is an unsatisfying conclusion to a study, especially given the costs of designing, planning, and implementing[One should realistically add to this list of costs, the expected cost of failure.Researchers have not always succeeded in salvaging scientific knowledge from experiments and complexity of field experiments is associated with greater risk of unexpected problems.] randomized field experiments. Therefore it is common for empirical researchers to conduct more extensive analysis of experimental data or to use it as input to models or simulations.We have found that the results of field experiments, though exhibiting high internal and external validity, often motivate deeper questions about the underlying mechanism and alternative counterfactual questions that can be explored through modeling or simulation.In Section <ref>, we made assumptions about the structure of social influence and specified models or inferential procedures to detect or estimate it.In most of these experiments we are more interested in how the effect scales with number or proportion of friends who engage in a particular behavior.But beyond estimation of that response curve, there are two other broad types of questions that can be answered by experiments.The first is treatment effect heterogeneity — the subpopulations of products, people, or social connections where social influence is stronger or weaker.By fitting more complex models researchers can estimate heterogeneous treatment effects and these estimates can help suggest causal mechanisms or guide design of marketing efforts or public policies, analogous to finding predictors of positive response to clinical treatments in medicine.The second is understanding optimal policies by simulating alternative policies.Policy simulations can be used to extrapolate the results of randomized experiments to alternative policies which were never directly tested.They are most commonly used by economists who have a rich history of using structural[Here we mean structural in the sense of imposing economic “structure,” meaning that assumptions about human behavior derived from theory are imposed in the models.] models in order to measure the effects of potential policy changes. §.§ Heterogeneous treatment effects One obvious type of effect heterogeneity is what clinical researchers might call a dose-response function, which characterizes how effects tend to scale as the number of friends who are influencing the person varies <cit.>.<cit.> look at a slightly different dose-response function:when an influential social cue from a single peer is present, how does the effect size vary with the tie-strength of that individual?This heterogeneity is important to understand because it can inform advertising strategies.For instance, knowing that close friends are far more influential than random friends, we might design a marketing campaign to encourage people to share with a small number of select friends rather than many of them.Another common analysis is measuring how influence may be moderated by the demographic characteristics of the pair of people involved.For instance, <cit.> observe pairwise demographic attributes of the message sender and recipient and use this information to measure how the relative effectiveness of viral messages (Facebook notifications generated by app usage) varies as a means of identifying more influential or susceptible members of social networks.It is completely plausible that the average treatment effect can be zero, yet obscure significant positive and negative treatment effects for many large subgroups that happen to cancel out.<cit.> show that the presence of some people's identity cues causes their content to receive higher and lower ratings than when their content is rendered anonymously. A distribution of effects that contains both positive and negative values is plausible in many social environments with fixed resources, such as status, reputation, or attention.In all three of the aforementioned papers in this section, we would like to point out the effect heterogeneity does represent a “free” causal estimand.If the experiment is designed to measure the ATE (the average treatment effect over the population), the effect heterogeneity we measure is simply an association between certain subgroups of the experiment and differential effects.Researchers cannot make the claim that an intervention designed to move a subject from one subgroup to another would change their treatment effect.For instance in <cit.>, the experiment tests the effects of anonymization of a commenter's identity on ratings, but is unable to answer questions about what rating a person's content would receive if she had some alternate identity.This type of counterfactual question is precisely the type of treatment effect heterogeneity that would drive policy decisions in the social advertising space. Such a finding can only be measured by a more complicated experiment which randomizes which identity is presented among some set of choices – a much more difficult experiment to design and implement.We end this section with a note of warning about seeking results based on treatment effect heterogeneity.As researchers search dimensions by which treatment effects may exhibit differential effects, they may increase the rate of false discovery.Independently testing many heterogeneity on many possible dimensions, or for many subgroups of the experiment will invariably result in false positive results as one of the subgroups may be “lucky.”[See for example: <https://xkcd.com/882/>]There are reasonable methods to control this risk while still detecting interesting heterogeneity, see <cit.> for a detailed discussion and recent methodological development. §.§ Policy simulations Given the obvious importance of experiments for effective policy decisions, it is natural to ask for a policy recommendation at the conclusion of a study based on a randomized experiment.Often a policy recommendation is not directly recoverable from causal quantities of interest.For instance, the average treatment effect (ATE) might tell you that the treatment has a positive effect on some outcome on average, but it does not necessarily follow that everyone should receive the treatment.Treatments have costs which might need to be weighed and nature of social spillovers means that treating a friend of an individual can be a substitute or a complement for treating that individual directly.<cit.> report policy simulations, employing models containing economic structure based on assumptions about individual behavior in the presence of peer effects <cit.>.The key idea behind the policy simulation approach is that the experiment is used to estimate parameters of the model and then the model can be used to extrapolate the findings to more complex or interesting policies than those randomly set in the original data set. Policy simulations are often used in conjunction with natural experiment, where the researcher did not ex ante decide the most informative randomization and would like to answer some additional questions at the cost of imposing additional assumptions through a model.Another example of reporting policy simulations is <cit.>, which applies experimental estimates from <cit.> to form the basis of an optimal seeding strategy in networks.As a key feature, their experiment estimated the degree to which influence and susceptibility to influence were clustered in the network, which is an important feature for understanding diffusion processes.§ CONCLUSIONWe believe that valid causal inference is an important goal in the practice of social science.There is obvious utility in knowing causal structure — good policy decisions require that the policy-maker at least know the sign of a causal effect.But also from a purely scientific perspective, measurements which do not have causal interpretation lack usefulness and insight because they afford multiple explanations. Correlations are interesting, but they usually cannot uniquely identify an explanation for a social phenomenon.We admit there is perhaps a bit of experimental dogma present in the social sciences and it is often possible to satisfyingly answer questions through some combination of reasonable assumptions, models, and observational data.However, the realm of social influence is one where alternative explanations are difficult to rule out without some exogenous variation which can identify causal effects. <cit.> refers to this problematic aspect of human behavior as “high causal density.”In domains of high causal density, where there are highly dense causal graphs that can explain the observed associations in data we collect, credible causal claims require randomization, either by the researcher or by nature. We would like to thank Lada Adamic, Norberto Andrade, Eytan Bakshy, and George Berry for helpful discussions in preparing this review. spbasic | http://arxiv.org/abs/1709.09636v1 | {
"authors": [
"Sean J. Taylor",
"Dean Eckles"
],
"categories": [
"cs.SI",
"physics.soc-ph",
"stat.ME"
],
"primary_category": "cs.SI",
"published": "20170927172032",
"title": "Randomized experiments to detect and estimate social influence in networks"
} |
Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Waterloo Institute for Nanotechnology, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario, N2L 2Y5, Canada We investigate the validity of two common assumptions in the modelling ofsuperconducting circuits: first, that the superconducting qubits arepointlike, and second, that the UV behaviour of the transmission line is not relevant to the qubit dynamics. We show that in the experimentally accessible ultra-strong coupling regime and for short (but attainable) times, the use of an inaccurate cutoff model (such as sharp, or none at all) could introduce very significant inaccuracies in the model's predictions. Finite sizes and smooth cutoffs in superconducting circuits Eduardo Martín-Martínez December 30, 2023 =========================================================== § INTRODUCTION Implementations of superconducting circuits have made significant advances in recent years, especially those employed in quantum information processing. Among other things, superconducting circuits have provided a platform for the implementation and simulation of most of the features of the light-matter interaction of quantum optics. Remarkably, superconducting circuits can reach coupling regimes far beyond the strength with which atomic systems interact with the electromagnetic field <cit.>. Furthermore, superconducting qubits can be ultra-strongly coupled to quantum fields in a controlled, time-dependent way <cit.>. In these regimes of fast ultra-strong coupling of light and matter, some of the assumptions usually made in conventional quantum optics may need to be reassessed. For example, circuits with (finite-length) resonators have been analyzed using the Jaynes-Cummings model <cit.>, which often includes making the rotating wave approximation (RWA) and assuming that the qubit interacts with a single or just a few modes of the electromagnetic field <cit.>. Ultrastrong coupling research has reached regimes where so-called counter-rotating terms are no longer negligible—thus the RWA is no longer valid <cit.>. Also, many works in the ultra-strong coupling regime consider only a few modes of the electromagnetic field <cit.>. In contrast, studies of open resonators have delved into analysis of qubit-line coupling over a continuum of modes, and are in the early stages of considering ultrastrong regimes <cit.>. It is therefore relevant to investigate the regime of ultrastrong coupling to a continuum. In particular, this regime opens up analysis of the effect on qubit dynamics ofthe mode-dependent interaction strength introduced by the breakdown of Cooper pairs at high frequencies—the natural ultraviolet cutoffs of superconducting materials. Mode-dependence has been studied before in terms of the increase of coupling strength for higher resonant frequencies <cit.>, but it is relevant to also understandthe decay of coupling strength as the frequency of the modes increases.When frequency cutoffs have been implemented, they are often introduced without a first-principle justification <cit.>. In this light, one may wonder whether the specific model of cutoff leads to significant differences in predictions and thus whether it is justified to use an effective cutoff model without deriving it from the microscopic physics of the superconducting setup.Related to the existence of a UV cutoff is the finite size and nontrivial shape of the qubit. Both the UV cutoff and the physical shape of quantum systems that couple to thefield can have non-trivial effects on the predictions of light-matter interaction models and their causal behavior <cit.>. In the particular case of superconducting qubits, the shape of the circuit has not yet been fully considered in the literature as a contributor to observable dynamics. We will also analyze whether the shape of the qubit can contribute to its dynamics in a manner similar to the ultraviolet cutoff. As we will see in this paper, short-time dynamics are especially affected by the way in which effective UV cutoffs are implemented. This is particularly interesting since short-time qubit-line interactions are desired for the generation ofso-called single-photon states (i.e. states with low expectation values of the number operator) <cit.>. Additionally, theoretical predictions of states generated through short interaction times have very wide distributions in frequency space, so it is important to take into account how the interaction strength behaves at high frequencies.In this paper, we will study the system of a qubit and open transmission line in the ultrastrong coupling regime with non-adiabatic switching, taking into account the presence of a natural ultraviolet cutoff in the continuum of modes and the finite size of the qubit. We will use a simple but flexible model to assess the impact of specific forms and scales of qubit shape and ultraviolet cutoff. Specifically, we calculate the probability of vacuum excitation of the qubit and of spontaneous emission into the line, and investigate how different shapes, sizes, cutoff behaviours, and magnitudes of the cutoff scale affect these probabilities. We will show that in the experimentally attainable regime of short interaction, disregarding the behaviour of the cutoff can lead to significant inaccuracies in the model's predictions.The paper is organized as follows: in section II, we set up our model of a qubit coupled to an infinite 1+1D transmission line and use this to calculate the state of the qubit after an interaction. In section III, we detail the study of how significantly the factors of qubit shape and size and cutoff model and scale impact observable physics of the qubit for different switching times. We show that the cutoff model and scale have a significant impact, which is increased for short and non-adiabatic switchings.§ SETUP§.§ A simple model for superconducting qubits coupled to a transmission line Let us consider a very simple superconducting setup: a superconducting flux qubit coupled to an infinite 1+1D transmission line.We can identify the following three characteristic parameters of such a setup, summarized in table <ref>. The first is the energy gap Ω between the two energy levels of an idealized superconducting qubit. Typically, superconducting qubits have energy gaps between between 1 and 20 GHz <cit.>. The second relevant scale is the frequency at which microwave modes no longer propagate through the transmission line. Most common superconductive materials stop behaving as superconductors for electromagnetic frequencies of the order of 100 GHz. More specifically, the propagation of electromagnetic signals in superconducting transmission lines is affected by attenuation, which increases dramatically at frequencies corresponding to the superconducting gap. For aluminum, a material commonly used for superconducting transmission lines, the superconducting gap is 75 GHz <cit.>. Experiments in Ref. <cit.> showed renormalization of the transition frequency of a qubit coupled to an aluminum transmission line, consistent with a cutoff scale of 50 GHz chosen in this paper, which is close to the value of the superconducting gap for aluminum. In this fashion, there is a natural UV cutoff scale in the problem associated with dissipative effects at higher frequencies. We refer to that UV cutoff scale as ε_0.The third scale is the physical size of the qubit σ_0, typically on the order of 10 μm <cit.>. In previous literature, this scale is often neglected by considering the qubit to be pointlike in its coupling to the transmission line. However, the finite size of the physical qubit and the long range nature of the interaction of the qubit with the field in the line could, in principle, affect the dynamics of the qubit-line system. Hence, we will not assume a pointlike flux qubit by default in this work.As a theoretical model of the qubit-field interaction, we consider the following interaction Hamiltonian:Ĥ_int = λ χ(t) μ̂(t) ∫x F_σ(x) π̂(x,t). Here, λ is the coupling strength (related to the coupling constant of spin-boson models in appendix <ref>), χ(t) is the switching function (analyzed in detail below),is the qubit's internal degree of freedom, and F_σ(x) is the spatial distribution of the qubit (which for now we keep arbitrary—it will be made concrete later on in the paper) and the parameter σ is the characteristic lengthscale of the qubit. Note that F_σ(x) describes the `shape' of the qubit and σ its `size'—as mentioned above, we can assume these will be related to the physical size of the qubit σ_0, and could be affected by factors such as the range of the interaction (i.e. the nontrivial decay of the interaction away from the qubit). The observable of the field that the qubit couples to in this simplified model is the conjugate momentum π̂(x,t) of a scalar field ϕ̂.This corresponds to the qubit being coupled to the current in the 1D wire. We can expand π̂ in plane waves asπ̂(x,t) = 1/2√(π)∫_-∞^∞kC_ε(k)√(|k|)(â^†_k ^(|k|t-kx) + H.c.),where â^†_k and â_k are creation and annihilation operators for each mode k. This field plays the role of the Ohmic environment in the spin-boson model. See Appendix <ref> for the relationship of this model and the usual spin-boson coupling often used in the literature. Notice that we have included a weight function C_ε(k), which implements a cutoff scale ε. This is done to model the dissipation in higher frequency modes coming from the breakdown of superconductivity and radiative loss. One simple way to visualize the effect of this soft cutoff function is to think that the qubit `sees' the effective coupling strength to different modes monotonically decreasing as the mode frequency increases. Different microscopic models for the dissipation mechanisms would give different shapes and scales for this cutoff function. In a similar manner to F_σ(x), C_ε(k) models the `shape' of the UV cutoff model and ε sets the frequency scale at which the cutoff of higher modes is performed. Note that this cutoff function generalizes the common practice of simply considering a single (or a few) modes in the transmission line, in which case C_ε(x) would just be a compactly supported function with a sharp decay at ε.The switching function χ(t) describes the way we turn the interaction on and off. We will consider a compactly supported switching function (one can think of a switchable coupling for a finite time <cit.>). We use a smooth switching to avoid extreme non-adiabatic effects <cit.>. Additionally, we would like to consider a function that could feasibly be implemented experimentally. To describe such a function, we will employ two time parameters: a ramp-up time r—controlling how fast the interaction is switched on and off—and a duration time T—controlling how long the interaction strength remains at its maximum value. We assume a switching function of the following form:χ(t) =1/2 + 1/2cos (π/r (t + T/2) ) t ∈ [-T/2-r,-T/2)1 t ∈ [-T/2,T/2] 1/2 + 1/2cos (π/r (t - T/2) ) t ∈ (T/2,T/2+r] .The ramp-up and ramp-down periods are taken to be half periods of a cosine function. From an experimental perspective, any ramping function will have to be constructed digitally through approximately discrete intervals of constant voltage, then transformed into an analog signal through a filter. Thus an experimentally amenable ramping function is one which can be constructed accurately with reasonable long intervals of constant voltage and an achievable analog bandwidth. This is important in view of lower availability of high bandwidth pulse generators and difficulties in preserving pulse shapes due to distortion during transmission from the generator to the switchable coupler between the qubit and transmission line. The cosine ramping fits these requirements. In addition, this pulse shape is more easily amenable to analytic expressions for the transition probabilities than other possible shapes. The particular shape of the switching function is not as important as the presence of two scales, one controlling the adiabaticity of the switching (ramp-up/down time) and another the duration of the interaction. We expect the dependence of the transition probabilities on the ramp-up and ramp-down times to provide insight into experiments with other pulse shapes. Both the ramp-up time r and the interaction duration T are tunable and can, in principle, be as short as 0.01 ns in experiment. Recently, a new generation of pulse generators with a time resolution of the order of 10 ps and analog bandwidth of the order of 10 GHz were used for direct digital synthesis of pulses for control of flux <cit.> and transmon qubits <cit.>.In this paper, we will explore a range of both r and T to characterize the behaviour of the model in the short/long and non-adiabatic/adiabatic interaction regimes. Based on previous results on the light matter interaction <cit.>, we expect the shape of the qubit and the cutoff model to have a more significant effect in the non-adiabatic and short interaction timescales. §.§ Time evolution We use perturbation theory to calculate the state of the qubit after its interaction with the field as mediated by the Hamiltonian (<ref>) with the cosine trapezoidal switching function (<ref>). We begin with the field and qubit in their respective free ground states, |0⟩ and |g⟩: ρ̂_in = ρ̂_π,in⊗ρ̂_q,in= |0⟩⟨0|⊗|g⟩⟨g|.Note that we assume that the initial state of the qubit is the ground state for the calculation of vacuum excitation probability below. However, if we let the qubit energy gap Ω acquire negative values, these computations also give the probability of spontaneous decay to the ground state starting from the excited state for values Ω<0. In the interaction picture, the final state after the interaction ρ̂_out is related to the initial sate ρ̂_inthrough the time evolution operatorÛ =𝒯exp(∫_-∞^∞ t Ĥ_I(t)) =𝒯exp(∫_-T-r^T+r t Ĥ_I(t)),where 𝒯 represents the time-ordering operation. Note that χ(t) has compact support, so the limits of integration can be written equivalently as [-T-r,T+r] or (-∞,∞). The Dyson expansion associated with this time evolution operator is given byÛ = -∫_-∞^∞tĤ_I(t)_Û^(1)- ∫_-∞^∞t∫_-∞^tt'Ĥ_I(t)Ĥ_I(t')_Û^(2) + 𝒪(λ^3).Considering this, the qubit-field state after the interaction becomesρ̂_out = Ûρ̂_inÛ^†= ρ̂_in + ρ̂_out^(1) + ρ̂_out^(2) + 𝒪(λ^3),whereρ̂_out^(1) = Û^(1)ρ̂_in + ρ̂_inÛ^(1)†, ρ̂_out^(2) = Û^(1)ρ̂_inÛ^(1)† + Û^(2)ρ̂_in + ρ̂_inÛ^(2)†.The state of the qubit after the interaction is obtained by tracing out the field degrees of freedom from ρ̂_out:ρ̂_q,out = _π[ρ̂_in] = ρ̂_q,in + ρ̂_q,out^(1) + ρ̂_q,out^(2)+ 𝒪(λ^3).The full detail of these calculations can be found in Appendix <ref>. For the initial state (<ref>) the first order term vanishes and the second order correction to the initial state yieldsρ̂_q,out^(2)= -P_e |g⟩⟨g| + P_e|e⟩⟨e|.P_e is the leading order contribution to the probability of vacuum excitation of the qubit, and is given byP_e= λ^2/4π∫_-∞^∞kF̃_σ^2(k) C_ε^2(k) |k| ×∫_-∞^∞t∫_-∞^∞t'χ(t)χ(t') ^ (|k|+Ω) (t-t'),where F̃_σ(k) = ∫_-∞^∞xF_σ(x)^ k xis the Fourier transform of the qubit's spatial smearing.Although the spatial smearing F̃_σ(k) (associated with the physical shape, size, and interaction range of the qubit) and the cutoff function C_ε(k) (associated with the effective cutoff imposed by the dissipation for high frequencies) are very different in origin, they affect the excitation probability on equal footing. In view of this, we can regard the product F̃^2_σ(k) C^2_ε(k) as an effective form factor of the qubit.The t and t' integrals can be expressed in closed form. Substituting in the cosine trapezoidal switching function (<ref>), we get the following expression for P_e:P_e= λ^2/4π∫_-∞^∞kF̃_σ^2(k)C_ε^2(k)|k|×-π^4/2^-1/2 (k+Ω ) (2 r+T)/(k + Ω)^2 (π - r(k + Ω))^2 (π + r(k + Ω)^2 × (1+^ r (k+Ω ))(-1+^ (k+Ω ) (r+T))× [sin(1/2 (k+Ω ) (2 r+T))+sin(1/2 T (k+Ω ))].Notice that the probability of de-excitation of a qubit initialized in the excited state (i.e. spontaneous emission) is given by the same expression (<ref>) once we perform the substitution Ω→-Ω.§ RESULTS AND DISCUSSION We analyze the effects on the qubit dynamics of variations of shape and size of the qubit and of different cutoff models and scales. In particular, we will characterize the impact of the effective form factor of the qubit on the probability of vacuum excitation and spontaneous emission (<ref>) in terms of the interaction ramp-up time r and the interaction duration T.For the shape of the qubit we consider four different models—Gaussian, Lorentzian, quartic decay, and sharp decay—and a range of sizes for each. For the specific form of these shape functions, see table <ref>. As for the cutoff model, we also consider four different models—Gaussian, Lorentzian, exponential, and sharp—and a range of cutoff scales, including asymptotic behaviour. Notice that we assume that there is no cutoff weight for frequencies below the qubit gap. For the specific form of these UV-cutoff functions, see table <ref>.The analysis is conducted by calculating the relative difference, referred to as Δ_ab, between shape or cutoff models A and B over a range of sizes, cutoff scales, ramp-up times, and interaction times:Δ_ab = |P_e,a - P_e,b|/max( P_e,a, P_e,b).where A and B denote which models are being compared, e.g. the relative difference between the exponential and sharp cutoffs is denoted as Δ_es. The relative differences Δ_ab quantify just how much each model affects the probability of vacuum excitation or spontaneous emission, and in particular estimates the error generated by analyzing data using an incorrect model. Notice that, since Δ_ab is a relative difference, it is independent of the magnitude of the coupling strength λ. In the following subsections, we assess the effect on vacuum excitation and spontaneous emission of the components of the effective form factor (shape and cutoff model) and of the two time scales of the interaction (ramp-up time and duration). In section <ref>, we detail how the cutoff model dominates over the shape model in the effective form factor. In section <ref>, we show that the effect of the size is negligible, while the effect of the cutoff scale is significant. In section <ref>, we detail the dependence of the cutoff model on the length of the interaction time and the ramp-up time of the switching, and how this dependence differs between excitation and emission.§.§ Sensitivity to the shape model and cutoff modelIn this section we compare the influence of the cutoff model and the qubit's shape model on the probability of vacuum excitation and spontaneous emission over a range of sizes and cutoff scales, for fixed interaction time scales r=r_0 and T=T_0. We refer to results displayed in Fig. <ref> for probability of spontaneous emission; the results for vacuum excitation are similar.Remarkably, the difference effected by the choice of cutoff model is at least two orders of magnitude larger than that effected by the choice of shape for cutoff scales up to at least 20ε_0, as seen by comparing panels b and c of Fig. <ref>. Thus, for all cutoff scales within at least an order of magnitude of ε_0, the choice of cutoff model has a much larger impact on the qubit dynamics than the choice of shape. The cutoff model dominating over the shape model persists for a range of sizes σ within several orders of magnitude of the qubit's physical size σ_0, shown by comparison of panels a and d of Fig. <ref>.The dominance of the cutoff model is expected from Eq. (<ref>) for emission/excitation probability. The Fourier transform of the shape F̃_σ(k), as in Eq. (<ref>) and the cutoff model C_ε(k), as noted earlier, contribute equally to the k integrand. The scale of the cutoff is, however, about 4 orders of magnitude larger than that of the shape. We thus anticipate that changes to the cutoff model will have a much larger impact on qubit dynamics than changes to the shape model.Thus the effect of the shape function onthe qubit dynamics is negligible as compared with the effect of the cutoff model. Given this irrelevance of the qubit shape for the regime studied, for the rest of this investigation, we choose the shape to be Gaussian for purely aesthetic reasons.§.§ Sensitivity to size and cutoff scale In this subsection we focus on assessing the impact on the qubit dynamics of the physical size of the qubit σ and the cutoff scale ε. We again analyze the processes of spontaneous emission and vacuum excitation, when the finite time scales of the interaction are fixed. It can be seen in Fig. <ref>d that the relative difference between different cutoff models is effectively insensitive to the size of the qubit σ—in fact, a relative difference of just 10^-6 is seen between sizes of 0.1σ_0 and 10σ_0. Thus, as long as the size of the qubit as seen by the transmission line σ is within a couple of orders of magnitude of the physical size of the qubit σ_0, the specific value will have a negligible effect on the dynamics. In view of the fact that the effective form factor of the qubit is largely independent of σ for a wide range of values, we will set σ=σ_0 from now on.In contrast, Fig. <ref>c shows high sensitivity to the cutoff scale—a relative difference of 0.7 is seen between cutoff scales of 0.1ε_0 and 10ε_0.Note that the relative difference between pairs of cutoff models Δ_AB decreases as ε increases. This is of course as expected, i.e. as the cutoff scale goes to infinity, the probabilities calculated with different cutoff models converge to the cutoff-free value. Observe here that P_e with an exponential cutoff is slower to converge than the other models considered. Unlike the Gaussian or the Lorentzian models, the exponential cutoff function suppresses the lower frequencies at a higher rate than the Gaussian and Lorentzian cutoffs (which present inflection points at ε+Ω) or the sharp cutoff. From this set of observations we can conclude that for the two scales of the effective form factor, the specific value of the size is irrelevant to qubit dynamics in comparison to the cutoff scale.We can thus say that, for the parameter regimes in table <ref>, the effective form factor of the qubit is dominated by the cutoff model and cutoff scale, and that the effects of the shape model and size are negligible. Intuition for this can be drawn from the difference of several orders of magnitude of the cutoff scale ε_0∼ 5Ω and the physical size of the qubit σ_0∼10^-4Ω^-1. This corroborates the current practice of treating the shape of the qubit as negligible. §.§ Sensitivity to cutoff models as a function of the switching time scalesWe have now established that for typical superconducting qubit setups in the USC regime (see table <ref>), the cutoff model and cutoff scale dominate the effective form factor of a qubit interacting with a superconducting transmission line. As discussed above, superconducting qubits and transmission lines cannot support arbitrarily high frequency modes. This can be traced back to the break down of superconductivity (effecting dissipation in the transmission line) at frequencies above the superconducting gap. The cutoff scale can thus be established, but the exact functional form of the UV cutoff function in a realistic scenario is complex to obtain from first principles, as it involves the complicated interplay of electrodynamics and quasi-particle physics. The question then arises when we make predictions as to how much we should care about the particular way in which the transmission line loses its ability to trap higher frequency modes. In other words, will an experiment be sensitive to the particular shape of the cutoff function or just its scale? How much do the microscopics of the superconductor impact the outcome of experiments? Does it really matter if the effective coupling strength decays exponentially with the mode frequency or with any other shape?To answer these questions, we are going to assess the impact of the shape of the cutoff function on superconducting experiments with fast finite-time switchable coupling. For this, we will take ε=ε_0 as a given quantity and explore only the effect of cutoff model. More concretely, we will analyze how the cutoff function influences the probability of vacuum excitation and spontaneous emission for different switching times. Recall that the switching is described by (<ref>), with two very different time parameters: the ramp-up time r, describing the adiabaticity of the switching process, and the interaction time T, describing the length of constant interaction time between switching on and switching off.The relative difference in the transition probability (both for spontaneous emission and vacuum excitation) between pairs of cutoff models is plotted as a function of r and T in Fig. <ref> and <ref>. Here we note a difference in behaviour between spontaneous emission and vacuum excitation processes: in emission, the ramp-up and constant interaction time have similar impacts, while in excitation, the ramp-up time is much more important than the constant interaction time. In Fig. <ref>, we see that in vacuum excitation, if the switching is sudden (r≲Ω^-1), having the interaction switched on for a long time T≫Ω^-1 does not remove the model dependence of the prediction. In other words, the impact of the cutoff model on excitation is dictated by the switching speed: whether the prediction is sensitive to the particular cutoff model is not dependent on how long the interaction is on, but rather on how suddenly the interaction was switched. In particular, for values of r≈Ω^-1, the relative difference becomes very significant even for large values of the interaction duration T (of the order of 15%-40% for the parameter values in table <ref>). Indeed we see that in the adiabatic limit r≫Ω^-1, the prediction is insensitive to the particular cutoff model. If we were trying to anticipate the behaviour of a superconducting qubit coupled to a transmission line in the adiabatic regime we could use any model that facilitates calculations without having to worry about the microscopic mechanism of the dissipation process. In comparison, in either the adiabatic or long interaction regimes, the dependence of predictions of spontaneous emission on the cutoff model is negligible. It is only for adiabatic and short interactions that the microscopic mechanisms of dissipation of high frequency modes makes a difference in the predictions.A few experimental aspects have to be considered for the implementation of fast switching times in superconducting circuits. The fastest arbitrary waveform generators can output switching waveforms with ramp-up and ramp-down times of the order of 20 ps. Other types of pulse generators have even shorter rise/fall times, but lack control of the pulse shape. Propagation of the pulses from room temperature equipment to a tunable coupler between the qubit and the transmission line is affected by distortion, due to dispersion and reflection of pulses at interconnects. Experiments in Ref. <cit.> indicated that pulses could be transmitted between a fast arbitrary generator and a flux qubit with less than 100 ps of edge distortion. Reduction of pulse distortion effects significantly below 100 ps may be possible with significant investments in customized setups. Finally, one needs to consider that coupling elements are usually assumed to work in the adiabatic regime. These coupling elements usually have transition frequencies of the order of tens of GHz and therefore their design has to be optimized to reduce non-adiabatic transitions.We would like to highlight that the values of r and T for the observation of relevant non-adiabatic dependence of the qubit response on the effective shape of the qubit (and therefore the specific form of the cutoff function) are well within reach of current superconducting qubit technology. A poor choice of cutoff model in these regimes can introduce errors in the theoretical analysis of the setup that are comparable to or larger than other sources of errors that are not neglected in previous analysis. § CONCLUSION In this work, we have introduced a simple model for an ideal superconducting circuit consisting of a flux qubit coupled to an infinite 1+1D transmission line. We specifically address the physics of ultrastrong coupling of a qubit to a continuum of modes by accounting for the mode dependence of superconducting behaviour, the finite size of the qubit, and the long-range interaction between the qubit and transmission line.We have investigated the effect of the shape model, size, frequency dependence of the coupling strength (cutoff model), and cutoff scale on the probability of the qubit undergoing spontaneous emission and vacuum excitation. We have determined that the cutoff scale and model are the dominant factors in determining the effective form factor of the qubit, while the physical size and shape of the qubit do not make a noticeable difference in comparison. This is consistent with the common practice of not considering the shape of qubits as an important factor in superconducting circuit models.Concretely, we have analyzed the effect of the coupling strength decaying with frequency (i.e. a cutoff model) in experiments with finite-time switching of the interaction. We have found that, if the switching process is fast (short-lived) and non-adiabatic (rapidly-switched), the cutoff model has a very significant effect on the probability of both vacuum excitation and spontaneous emission. This effect can be comparable to or larger than other sources of error in the experiment. The relative difference in observable quantities between several cutoff models can even be of the order of 10% in experimentally attainable regimes. For long interaction times, the probability of spontaneous emission becomes insensitive to the cutoff model. However, for vacuum excitation, the large differences can remain for arbitrarily long times if the switching ramp-up is non-adiabatic (i.e., the maximum interaction strength is reached in times comparable to the inverse of the qubit gap, which again is experimentally feasible <cit.>). We conclude therefore that assuming a particular cutoff model (e.g. ignoring all frequencies of the field above a specified cutoff) without being careful about the specific way in which this UV cutoff is implemented may lead to inaccurate predictions with these kinds of models with fast switchings and, depending of the particular process studied, for long evolution times.To end with a bit of a philosophical note, it is perhaps interesting to think of our result in terms of relational ontology: in this work we have arrived to an effective form factor in the qubit-line interaction which constitutes a shape that emerges from the particularities of the interaction. In other words, the shape of the qubit cannot be determined just with an individual description of the qubit itself. Rather, this shape belongs neither to the qubit nor to the line but to the both of them in interaction with each other, constituting a property that becomes evident and relevant in and through interactions between the relevant quantum systems. For further discussion on how properties of quantum systems are difficult to individualize, see, for instance, <cit.>.§ RELATION OF COUPLING CONSTANT TO THE SPIN-BOSON MODEL In this appendix we relate the coupling constant λ of our model (<ref>) to that of the spin-boson model α_SB, as used in Ref. <cit.>. We start by considering the interaction Hamiltonian Ĥ_int in the Schrödinger picture given in the Supplementary Information of Ref. <cit.>:Ĥ_int = σ̂_x ∑_k g_k (b̂_k + b̂^†_k),where σ̂_x is the qubit Pauli operator expressed in the energy eigenbasis and we sum over all bosonic modes. b̂_k and g_k are respectively the annihilation operator and the coupling constant for mode k. In the language of the spin-boson model, the field, which isacting as the environment to the qubit, is characterized by the spectral density functionJ(ω)= 2π/ħ^2∑_k g^2_kδ(ω-ω_k),as in Eq. (35) of the SI of <cit.>, where ω_k = c k is the frequency of mode k, with c the speed of light. In 1+1D, the spectral density of the electromagnetic field is Ohmic, that is, the power is proportional to the frequency:J(ω)=πωα_SBEquating spectral densities (<ref>) and (<ref>) givesg_k = ħ/√(2)√(α_SB ω_kΔω_k)where Δω_k is the frequency difference between neighbouring modes. Writing this expression for g_k into Eq. (<ref>) givesĤ_int = σ̂_x ∑_k Δω_k ħ/√(2)√(α_SBω_k)(b̂_k/Δω_k + b̂^†_k/Δω_k).Taking the continuum limit of Δω_k→ ω_k allows us to writeĤ_int = σ̂_x ħ/√(2)∫_-∞^∞ω_k√(α_SBω_k)(â_k + â^†_k).where the â_k are new annihilation operators, now defined over a continuum of modes k. Now, we can directly compare this with our Hamiltonian (<ref>), recalling a few facts: our Hamiltonian is written in the interaction picture; our qubit's monopole moment μ̂(t) is given by σ̂_x; ω_k = c k; and in the body of this paper, we have taken ħ=c=1. We can thus write the coupling constant λ of Eq. (<ref>) in terms of quantities of the spin-boson model (<ref>) asλ = √(2π α_SB).We end this appendix by noting that in Ref. <cit.>, the spin-boson coupling parameter α_SB reached values of order 1, indicating that the coupling constant λ can also reach values of order 1.§ CALCULATION OF THE STATE In this appendix we detail the calculation of the state of the qubit as in equation (<ref>) after interaction with the field as described by the Hamiltonian (<ref>). We treat the interaction Hamiltonian as a perturbation to the qubit free dynamics, using the Dyson expansion (<ref>) to calculate the first and second order evolution operators Û^(1) and Û^(2). The state of the qubit after the interaction is given by tracing out the field from the state ρ̂_out as in equation (<ref>):ρ̂_q,out = _π[ρ̂_in] = ρ̂_q,in + ρ̂_q,out^(1) + ρ̂_q,out^(2)+ 𝒪(λ^3).If the field's initial state is the vacuum, the one-point function_π ( [ρ̂_π,0,π̂(x,t)])=0. This means that the leading order contribution to the qubit dynamics will be given by terms of order 𝒪(λ^2). We can thus begin by writing out the second order term in full:ρ̂_q,out^(2) = - λ^2 (∫_-∞^∞t∫_-∞^∞t'χ(t)χ(t') μ̂(t)ρ̂_q,0μ̂(t').∫x∫x'F_σ(x)F_σ(x') π̂(x,t)00π̂(x',t') +∫_-∞^∞t∫_-∞^tt'χ(t)χ(t')μ̂(t)μ̂(t')ρ̂_q,0.∫x∫x'F_σ(x)F_σ(x') π̂(x,t)π̂(x',t')00 +H.c.).Using the smeared Wightman function W_σ[t,t'], we can rewrite the above:ρ̂_q,out^(2) = - λ^2 ∫_-∞^∞t∫_-∞^∞t'χ(t)χ(t') μ̂(t)ρ̂_q,0μ̂(t')W_σ[t',t] -λ^2∫_-∞^∞t∫_-∞^tt'χ(t)χ(t')μ̂(t)μ̂(t')ρ̂_q,0W_σ[t,t'] +H.c.,whereW_σ[t,t'] =∫x∫x'F_σ(x)F_σ(x') [ π̂(x,t)|0⟩⟨0|π̂(x',t') ].Since the state of the field is initially pure, we can rewrite the term [ π̂(x,t)|0⟩⟨0|π̂(x',t') ] as the expectation value of π̂(x,t)π̂(x',t') on the state |0⟩:W_σ[t, t'] =1/4π∫x∫x' F_σ(x)F_σ(x') ∫k∫k' C_ε(k) C_ε(k') √(|k|)√(|k'|)⟨0|(â_k^†^(|k|t-kx) - H.c.) (â_k'^†^(|k'|t'-k'x') - H.c.) |0⟩.Using that , we arrive at:W_σ[t,t']=1/4π∫k∫x F_σ(x)e^ kx∫x' F_σ(x')e^- kx' C_ε^2(k) |k| ^-|k|(t-t').We can write this more concisely asW_σ[t,t']= 1/4π∫kF̃_σ^2(k) C_ε^2(k) |k| ^- |k|(t-t'),where F̃_σ (k) is the Fourier transform of the spatial distribution, i.e., F̃_σ(k) = ∫xF_σ(x)^ k x. Note also that our choice of real symmetric smearing means that F̃_σ(k)=F̃_σ(-k).We can now write (<ref>) in a more compact fashion:ρ̂_q,out^(2) =∫_-∞^∞t∫_-∞^∞t'χ(t) χ(t') W_σ[t',t] ^Ω(t-t')|e⟩⟨e| -2∫_-∞^∞t∫_-∞^tt'χ(t) χ(t') Re[W_σ[t,t']]cos(Ω(t-t'))|g⟩⟨g|.Substituting (<ref>) in, we finally obtain an expression for the final state of the qubit that is general in choice of shape, cutoff, and switching functions, as in equations (<ref>) and (<ref>):ρ̂_q,out^(2) = 1/4π∫kF̃_σ^2(k) C_ε^2(k) |k| ( ∫_-∞^∞t∫_-∞^∞t'χ(t) χ(t'). ^(|k|+Ω)(t-t')|e⟩⟨e| - 2 ∫_-∞^∞t∫_-∞^tt'χ(t) χ(t') cos(-|k|(t-t'))cos(Ω(t-t'))|g⟩⟨g|). | http://arxiv.org/abs/1709.09684v1 | {
"authors": [
"Emma McKay",
"Adrian Lupascu",
"Eduardo Martin-Martinez"
],
"categories": [
"quant-ph",
"cond-mat.supr-con"
],
"primary_category": "quant-ph",
"published": "20170927181203",
"title": "Finite sizes and smooth cutoffs in superconducting circuits"
} |
𝐔𝐇̋𝐒𝐁𝐀𝐈𝐅𝐜̧𝐓𝐃𝐌𝐕constØ𝒪𝒩Re Imsech|#⟩1| #1 ⟩⟨#|1⟨ #1 |⟨#|1⟩#2⟨ #1 | #2 ⟩rms æeåacolor cmyk 1 0.5 0 0color cmyk 0 0.81 1. 0.60color cmyk 1. 0 0 0color cmyk0 0.10 0.84 0color cmyk 1. 0 1. 0color cmyk 0 1. 1. 0color cmyk 0 0 0 1.color cmyk 0 0.8 0.5 0Dipartimento di Fisica, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, I-20133 Milano, [email protected] a seminal work, S.A.R. Horsley and collaborators [S.A.R. Horsleyet al., Nature Photon.9, 436 (2015)] have shown that, in the framework of non-Hermitian extensions of the Schrödinger and Helmholtz equations, a localized complex scattering potential with spatial distributions of the real and imaginary parts related to one another by the spatial Kramers-Kronig relations are reflectionless and even invisible under certain conditions. Here we consider the scattering properties ofKramers-Kronig potentials for the discrete version of the Schrödinger equation, which generally describes wave transport on a lattice. Contrary to the continuous Schrödinger equation, on a lattice a stationary Kramers-Kronig potential is reflective. However, it is shown that a slow drift can make the potential invisible under certain conditions.Kramers-Kronig potentials for the discrete Schrödinger equation Stefano Longhi^*=============================================================== § INTRODUCTIONReflection is an ubiquitous phenomenon of wave physics which is found both in classical and quantum systems <cit.>. Reflection of electromagnetic (optical) waves in dielectric media with sharp refractive index changes and scattering of non-relativistic particles from a quantum potential provide important examples of wave reflection which share strong similarities <cit.>. However, it is known since long time <cit.> that reflection can be avoided in special classes of scattering potentials, the so-called reflectionless potentials <cit.>. Recently, wave reflection and scattering from complex potentials in non-Hermitian systems has sparked a great interest with the prediction of intriguing physics forbidden in ordinary Hermitian models, such as asymmetric scattering and unidirectional or bidirectional invisibility of the potential <cit.>. In a seminal paper, S.A.R. Horsleyand collaborators have introduced the class of Kramers-Kronig complex potentials <cit.>,in which the spatial profiles of the real and imaginary parts of the potentials are related one another by a Hilbert transform. The properties of such newly discovered potentials, i.e. unidirectional or bidirectional transparency, invisibility and some sublets related to the slow decay of the potentials,have been theoreticallyinvestigated in a couple of subsequent works <cit.>, with recent attempts to experimentally realize such a kind of complex potentials <cit.>. In all previous studies, wave propagation was formulated in the framework of the Helmholtz or the stationary Schrödinger equations, which are suited to describe scattering phenomena of waves in continuous systems. However, in several physical contexts, such as in quantum or classical transport on a lattice <cit.> or in quantum mechanical models with discretized space <cit.>, wave transport is better described by the discrete version of the Schrödinger equation. Like for the continuous Schrödinger equation, reflectionless potentials can be constructed for the discrete Schrödinger equation as well <cit.>, for example usingthe discrete version of supersymmetry or the Darboux transformation. Such previous works could not find any substantial different behavior of supersymmetric-synthesizedscatteringless potentials when space is discretized. However, the continuous and discrete versions of the Schrödinger equation may show distinctly different behaviors, which arise mainly for the limited energy band imposed by the lattice as opposed to parabolic dispersion curve in the continuous limit.In this work we consider Kramers-Kronig potentials for the discrete version of the Schrödinger equation and highlight some very distinct features of wave scattering on a lattice as compared to the continuous Schrödinger equation. While in the latter case a Kramers-Kronig potential is unidirectionally or bidirectionally reflectionless, a stationary Kramers-Kronig potential on a lattice is reflective, i.e. discretization of space breaks the reflectionless property of the Kramers-Kronig potentials. However, we show that a class of slowly drifting Kramers-Kronig potentials on a lattice can become invisible. Our results disclose a very distinct scattering behavior of Kramers-Kronig potentials in continuous and discreteSchrödinger equation models, and are expected to stimulate further theoretical and experimental investigations of such an important class of recently discovered complex potentials.§ WAVE REFLECTION FROM A MOVING POTENTIAL ON A LATTICE§.§ Drifting potential on a lattice: Basic equations We consider wave reflection from a drifting potential on a one-dimensional lattice, which is described by the discrete Schrödinger equation for the wave amplitude ψ(x,t)<cit.> i ∂ψ/∂ t= -2 κcos (a p̂_x) ψ+ V(x+vt) ψi.e.i ∂ψ/∂ t= -κ [ψ(x+a,t)+ψ(x-a,t) ]+ V(x+vt) ψ,where T̂=-2 κcos(a p̂_x)=- κ [exp(a ∂_x)+exp(-a ∂_x)] is the kinetic energy operator, a is the lattice period, x is the spatial variable defined on the discrete sites x=na (n=0, ±1 , ±2, ...), p̂_x=-i ∂_x is the momentum operator, V(x) is the scattering potential and v is the drift velocity. The parameter κ entering in the kinetic energy operator is the hopping rate which determines the width of the tight-binding lattice band. The dispersion relation of the lattice band is sinusoidal and given by E(q)=-2 κcos(qa), where q is the Bloch wave number. The continuous limit is obtained for a small lattice period a after setting cos( a p̂_x) ≃ 1-(a^2/2) p̂_x^2 in Eq.(1). In this limit the discreteness of space is lost and one obtains the continuous Schrödinger equationi ∂ψ/∂ t=- κ a^2 ∂^2 ψ/∂ x^2-2κψ+V(x+vt) ψwith the usual parabolic dispersion relation E(q)=-2κ+ κ a^2q^2 of the kinetic energy term.The scattering potential V(x) is assumed to vanish as x →±∞ sufficiently fast so as the asymptotic solutions to Eqs.(2) and (3) far from the scattering potential are plane waves. To study the scattering problem, it is convenient to write Eq.(2) in the reference frame of the drifting potential via the Galileian transformationX=x+vt ,T=t. This yields the transformed equationi ∂ψ/∂ T=- κ [ ψ(X+a,T)+ψ(X-a,T)]+V(X) ψ -iv ∂ψ/∂ X which differs from Eq.(2) owing to the drift term on the right hand side of Eq.(5). Note that, after the Galileian transformation (4), the continuous limit of the Schrödinger equation [Eq.(3)] takes the form i ∂ψ/∂ T=-2 κψ -κ a^2 ∂^2 ψ/∂ X^2+V(X) ψ-i v ∂ψ/∂ X which again differs from the original equation because of a drift term [the last term on the right hand side of Eq.(6)]. In such a continuous limit, the drift term can be removed via a gauge transformation and the continuous Schrödinger equation is thus invariant under a Galileian transformation. In fact, after the gauge transformation ψ(X,T)=ϕ(X,T) exp(-i β T+i γ X) with γ=-v/(2 κ a^2) and β=-v^2/(4κ a^2), one can readily show that ϕ(X,T) satisfies Eq.(6) but without the drift term on the right hand side.This result is basically due to the fact that the continuous Schrödinger equation is a non-relativistic wave equation, and it is therefore invariant under a Galileian transformation <cit.>. Such an invariance ensures that the scattering properties of the potential V(x) are not changed when it drifts at a uniform speed v: in the laboratory reference frame (x,t), the main effect of the moving potential is a Doppler shift of the frequency of the scattered (reflected) wave. However, for the discrete Schrödinger equation (5) in the moving reference frame the drift tern can not be removed via a gauge transformation, i.e. the discrete Schrödinger equation is not invariant under a Galileian transformation. This result basically stems from the discrete translational symmetry of the lattice, so that in the reference frame (X,T) the scattering potential is at rest however the lattice is drifting in the opposite direction. A major impact of the breakdown of Galileian invariance for the discrete Schrödinger equation is that the scattering properties of a potential V(x) on a lattice are modified when the potential drifts, as we are going to show in the following analysis. §.§ Reflection and transmission coefficients in the moving reference frame Let us first consider the case of a vanishing scattering potential V(X)=0. Then in the moving reference frame the scattering solutions to Eq.(5) are plane waves ψ(X,T) ∝exp(iqX-iET) with Bloch wave number q and energy E=E(q) defined by the dispersion relation E(q)=-2 κcos (qa)+qv and group velocity v_g(q)=∂ E/∂ q=2 a κsin (qa)+v. A typical behavior of the dispersion curve E=E(q) is shown in Fig.1 for increasing values of the drift velocity v. Note that, in the moving reference frame, the energy dispersion curveacquires a linear ramp term qv which breaks the periodicity of E(q).Let us now consider a scattering potential V(X) which vanishes sufficiently fast as |X| →∞ so that the scattering solutions to Eq.(5) with energy E are asymptotically plane waves.To study the scattering problem, for the sake of definiteness we will assume v>0and willconsider a forward-propagating plane wave with Bloch wave number q_0 and positive group velocity v_g(q_0)>0 (left incidence side), however the analysis can be readily extended to the v<0 case or to the right incidence side. Note that the limit of a non-driftingpotential is obtained by letting v=0. Since in the moving reference frame (X,T) the scattering potential V(X) is at rest, scattering of a plane wave with defined energy E_0 is elastic, i.e. it conserves the energy, and the solution to Eq.(5) corresponding to an incoming plane wave from the left side with wave number q_0 is then of the form ψ(X,T)=f(X) exp(-i E_0 T), where E_0=E(q_0) and f(X) satisfies the stationary differential-difference equation E_0 f(X)=-κ [f(X+a)+f(X-a)]+V(X)f(X)-iv df/dX with the asymptotic behavior f(X)∼{[ exp(i q_0 X)+ ∑_α r_α(q_0) exp(iQ_α X)X → - ∞;∑_β t_β(q_0) exp(i q_β X) X →∞ ]. In Eq.(10), the wave numbers Q_α and q_β are defined as the real roots of the equation -2 κcos (qa)+vq=E_0 with v_g(q_β) ≥ 0 and v_g(Q_α)<0; see Fig.1(b). They correspond to the wave numbers of reflected and transmitted plane waves with the same energy E_0 than the incident wave, r_α and t_β being the reflection and transmission amplitudes, respectively. Note that, for β=0, q_β=q_0 is precisely the wave number of the incident wave. The number of the roots Q_α and q_β depends sensitively on the drift velocity v, and increases as v → 0, as schematically shown in Fig.1 <cit.>. For a drifting potential with a speed v larger than the critical velocity v_c ≡ 2 κ a, {Q_α} is empty, whereas {q_β} is composed solely by the wave number q_0 of incident wave [Fig.1(c)]: this means that elastic scattering forbids wave reflection from any potential <cit.>. Here we focus our analysis to a slowly drifting potential v< v_c, for which elastic scattering permits wave reflection. § SCATTERING FROM A KRAMERS-KRONIG POTENTIAL ON A LATTICEUnlike for the continuous Schrödinger equation, a Kramers-Kronig potential at rest on a lattice is not reflectionless. The main physical reason of such a result is schematically illustratedin Fig.2 and can be explained as follows. Let us consider a plane wave with wave number q_0, corresponding to a positive group velocity (progressive wave) which is scattered off by a Kramers-Kronig potential V(x) which is holomorphic, for the sake of definiteness, in the upper half complex plane Im(x) ≥ 0. The analyticity of the potential in the half complex plane ensures that its Fourier spectrum V̂(k)=∫ dx V(x) exp(-ikx) vanishes for any k <0, i.e. it is composed solely by positive wave numbers, depicted by the solid thin arrows in Fig.2. Therefore, at any scattering order the scattered waves have wave numbers which can not be smaller than q_0. In the continuous limit, the Schrödinger equation shows a parabolic energy dispersion curve [Fig.2(a)], meaning that all scattered waves have a positive group velocity, i.e. reflection is cancelled. Conversely, for the discrete Schrödinger equation [Fig.2(b)] the energy dispersion curve is sinusoidal, so that scattered waves with a wave number larger than q_0 may correspond to a negative group velocity, i.e. reflection is allowed.Figure 3 shows, as an example, reflection of a Gaussian wave packet from a stationary Kramers-Kronig potential on a lattice as obtained by numerical simulations of Eq.(1), for both left and right incidence sides, in the (x,t) laboratory reference frame. The numerical method of integration is described at the end of the section. The scattering potential used in the simulations is given by V(x)=V_0(x) exp (i Ω x) with V_0(x)=V_0 / (x/a+i α)^2 and with parameter values V_0 / κ=i, Ω=10/a and α=0.3. Note that for both left and right incidence sides the potential is not reflectionless. The main result of the present work thatwe are going to demonstrate is that, under certain conditions, a class of Kramers-Kronig potentials which are reflective at rest become reflectionless (and even invisible) when drifting on the lattice. As discussed in the previous section, such a result stems from the fact that the discrete Schrödinger equation is not invariant under a Galileian transformation, so that the scattering properties of a potential on a lattice change with the drift velocity of the potential. Precisely, we can prove the following general theorem, which states a sufficient condition for a slowly-drifting Kramers-Kronig potential to be invisible:Let V(X) be a Kramers-Kronig potential of the form V(X)=V_0(X) exp(i Ω X), with V_0(X) holomorphic in the Im(X) ≥ 0 half complex plane, drifting on a lattice with a speed v smaller than the critical speed v_c= 2 κ a. Then for Ω≥ 4 κ /v the potential is bidirectionally invisible <cit.>.To prove the theorem, we follow a procedure similar to the one illustrated in Refs.<cit.> and based on the complex spatial displacement method. Let us consider, for the sake of definiteness, a progressive wave incident from the left side, so that the asymptotic form of the scattered solution is given by Eq.(10).Since V(X) is holomorphic in the half complex plane Im(X) ≥ 0, the solution to Eq.(9) can be analytically prolonged from the real X axis into such a half plane. In particular, let us indicate by f(ξ, δ)=f(X=ξ+i δ) the solution to Eq.(9) on the horizontal line Γ defined by the parametric equation X=ξ+i δ, with fixed δ>0 and -∞ < ξ < ∞, and with the asymptotic form defined by Eq.(10) as δ→ 0^+. The main idea of the complex spatial displacement method is to find suitable connection relations between reflection and transmission amplitudes of scattered waves on the real X axis, i.e. for δ=0, and on the line Γ, i.e. for δ>0. Since for δ→∞ the scattering potential V(X) vanishes, the reflection and transmission amplitudes of scattered waves on the line Γ can be readily determined by perturbative methods for δ large. The connection formulas can then be used to compute reflection and transmission amplitudes of the original problem, i.e. on the real axis δ=0. For δ >0, the asymptotic form of f(ξ, δ) as ξ→±∞ is given byf(ξ, δ)∼{[ A(δ) [ exp(i q_0 ξ)+ ∑_α r_α(q_0,δ ) exp(iQ_αξ) ];ξ→ - ∞; A(δ) ∑_β t_β(q_0,δ) exp(i q_βξ); ξ→∞ ]. with A(δ=0^+)=1. As in Eq.(10), in Eq.(11) r_α(q_0,δ ) and t_β(q_0,δ) are the reflection and transmission amplitudes of scattered waves on the line Γ, which reduce tor_α(q_0) and t_β(q_0) entering in Eq.(10) in the δ→ 0^+ limit. Since f(X) is an analytic function of X=ξ+i δ, the following relation holds ∂ f/∂δ=i ∂ f/∂ξ.Using Eqs.(11) and (12), it readily follows that A(δ)= exp(-q_0 δ). Moreover, the following connection formulas for reflection and transmission amplitudes on the real X axis and on the line Γ are foundr_α(q_0,0) = r_α(q_0,δ) exp [-(q_0-Q_α) δ] t_β(q_0,0) = t_β(q_0,δ) exp [-(q_0-q_β) δ] . For δ→∞, the potential V(X= ξ+i δ) vanishes and the order of magnitude of r_α(q_0,δ),t_β(q_0,δ) can be estimated by first-order Born approximation <cit.>. As shown in the Appendix, r_α(q_0,δ) → 0 and t_β(q_0,δ) → 0 (β≠ 0) at least like ∼exp(-δΩ), whereas t_0(q_0,δ) → 1. Provided that the condition Ω≥4 κ / v is met, Ω is always larger than any difference |q_0-q_β| and |q_0-Q_α|. Therefore from Eqs.(13) and (14) one obtains r_α(q_0,0)=0, t_β(q_0,0)=δ_β,0, which means that the scattering potential V(X) is invisible for left incidence side. A similar proof can be done assuming a wave incident from the right side, i.e. the potential V(X) is bidirectionally invisible. We checked the bidirectional invisibility of moving Kramers-Kronig potentials by direct numerical simulations of the discrete Schrödinger equation (1) in the laboratory reference frame (x,t). By letting x=na and c_n(t)=ψ(x=na,t), the differential-difference equation (2) is equivalent to the following set of linear coupled equations for the complex amplitudes c_n(t) on the latticei dc_n/dt=-κ(c_n+1+c_n-1)+V(na+vt) c_nwith time-dependent coefficients. The coupled equations (15) are numerically solved using an accurate variable-step fourth-order Runge-Kutta method assuming open boundary conditions. The lattice size, i.e. number of lattice sites, has been set large enough (typically -100 ≤ n ≤ 100) to avoid edge effects. As an example, Figs.4 and 5 show numerical results of bidirectional invisibility for a propagating Gaussian wave packet scattered off by the same Kramers-Kronig potential as in Fig.3, but when the potential drifts on the lattice with a velocity v= ± 0.4a κ. The figures depict the temporal evolution of the wave packet amplitude |ψ(x,t)|, for either left and right incidence sides, and compare the wave packet distributions after the scattering process with the ones observed in the absence of the scattering potential. The coincidence of the distributions is the clear signature that the drifting Kramers-Kronig potential is invisible, while it is reflective at rest.§ CONCLUSIONS AND DISCUSSIONWave scattering from complex potentials in the framework of non-Hermitian extensions of the Schrödinger or Helmholtz equations has received a great and increasing interest in the past few years, with the discovery of intriguing physics forbidden in ordinary Hermitian scattering problems, such as asymmetric reflection and unidirectional or bidirectional invisibility of the potential. An important class of complex potentials, which do no reflect waves from one or both incidence sides, in provided by so-called Kramers-Kronig potentials <cit.>, in which the real and imaginary spatial profiles of the potential are related one to another by a Hilbert transform. Most of recent studies focused on wave scattering from non-Hermitian potentials in continuous wave equations, however in several physical systems wave transport is better described by discrete wave equations. A paradigmatic equation describing discrete wave transport is provided by the discrete Schrödinger equation, which is encountered in models of quantum or classical transport on a lattice or in quantum mechanical models with discretized space. In this work we have shown that discretization of space and breaking of the continuous translational spatial invariance deeply change the scattering properties of Kramers-Kronig potentials on a lattice. In particular, the physical mechanism that prevents wave reflection of a Kramers-Kronig potential in the continuous Schrödinger equation breaks down when scattering occurs on a lattice with discrete translational invariance. Therefore, a Kramers-Kronig potential on a lattice is rather generally a reflective potential. However, we have shown that if the potential slowly drifts on the lattice, under certain conditions it can become bidirectionally invisible. Our study sheds new light into the important and broad field of wave scattering in non-Hermitian physical models and highlights important distinctive features of wave scattering in discrete versus continuous wave equations. In particular, we revealed that breakdown of Galileian invariance in discrete wave equations can enable a reflective potential to become reflectionless when drifting on the lattice. Physically, Kramers-Kronig potentials on a lattice and their reflection properties could be implemented in optics using arrays of evanescently-coupled optical waveguides or chains of microring resonators with tailored gain and loss profiles <cit.>. For example, it is known that spatial light propagation along the longitudinal z axis in a lattice of coupled dielectric optical waveguides is described by coupled-mode equations analogous to Eq.(15), in which time t is replaced by the spatial coordinate z and n is the waveguide number <cit.>. The real and imaginary parts of the potential V can be tailored by controlling, along the propagation distance z, the effective mode index of the waveguides, i.e. propagation constant offset of the waveguide mode and optical amplification/attenuation. If optical amplification (gain) is not available, one can resort to a purely dissipative (lossy) structure <cit.>. For example, using waveguide arrays written in a glass with the femtosecond laser writing technique <cit.> an effective propagation constant mismatch can be introduced by varying the writing speed of the focused laser beam in the glass <cit.>, whereas selective optical losses can be obtained by patterning a selective layer of absorptive material on the top of the array or by suitable bending of waveguides <cit.>. Static and moving potentials are readily obtained by manufacturing different waveguide arrays with straight or transversely tilted perturbation V of the effective mode index. Excitation of the array by a tilted Gaussian beam and monitoring its propagation along the z axis using fluorescence imaging methods <cit.> enables one to visualize the wave packet dynamics in the different regimes. It is envisaged that our results could stimulate further theoretical and experimental investigations on discrete wave transport and scattering by non-Hermitian potentials. Optical waveguide arrays could provide an experimentally accessible laboratory tool for the observation of the scattering properties of spatial Kramers-Kronig potentials on a lattice. On the theoretical side, the analysis could be extended to a two-dimensional lattice, in which the band structure and transport are known to be deeply modified by synthetic gauge fields. In principle, in a two-dimensional lattice the scattering properties of non-Hermitian potentials could be controlled by synthetic gauge fields, which is not feasible in continuous wave scattering settings. The interplay of non-Hermitian potential scattering and gauge fields could be a subject of future research. § REFLECTION AND TRANSMISSION AMPLITUDES OF THE SPATIALLY DISPLACED POTENTIALLet us indicate by G(ξ) the potential V(X)=V_0(X) exp(i Ω X) on the line Γ of the upper complex plane, i.e. for X= ξ+i δ, with δ>0 large enough and -∞ < ξ < ∞. The displacement δ on the imaginary axis of the potential pushes all the possible singular behavior ofV_0(X) further down into the lower complex plane, simultaneously reducing its magnitude. In particular, since G(ξ)=G_0(ξ) exp(i Ωξ -δΩ) with G_0(ξ) ≡ V_0(ξ+i δ), for δ→∞G(ξ) is exponentially small, uniformly over the entire line Γ, of order smaller than ∼exp( -δΩ). Therefore, the weak scattering introduced by the vanishingly potential G(ξ) can be computed by first-order (Born) approximation <cit.>. The solution to Eq.(9) on the line Γ, corresponding to an incident plane wave from the left side of wave number q_0, is thus given byf(ξ,δ) = A(δ) [ f^(0)(ξ)+ϕ (ξ)]where f^(0)(ξ)= exp(iq_0 ξ) is the unperturbed incident plane wave of amplitude A(δ) and ϕ(ξ) is a small correction introduced by the weak scattering potential G(ξ). At first-order (Born) approximation, ϕ(ξ) is the solution of the forced linear equationE_0 ϕ(ξ)+κ [ϕ(ξ+a)+ϕ(ξ-a)]+iv dϕ/dξ= G_0(ξ) exp(-δΩ) exp(i Ωξ+i q_0 ξ).The solution to Eq.(A2) is formally given by ϕ(ξ)=1/2 π(∫_-∞^∞ dk Ĝ_0(k-q_0-Ω) exp(i k ξ )/E_0+2 κcos (ka)-vk) exp(-δΩ)where Ĝ_0(k)=∫ dξ G_0(ξ) exp(i k ξ) is the Fourier transform of the potential G_0(ξ). Note that, since G_0(ξ)=V_0(ξ+i δ) and V_0(X) is holomorphic for Im(X) ≥ 0, Ĝ_0(k) vanishes for k<0, so that the integral on the right hand side of Eq.(A3) is actually extended from k=q_0+ Ω to k= ∞. In such a range, the function under the sign of the integral is not singular, since its poles q_β and Q_α lie in the range k<Ω+q_0. For δ→∞, Eq.(A3) thus shows that ϕ(ξ) is exponentially vanishing, at least like ∼exp(-δΩ). A comparison of Eqs.(11) and (A1) indicates that r_α(q_0, δ) and t_β(q_0, δ) (β≠ 0) are vanishing at least like ∼exp(-δΩ) as δ→∞, whereas t_0(q_0, δ) → 1.31r1 J. 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Rev. A89, 012709 (2014). r15 A. Mostafazadeh, J. Phys. A49, 445302 (2016). r16 M. Kulishov, H. F. Jones, and B. Kress, Opt. Express23, 9347 (2015). r17 M. Sarisaman, Phys. Rev. A 95, 013806 (2017). r18 S.A.R. Horsley, M. Artoni, andG.C. La Rocca, Nature Photon. 9, 436 (2015). r19 S. Longhi, EPL112, 64001 (2015). r20 S.A.R. Horsley, C.G. King, and T.G. Philbin, J. Opt.18, 044016 (2016). r21 S. Longhi, Opt. Lett.41, 3727 (2016). r22T.G. Philbin, J. Opt.18, 01LT01 (2016).r23 S.A.R. Horsley, M. Artoni, and G.C. La Rocca, Phys. Rev. A94, 063810 (2016).r24 C.G. King, S.A.R. Horsley, and T.G. Philbin, Phys. Rev. Lett.118, 163201 (2017). r25 S.A.R. Horsley and S. Longhi, Am. J. Phys.85, 439 (2017). r26 C. G. King, S. Horsley, and T. Philbin, J. Opt.19, 085603 (2017). r26bis S.A.R. Horsley and S. Longhi, Phys. Rev. A96, 023841 (2017). r27 W. Jiang, Y. Ma, J. Yuan, G. Yin, W. Wu, and S. He, Laser & Photon. Rev.11, 1600253 (2017). r27bis D. Ye, C. Cao, T. Zhou, J. Huangfu, G. Zheng, and L. Ran, Nat. Commun.8, 51 (2017). r28 E. Papp andC. Micu, Low-dimensional Nanoscale Systems on Discrete Spaces (World Scientific, New Jersey, 2007). r29 D. C. Mattis, Rev. Mod. Phys.58, 361 (1986). r30 T.B. Boykin, Eur. J. Phys.25, 503 (2004).r31 D. N. Christodoulides, F. Lederer, and Y. Silberberg,Nature424, 817 (2003). r32I.L. Garanovich, S. Longhi, A.A. Sukhorukov, and Y.S. Kivshar, Phys. Rep.518, 1 (2012).r33 F.T. Wall, Proc. Nat. Acad. Sci.83, 5360 (1986).r34 F.T. Wall, Proc. Nat. Acad. Sci.83, 5753 (1986).r35 S. Odake and R. Sasaki, J. Phys. A44, 353001 (2011).r36 V. Spiridonov and A. Zhedanov,Ann. Phys.237, 126 (1995). r37 S. Longhi, Phys. Rev. A82, 032111 (2010).r38 A. A. Sukhorukov, Opt. Lett. 35, 989 (2010). r39A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, and A. A. Sukhorukov, Phys. Rev. Lett.106, 193903 (2011).r40 S. Odake and R. Sasaki, J. Phys. A48, 115204 (2015). r41 S. Longhi, Phys. Rev. A77, 035802 (2008). r42 S. Longhi, Int. J. Mod. Phys. B30, 1650189 (2016). r43 G. Rosen, Am. J. Phys.40, 683 (1972). note0 For a potential at rest, i.e. for v → 0, the roots of the equation E(q_0)=E_0 are infinite and given q_β=q_0+2 βπ/a and Q_α=-q_0+2 απ /a (α, β=0 ± 1, ± 2, ...); see Fig.1(a). However, since on the lattice space x is discretized by the lattice period a, such waves are degenerate and there is actually only one transmitted wave with wave number q_0 and one reflected wave with wave number Q_0=-q_0. The degeneracy is generally lift when v is nonvanishing, as shown in Fig.1(b). note The implications of such an interesting result, i.e. absence of wave scattering for fast-moving potentials on a lattice, is discussed elsewhere [S. Longhi, Opt. Lett.42, 3229 (2017)].note2 Likewise, the invisibility property holds for a drifting Kramers-Kronig potential of the form V(X)=V_0(X) exp(-i Ω X), with V_0(X) holomorphic in the Im(X) ≤ 0 half complex plane and Ω≥ 4 κ / v.refer1 S. Longhi, Laser & Photon. Rev.3, 243 (2009).refer3 A. Guo, G.J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett.103, 093902 (2009).refer4 A. Szameit and S. Nolte, J. Phys. B43, 163001 (2010).refer5 F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, A. Tünnermann, and S Longhi, Phys. Rev. Lett.101, 143602 (2008).refer6 T. Eichelkraut, S. Weimann, S. Stützer, S. Nolte, and A. Szameit, Opt. Lett.39, 6831 (2014).refer7 G. Corrielli, A. Crespi, G. Della Valle, S. Longhi, and R. Osellame, Nat. Commun.4, 1555 82013). | http://arxiv.org/abs/1709.09344v1 | {
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"published": "20170927055625",
"title": "Kramers-Kronig potentials for the discrete Schrödinger equation"
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pckpcMpcM_⊙ ⊙ .3ex<-.75em1ex∼ .3ex>-.75em1ex∼ M_⊙ L_⊙ L_⊙ km/scm^2/scm^2/gdis | http://arxiv.org/abs/1709.09717v3 | {
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"Manoj Kaplinghat",
"Zhen Pan",
"Hai-Bo Yu"
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"published": "20170927195959",
"title": "Signatures of Self-Interacting Dark Matter in the Matter Power Spectrum and the CMB"
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A Bimodal Network Approach to Model Topic DynamicsLuigi Di Caro ^1,3, Marco Guerzoni ^1,2,Massimiliano Nuccio ^1,2, Giovanni Siragusa ^1,3^1 Despina, Big Data Lab ^2 Department of Economics and Statistics "Cognetti de Martiis", University of Turin, Italy ^3 Department of Computer Science, University, of Turin, Italy =====================================================================================================================================================================================================================================================================================================To the memory of Vladimir Igorevich Arnold§ INTRODUCTION A partial order ongroups of contact diffeomorphisms wasintroduced in <cit.> as a contact analog of Hofer's geometry for groups of Hamiltonian diffeomorphisms of symplectic manifolds. In this paper webegin studying theremnants of this order on the conjugacy classes of contactomorphisms. Our main interestin this paper are non-compact contact manifolds, and more specifically a special class of non-compact contact manifolds which we call convex at infinity, see Section <ref> below. While orderability problems for closed manifolds have obvious answers on the level of Lie algebra of contact vector fields, the situation for non-compact manifolds is quite subtle already on the Lie algebra level. Problems of this kind naturally arisen in connection with constructions of contact structures in <cit.>. The goal of the paper is to illustrate the arising phenomenaon a restricted class of examples, leaving a more general study, both in the Lie algebra and the group cases, to our forthcoming paper <cit.>. §.§ Groups of contactomorphisms and their Lie algebras Let (U,ξ) be a coorientable noncompactcontact manifold.We fix a contact form α for ξ and denote by R its Reeb vector field. LetG:=_c(U,ξ)be the identity component of the group of contactomorphisms of (U,ξ) with compact support. The Lie algebraof G, which consists of compactly supported contact vector fields,can be identified with the space C^∞_c(U) of smooth functions with compact support by associating to each function K its contact vector fieldY_K=KR + Z_K, Z_K∈ξ, (dK+i_Z_Kdα)|_ξ=0.Note thatdK(Z_K)=0, L_Y_Kα=dK(R)α.Conversely, given a contact vector field Y its contact Hamiltonian is defined by the formulaK(x)=α(Y(x)), x∈ U. Let us stress the pointthat to identify the Lie algebrawith the function space C^∞_c(U) one needs to fix a contact form.The adjoint action of ψ∈ G on K∈ computes to_ψ K = (c_ψ K)∘ψ^-1,where c_ψ:U→ is the positive function satisfying ψ^*α = c_ψα. The Lie bracket onis given by{H,K} = dK(X_H)-KdH(R).The Lie algebra carries a canonical partial order defined by H≤ K if H(x)≤ K(x) for all x∈ M, which is -invariant by equation (<ref>).§.§ Dominating positive cones Denote by ^≥0 the cone in the Lie algebra ≅ C^∞_c(U) consisting of nonnegative functions. A subcone ⊂^≥0∖ 0 is called a dominating (positive) cone if the following hold: *is -invariant;*is relatively open in ^≥0∖{0}; * for each H∈ there exists K∈ with H≤ K;* for all H∈, K∈ there exist t>0 and g∈ G such that t_gH≤ K;* for each H∈^≥0∖{0} there exist g_1,…,g_k∈ G such that _g_1H+…+_g_kH∈. Property (v) is not needed in this paper, but will become relevant for the discussion of partial orders on contactomorphism groups in <cit.>. Clearly, if the manifold U is closed then the only dominating cone inis the cone ^>0 consisting of everywhere positive functions. If U is not closed, then a dominating cone in generalneed not exist. For instance, S^1×^2 with the contact form dt+1/2(xdy-ydx) does not admit any dominating cone because if (H)⊃ S^1× D_R and (K)⊂ S^1× D_r with r<R, then there is no contactomorphism g⊂ G such that g((H))⊂(K), see <cit.>.However, there is an important class of noncompact contact manifolds, called convex at infinity, for which a dominating cone always exists. We discuss this class in the next subsection.§.§ Contact manifolds convex at infinityA noncompact contact manifold (U,ξ) is called convex at infinity if there exists a contact embedding σ:U U which is contactly isotopic to the identity such that σ(U)⋐ U, i.eσ(U)has a compact closure in U. The space of all embeddings σ with this property will be denoted by=(U,ξ).Note that by cutting off a contact isotopy, the restriction σ|_K toany compact set K⊂ U can be extended to a contactomorphism in the group G=_c(U,ξ).The notion of contact convexity for hypersurfacesin a contact manifolds was introduced in <cit.> and studied in detail in <cit.>. Let us recall that a hypersurface in a contact manifold is called convex if it admits a transverse contact vector field. The coorientation of this vector field is irrelevant because if Y is contact then -Y is contact as well. (i) A major class of contact manifolds convex at infinity is provided by interiors of compact manifolds with convex boundary. Indeed, as the required embedding σ one can take the flow for small positive time of an inward pointing contact vector field transverse to the boundary. (ii) More generally, suppose a contact manifold (U,ξ) admits a (not necessarily complete) contact vector field Y without zeroes at infinity, which outside a compact set is gradient-like for an exhausting function ϕ:U→. Then(U,ξ) is convex at infinity.Indeed, firstuse <cit.> to conclude thatfor a sufficiently large c the end({ϕ≥ c},ξ) is contactomorphic to (Σ×[0,∞),ξ) such thatthe vector field / s is contact. Here we set Σ:={ϕ=c} and denoted by s the coordinatecorresponding to the second factor. There is a contact isotopyh_t:(Σ×[0,∞),ξ)→(Σ×[0,∞),ξ), t∈[0,1], such that that h_0=𝕀, h_t=𝕀 near Σ×0 for all t∈[0,1] and h_1(Σ× n)=Σ×n-1/n, n=1,…, which implies that h_1(Σ×[0,∞))=Σ×[0,1).(iii) In a 3-dimensional contact manifolda generic surface is convex, see <cit.>, hence the interior of a genericconnected compact contact manifold with non-empty boundary is convex at infinity. If the boundary components of a 3-manifold are 2-spheres and the manifold is tight near the boundary, then it is convex at infinity, even when the boundary components are not convex, see <cit.>.(iv) On the other hand, the contact manifold S^1×^2=(/)×^2 with the tight contact form dt+1/2(x dy-y dx) is not convex at infinity, see <cit.>.Now we will introduce our main example. Let λ_=1/2∑_1^n(x_idy_i-y_idx_i) be the standard Liouville form on ^2n with itsLiouville vector fieldZ=1/2∑_1^n(x_i/ x_i+y_i/ y_i),and let α_=λ_|_S^2n-1 be the standard contact form on the unit sphere S^2n-1⊂^2n. Let us order coordinates in ^2n as (x_1,…, x_n,y_1,…, y_n) and denote by Π_k a k-dimensional coordinate subspace of ^2n which is spanned by the last k vectors of the basis. For instance, Π_1 is the y_n-coordinate axis, while Π_2n-1 is the hyperplane {x_1=0}. Note that Π_k is isotropic when k≤ n, and coisotropic otherwise. We denote by Π_k^⊥ the orthogonal subspace spanned by the first 2n-k basic vectors. (a) For each k=1,…,2n-1 the contact manifold (S^2n-1∖Π_k,ξ_) is convex at infinity. Moreover, it can be contracted by an element ofto an arbitrarily small neighborhood of the equatorial sphere S^2n-1∩Π_k^⊥.(b) For k≥ n the manifold (S^2n-1∖Π_k,ξ_) is contactomorphic to J^1(S^2n-k-1)×^2k-2n=T^*(S^2n-k-1×^k-n)×.Recall that r=|x|^2+|y|^2 induces the canonical isomorphism (^2n∖{0},λ_)≅(_+× S^2n-1,rα_) under which the Liouville vector field Z corresponds to r/ r. Thus contact vector fields on S^2n-1 are in one-to-one correspondence with Hamiltonian vector fields on ^2n∖{0} which commute with Z. Note that each linear vector field on ^2n automatically commutes with Z.(a) Firstconsiderthe case k≤ n. The linear vector fieldY_k := ∑_j=n-k+1^n(-x_j/ x_j+y_j/ y_j)is the Hamiltonian vector field of the function ∑_n-k+1^nx_jy_j. It commutes with Z, so it descends to a contact vector field Y_k to S^2n-1. On ^2n∖(Π_k∪Π_k^⊥) the field Y_k is gradient-like for the Z-invariant function -ln(1-∑_n-k+1^ny_j^2/|x|^2+|y|^2), hence on S^2n-1∖(Π_k∪Π_k^⊥) the field Y_k has no zeroes and is gradient-like for the exhausting function -ln(1-∑_n-m+1^n y_j^2). Now convexity at infinity follows from Example <ref>(ii), and the flow of Y_k for very negative times contracts any compact set S^2n-1∖Π_k to a neighborhood of S^2n-1∩Π_k^⊥.The case k>n follows from part (b) andExample <ref>(ii).(b) For k ≥ n let Z_k := ∑_j=1^2n-k(x_j/ x_j-y_j/ y_j)bethe Hamiltonian vector field of the function -∑_1^2n-k x_jy_j. Itdescends to a complete contact vector field Z_k onS^2n-1. The flow of Z_k contracts every compact set in S^2n-1∖Π_k to a neighborhood of the isotropic sphere S^2k-1∩Π_k^⊥ and the field -Z_k is gradient like for the function (∑_1^n y_j^2+∑_2n-k+1^n x_j^2) on S^2n-1∖Π_k. By Weinstein-Darboux theorem contact structureson S^2n-1∖Π_k and J^1(S^2n-k-1)×^k-n are isomorphic on tubular neighborhoods of isotropic sphereS^2n-k-1=S^2n-1∩Π_k^⊥and the 0-section S^2n-k-1× 0⊂ J^1(S^2n-k-1)×^2k-2n. This isomorphism then extends to a contactomorphism between S^2n-1∖Π_k and J^1(S^2n-k-1)×^2k-2n by matching the corresponding trajectories of the contactvector field -Z_kwith trajectories of the canonical contact vector field onJ^1(S^2n-k-1)×^2k-2n contracting this manifold to its 0-section.§.§ The maximal dominating cone ^+ Let (U,ξ) be a contact manifold convex at infinity.For (U,ξ) connected and convex at infinity the cone^+ := {H∈^≥0| H|_σ(U)>0for some σ∈(U,ξ)}.is dominating and maximal, (i.e., all other dominating cones are subcones of ^+).Properties (i), (ii) and (iii) in Definition <ref> are clear.For (iv), consider H∈, K∈^+. Then C:= (H) is compact and K is positive on σ(U) for some σ∈(U,ξ). By cutting off the contact isotopy from the identity to σ outside C we find g∈ G with g|_C=σ|_C. Then (_gH)=g( (H))=σ(C)⊂σ(U). Since K|_σ(U)>0, it follows that t_gH≤ K for t sufficiently small.For (v), let H∈^≥0∖{0} be strictly positive on some open set V⊂ U. Pick any σ∈(U,ξ). Since the group G acts transitively on U and σ(U) is relatively compact, there exist g_1,…,g_k∈ G such that σ(U)⊂ g_1(V)∪⋯∪ g_k(V). Then _g_iH is nonnegative and strictly positive on g_i(V), hence _g_1H+…+_g_kH is strictly positive on σ(U) and therefore belongs to ^+.To prove maximality of ^+, letbe any other dominating cone. Take anyH∈ and F∈^+. Then by Definition <ref> there exists K∈ such that F≤ K, and there exist g∈ G and t>0 such that t_gK≤ H. It follows that t_gF≤ H. Since F∈^+, this implies that H is positive on σ(U) for some σ∈, and therefore H∈^+.(i) For the standard contact structure on ^2n+1 we have ^+=^≥0∖ 0.(ii) For the 1-jet space U=J^1(M) of a closed manifold M endowed with its standard contact structure, the maximal dominating cone ^+ consists of all nonnegative functions whose support contains a neighborhood of a Legendrian submanifold isotopic to the zero section.(iii) For (S^2n-1∖Π_k,ξ_) as in Example (iii) in Section <ref>, the cone ^+ consists of all nonnegative functions which are positive on an image of the equatorial sphere S^2n-1∩Π_k^⊥ under a contactomorphism isotopic to the identity.(iv) In the special case(S^3∖Π_1,ξ_), which is the same as ^3∖ 0with the standard contact structure inherited from ^3, the cone ^+ can also be characterized as consisting of all nonnegative functions whose support contains aneighborhoodhomologically non-trivial 2-sphere.Both contact manifolds in (i) and (ii) admitcomplete contactvector fields which contract every compact subset to an arbitrarily small neighborhood of the origin incase (i), andto an arbitrarily smallneighborhood of the zero section in case (ii).For (^2n-1,dt+∑_1^n-1(x_jdy_j-y_jdx_j)) this is the vector field -2/ t-∑_1^n-1(x_j/ x_j+y_j/ y_j), and for (J^1(M), dz+pdq) this is the vector field -/ z-p/ p. But the group G acts transitively on points and on Legendrian submanifolds isotopic to the zero section, respectively. In (iii), according to Lemma <ref> the space S^2n-1∖Π_k can be contracted by an element into a neighborhood of the equatorial sphere S^2n-1∩Π_k^⊥. For (iv), we note in addition that any two smoothly isotopic 2-spheres in a tight contact manifold can be C^0-approximated by spheres which are contactly isotopic, see <cit.>.If U contains a compact subset which is notcontractible in U, then the cone ^+ never coincides with ^≥0∖{0}. To see this, pick H,K∈^≥0∖{0} such that (H) is noncontractible in U and (K) is contractible in U. Suppose there exists g∈ G and t>0 with t_g H≤ K. Then we must have g( (H))⊂ (K), which is impossible if (H) is noncontractible and (K) is contractible in U. §.§ Partial order on ^+ up to conjugation Let us denoteby Θ:=^+/∼ the quotient space of ^+ by the adjoint action of G on . The partial order H≤ K on ^+ descends to a possibly degenerate partial order ≼ on Θ defined on h,k∈Θ byh≼ k :⟺there existsH∈ h, K∈ k such that H≤ K. The following are equivalent: * there exists H∈^+ and g∈ G such that _gH≤ sH for some 0<s<1;* for all K_1,K_2∈^+ there exists h∈ G such that _hK_1≤ K_2. Clearly (b) implies (a). Conversely, suppose that (a) holds for elements H, g and let K_1,K_2∈^+ be given. By Definition <ref> there exist t_i>0 and h_i∈ G such thatt_1_h_1K_1≤ H≤1/t_2_h_2K_2.Applying _g^Nfor some N∈ to these inequalities, we obtaint_1_g^N_h_1K_1≤_g^NH≤ s^NH≤s^N/t_2_h_2K_2.Applying _h_2^-1 to both sides and dividing by t_1, we obtain_h_2^-1_g^N_h_1K_1≤s^N/t_1t_2K_2.Hence _hK_1≤ K_2 with h:=h_2^-1g^Nh_1, provided that N is chosen so large that s^N≤ t_1t_2. We call the positive cone ^+ non-orderable up to conjugation if the equivalent conditions in Lemma <ref> hold, and orderable up to conjugation otherwise. Thus to prove orderability up to conjugation of ^+, it suffices to find some pair K_1,K_2∈^+ for which there exists no h∈ G with _hK_1≤ K_2.a) Even if^+ is orderable up to conjugation this does not imply that the induced binary relation on Θ is a genuine order. However, we do not know any counterexamples to this implication. We will discuss the arising structures in more detailin Section <ref> below.b) If the manifold U is closed then the cone ^+ is always orderable up to conjugation for the following trivial reason: the volume integralI(H) := ∫_U (α/H)∧d(α/H)^n-1satisfies I(_gH)=I(H) for all g∈ G, so one can never have _gH≤ sH for some 0<s<1. Note that the strict order H>G does descend in the caseof a closed U to a genuine order on Θ, as it follows from the same preservation of volume argument.(a) If (U,ξ) is the standard contact ^2n+1 or J^1(M), as in Lemma <ref> (i) and (ii), then ^+ is non-orderable up to conjugation.(b) More generally, let (V,λ) be the completion of a Liouvilledomain (see <cit.>). Then for its contactization (U=V×,(λ+dt)) the maximal dominating cone ^+(U,ξ) is non-orderable up to conjugation.Since (a) follows from (b), it suffices to prove (b). The Liouville flow ϕ_s on V induces a contactdiffeotopy ψ_s(x,t)=(ϕ_s(x),e^st) of V× satisfying ψ_s^*(λ+dt)=e^s(λ+dt). Let C⊂ V× be the attractor of the flow ψ_s when s→-∞.Take K_1,K_2∈^+,K_1≥ K_2. By the definition of the cone ^+ there exists a contacomorphism h ∈ G such that (_h K_2) contains a neighborhood of C.The flow ψ_s when s→-∞ moves (K_1) into an arbitrarily small neighborhood of C, and hence for sufficiently large -s we have (_ψ_s(K_1)=(K_1∘ψ_s^-1)⊂(_h K_2). Therefore, (_ψ_s K_1)(x)=e^sK_1(ψ_s^-1(x))≤_hK_2(x) forfor sufficiently large -s, or(_h^-1∘ψ_s K_1)(x) ≤K_2(x), which means that ^+ satisfies condition (b) in Lemma <ref>. Proposition <ref>(b) combined with Lemma <ref>(b) yields If k≥ n, then for (S^2n-1∖Π_k,ξ_) from Section <ref> the cone ^+ is non-orderable up to conjugation. □ By contrast, we will show below in Section <ref> If k<n, then for (S^2n-1∖Π_k,ξ_) the cone ^+ is orderable up to conjugation: there exist H,K∈^+ for which there is no g∈ G with _g H≤ K. More precisely, there exists a surjective map w:^+→(0,∞) such that w(_gH)=w(H) for g∈ G, w(sH)=s^-2w(H) for any s>0 and such that H≤ K implies w(H)≥ w(K). § ORDERABILITY AND SYMPLECTIC NON-SQUEEZINGIn this section we will rephrase Theorem <ref> as a non-squeezing result for suitable unbounded domains in the standard symplectic space (^2n,=∑_1^n dx_j∧ dy_j). Throughout this section we fix k with 1≤ k≤ n and denoteU_k:=S^2n-1∖Π_k, G_k:=_c(U_k,ξ_).§.§ The class _k of unbounded domains in ^2n Introduce “polar coordinates", r=|x|^2+|y|^2∈, θ=r^-1/2(x,y)∈ S^2n-1, so that the standard Liouville form λ_ can be written asλ_=1/2∑_1^n(x_jdy_j-y_jdx_j) = rα_,where α_ is the standard contact form on the unitsphere S^2n-1 which defines the standard contact structure ξ_=α_. The coordinates (r,θ) identify (^2n∖ 0,λ_) with the symplectization (_+× S^2n-1, rα_) of the standard contact structure on S^2n-1. Thus the symplectization of U_k=S^2n-1∖Π_k gets identified with ^2n∖Π_k.Note that any contactomorphism ϕ of (S^2n-1,ξ_) defines a symplectomorphism Sϕ:^2n→^2n, singular at the origin, by the formulaSϕ(r,θ) := (r/c_ϕ(θ),ϕ(θ)),where ϕ^*α_ = c_ϕ(θ)α_. If ϕ is contactly isotopic to the identity, then there exists a constant K_ϕ>1 such that for any >0 there exists a smooth symplectomorphism S_ϕ of ^2n which equals the identity on the -ball around the origin and which coincides with Sϕ outside the (K_ϕ)-ball.Moreover, if ϕ as well as its isotopy to the identity equal the identity near some compact subset C⊂ S^2n-1, then S_ϕ can be chosen equal to the identity on the cone over C.Let ϕ_t, t∈[0,1] be a contact isotopy of (S^2n-1,ξ_) connecting ϕ_0=𝕀 to ϕ_1=ϕ with ϕ_i=𝕀 on C. Let Sϕ_t:^n∖ 0→^n ∖ 0 be its symplectization. SetM:=max_t∈[0,1]max (c_ϕ_t),m:=min_t∈[0,1]min (c_ϕ_t)and note that m≤ 1≤ M. Then for any δ>0, t∈[0,1] we have{r≤δ/M}⊂Sϕ_t({r≤δ})⊂{r≤δ/m}.Therefore, we can extend the Hamiltonian isotopy Sϕ_t|_{r≥δ} to a Hamiltonian isotopy g_t:^n→^n such that g_0=𝕀 andfor all t∈[0,1]g_t|_{r≤δ/2M}∪{θ∈ K}=𝕀,g_t|_{r≥2δ/m}=Sϕ_t.Then for δ =2Mthe symplectomorphism S_ϕ:=g_1 satisfies the required conditions with K_ϕ=4M/m. Given a nonnegative compactly supported contact Hamiltonian H:U_k→ we extend it by 0 to S^2n-1 and will keep the notation H for this extension. Recall that according to Lemma <ref> the cone ^+=^+(U_k) consists of all functions satisfying the conditions * H =0 near S^2n-1∩Π_k=S^2n-1∩{x=0,y_1=…=y_n-k=0};* H is positiveon theimage g(S^2n-1∩Π_k^⊥) of the equator under a contactomorphismg∈ G_k. We now define a class _k of domains in ^2n which, in particular (see Lemma <ref> below), contains all the domains of the formV(H) := {(r,θ)∈_+× S^2n-1|rH(θ)<1}, H∈^+(U_k). First, weadd to _k all hyperboloidsV_k^a,b := {1/a^2(∑_1^nx_i^2+∑_1^n-k y_i^2)-1/b^2∑_n-k+1^n y_i^2 <1}, a,b>0. Let ^a,b_k denote the identity component of the group of Hamiltonian diffeomorphisms of ^2n supported away from V^a,b_k and set_k := ⋃_a,b>0_k^a,b.It follows from Lemma <ref> above that for any contactomorphism ϕ∈ G_k the smoothed symplectomorphism S_ϕ:^2n→^2n belongs to ^a,b_k if a and a/b are small enough. Moreover, S_ϕ agrees with Sϕ outside V^a',b'_k for any b'>0 and a'> K_ϕ, where K_ϕ is the constant from Lemma <ref>. Thus, although the smoothing S_ϕ is not canonical, its action on domains which contain V^a',b'_k with a'>K_ϕ is independent of the choice of the smoothing.Now we are ready to give the general definition of the domains which form the class _k.A connected open domain V∈^2n belongs to _k if there exist a_1,b_1,a_2,b_2>0 and a symplectomorphism Φ∈_k such thatV_k^a_1,b_1⊂ V⊂Φ(V_k^a_2,b_2). The group _k, and hence the group G_k, acts on _k by symplectomorphisms. (i) For H∈^+(U_k) we have V(H)∈_k. (ii) For H∈^+(U_k), ϕ∈ G_k andsufficiently small we haveS_ϕ(V(H)) = V(_ϕ H).(iii) If H,K∈^+(U_k) satisfy H≥ K, then V(H)⊂ V(K).Claims (ii) and (iii) are straightforward. To prove(i) we first observe that, since the class _k is invariant under the action of the group _k, we can replace H by _g(H) for any g∈ G_k. Hence we can assume without loss of generality that H|_S^2n-1∩Π_k^⊥>0. For (x,y)∈^2n we denote u:=|x|^2+∑_1^n-ky_j^2,v:=∑_n-k+1^ny_j^2, r=u+v=|x|^2+|y|^2,ρ:=v/u.Then ρ=tan^2α, where α is the angle between the vector (x,y)∈^2n and the subspace Π_k^⊥.We canview ρ as a function on S^2n-1. Take H∈^+ and set M:=max (H). Then H|_ρ≥ρ_1 =0 for a sufficiently large ρ_1, and for a sufficiently small ρ_0 we have m:=min_ρ≤ρ_0(H)>0. Figure <ref> shows thatV(H)⊂Ω:={ v≤ρ_1u, u+v≥1/M}∖{ v<ρ_0 u, u+v >1/m}. Consider the hyperboloids V^a,b_k={u/a^2-v/b^2<1},V^a',b'_k={u/(a')^2-v/(b')^2<1}.An elementary geometric argument illustrated by Figure <ref> [We thank V. Stojisavljević for preparing this figure.] shows thatΩ⊂ V^a',b'_k∖ V^a,b_k, provided thata^2 < T:= (M(1+ρ_1))^-1, (b/a)^2 > ρ_1T/T-a^2and(b'/a')^2<ρ_0, (a')^2> 1/m .These inequalitiesexpressthe conditionthat the dotted lines u/(a)^2-v/(b)^2=1 and u/(a')^2-v/(b')^2=1, representing the boundaries of the domains V^a,b_k and V^a',b'_k, do not intersect the shaded region representing Ω.Thus, thischoice of a,b,a', b' guarantees thatV_k^a,b⊂ V(H)⊂V_k^a',b', hence V(H)∈_k.§.§ Capacity-like function on _k and proof of Theorem <ref> The following theorem will be proved in Section <ref> below. There exists a capacity-like function w:_k→(0,∞) with the following properties: * w(Ψ(V))=w(V) for all Ψ∈_k and V∈_k;* V⊂ V'implies w (V)≤ w(V') for all V,V'∈_k;* w(sV)=s^2w(V) for all s>0;* w(V_k^a,b)=π a^2 for all a,b>0. For any domain V∈_k and s>1 there is no Ψ∈_k such that Ψ(sV)⊂ V.By Theorem <ref>(iii) we havew(sV)=s^2w(V)>w(V),and the result follows from Theorem <ref>(i) and (ii).We define therequired function w:^+→ by the formula w(K):=w(V(K)). For g∈ G_k we have V(_g(K))=S_gϕ(V(K)), so Theorem <ref>(i) implies that the function w is constant on orbits of the adjoint action. We have V(sK)=s^-2V(K), hence w(sK)=s^-2w(K) in view of Theorem <ref>(iii). This also yields surjectivity of w, and Theorem <ref>(ii) implies that if w(H)<w(K) then there is no contactomorphism g∈ G such that _gH≤ K.§ INVARIANTS OF DOMAINS FROM _K§.§ Floer-Hofer symplectic homology of bounded domains in ^2nFiltered symplectic homology SH^(a,b)(U) of a bounded open set U in the standard symplectic ^2n was introduced by A. Floer and H. Hofer in <cit.> as a far-reaching generalization of earlier symplectic invariants, such as Gromov's symplectic width <cit.>, and later symplectic capacities, see <cit.>. Since then the invariant has been greatly generalized and expanded, but for the purposes of this paper we will use the original Floer-Hofer version up to the following slight modification. Instead of taking as in <cit.> a direct limit over Hamiltonians which are negative on U and equal a positive definite quadratic form at infinity, we will take an inverse limit over nonpositive Hamiltonians with compact support in U. This version enjoys the same functorial properties as the one in <cit.> but will be more convenient for the computations below. For domains with smooth boundary of restricted contact type, our version of symplectic homology differs from the one in <cit.> only by a degree shift of -1 (this follows e.g. from the duality results in <cit.>). We use /2-coefficients and grade all groups by Conley-Zehnder index.Letdenote the group of (not necessarily compactly supported) Hamiltonian diffeomorphisms of ^2n. The following proposition summarizes some relevant properties of symplectic homology, see <cit.>. Filtered symplectic homology assigns to each bounded open subset U⊂^2n and numbers 0≤ a<b<∞ a -graded /2-vector space SH^(a,b)(U) with the following properties.(Functoriality) Each Ψ∈ induces isomorphismsΨ_*:SH^(a,b)(U) ≅⟶ SH^(a,b)(Ψ(U)).(Transfer map) Each inclusion ι:U V induces a homomorphismι_!:SH^(a,b)(V)→ SH^(a,b)(U)It follows that for Ψ∈ with Ψ(U)⊂ V, the inclusion ι:Ψ(U) V together with Ψ induces a homomorphismΨ_!:=Ψ_*^-1∘ι_!:SH^(a,b)(V)→ SH^(a,b)(U).(Isotopy invariance) For a smooth family Ψ^s∈ withΨ^s(U)⊂ V for all s∈[0,1], the mapsΨ^s_!:SH^(a,b)(V)→ SH^(a,b)(U) are independent of s.(Window increasing homomorphism) For 0≤ a<b and 0≤ a'<b' with a≤a' and b≤ b' we have natural homomorphismsSH^(a,b)(U) → SH^(a',b')(U).(Symplectic homology of a ball) The symplectic homology in the action window (0,c) of the ball B_a^2n of radius a in ^2n is given bySH_j^(0,c)(B_a^2n) ≅/2, c>π a^2 and j=n,/2, c>π a^2 and j=n(2⌊c/π a^2⌋-1)-1, 0,otherwise.§.§ Symplectic homology for domains from _kWe extend the definition of symplectic homology to unbounded open domains V⊂^2n bySH^(a,b)(V) := lim_⟵SH^(a,b)(U),where the inverse limit is taken over all bounded open subsets U⊂ V. We also define symplectic homology in the infinite action window (a,∞) asSH^(a,∞)(V) := lim_c→∞ SH^(a,c)(V).The extended symplectic homology still satisfies the properties in Theorem <ref>. However, the invariants one can extract from these general properties are not sufficient for our purposes. Instead, we will concentrate on the special class _k of unbounded domains introduced in Section <ref> above and study their invariants under the smaller group _k which preserves this class.We begin with the computation of symplectic homology of the hyperboloids V_k^a,b. Let 1≤ k<n and 0<c<∞. Then:(a) For a,b>0 we haveSH_j^(0,c)(V_k^a,b) ≅/2, c>π a^2 and j=n-k,/2, c>π a^2 and j=(n-k)(2⌊c/π a^2⌋-1)-1, 0,otherwise.(b) For a>a, b/ a<b/a and c>π a^2 the transfer map/2=SH_n-k^(0,c)(V_k^ a, b)→SH_n-k^(0,c)(V_k^a,b)=/2is an isomorphism.(c) For c>π a^2 the window increasing homomorphism/2=SH_n-k^(0,c)(V_k^a,b)→SH_n-k^(0,∞)(V_k^a,b)=/2is an isomorphism. Heuristically, this result is easy to understand: All closed characteristics on the boundary of V_k^a,b satisfy z_n-k+1=⋯=z_n=0, so they agree with the closed characteristics on the boundary of the ball B_a^2(n-k) of radius a in ^2(n-k) whose symplectic homology is given in Theorem <ref>. The actual proof of Proposition <ref> will be given in Section <ref> below.For any domain V∈_k there exists c_0 such that for each c≥ c_0 and for all a,b with V^a,b_k⊂ V, the compositionα_V,c:SH_n-k^(0,c)(V) → SH_n-k^(0,c)(V_k^a,b) → SH_n-k^(0,∞)(V_k^a,b)=/2(where the first map is the transfer map and the second one the window increasing homomorphism) is surjective. Let V∈_k and V^a,b_k⊂ V. By definition, there exist a_1,b_1,a_2,b_2>0 and Φ∈_k such that V_k^a_1,b_1⊂ V⊂Φ(V_k^a_2,b_2). We choose a_1 so small thatV_k^a_1,b_1⊂ V^a,b_k⊂ V⊂Φ(V_k^a_2,b_2)and Φ_s=𝕀 on V_k^a_1,b_1 for all s∈[0,1], where Φ_s is the isotopy in _k from the identity to Φ_1=Φ. Set c_0:=π a_2^2 and consider c>c_0. Then Φ_s^-1(V_k^a_1,b_1)=V_k^a_1,b_1⊂ V_k^a_2,b_2, so by isotopy invariance the map(Φ_s^-1)_!:/2=SH_n-k^(0,c)(V_k^a_1,b_1)→ SH_n-k^(0,c)(V_k^a_2,b_2)=/2is independent of s∈[0,1], hence an isomorphism by Proposition <ref>(b). It follows that the composition of the obvious maps/2= SH_n-k^(0,c)(V_k^a_2,b_2) ≅ SH_n-k^(0,c)(Φ(V_k^a_2,b_2))→ SH_n-k^(0,c)(V)→ SH_n-k^(0,c)(V_k^a,b) → SH_n-k^(0,c)(V_k^a_1,b_1)=/2is an isomorphism. This implies that SH_n-k^(0,c)(V_k^a,b)=/2 and the map SH_n-k^(0,c)(V) → SH_n-k^(0,c)(V_k^a,b) is surjective, which combined with Proposition <ref>(c) proves the corollary. §.§ A capacity for domains from _k For any c>0 and V∈_k we define the augmentationα_V,c:SH_n-k^(0,c)(V) → SH_n-k^(0,∞)(V_k^a,b)=/2as in Corollary <ref>, where V^a,b_k⊂ V. For another hyperboloid V^a',b'_k⊂ V we find a hyperboloid V^a_1,b_1_k⊂ V^a,b_k∩ V^a',b'_k, and the commuting diagram6pcSH^(0,c)_n-k(V) [r] [d]SH^(0,∞)_n-k(V^a,b_k)=/2 [d]^≅ /2=SH^(0,c)_n-k(V_k^a',b')[r]^≅ SH^(0,∞)_n-k(V_k^a_1,b_1)=/2 .shows that α_V,c does not depend on the choice of a,b. Corollary <ref> shows that a_V,c is trivial for sufficiently small c and surjective for sufficiently large c. The following corollary now is immediate from the properties of symplectic homology in Theorem <ref>. (i) For Ψ∈_k and a,b such that Ψ=𝕀 on V_k^a,b we have the commutative diagram6pcSH^(0,c)_n-k(V) [r]^a_V,c[d]^Ψ_* SH^(0,∞)_n-k(V^a,b_k)=/2 [d]^𝕀 SH^(0,c)_n-k(Ψ(V))[r]^a_Ψ(V),c SH^(0,∞)_n-k(Ψ(V^a,b_k)=V^a,b_k)=/2 .(ii) For an inclusion ι:V V' we have the commuting diagram6pcSH^(0,c)_n-k(V') [r]^a_V',c[d]^ι_! /2 [d]^𝕀 SH^(0,c)_n-k(V)[r]^a_V,c /2 .§.§ Proof of Theorem <ref> As in <cit.>, we define the capacity-like function w:_k→(0,∞] on V∈^k byw(V) := inf{c | a_V,c:SH^(0,c)_n-k(V)→/2issurjective}.Let us verify that the function w has all the properties in Theorem <ref>. Properties (i) and (ii) follow from the commuting diagrams in Corollary <ref>, and (iv) follows from Proposition <ref>. Property (iii) follows as in <cit.>, noting that the conformal rescaling z↦ sz on ^2n preserves the class of domains _k and multiplies actions by s^2.§.§ Proof of Proposition <ref>To compute symplectic homology of the unbounded domain V=V^(a,b)_k, we first need to choose an increasing family V_C⊂ V of bounded subdomains exhausting V. By definition of the inverse limit it is up to us which sequence to choose, and we do itcarefully to control the dynamics of the arising Hamiltonian vector fields.For C>0 set g_C(t):=max(-t^2, 3t^2-4C^2) and let g_C:→ be a smoothing of the function g_C. We do the smoothing such that for small >0g_C(t) = -t^2, |t|≤ C-, 3t^2-4C^2,|t|≥ C+,and g_C has minima at the points ±C with the minimal value g_C(C)=-C^2+. Moreover, we can choose g_C such thatt/2 g_C(t)≥ g_C(t)for all t∈.Consider the function H_C(x,y):= x^2/a^2+ g_C(y)/b^2 on ^2, thusH_C(x,y) = x^2/a^2-y^2/b^2, |y|≤ C-,-4C^2/b^2+x^2/a^2+3y^2/b^2, |y|≥ C+.The critical points of H_C are a saddle point at the origin and two minima at (0,± C) with the minimal value C^2+/b^2. Note that H_C(x,y)≥x^2/a^2-y^2/b^2. Moreover, condition (<ref>) implies1/2(x H_C/ x+y H_C/ y)= x^2/a^2+y g_C'(y)/2b^2≥ H_C(x,y).The Hamiltonian system of H_C on the plane (^2,dx∧ dy) has the folowing properties: * all its trajectories except the two homoclinic orbits at the origin are closed;* for every closed trajectory γ we have ∫_γ xdy≥ 0;* for every nonconstant closed trajectory γ with H_C|_γ≥-C^2/4b^2 we have∫_γ xdy≥ C^2A_a,b, with a constant A_a,b depending only on a and b.The zero level set of H_C forms a figure eight consisting of the two homoclinic orbits at the origin and enclosing the two minima. Assertions (i) and (ii) follow from this picture. For (iii), consider the closed trajectories γ_± of value H_C|_γ_±=-C^2/4b^2 enclosing the points (0,± C). Let V_± be the region bounded by γ_±. Then V_±∩{± y≥ C+}=E∩{± y≥ C+} for the ellipse E={x^2/a^2+3y^2/b^2≤15 C^2/4b^2}, and rescaling by C shows that the area of V_±∩{± y≥ C+} equals C^2A_a,b with a constant A_a,b depending only on a and b. This proves ∫_γ_± xdy≥ C^2A_a,b, and the area of each nonconstant closed trajectory γ with H_C|_γ>-C^2/4b^2 is larger than this one. Define now the open domainV_C := {G < 1}⊂^2nwith the HamiltonianG(x_1,…, x_n, y_1,…, y_n) :=∑_1^n-kx_j^2+y_j^2/a^2 + ∑_n-k+1^n H_C(x_j,y_j).V_C is a bounded domain contained in V_k^a,b with smooth restricted contact type boundary V_C. The closed characteristics on V_C fall into two groups:* closed characteristics on the sphere S of radius a in the subspace ^2(n-k)⊂^2n, of actions kπ a^2 for k∈;* all other closed characteristics have actions ≥ C^2B_a,b, with a constact B_a,b depending only on a and b. V_C is a bounded because G is exhausting, and V_C⊂ V_k^a,b follows from H_C(x,y)≥x^2/a^2-y^2/b^2. Its boundary is of restricted contact type because it is transverse to the Liouville vector field Z=1/2∑_1^n(x_i/ x_i+y_i/ y_i), which in turn follows from (<ref>) by computation at points of V_C={G=1}:Z· G = ∑_1^n-kx_j^2+y_j^2/a^2 + ∑_n-k+1^n1/2(x_j H_C/ x_j+y_j H_C/ y_j)= 1 + ∑_n-k+1^n(1/2(x_j H_C/ x_j+y_j H_C/ y_j)-H_C(x_j,y_j))≥ 1.Let us study the periodic orbits on V_C. First, we observe that the corresponding Hamiltonian system with the Hamiltonian G is completely integrable and has integralsG_j(x,y):=x_j^2+y_j^2/a^2,j=1,…, n-k, H_C(x_j,y_j),j=n-k+1,…, n.Hence the simple periodic orbits are given by equations G_j=c_j for j=1,…,n, with ∑_1^n c_j=1.Let us make an accounting of periodic orbits. First of all we have orbits {x_n-k+1=y_n-k+1=… x_n=y_n=0,G_j=c_j≥ 0, j=1,…, n-k} with ∑_1^n-k c_j=1 which foliate the sphere S of radius a in the subspace ^2(n-k)⊂^2n. These orbits and their multiples correspond to group (i) in the lemma and their actions are kπ a^2 for k∈.Consider now a simple periodic orbit γ which is not in group (i). Note that γ is a product of periodic orbits γ_j for the Hamiltonians G_j, and each γ_j has nonnegative action by Lemma <ref>(ii). By assumption, at least one of the orbits γ_j, j=n-k+1,…,n, is not the constant orbit at the origin.If at least one of the constants c_j, j=n-k+1,…,n is positive, then the action of the orbit γ_j, and hence of γ, is ≥ C^2A_a,b by Lemma <ref>(iii). Otherwise, set δ:=-∑_n-k+1^n c_j>0. Then ∑_1^n-k c_j=1+δ, and therefore ∑_1^n-k∫_γ_jx_jdy_j=π (1+δ)a^2. If δ>C^2/4b^2, then the action of the orbit γ is >π(1+C^2/4b^2)a^2. Otherwise, all the constants c_j for j=n-k+1,…,n satisfy c_j≥-C^2/4b^2. By assumption, at least one of the corresponding orbits γ_j is nonconstant, so by Lemma <ref>(iii) the action of this γ_j, and hence of γ, is ≥ C^2A_a,b. This proves Lemma <ref>.§.§ Deformation to a split HamiltonianWe write G=F_1+F_2 with the HamiltoniansF_1:=∑_1^n-kG_j:^2(n-k)→, F_2:=∑_n-k+1^nG_j:^2k→.Recall that F_1,F_2 have the following properties: * F_1 and F_2 are exhausting with F_1≥ -(n-k)C^2+/b^2 and F_2≥ 0;* Z_i· F_i≥ F_i for the respective Liouville fields Z_i;* all periodic orbits of F_i have action ≥0;* all periods of nonconstant periodic orbits of F_i are bounded below by some δ>0;* all second partial derivatives of F_i are uniformly bounded;* the (non-Hamiltonian) action of each k-fold covered periodic orbit z_2 of F_2 satisfies Å(z_2)=π ka^2F_2(z_2)≥π a^2F_2(z_2).Consider a family of Hamiltonians H_s:^2n→, s∈, of the formH_s(z_1,z_2) = h_s(F_1(z_1),F_2(z_2))with a smooth family of function h_s:^2→ satisfying the following properties: * h_s is locally constant in s outside a compact subset of ;* outside a compact subset of ^2 we have 0< h_s/ F_1<δ or 0< h_s/ F_2<δ (or both);* all second partial derivatives of the function ^3→, (s,F_1,F_2)↦ h_s(F_1,F_2) are uniformly bounded.For H_s as above all 1-periodic orbits are contained in a compact set, and for each c>0 the Floer homology FH^(0,c)(H_s) is well-defined and independent of s.For each s the 1-periodic orbits of H_s are of the form z=(z_1,z_2), where the z_i satisfy ż_i= h_s/ F_iX_F_i(z_i). Hence F_1,F_2 are constant along z and z_i(t)=γ_i( h_s/ F_it) for periodic orbits γ_i of X_F_i of period h_s/ F_i. Therefore, conditions (iv) on F_i and (ii) on h_s imply that all 1-periodic orbits of X_H_s are contained in a compact set.Next, let u:× S^1→^2n be a Floer cylinder connecting 1-periodic orbits. It satisfiesu_s+iu_t+∇ H_s(u)=0,where u_s,u_t denotes the partial derivatives with respect to the coordinates (s,t)∈× S^1. The bounds on the second derivatives of F_i and h yield uniform bounds |D^2H_s|≤ A on the Hessian of H_s and |∇_sH_s(u)|≤ B|u| on the gradient of the s-derivative _sH_s. Using this, a standard computation shows that the function ρ(s,t):=|u(s,t)|^2 satisfies the estimateΔρ = |u_s|^2+|u_t|^2+ u,iD^2H(u)u_t-D^2H(u)u_s+∇_sH_s(u) ≥ |u_s|^2+|u_t|^2-A|u|(|u_s|+|u_t|) - B|u|^2≥ -(1/2A^2+B)ρ.By an argument in <cit.>, this estimate implies that Floer cylinders for the s-dependent Hamiltonian H_s remain in a compact set, hence the Floer homology of H_s is well-defined and independent of s. Pick a nondecreasing function ϕ:→ satisfying ϕ(t)=δ t/2 for t≥ 0 and ϕ(t)≡ -m for t≤δ with some large constant m. For s∈[0,1] consider the HamiltonianH_s = h_s(F_1,F_2) := (1-s)ϕ(F_1+F_2-1)+sϕ(F_1)+sϕ(F_2-1).For ϕ as above with m>c each 1-periodic orbit (z_1,z_2) of H_s with action in the interval (0,c) satisfies z_1≡0, and z_2 is a 1-periodic orbit of ϕ(F_2-1) of action π a^2k for k=1,…,⌊c/π a^2⌋.Consider a 1-periodic orbit z=(z_1,z_2) of H_s with action in the interval (0,c). Its components satisfies the equationsż_1= ((1-s)ϕ'(F_1+F_2-1)+sϕ'(F_1))X_F_1(z_1),ż_2= ((1-s)ϕ'(F_1+F_2-1)+sϕ'(F_2-1))X_F_2(z_2).We distinguish three cases.Case 1: z_1 is not constant. Then by Lemma <ref>, for the action to be below c, each nonconstant component of z_1 must have value H_C≤-C^2/4b^2. Since each constant component has value ≤ 0 and at least one component is nonconstant, we deduceF_1(z_1)≤ -C^2/4b^2.It follows that ϕ(F_1)=-m and ϕ'(F_1)=0, so z_1 satisfies the equation ż_1 = (1-s)ϕ'(F_1+F_2-1)X_F_1(z_1). Since ż_1≠ 0, we must have s<1 and F_1+F_2-1>-δ. Together with the preceding displayed equation this yields F_2(z_2)>1-δ-F_1(z_1)≥ 1-δ+C^2/4b^2, which in view of property (vi) of F_2 implies Å(a_2)>π a^2(1-δ+C^2/4b^2)>c. So Case 1 cannot occur.Case 2: z_1 is constant but not all its component are zero. We rule this out by distinguishing several cases.(i) If X_F_1=0, then each components of z_1 is a critical point of H_C, with at least one of them being nonzero. Since H_C(0)=0 and H_C= at the minima we conclude F_1(z_1)≤ -C^2+/b^2≤ -C^2/4b^2, which is ruled out as in Case 1.(ii) If X_F_1≠0 and 0<s<1, then we must have ϕ'(F_1+F_2-1)=ϕ'(F_1)=0, hence ϕ(F_1+F_2-1)=ϕ(F_1)=-m. Then z_2 satisfies ż_2=sϕ'(F_2-1)X_F_2(z_2), so by the choice of ϕ it can only be 1-periodic if ϕ(F_2-1)≤ 0. Thus H_s(z)≤ -m and the Hamiltonian action of z satisfies Å_H_s(z)≥ m>c.(iii) If X_F_1≠0 and s=0, then ϕ'(F_1+F_2-1)=0, hence H_0(z)=ϕ(F_1+F_2-1)=-m and again Å_H_0(z)≥ m>c.(iv) If X_F_1≠0 and s=1, then ϕ'(F_1)=0, hence ϕ(F_1)=-m. Then ż_2=ϕ'(F_2-1)X_F_2(z_2) implies ϕ(F_2-1)≤ 0, thus H_1(z)≤ -m and again Å_H_1(z)≥ m>c.Case 3: z_1≡0. Then F_1(z_1)=0 and z_2 is of the form described in the lemma.Lemma <ref> and Lemma <ref> together imply (after replacing H_s by H_σ(s) for a nondecreasing function σ:→[0,1] which equals 0 for s≤0 and 1 for s≥1) that the Floer homology FH^(0,c)(H_s) is independent of s∈[0,1]. By definition, the Hamiltonian H_0(z)=ϕ(G(z)-1) computes the symplectic homology of V_C,FH^(0,c)(H_0)≅ SH^(0,c)(V_C).The HamiltonianH_1(z_1,z_2) = ϕ(F_1(z_1))+ϕ(F_2(z_2)-1)is split, as well as the corresponding Floer equation. It follows that all its Floer cylinders are contained in the subspace ^2(n-k), so H_1 computes the symplectic homology of the ball B_a^2(n-k) of radius a in ^2(n-k). This symplectic homology is computed in <cit.>, up to an index shift by 1 due to our different conventions, to beSH_j^(0,c)(B_a^2(n-k)) ≅_2, c>π a^2 and j=n-k,_2, c>π a^2 and j=(n-k)(2⌊c/π a^2⌋-1)-1, 0,otherwise.This proves part (a) of Proposition <ref>. Parts (b) and (c) follow by similar arguments. § GEOMETRY OF THE DOMINATING POSITIVE CONELet ^+ be the maximal dominating cone of an open contact manifold (U,ξ). The group _+ acts on Θ:= ^+/∼ by multiplication. We assume that ^+ is orderable up to conjugation and consider the binary relation ≼ on Θ from Section <ref>. For a pair of classes f,h ∈Θ defineρ(f,h) := inf{s>0| f ≼ sh} .The fact that ^+ is dominating implies that the set on the right hand side is nonempty, and orderability up to conjugation means that there exist a,b ∈Θ such that ρ(a,b) ≠ 0. Furthermore, clearly we have sub-multiplicativityρ(f,h) ≤ρ(f,g)ρ(g,h) for all f,g,h ∈Θ .Observe that ρ(h,h) ≥ 1 for all h ∈Θ by Lemma <ref> (a). On the other hand, obviously ρ(h,h) ≤ 1 and hence ρ(h,h)=1.We claim that ρ(f,g) ≠ 0 for all f,g ∈Θ. Indeed, take a,b with ρ(a,b)≠ 0 and write, by sub-multiplicativity,0< ρ(a,b) ≤ρ(a,f)ρ(f,g)ρ(g,b) ,yielding ρ(f,g) ≠ 0. The claim follows.Define now a function d: Θ×Θ→ byd(g,h) = max (|logρ(g,h)|,|logρ(h,g)|) .The above discussion shows that d is a pseudo-metric on Θ: it is symmetric, nonnegative and satisfies the triangle inequality. It is unknown whether d is a genuine distance on Θ (and sounds unlikely that it is). Introduce the equivalence relation ≈ on Θ by f ≈ g whenever d(f,g)=0. This relation measures the deviation of d from a genuine metric. Interestingly enough, it also measures the deviation of the binary relation ≼ from a genuine partial order. Indeed, if f ≼ g and g ≼ f, then we have ρ(f,g) ≤ 1 and ρ(g,f) ≤ 1. By(<ref>),1 = ρ(f,f) ≤ρ(f,g)ρ(g,f) ,and henceρ(f,g)=ρ(g,f) =1 .This yields d(f,g)=0 and hence f ≈ g. Denote Ξ:= Θ/≈, and note that the pseudo-metric d descends to Ξ as a genuine metric D. What about the partial order? Define a relation ≪ onΞ as follows: p ≪ q if there exist f,g ∈Θ and a sequence of positive numbers _i → 0 so that p=[f], q=[g] and f ≼ (1+_i)g for all i. (The _i>0 are needed because of the infimum in (<ref>)). Note that this is a transitive and reflexive relation. We claim that ≪ is a genuine partial order.Indeed if p ≪ q and q ≪ p, then D(p,q)=0 and hence p=q. It would be interesting to explore the geometry of Ξ with respect to D. The quantity ρ and the pseudo-metric d has cousins in the earlier literature. On the one hand, they can be considered Lie algebra counterparts of the relative growth between positive contactomorphisms, and the corresponding pseudo-metric, respectively, studied in <cit.>. On the other hand, each compactly supported non-negative function H on U defines a “contact form" α/H on U, where the quotation marks stand for the fact that this form could be infinite and certainly is infinite outside a compact set. With this language, the adjoint action of contactomorphisms on functions corresponds to the action of contactomorphisms on contact forms, and the metric d is an analogue, for open manifolds, of the contact Banach-Mazur distance introduced by Yaron Ostrover and the third-named author and discussed in <cit.> in the context of closed contact manifolds. Consider (S^2n-1∖Π_k,ξ_st) as above with k<n. The function w: ^+ → (0,∞) from Theorem <ref> descends to a function w: Θ→ (0,∞) with the following properties: (i) h ≼ f ⇒w(h) ≥w(f);(ii) w(sh)= s^-2w(h)for all s >0, h ∈Θ .These properties readily yield the following inequality:d(f,h) ≥1/2|logw(f)/w(h)| .For instance, this shows that d(g,sg)= |log s|, and in particular the restriction of the pseudo-metric d to each orbit of the _+-action on Θ is isometric to the Euclidean line. It would be interesting to explore whether Θ admits a quasi-isometric embedding of the Euclidean ^N for N ≥ 2. Let us mention also that the above conclusions continue to hold verbatim for the metric space (Ξ, D).§ DISCUSSION §.§ Partial order on groups of contactomorphismsLet (U,ξ) be a possibly non-compact contact manifold, G the identity component of the group of compactly supported contactomorphisms, G its universal cover, and g^≥ 0⊂ isthe non-negative cone in the Lie algebra.A path g_t∈ G, t∈[0,1] is called non-negative) if g_t^-1ġ_t∈^≥ 0 for all t∈[0,1].A contactomorphism g is called non-negative, or g≥𝕀 if there exists a positive path g_t connecting g_0=𝕀 with g_1=g. We say that g≤ h of hg^-1≥𝕀.The group G or G is called orderable if ≥ defines a genuine order on the group, i.e. whetherg≥ h and g≤ h imply g=h. The order was first studiedin <cit.>, wherethere were constructed first examples ofcontact manifolds withorderable and non-orderable groups G or G.[We note that orderability of G implies orderability of G.]Since these papers had appeared there were found many largeclasses of manifolds with orderableG and G. However,there were found nonew examples of non-orderability. In particular, we do not know a single example of a non-compact manifold with non-orderable G or G. In particular, for all domains U in the standard contact ^2n+1 the group G is orderable, see <cit.>.In the spirit of this paper we are interested here in the remnants of the partial order on the conjugacy classes of contactomorphisms of non-compact manifolds. §.§ Partial order up to conjugationLet (U,ξ=α) be a contact manifold convex at inifinity, and ^+⊂^≥ 0 the maximal dominating cone.Supposethere is a non-negative path g_tconnecting 𝕀 with g, such that for some t_0∈[0,1]we have g_t_0^-1ġ_t_0∈^+. Then there is a homotopy of g_tto a positive paththrough non-negative paths with fixed ends.It is sufficient to show that the concatenation of a positive path f_t and a non-negative pathg_t can be deformed to a positive path.Denote g_t:=g_tg_0^-1. Suppose that f_t and g_t are generated by contact Hamiltonians F_t and G_t, respectively. Then path h_t= g_tf_t is generated by the contact Hamiltonian H_t(x)=G_t(x)+ϕ_t(g_t^-1(x))F_t(g_t^-1(x)), where g_t^α=ϕ_tα. Hence, (H_t) ⊃(F_t∘ g_t^-1), and hence H_t∈^+, i.e. the path h_t is positive. It remains to observethat the concatenation of paths f_t and g_t and the path h_t are homotopic with fixed end-points via a non-negative homotopy. The required homotopy is h_s,t is givenby concatenating paths g_tsf_t, t∈[0,1],and g_tg_s^-1, t∈ [s,1]. We say in this case that g>𝕀, and respectively we say that g>h if gh^-1>𝕀.Let G^+ be the set of all positive elements of G. It is closed underthecomposition of contactomorphisms. We call G_+ the dominating positive cone in G. With any element g∈ G^+ its conjugacy class [g] is contained in G^+, and we willdenote by G^+/∼ the set of conjugacy classes of contactomorphisms from G^+.The partial order> descends to a possibly degenerate partial order ≻ onG^+/∼. The following lemma justifies this term “dominating":For any g∈ G and h∈ G^+ there exists an integer m>0such that [g]≺ [h^m]. The argument in the proof of Lemma <ref>shows that there exists a positive pathh_t, t∈[0,1], connecting 𝕀 with h generated a contact Hamiltonian H_t withthe property (H_t)⊃(H_0). We can also assume that H_t is time 1 periodic.Let G_t be the contact Hamiltonian generating g. By the definition of ^+ there exists a contactomorphism ϕ∈ G such that (_ϕ H_0)⊃ (G_t) for all t∈[0,1], and hence (_ϕ H_s)⊃ (G_t) for all s,t∈[0,1]. But then h^m is the time 1 map of the Hamiltonian mH(mt), and hence it is bigger G_t for a sufficiently large m. For a closed contact manifold (U,ξ) Lemma <ref> holds in a much stronger form, without passing to conjugacy classes:there exists an integer m>0such that g≺ h^m.In analogy with Lemma <ref>we haveThe following conditions are equivalent: * there exists f∈ G^+ such that f^m≺ f for some m>1;* there exists f∈ G^+ such that f^m≺ f for any m>1;* for any h,g∈ G^+ such that h<g there exists m such that h^m≻ g^m. Clearly (iii) implies (i) and (ii). Conversely, supposef satisfies (i), i.e. there exists ϕ∈ G such that ϕ^-1fϕ>f^m. Then ϕ^-2fϕ^2>(ϕ^-1fϕ)^m>f^2m, i.e. f≻ f^2m≥ f^kfor any k<2m.Iterating this argument we prove (ii). To prove (iii) chooseg<h. By Lemma <ref>there existk, lsuch that h≺ f^l and f≺ g^k.Then for any N we haveh^N≺ f^lN≺ f ≺ g^k,which proves the claim if N is large enough. The cone G^+ is called non-orderable up to conjugation if any of the equivalent conditions (i)-(iii) holds.Otherwise, G_+ iscalled orderable.If ^+ is non-orderable up to conjugation then so does G^+.Indeed,property<ref>(i) yields<ref>(i)by integration. In particular, for contactmanifolds in examples from Proposition <ref> the cone G^+ is not orderable up to conjugation. However, it will be shown in <cit.> that for (U=S^2n-1∖Π_k,ξ_) for k<n the dominating cone G^+ is orderable up to conjugation. When G^+ is orderable one can, following a construction in <cit.> define the relative growth γ(f,g) of two elements f,g∈ G^+: γ(f,g):=lim_q→∞inf_p{p/q;[f^p]≻ [g^q]}. As it was shown in<cit.> the limitalways exists and one has an inequality γ(f,g)γ(g,f)≥ 1.This inequality impliesthatd(f,g):=max(ln(γ(f,g))+lnγ(g,f))defines the metric spacestructure on the quotient Z of G^+/∼ by the equivalence relation[f]≡[g] if d(f,g)=0. In our forthcoming paper <cit.> we will explore geometry of the space Z for a number of examples of non-compact contact manifolds, including(U=S^2n-1∖Π_k,ξ_) for k<n. 99. BEM M.S. Borman, Y. Eliashberg and E. Murphy, Existence and classification of overtwisted contact structures in all dimensions, Acta Math. (2) 215(2015), 281–361.Bhupal M. Bhupal, A partial order on the group of contactomorphisms of ^2n+1 via generating functions, Turkish J. Math. 25 (2001), 125–135.Cie94 K. Cieliebak, Pseudo-holomorphic curves and periodic orbits on cotangent bundles, J. Math. Pures Appl. (9) 73 (1994), no. 3, 251–278.CE12 K. Cieliebak and Y. Eliashberg, From Stein to Weinstein andBack. Symplectic geometry of affine complex manifolds, AmericanMathematical Society Colloquium Publications 59 (2012).CEP2 K. Cieliebak, Y. Eliashberg and L. Polterovich, in preparation.CFH96 K. Cieliebak, A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology II: Stability of the action spectrum, Math. Z. 223 (1996), no. 1, 27–45.CO K. Cieliebak and A. Oancea, Symplectic homology and the Eilenberg-Steenrod axioms, arXiv:1511.00485.eliash-shapes Y. Eliashberg, New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc. 4 (1991), no. 3, 513–520.eliash-20Martinet Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet's work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165–192.Eliash-Weinstein-revisited Y. Eliashberg, Weinstein manifolds revisited, arXiv:1707.03442.EG-convex Y. Eliashberg and M. Gromov, Convex symplectic manifolds, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 135–162, Proc. Sympos. Pure Math., 52, Part 2, Amer. Math. Soc., Providence, RI, 1991.EKP Y. Eliashberg, S. Kim and L. Polterovich, Geometry of contact transformations and domains: orderability versus squeezing, Geom. Topol. 10 (2006), 1635–1747.EP Y. Eliashberg and L. Polterovich, Partially ordered groups and geometry of contact transformations Geom. Funct. Anal. 10 (2000), no. 6, 1448–1476. FH94 A. Floer and H. Hofer, Symplectic homology I: Open sets in ^n, Math. Z. 215 (1994), no. 1, 37–88.FHW94 A. Floer, H. Hofer and K. Wysocki, Applications of symplectic homology I, Math. Z. 217 (1994), no. 4, 577–606.Giroux-convex E. Giroux, Convexité en topologie de contact (French) Comment. Math. Helv. 66 (1991), no. 4, 637–677.Gr85 M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347.HZ94 H. Hofer and E. Zehnder, Symplectic invariants and Hamiltonian dynamics, Birkhäuser (1994).PZ L. Polterovich and J. Zhang, in preparation.SZ V. Stojisavljević and J. Zhang, in preparation.Viterbo99 C. Viterbo, Functors and computations in Floer homology with applications I, Geom. Funct. Anal. 9 (1999), no. 5, 985–1033. | http://arxiv.org/abs/1709.09358v1 | {
"authors": [
"Kai Cieliebak",
"Yakov Eliashberg",
"Leonid Polterovich"
],
"categories": [
"math.SG",
"math.GT",
"53D10, 53D40, 53D35, 53D50"
],
"primary_category": "math.SG",
"published": "20170927065745",
"title": "Contact orderability up to conjugation"
} |
Manifold Ways to Darboux–Halphen SystemManifold Ways to Darboux–Halphen System[This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.htmlhttp://www.emis.de/journals/SIGMA/modular-forms.html]John Alexander Cruz MORALES ^†^1, Hossein MOVASATI ^†^2, Younes NIKDELAN ^†^3,Raju ROYCHOWDHURY ^†^4 and Marcus A.C. TORRES ^†^2J.A.C. Morales, H. Movasati, Y. Nikdelan, R. Roychowdhury and M.A.C. Torres^†^1 Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia mailto:[email protected]@unal.edu.co^†^2 Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil mailto:[email protected]@impa.br, mailto:[email protected]@impa.br <http://w3.impa.br/ hossein/>^†^3 Instituto de Matemática e Estatística (IME), Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, Brazil mailto:[email protected]@ime.uerj.br <https://sites.google.com/site/younesnikdelan/>^†^4 Instituto de Física, Universidade de São Paulo (IF-USP), São Paulo, Brazil mailto:[email protected]@if.usp.brReceived September 29, 2017, in final form January 03, 2018; Published online January 08, 2018Many distinct problems give birth to Darboux–Halphen system of differential equations and here we review some of them. The first is the classical problem presented by Darboux and later solved by Halphen concerning finding infinite number of double orthogonal surfaces in ℝ^3. The second is a problem in general relativity about gravitational instanton in Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called Gauss–Manin connection in disguise developed by one of the authors and finally in the last problem Darboux–Halphen system emerges from the associative algebra on the tangent space of a Frobenius manifold.Darboux–Halphen system; Ramanujan system; Gauss–Manin connection; relativity and gravitational theory; Bianchi IX metric; Frobenius manifold; Chazy equation34M55; 53D45; 83C05 § INTRODUCTIONThe Darboux–Halphen system of differential equationsṫ_1=t_1(t_2+t_3)-t_2t_3, ṫ_2= t_2(t_1+t_3)-t_1t_3,ṫ_3= t_3(t_1+t_2)-t_1t_2, =∂ /∂τ,where τ is a free parameter, first came to existence when Darboux <cit.> was studying the existence of an infinite number of double orthogonal system of coordinates. He formulated the problem as follows: Let A and B be two fixed surfaces in the 3-dimensional Euclidean space ℝ^3 and suppose that Σ is the family of surfaces which are the locus of the points such that the sum of their distances from the surfaces A and B are constant; and Σ' is the family of surfaces which are the locus of the points so that the difference of their distances from the surfaces A and B are constant. Is there a third family of surfaces intersecting Σ and Σ' orthogonally? When we restrict the third family to the surfaces given by second degree equations, we find the Darboux–Halphen system. In Section <ref> we present Halphen's solution to this problem.The Darboux–Halphen system also emerge from a direct map from Ramanujan relations (Section <ref>).In 1979, Gibbons and Pope <cit.> found the Darboux–Halphen equations while studying gravitational instanton solutions in Bianchi IX spaces without having noticed it. Couple of decades later, Ablowitz et al. <cit.> pointed it out and recently one of the authors <cit.> explored its integrability aspects. A gravitational instanton is simply the (anti-)self-duality condition imposed on the curvature of a Einstein manifold with asymptotic locally Euclidian boundary conditions.Hitchin <cit.> and Tod <cit.> realized that (anti-)self-duality in Bianchi IX metric has a more general solution envolving to a Darboux–Halphen system coupled to another system of linear differential equation similar to Darboux–Halphen. A revised and simplified proof of the results of Tod and Hitchin can be found in <cit.>. See <cit.> for a physical application in cosmology. We review these works in Section <ref>.Another author <cit.> among us met the Darboux–Halphen system while exploring the Gauss–Manin connection of a universal family of elliptic curves. This method is called Gauss–Manin connection in disguise, which also name the vector field in this method that gives rise to the Darboux–Halphen equations and we present it in Section <ref>.The last interesting problem where Darboux–Halphen system appears is in the context of a 3-dimensional Frobenius manifold with a certain potential function F(t). Frobenius manifold arose as a geometrization of Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations <cit.>, an overdetermined system of differential equations that appear in the physics of topological field theories in 2 dimensions. In this particular case of dimension 3, the WDVV equation is known as Chazy equation, which has a close tie with the solutions of Darboux system. We present it in Section <ref>, where we follow Dubrovin's notes <cit.>.We conclude this article, crossing information between problems displayed here, which led us to interesting remarks and an evidence that leads to a new way on how to examine spectral curves from monopoles using Gauss–Manin connection in disguise, further explored in <cit.> by one of the authors.Throughout this text we make extensive use of Einstein summation convention where the sum over identical upper and lower indices is implicit.§ THE DARBOUX PROBLEMThe above Darboux problem given in Section <ref> is equivalent to the following problem: Let A and B be as before and suppose that Σ is a family of surfaces parallel to A which is parameterized by v, and Σ ' is a family of surfaces parallel to B that is parameterized by w. Is there a third family of surfaces parameterized by τ such that it intersects Σ and Σ ' orthogonally? Note that, two surfaces A_1 and A_2 are said to be parallel, if there exist a constant c∈ℝ_ 0 and a continuous one to one map between points a_1∈ A_1 and points a_2∈ A_2, such that the tangent planes at these points are parallel and the position vector a_2=a_1+ cN̂, where N̂ is the unitary vector normal to the surface A_1 at a_1. We say that a family of surfaces is parameterized by φ=φ(x,y,z), if any surface belonging to this family is given by φ(x,y,z)= const, in which x, y, z are the standard coordinates of ℝ^3. If for a function φ=φ(x,y,z), we define φ_x=∂φ/∂ x , φ_y=∂φ/∂ y , φ_z=∂φ/∂ z ,then in the latter problem, Darboux chose a case of parametrization of parallel surfaces that gives the Gauss map for points on the parallel surfacesv_x^2+v_y^2+v_z^2=1, w_x^2+w_y^2+w_z^2=1,and the condition of orthogonality at points in the intersection of v and τ and w and τ, respectively, is given byτ_xv_x+τ_yv_y+τ_zv_z=0,τ_xw_x+τ_yw_y+τ_zw_z=0.So the problem is equivalent to the following system of equations,v_x^2+v_y^2+v_z^2=1, w_x^2+w_y^2+w_z^2=1,τ_xv_x+τ_yv_y+τ_zv_z=0,τ_xw_x+τ_yw_y+τ_zw_z=0.If for a function φ of three variables (x,y,z) we define the operator∂_φ:=φ_x∂/∂ x+φ_y∂/∂ y+φ_z∂/∂ z,then equations (<ref>), (<ref>), (<ref>) and (<ref>), respectively, are given by ∂_vv=1, ∂_ww=1, ∂_τ v=∂_vτ =0 and ∂_τ w=∂_wτ =0, respectively. These equations imply ∂_τ∂_vv=0, ∂_τ∂_ww=0, ∂_v∂_vτ =0, ∂_w∂_wτ =0. Hence we get2∂_v∂_vτ -∂_τ∂_vv=0, 2∂_w∂_wτ -∂_τ∂_ww=0. The situation is more interesting when the family (τ ) is of second degree. Hence let us suppose that the family (τ ) is given byax^2+by^2+cz^2=1,where a, b, c are functions of the parameter τ. By this assumption, equations (<ref>) and (<ref>) yieldaxv_x+byv_y+czv_z=0,axw_x+byw_y+czw_z=0.As well, from equation (<ref>) we getav_x^2+bv_y^2+cv_z^2=0, aw_x^2+bw_y^2+cw_z^2=0.Equations (<ref>), (<ref>), (<ref>) and (<ref>) imply(a^2b'+b^2a')(xv_y-yv_x)^2+(b^2c'+c^2b')(yv_z-zv_y)^2 +(c^2a'+a^2c')(zv_x-xv_z)^2=0, ab(xv_y-yv_x)^2+bc(yv_z-zv_y)^2+ca(zv_x-xv_z)^2=0,in which '=d/dτ. Analogously for w we find(a^2b'+b^2a')(xw_y-yw_x)^2+(b^2c'+c^2b')(yw_z-zw_y)^2 +(c^2a'+a^2c')(zw_x-xw_z)^2=0,ab(xw_y-yw_x)^2+bc(yw_z-zw_y)^2+ca(zw_x-xw_z)^2=0.The two equations in (<ref>) and the two equations in (<ref>) become equivalent ifa^2b'+b^2a'/ab=b^2c'+c^2b'/bc=c^2a'+a^2c'/ac.If in (<ref>) we substitute a, b, c respectively by 1/t_1, 1/t_2, 1/t_3, then from (<ref>) we get that the familyx^2/t_1+y^2/t_2+z^2/t_3=1,is orthogonal to both Σ and Σ' if t_1, t_2, t_3 satisfy the followingt_3( dt_1/ dτ+ dt_2/dτ)=t_2( dt_1/ dτ+ dt_3/ dτ)=t_1( dt_2/ dτ+ dt_3/ dτ).A particular case of the equation (<ref>), which is known as Darboux–Halphen system, is given in (<ref>). In 1881, G. Halphen <cit.> studied this system of differential equations and expressed a solution of it in terms of the logarithmic derivatives of the theta functions; namely,t_1= 2 (lnθ_2(τ ))', t_2=2 (lnθ_3(τ ))',t_3=2 (lnθ_4(τ ))'.withθ_2(τ ):=∑_n=-∞^∞ q^1/2(n+1/2)^2,θ_3(τ ):=∑_n=-∞^∞ q^1/2n^2, θ_4(τ ):=∑_n=-∞^∞ (-1)^nq^1/2n^2, q=e^2π i τ,τ∈ℍ.These theta functions can be written in terms of the more general theta functions with characteristics r and s and arguments z and σ:ϑ[r,s](z,σ)=∑_m∈ℤexp{π i(m+r)^2σ+2π i (m+r)(z+s)}, z,r,s∈ℂ,σ∈ℍ,such thatθ_2(τ )= ϑ[1/2,0](0,τ ),θ_3(τ )= ϑ[0,0](0,τ ),θ_4(τ )= ϑ[0,1/2](0,τ ). § RAMANUJAN RELATIONS BETWEEN EISENSTEIN SERIESThe following differential equationq∂ E_2/∂ q=1/12(E_2^2-E_4),q∂ E_4/∂ q=1/3(E_2E_4-E_6),q∂ E_6/∂ q=1/2(E_2E_6-E_4^2),where E_i's are the Eisenstein seriesE_2i(q):=1+b_i∑_n=1^∞ (∑_d| nd^2i-1 )q^n, i=1,2,3,and (b_1,b_2,b_3)=(-24, 240, -504), was discovered by Ramanujan in <cit.> and it is mainly known as Ramanujan's relations between Eisenstein series. Ramanujan was a master of formal power series and had a very limited access to the modern mathematics of his time. In particular, he and many people in number theory didn't know that the differential equation (<ref>) had already been studied by Halphen in his book <cit.>, thirty years before S. Ramanujan. The equalities of the coefficients of x^i in4(x-t_1)(x-t_2)(x-t_3)=4(x-a_1E_2)^3-a_2E_4(x-a_1E_2)-a_3E_6,where(a_1,a_2,a_3):=(2π i/12, 12(2π i/12)^2, 8(2π i/12)^3),gives us a map from ℂ^3 into itself which transforms Darboux–Halphen into Ramanujan differential equation, see <cit.>.§ SELF-DUALITY IN BIANCHI IX METRICSAn instanton is a field configuration that vanishes at spacetime infinity. It is the quantum effect that leads metastable states to decay into vacuum. It is a phenomenon that takes place in usual spacetime with signature (-,+,+,+) but in order to perform physical calculation we use its equivalence with a soliton solution (static and energetically stable field configuration) in Euclidean spacetime.In Yang–Mills theory, self-duality of the field strength F_μν=ϵ_μνρσF^ρσ in four spacetime dimensions is a widely known instanton configuration <cit.>. Similarly, self-duality constraint on the curvature two-form (and connection 1-form) in Cartan's formalism of general relativity characterizes a gravitational instanton. An important feature of self-duality of the curvature is that the Ricci-tensor vanishes and it is a solution of the vacuum Einstein equations. Also, self-dual curvature leads to solving a linear differential equation, a task much easier than solving the full non-linear Einstein equations. Gravitational instantons were found in Bianchi IX metrics, by Gibbons and Pope <cit.>. Without realizing it, they arrived at Darboux–Halphen system from self-duality constraints.In <cit.>, L. Bianchi studied continuous isometries of 3-dimensional spaces. He noticed that the continuous isometries (continuous motion that preserve ds^2) of a space form a finite-dimensional Lie group and he classified such spaces according to the corresponding group of isometries. Bianchi IX corresponds to a 3-dimensional space with SO(3) or SU(2) as Lie group of isometries. When we consider it in the context of 4-dimensional cosmology, the isometries lie in the 3 spacial directions <cit.>, but since we are working in Euclidean signature we consider the isometry group SO(3) as a subgroup of SO(4). In this configuration, as the instanton vanishes at infinity, Lorentz symmetry is recovered and the space is called asymptotically locally Euclidean (ALE). This same manifold describes the reduced[Moduli of charge 2 monopoles reduced by quotient by ℝ^3 action.] moduli M^0_2 of charge 2 monopoles in a SU(2) Yang–Mills–Higgs theory.A magnetic 2-monopole is a soliton solution of charge 2 of Bogomolny equations in the Yang–Mills–Higgs theory in ℝ^3, where SU(2) Yang–Mills is a gauge theory of 1-form connections A on a principal SU(2)-bundle while the Higgs field Φ correspond to a section of an associated 𝔰𝔲(2)-bundle <cit.>. In <cit.>, Atiyah and Hitchin showed that the reduced moduli M^0_2 of 2-monopoles is a 4-dimensional hyperkähler manifold and an anti-self-dual (curvature-wise) Einstein manifold. Since M^0_2 admits SO(3) isometry, the metric is a Bianch IX[Note that here the four coordinates of the moduli are not spacetime directions, but internal parameters of a 2-monopole solution.] (<ref>). This is a consequence of the hyperkähler structure of M^0_2 which has an S^2-parameter family of complex structures, i.e., if I, J, K are covariantly constant complex structures in M^0_2 then aI+bJ+cK is also a covariantly constant complex structure in M^0_2 given that a^2+b^2+c^2=1.Here we present a detailed derivation of the Darboux–Halphen system starting from the Euclidean Bianchi IX metric with SO(3) symmetry with an imposition of the constraints of self-duality at the level of Riemann curvature. The constraint of anti-self-duality yields an anti-instanton, a solution with negative instanton number and we present this solution together by using ±sign. We follow the steps of <cit.> and <cit.>, see also <cit.>. §.§ Geometric analysisA metric for a 4-dimensional spacetime with coordinates (x^1,x^2,x^3,x^4), Euclidean time x^4 and SO(3)⊂ SO(4) isometry is written in terms of invariant 1-forms σ^i on SO(3), dual to the standard basis X_1, X_2, X_3 of its Lie algebraσ^i = - 1/r^2η^i_μν x^μ dx^ν,where μ, ν = 1,2,3,4 and i= 1,2,3 and η^i_μν is a 't Hooft symbol given byη^i_μν = ε_iμν+δ_iμδ_ν 4-δ_μ 4δ_iνor, ε_iμν-δ_iμδ_ν 4+δ_μ 4δ_iν.according to two different choices of 𝔰𝔬(3) generators in the Lie algebra of the group SO(4)=SU(2)× SU(2). Among the symbols presented above, ε_iμν is the Levi-Civita symbolε_123=ε_231=ε_312=-ε_213=-ε_321=-ε_132=1,and zero elsewhere,and δ_iμ refers to the Kronecker delta. The σ^i's obey the structure equation:dσ^i = - ε_ijkσ^j ∧σ^k ,where we use Einstein summation in the repeated upper and lower indices here and what follows below. This choice of SO(3) isometry leads to a 4D spherically symmetric Bianchi IX metric ds^2 = c_0(r)^2dr^2 + c_1^2 (r) (σ^1)^2 + c_2^2 (r) (σ^2)^2 + c_3^2 (r) (σ^3)^2 ,with r=√(x_1^2+x_2^2+x_3^2+x_4^2), c_0 (r) = c_1 (r) c_2 (r) c_3 (r) and c_1, c_2, c_3 being functions of r.We can impose self-duality in Bianchi IX metric in two ways: =0pt1) connection wise self-duality,2) curvature wise self-duality.The connection wise self-duality is a stronger form of self-duality that leads to self-dual curvature tensor <cit.>. This form of self-duality does not present Darboux–Halphen system, but the Lagrange or Euler-top system <cit.>. It is not in our goal to describe it here. One can perform a standard analysis using vierbeins, leading to Cartan's structure equation. The vierbeins could be chosen as e^0 = c_0dr,e^i = c_i σ^i(no sum in i),i = 1, 2, 3, and the connection 1-form can be obtained from the structure equation de^a= e^b∧ω^a_b, where a, b=0,1,2,3. Obviously, e^0 produces no connections while other three does d e^0 = 0, d e^i = ∂_r c_idr ∧σ^i - c_i ε_ijkσ^j ∧σ^k.The first term on the r.h.s. above gives ω^i_0 while the second term needs to be rewritten in order to produce a antisymmetric connection 1-form ε_ijkc_i^2/c_iσ^j ∧σ^k = ε_ijk2c_i^2 + ( c_j^2 - c_k^2 ) - ( c_j^2 - c_k^2 )/ 2c_iσ^j ∧σ^k = ε_ijkc_i^2 + c_j^2 - c_k^2 / 2c_i c_j e^j ∧σ^k + ε_ikjc_i^2 + c_k^2 - c_j^2 / 2c_i c_k e^k ∧σ^j = ε_ijkc_i^2 + c_j^2 - c_k^2 / c_i c_j e^j ∧σ^k . Rewriting (<ref>), d e^i = - ∂_r c_i/c_0σ^i ∧ e^0 - ε_ijkc_i^2 + c_j^2 - c_k^2 / c_i c_j e^j ∧σ^k .Hence, ω^i_0 = ∂_r c_i/c_0σ^i(no sum in i) ,ω^i_j = - ε_ijkc_i^2 + c_j^2 - c_k^2 / c_i c_jσ^k.Here the connection 1-form components are anti-symmetric under permutation of its indices. §.§ Curvature wise self-duality and Darboux–Halphen system Curvature-wise self-duality was first studied in search of gravitational instantons. It is a more general solution than imposing self-duality on connection 1-forms. The Cartan-structure equation for Ricci tensor isR_ij =dω_ij + ω_im∧ω^m_j.The (anti-)self-duality of curvature demands thatR_0i =±1/2ε_0ilm R^lm = ± R_jk,where {i,j,k}, in this order, are a cyclic permutation of {1,2,3} and we used the fact that Euclidean vierbein indices are raised and lowered with Kronecker deltas δ^i_j. Comparing the l.h.s. and r.h.s. of (<ref>), we haved(ω_0i∓ω_jk)=±(ω_0k∓ω_ij)∧(ω_0j∓ω_ki),d(λ_1(r)σ^i)=±(λ_3(r)σ^k)∧(λ_2(r)σ^j)=λ_3(r)λ_2(r)σ^k∧σ^j,∂_rλ_1 dr∧σ^i+λ_1dσ^i=∓λ_2λ_3σ^j∧σ^k,where the second line comes from equation (<ref>) with λ_1, λ_2, λ_3 being functions of r. But the third line and (<ref>) show that λ_i's are constants and λ_1=±12λ_2λ_3. From cyclicity of i, j, k we obtain two more copies of (<ref>). Therefore, =0pt 1) λ_1=λ_2=λ_3=0 or 2) (λ_1)^2=(λ_2)^2=(λ_3)^2=4 with λ_1λ_2λ_3=± 8.The first case leads to self-dual connection 1-forms and Euler-top system, while the second case can be resumed to λ_1=λ_2=λ_3=±2 by an appropriate change of sign in c_i <cit.>. Therefore, from equations (<ref>) and (<ref>) we get( ∂_r c_i/c_0) = ∓(c_j^2 + c_k^2 - c_i^2 / c_j c_k -2), ∂_r ( ln c_i^2 ) = ∓ 2( c_j^2 + c_k^2 - c_i^2 - 2c_j c_k ) .One may suppose that we must parametrize the l.h.s. to match the linear form in c^2_i, c^2_j and c^2_k of the r.h.s. in the equation above. Essentially, the derivative operator aside, c_i^2 must be parametrized such thatln c_i^2= lnΩ_j + lnΩ_k - lnΩ_i +const= ln( Ω_j Ω_k/Ω_i)+const. We choose new parametrization ( c_i )^2 = Ω_j Ω_k/2Ω_i ⇒ Ω_i = 2c_j c_k. which enable us to decouple the individual parameters into their own equations turning into simpler expressions. This allows us to continue our analysis ∂_r [ ln( Ω_j Ω_k/2Ω_i) ] = Ω̇_j/Ω_j + Ω̇_k/Ω_k - Ω̇_i/Ω_i = ∓( Ω_k Ω_i/Ω_j + Ω_i Ω_j/Ω_k - Ω_j Ω_k/Ω_i -2Ω_i). Adding up the above equation with cyclic permutations of i, j, k we will find that (anti-)self-dual cases of the Bianchi IX metric gives us Ω̇_j/Ω_j + Ω̇_k/Ω_k - Ω̇_i/Ω_i = ∓( Ω_k Ω_i/Ω_j + Ω_i Ω_j/Ω_k - Ω_j Ω_k/Ω_i -2Ω_i) +Ω̇_k/Ω_k + Ω̇_i/Ω_i - Ω̇_j/Ω_j = ∓( Ω_i Ω_j/Ω_k + Ω_j Ω_k/Ω_i - Ω_k Ω_i/Ω_j -2Ω_j)↓ ⇒ Ω̇_k/Ω_k = ∓ 2 (Ω_i Ω_j/Ω_k-Ω_i-Ω_j)⇒ Ω̇_k = ∓ (Ω_i Ω_j-Ω_kΩ_i-Ω_kΩ_j),where throughout derivative (denoted by dot) is taken with respect to r. Self-duality proceeds to give us the classical Darboux–Halphen systemΩ̇_i+Ω̇_j = 2Ω_i Ω_j .§.§ General Bianchi IX self-dual Einstein metric Following <cit.>, we rewrite the Bianchi IX by adding a conformal scaling term F in the metricds^2= F( dt^2+σ^2_1/Ω^2_1+σ^2_2/Ω^2_2+σ^2_3/Ω^2_3).where t is the cosmological time and different from before, here the isometry is SU(2) and (σ_i) are the corresponding SU(2) invariant forms along the spacial directions with structure constantdσ_1=σ_2∧σ_3, dσ_2=σ_3∧σ_1, dσ_3=σ_1∧σ_2.We define the new variables A_i(t) by the equations∂_t Ω_i=-Ω_jΩ_k+Ω_i(A_j+A_k),for distinct i, j and k taking values in the set {1,2,3}. The curvature-wise self-duality condition is expressed in terms of the new variables A_i in the form of the Darboux–Halphen system∂_t A_i= -A_jA_k+A_i(A_j+A_k).Therefore we find Ω_i's by first solving system (<ref>) and applying its solution in (<ref>). A non-trivial solution is given by (<ref>)A_1= 2 ∂/∂ t(lnθ_2(it)), A_2=2 ∂/∂ t(lnθ_3(it)),A_3=2 ∂/∂ t(lnθ_4(it)).For simplicity, we rename ϑ_2≡θ_2(it), ϑ_3≡θ_3(it), ϑ_4≡θ_4(it). The system (<ref>) thus becomes∂_tΩ_1=-Ω_2Ω_3+2Ω_1∂_tln (ϑ_3ϑ_4), ∂_tΩ_2=-Ω_3Ω_1+2Ω_2∂_tln (ϑ_4ϑ_2), ∂_tΩ_3=-Ω_1Ω_2+2Ω_3∂_tln (ϑ_2ϑ_3).There is a class of solutions of this system that satisfies vacuum Einstein equationsR_ab-1/2Rg_ab+Λ g_ab=0 ,once we choose the appropriate conformal factor F <cit.>. This class depend on the values of the cosmological constant Λ and satisfy the constraintϑ^4_2Ω_1^2-ϑ^4_3Ω_2^2+ϑ^4_4Ω_3^2=π^2/4ϑ^4_2ϑ^4_3ϑ^4_4,The general two-parametric family of solutions of the system (<ref>) satisfying condition (<ref>), is given by the following formulasΩ_1=-i/2ϑ_3ϑ_4 d/ dqϑ[p,q+12]/e^π i pϑ[p,q],Ω_2=i/2ϑ_2ϑ_4 d/ dqϑ[p+12,q+12]/e^π i pϑ[p,q], Ω_3=i/2ϑ_2ϑ_4 d/ dqϑ[p+12,q+12]/e^π i pϑ[p,q], where ϑ[p,q] denotes the theta function ϑ[p,q](0,ir), p,q∈ℂ. The corresponding metric is real and satisfies the Einstein equations for negative cosmological constant Λ if p∈ℝ and ℛ{q}=12 (real part of q) or for positive cosmological constant if q∈ℝ and ℛ{p}=12. In both the cases the corresponding conformal factor is given by F=2/πΛΩ_1Ω_2Ω_3/(d/ dqlnϑ[p,q])^2. There is another family of solutions Ω_1=1/t+q_0+2∂/∂ tlnϑ_2,Ω_2=1/t+q_0+2∂/∂ tlnϑ_3,Ω_3=1/t+q_0+2∂/∂ tlnϑ_4, with q_0∈ℝ, that defines manifolds with vanishing cosmological constant if F=C(t+q_0)^2Ω_1Ω_2Ω_3.§ GAUSS–MANIN CONNECTION IN DISGUISEIn this section we explain how one can derive the Darboux–Halphen equations from the Gauss–Manin connection of a universal family of elliptic curves. This has been taken from the references <cit.>. The family of elliptic curvesE_t y^2-4(x-t_1)(x-t_2)(x-t_3)=0, t∈ℂ^3\∪_i,j{t_i=t_j},is the universal family for the moduli of 3-tuple (E,(P,Q),ω), where E is an elliptic curve and ω∈ H^1_ dR(E)\ F^1. There is a unique regular differential 1-form in the Hodge filtration ω_1∈ F^1, such that ⟨ω,ω_1⟩=1 and ω, ω_1 together form a basis of H^1_ dR(E). P and Q are a pair of points of E that generate the 2-torsion subgroup with the Weil pairing e(P,Q)=-1. The points P and Q are given by (t_1,0) and (t_2,0) and ω=x dx/y and ω_1= dx/y. The Gauss–Manin connection of the family of elliptic curves E_t written in the basis dx/y, x dx/y is given as bellow∇[ dx/y; x dx/y ]= A [ dx/y; x dx/y ],whereA = dt_1/2(t_1-t_2)(t_1-t_3)-t_11t_2t_3-t_1(t_2+t_3)t_1 + dt_2/2(t_2-t_1)(t_2-t_3)-t_21t_1t_3-t_2(t_1+t_3)t_2 +dt_3/2(t_3-t_1)(t_3-t_2)-t_31t_1t_2-t_3(t_1+t_2)t_3.The reader who is not familiar with the Gauss–Manin connection must replace ∇ with d∫_δ_t, where t_i's are assumed to depend on some parameter τ, d=∂/∂τ and δ_t is a 1-dimensional homology class in E_t. In the parameter space of the family of elliptic curves E_t there is a unique vector field R, such that∇_R( dx/y)= -x dx/y,∇_R(x dx/y)= 0.The vector field R is given by the Darboux–Halphen system (<ref>) and it is called Gauss–Manin connection in disguise.§ FROBENIUS MANIFOLDS AND CHAZY EQUATIONFrobenius manifolds were developed in order to give a geometrical meaning to WDVV equations:∂^3 F(t)/∂ t^α∂ t^β∂ t^λη^λμ∂^3 F(t)/∂ t^μ∂ t^γ∂ t^δ =∂^3 F(t)/∂ t^δ∂ t^β∂ t^λη^λμ∂^3 F(t)/∂ t^μ∂ t^γ∂ t^α,where F(t), with t=(t^1,t^2,…,t^n), is a quasi-homogeneous function on its parameters. The above equations conceal properties of an associative commutative algebra on the tangent space of a manifold M of dimension n defined by the parameter space (t^1,t^2,…,t^n). That's the essence of a Frobenius manifold that we will detail below starting with the algebraic structure in TM. §.§ Frobenius algebra An algebra A over ℂ is Frobenius if * it is a commutative associative ℂ-algebra with unity e,* it has a ℂ-bilinear symmetric non-degenerate inner product⟨,⟩ A× A⟶ℂ,(a , b)↦⟨ a,b ⟩,which is invariant, i.e., ⟨ a.b,c ⟩ =⟨ a,b.c ⟩Properties: Let e_α, α=1,…, N, be any basis in A, such that e_1=e is the unity. By notation, we define η_αβ:=⟨ e_α,e_β⟩, which yields the matrix η:=[η_αβ]_1≤α, β≤ N and its inverse η^-1:=[η^αβ]_1≤α, β≤ N, and it follows η^αβη_βγ=δ^α_γ. By writing e_α· e_β in the given basis, we find the structure constants c_αβ^γ defined by e_α· e_β= c_αβ^γ e_γ. If we set c_αβγ=c_αβ^ϵη_ϵγ, then we get c_αβ^γ=c_αβϵη^ϵγ. Note that in all above expressions, and in what follows, Einstein summation of indices is implicit. Therefore, η_αβ and the structure constants c_αβ^γ satisfycommutativityη_αβ=η_βα,associativity (e_α. e_β).e_γ=e_α.( e_β.e_γ) ∴c_αβ^ϵc_ϵγ^δ=c_αϵ^δc_δβ^ϵ,normalization c_1β^α=δ_β^α, invariance&commutat. c_αβγ= ⟨e_αe_β, e_γ⟩=c_βαγ=c_αγβ. Now consider an n-parametric deformation of the Frobenius algebra A_t, t= (t^1,t^2,…,t^n), with structure constants c_αβ^γ(t) preserving relations (<ref>) to (<ref>). Such deformed algebra A_t can be seen as a fiber bundle with the space of parameters t∈ M as base space. We identify this fiber bundle with the tangent bundle TM to arrive at the definition of a Frobenius manifold. The requirements for this to happen are presented in the definition below. §.§ Frobenius manifold A Frobenius manifold M of dimension n, is an n-dimensional Riemannian manifold, such that for all t∈ M the tangent space T_tM contains the structure of a Frobenius algebra (A_t, ⟨ ,⟩_t), satisfying the following axioms: =0pt A.1. The metric ⟨ , ⟩_t on M is flat. The unit vector e must be flat, i.e., ∇ e = 0, where ∇ is the Levi-Civita connection for the metric. A.2. Let c be the 3-tensor c(x,y,z) = ⟨ x.y ,z ⟩, with x, y, z ∈ T_tM. Then the 4-tensor (∇_w c)(x, y,z) must be symmetric in x,y,z,w ∈ T_tM. A.3. A linear vector field E must be fixed on M, i.e., ∇( ∇ E) = 0 such that the corresponding one-parameter group of diffeomorphisms acts by conformal transformations of the metric ⟨,⟩ and by rescaling on the Frobenius algebras T_tM.The flatness of the metric ⟨ ,⟩ implies the existence of a system of flat coordinates t^1, …, t^n on M. In these flat coordinates the structure constants of A_t are given by∂/∂ t^α.∂/∂ t^β = c_αβ^γ(t) ∂/∂ t^γ. Potential deformation. If there is a function F(t), called potential, such that the structure constants of A_t, t∈ M, can be locally represented asc_αβγ(t) = ∂^3 F(t)/∂ t^γ∂ t^α∂ t^β, satisfying A.2 with unity vector e=∂∂ t^1, and the metric given byη_βγ=c_1 βγ = ∂^3 F(t)/∂ t^γ∂ t^1∂ t^βs.t.∂^4 F(t)/∂ t^α∂ t^γ∂ t^1∂ t^β=0, satisfying A.1, and the associativity property (<ref>) represented by the WDVV equations∂^3 F(t)/∂ t^α∂ t^β∂ t^λη^λμ∂^3 F(t)/∂ t^μ∂ t^γ∂ t^δ =∂^3 F(t)/∂ t^δ∂ t^β∂ t^λη^λμ∂^3 F(t)/∂ t^μ∂ t^γ∂ t^α, then M is a Frobenius manifoldand the Frobenius algebra A_t is called a potential deformation.Note that the condition A.3 is satisfied by a quasihomogeneous function F(t). Let M = 3, and consider the basis e = e_1=∂∂ t^1, e_2=∂∂ t^2 and e_3=∂∂ t^3 of the 3-dimensional algebra A_t. Then the multiplication law is given bye_2^2 = f_xxye_1 + f_xxxe_2 + e_3,e_2 e_3 = f_xyye_1 + f_xxye_2,e_3^2 = f_yyye_1 + f_xyye_2,where the funtion F(t) has the form F(t) = 1/2 (t^1)^2t^3 + 1/2 t^1(t^2)^2 + f(t^2,t^3) and the notation f_x= ∂_x f(x,y), f_y= ∂_y f(x,y). The associativity condition (e_2^2) e_3 = e_2 (e_2 e_3) implies the following PDE for f(x,y):f^2_xxy = f_yyy + f_xxxf_xyy. §.§ Chazy equation and Darboux–Halphen systemIn this section we explain how the Chazy equation arises from a 3-dimensional Frobenius manifold. We follow Dubrovin's notes <cit.>. Let dim M = 3, and consider the potential functionF(t)= 1/2(t^1)^2t^3 + 1/2 t^1(t^2)^2 - (t^2)^4/16γ(t^3),where γ(τ) is an unknown 2π-periodic function that is analytic at τ=i ∞. Then the associativity condition (<ref>) leads to the Chazy equationγ”'= 6γγ”-9(γ')^2.The solution, up to a shift in τ, is given by γ(τ)= π i/3E_2(τ), where E_2 is the weight-2 Eisenstein series. Notice that the Darboux–Halphen solution (<ref>) leads tot_1+t_2+t_3= π i/2E_2(τ),which can be easily checked from (<ref>) or by writing the theta functions in terms of Dedekind eta function, see <cit.>. Applying τ derivatives on both sides and using Darboux–Halphen equations, one can also check that the solution to Darboux–Halphen system (<ref>) are the roots of the cubic equationy^3-3/2γ(τ)y^2+3/2γ'(τ)y -1/4γ”(τ)=0. § CONCLUSIONThe study of Darboux–Halphen equations in several different problems in theoretical physics and mathematics raised more and more questions that eventually lead us to further studies.The problem involving Gauss–Manin connection in disguise lies at the center of some questions. It shows that the Darboux–Halphen system corresponds to a vector field in the moduli of an enhanced elliptic curve. As mentioned in Section <ref>, the Bianchi IX four-manifold (<ref>) also describes the reduced moduli of 2-monopoles and its self-dual curvature equations can be reparametrized to the Darboux–Halphen equations. Furthermore, in the problem of 2-monopoles it has been found that a 2-monopole solution relates to an elliptic curve as its spectral curve <cit.>. Therefore, we believe that Gauss–Manin connection in disguise is a new way to demonstrate the association of spectral curves and the curvature equations of the moduli of monopole solutions. Starting from these coincidences, in <cit.> one of the authors started to find more evidences to support this idea.Another interesting remark is the fact that potential functions and structure constants in Frobenius manifolds correspond to prepotentials (or genus zero topological partition function) and Yukawa couplings in topological string theory and Gauss–Manin connection in disguise has been used in the moduli of enhanced Calabi–Yau varieties to find polynomial expressions for Yukawa couplings and higher genus topological partition functions <cit.>. It would be interesting to find cases where the moduli of enhanced Calabi–Yau varieties are also Frobenius manifolds. In particular, the Frobenius manifold presented in Section <ref> is a case of modular Frobenius manifold where the prepotential is preserved under a inverse symmetry that acts as an S generator of the modular group SL(2,ℤ) in t^3 direction <cit.>. Such modularity is a desirable property that can establish a relation to Gauss–Manin connection in disguise and may be extended to the group of transformations of Calabi–Yau modular forms <cit.>. §.§ Acknowledgements During the period of preparation of the manuscript MACT was fully sponsored by CNpQ-Brasil. The research of RR was supported by FAPESP through Instituto de Fisica, Universidade de Sao Paulo with grant number 2013/17765-0. The work was initiated during the visit of RR to IMPA, he would like to thank IMPA for the hospitality during the course of this project.[1]Referencesref 99 =0ptMR1713583 Ablowitz M.J., Chakravarty S., Halburd R., On Painlevé andDarboux–Halphen-type equations, in The Painlevé Property, CRM Ser.Math. Phys., Springer, New York, 1999, 573–589.Alim:2014dea Alim M., Movasati H., Scheidegger E., Yau S.T., Gauss–Manin connection indisguise: Calabi–Yau threefolds, https://doi.org/10.1007/s00220-016-2640-9Comm. Math. 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"authors": [
"John Alexander Cruz Morales",
"Hossein Movasati",
"Younes Nikdelan",
"Raju Roychowdhury",
"Marcus A. C. Torres"
],
"categories": [
"math.DG",
"hep-th",
"math-ph",
"math.MP"
],
"primary_category": "math.DG",
"published": "20170927181140",
"title": "Manifold Ways to Darboux-Halphen System"
} |
#1⟨#1⟩ Re ImPhys. Rev. BPhys. Rev. APhys. Rev. Lett.∥#1#2d#1/d#2#1#2∂ #1/∂ #2#1#2#3∂^2 #1/∂ #2 ∂ #3#1∇ #1#1∇× #[email protected] of Physics, The University of Texas at Austin, Austin, TX 78712, USADepartment of Physics, Colorado State University, Fort Collins, CO 80523, USASchool of Advanced Materials Discovery, Colorado State University, Fort Collins, CO 80523, USADepartment of Physics, The University of Texas at Austin, Austin, TX 78712, USADepartment of Physics, The University of Texas at Austin, Austin, TX 78712, USAICQM and CICQM, School of Physics, Peking University, Beijing 100871, China To give a general description of the influences of electric fields or currents on magnetization dynamics, we developed a semiclassical theory for the magnetization implicitly coupled to electronic degrees of freedom. In the absence of electric fields the Bloch electron Hamiltonian changes the Berry curvature, the effective magnetic field, and the damping in the dynamical equation of the magnetization, which we classify into intrinsic and extrinsic effects. Static electric fields modify these as first-order perturbations, using which we were able to give a physically clear interpretation of the current-induced spin-orbit torques. We used a toy model mimicking a ferromagnet-topological-insulator interface to illustrate the various effects, and predicted an anisotropic gyromagnetic ratio and the dynamical stability for an in-plane magnetization. Our formalism can also be applied to the slow dynamics of other order parameters in crystalline solids. 75.78.-n,75.60.Jk,75.76.+jGeometric Dynamics of Magnetization: Electronic Contribution Qian Niu December 30, 2023 ============================================================Introduction—Magnetization dynamics is conventionally described by the phenomenological Landau-Lifshitz-Gilbert (LLG) equation, in which the effective magnetic field and the damping factor can be associated with various mechanisms such as dipolar interaction, exchange coupling, electron-hole excitations, etc., through microscopic theories. The recently discovered current-induced spin-orbit torques emerge as current-dependent modifications to the LLG equation, and can be consequently categorized as field-like and damping-like torques <cit.>. In systems with strong spin-orbit coupling and broken inversion symmetry, e.g. GaMnAs, heavy-metal/ferromagnet bilayers and magnetically doped topological insulator heterostructures, magnetization switching using electric current alone through the spin-orbit torque has been achieved experimentally <cit.>. Theoretical studies of spin-orbit torques have mostly adopted s-d type couplings between transport electrons and those contributing to magnetization <cit.>, or a self-consistent-field picture based on the spin density functional theory <cit.>. Then the spin-orbit torques can be understood as the modification to the effective exchange fields proportional to the current-induced spin densities in inversion symmetry breaking systems, known as the Edelstein effect <cit.>. However, in general neither the size of the exchange field nor its dependence on order parameter (magnetization) direction is known a priori<cit.>. It is thus more desirable to develop a theoretical framework that does not explicitly depend on the details of the coupling between transport electrons and the magnetization.In this Rapid Communication, we provide a semiclassical framework for the dynamics of magnetization implicitly coupled to electronic degrees of freedom, based on the wave-packet method. We found that the Bloch electrons yield a Berry curvature Ω_mm, acting as a magnetic field in the magnetization space, while the gradient of the electronic free energy with respect to the magnetization acts as a static electric field in the magnetization space, in agreement with previous adiabatic theory of magnetization dynamics <cit.>. These two fields thus govern the dynamics of magnetization as that of Lorentz force to a charged particle. In addition, we identified an extrinsic contribution to the magnetization dynamics, corresponding to the Gilbert damping in the LLG equation, which is not included in the adiabatic theory. A static electric field enters the magnetization equation of motion by modifying the Berry curvature Ω_mm, the effective field, and the damping factor as a first-order perturbation. In particular, the modification to the effective field includes a part proportional to the Berry curvature Ω_mk and having a geometric nature. We used a simplified model for the ferromagnet-topological-insulator interface to illustrate the various effects, and showed that the gyromagnetic ratio is renormalized anisotropically and that an in-plane magnetization can be dynamically stable under moderate electric fields.Formulation and general results—We start from a general Hamiltonian of Bloch electrons implicitly depending on the order parameter m, Ĥ_e(q;m), where q is the crystal momentum. External electromagnetic fields are described by the scalar and vector potentials (ϕ,A ) that enter the Hamiltonian through minimum coupling (ħ = 1, e = |e|), Ĥ=Ĥ_e(q + eA;m)-eϕ. Following Ref. sundaram:1999aa, a wave packet is constructed with center position x and center physical momentum k from the Bloch eigenstates of the local electronic Hamiltonian. The Lagrangian of a single wave packet reads as = ẋ·[k-e A(x,t)] + k̇· A_k + ṁ· A_m - [ε-eϕ(x,t)],with A_λ = i ⟨u|∇_λ u⟩(λ= k or m) the Berry connections of the Bloch state |u⟩, and ε the wave packet energy. For notational simplicity we have dropped the band index. The Lagrangian depends on (x,k) of the wave packets and magnetization m. Thus a set of coupled equations of motion for all three variables can be derived from the Lagrangian principle <cit.>: k̇ = -eE,ẋ =εk + k̇·Ω_kk+ṁ·Ω_mk,∫ [dk] f(ṁ·Ω_mm+ k̇·Ω_km+εm )=0, where the Berry curvaturesΩ_λ_iλ_j = -2Im⟨∂ u /∂λ_i| ∂ u/ ∂λ_j⟩, λ = k or m. Eq. <ref> is obtained by summing over all occupied states, and f is the distribution function for the electrons. Note the magnetization dynamics enters the electron equations of motion through Ω_kk in Eq. <ref>, and the terms in the square brackets of Eq. <ref> can be viewed as conjugates of the right hand side of Eq. (<ref>), by interchanging k and m. This is a manifestation of the reciprocity between charge pumping due to magnetization precession and electric-current-induced spin-orbit torque.The nonequilibrium response of the electrons to an external electric field and/or a dynamical m is accounted for using the semiclassical Boltzmann equation, according to which the deviation of the distribution function from the equilibrium Fermi-Dirac distribution f_0[ε(k,m)] is δ f = - τf_0ε (k̇·εk + ṁ·εm), where we have assumed a grand canonical ensemble with fixed temperature and chemical potential. τ is the relaxation time which we take as a constant for simplicity. Generalization to including more specific scattering mechanisms is straightforward but involved, and does not necessarily provide additional insight on the main issues considered in this work. The equations (<ref>-<ref>) complete our semiclassical description of coupled magnetization and electron dynamics in the presence of external electric fields, though they can be easily extended to including magnetic fields and other perturbations. In the absence of electric fields, k̇ = 0, and we can obtain from Eq. (<ref>) and Eq. (<ref>) the following equations of motion of the magnetization, ṁ· (Ω̅_mm+η_mm) - H= 0,in getting which we have ignored higher order ṁ^2 terms by assuming that the magnetization dynamics is slow compared to typical electronic time scales. The Berry curvature Ω̅, the damping coefficient η and the effective field H in the equation above are respectively Ω̅_mm = ∫ [dk] f_0 Ω_mm, η_mm =-τ∫ [dk] f_0εεmεm,H = -Gm,where G is the free energy of the electron system. For non-interacting electrons G = -β^-1∫ [dk] ln[1+e^-β(ε-μ)] for a single band, where β = 1/k_B T. Interaction effects may be included in G through different levels of approximations, which will also modify the way magnetization appears in G. At this point we will leave G as a general electron free energy depending on m implicitly.We only consider the transverse modes (ṁ perpendicular to m) of the magnetization dynamics in this work, although Eq. <ref> can be used for the longitudinal mode as well. The magnetization is thus described by the polar angle θ and the azimuthal angle ϕ. Eq. (<ref>) can then be converted to the familiar form of the LLG equation, ṁ = - γm×(H - η_mm·ṁ),where the gyromagnetic ratio γ is related to the Berry curvature through Ω̅ =m/ γ m^2, where Ω̅_i = ε_ijkΩ̅_jk/2 is the vector form of the Berry curvature tensor. Expressions similar to Eq. (<ref>), but without the damping term, have been derived using the adiabatic theory <cit.>. Since the damping term is explicitly dependent on the relaxation time, which is ultimately due to dissipative microscopic processes such as electron-phonon scattering and electron-impurity scattering, we call it extrinsic contribution to the magnetization dynamics. Note Eq. <ref> suggests η is positive definite, which means it always leads to energy dissipation through Eq. <ref>. The remaining terms are intrinsic contributions from the electron degrees of freedom. In particular, from Eq. (<ref>) one can see that the two intrinsic terms are formally similar to the Lorentz force of a charged particle, with the antisymmetric part of Ω_mm (or equivalently the vector form Ω_m) analogous to the magnetic field and H playing the role of the electric field. Electric fields enter our formalism through the equation of motion for k [Eq. (<ref>)], which makes the 2nd term in the integrand of Eq. <ref> nonzero and also contributes to the nonequilibrium distribution function δ f inEq. (<ref>). After some algebra, we arrive at the same equation as Eq. (<ref>), but with H, Ω̅_ mm, and η_ mm acquiring the following corrections proportional to the electric field:H^E=e E·∫ [dk]( Ω_kmf_0 -τεkεmf_0ε ), Ω̅_m_im_j^E =eτ E · ∫ [dk][εkΩ_m_im_j -( Ω_km_iεm_j)_ A ]f_0ε,η_m_im_j^E =eτ E ·∫ [dk]( Ω_km_iεm_j)_ Sf_0ε where subscript S (A) means the part of Ω_km_iεm_j that is symmetric (antisymmetric) under i↔ j. We next discuss the physical meanings of these results in detail.For the correction to the effective field, H^E, the first term in Eq. <ref> has a geometric nature and is an intrinsic contribution from the Fermi sea electrons. It is of Ω_mt type, where the time variation is due to the momentum change of a single wave packet driven by E: ∂_t = k̇·∂_k= -eE·∂_k. We note there is a nice identity connecting Ω_mt and the “magnetic field" in magnetization space Ω_ m:∂_t Ω_m + ∇_m×Ω_mt= 0.Since Ω_mt = Ω_m k· (-e E) is a correction to the static effective electric fieldH (Eq. <ref>) in the magnetization space, above equation is a magnetic analog of the Faraday's law for charged particles. The 2nd term in Eq. <ref> is extrinsic since it is proportional to τ, and does not have an electromagnetism analog.H^E also provides new insights on the charge pumping effect of a nonzero ṁ<cit.>. Since P≡ H^E·ṁ has the meaning of power density and is proportional to E, there is an electric current induced by ṁ as j_p = ∂( H^E·ṁ)/∂ E. The change of the polarization density (“pumping") after m completes a closed path in its configuration space is obtained by integrating j_p over this period. A finite charge pumping thus corresponds to a nonzero work density, and is related to the curl of H^E in the magnetization space throughW = ∮ j_p· E dt = ∮ H^E· d m = ∬∇_ m× H^E · dσ_ m,where we have used the Stokes theorem, and dσ_ m is the infinitesimal area in the magnetization space. Thus in order to have finite charge pumping, H^E must not be conservative, i.e., it cannot be written as a gradient of certain scalar free energy. We now move on to Ω̅_ mm^E and η_ m m^E, which are all Fermi surface contributions due to the non-equilibrium part of the distribution function δ f. They are important in magnetic metals and should be discussed on an equal footing as H^E for current-induced effects on magnetization dynamics. In the form of Eq. <ref>, Ω̅_ mm^E renormalizes the gyromagnetic ratio as γ^' = γ/(1+γ/γ^E), where γ^E ≡ 1/ m ·Ω̅^E, while η_ m m^E modifies the damping tensor as η^' = η + η^E. It is interesting to note that η^E does not have to be positive definite. A negative definite total damping will make the free energy minima dynamically unstable while the maxima dynamically stable. Thus in addition to the potential of switching the magnetization between different easy directions, a suitably chosen electric field can in principle switch the magnetization between easy and hard directions, which provides a new mechanism (though volatile) for current driven reading and writing processes in magnetic memory devices.Before ending this section, we translate our results Eq. (<ref>-<ref>) into the commonly used spin-orbit torque language. For small electric fields they can be converted to additional terms added to the right hand side of the LLG equation Eq. <ref>: ṁ = - γm×(H - η_mm·ṁ) - γτ_so, where τ_so = τ_so^H +τ_so^γ +τ_so^η with the separate terms beingτ_so^H=m×H^E, τ_so^γ=-γ/γ^Em× (H+ηγm×H), τ_so^η=γη^E m×(m×H).For the special s-d type coupling, H^E is proportional to the spin density response to electric fields since ∂Ĥ /∂m∼s, in agreement with previous studies <cit.>, though our formalism is not limited to this coupling form. Morever, there are additional torques τ_so^γ and τ_so^η that cannot be directly explained using spin density response to electric fields. They can, however, always be classified into either field-like or damping-like torques depending on whether there is a sign change upon m→ - m.Model example—As a concrete example, we consider a 2D toy model that can be used to describe the interface between a ferromagnetic insulator and a 3D topological insulator (TI) <cit.>: Ĥ(m) = ħ v(-k_y σ_x + k_xσ_y) + J m·σ, where m is the 2D magnetization of the ferromagnet, σ is the Pauli matrix vector for the spin operators, v is the Fermi velocity of the Dirac surface electrons of the TI, and J is the exchange coupling strength between m and σ. The exchange coupling opens a gap proportional to the z component of m. We consider zero temperature and set the chemical potential μ=0. The Berry curvature of the lower band is calculated similarly as the k⃗·σ⃗ model <cit.>Ω̅^s_θϕ = α^2|sin2θ|/8π a^2 sgn(α),where α = Jma/ħ v is the exchange energy measured in typical scales of the kinetic energy ϵ_0 = ħ v /a (a is the lattice constant). Using relation Eq. (<ref>), the Berry curvature gives an anisotropic gyromagnetic ratio γ_s(θ) = 4π ma^2/ħα^2|cosθ| sgn(α).We should note that the ferromagnet by itself has a gyromagnetic ratio, denoted as γ_f, and the overall gyromagnetic ratio γ is corrected asγ^-1 =γ_f^-1 + γ_s^-1, or equivalently γ = γ_f ·1/1+γ_f/γ_s(θ).The variation of γ for m moving across the Bloch sphere is shown in Fig. fig:1(a). On the equator (θ = π/2), γ= γ_f; at the north and south poles, γ = γ_f/(1+γ_f ħα^2 /4π ma^2 sgn(α)). This angular dependence of gyromagnetic ratio should be able to be detected by ferromagnetic resonance experiments in such systems. The free energy density at zero temperature is calculated by integrating the energy of the lower bands. Ignoring a constant term, we getG^s= - J_0 m_z^2where J_0= ϵ_0 k_cα^2/4π m^2a and k_c is the momentum cutoff. G^s has two minima at the north and south poles, as shown in Fig. fig:1(b). Thus the surface states provide a perpendicular magnetic anisotropy for the ferromagnet. For simplicity we ignored the magnetic anisotropy energy of the ferromagnet itself. For nonzero m_z there is no contribution from the surface state electrons to η because of the finite gap, and if the intrinsic damping of the ferromagnet is ignorable the magnetization should move along equal-energy lines without driving forces, along the directions determined by - γm×H [Eq. (<ref>)], as illustrated in Fig. fig:1(b).We now consider the effect of an electric field along x direction on the magnetization dynamics. For nonzero m_z all Fermi surface contributions in Eqs. (<ref>-<ref>) are zero, and the only finite term is the Fermi sea contribution in H^E: H^E = -eE|α|/4π ma sgn(m_z) x̂. It has constant magnitude but opposite directions depending on the sign of m_z. The curl of H^E is thus zero everywhere except on the equator, which also means nonzero charge is pumped by magnetization dynamics when the precession axis is in plane <cit.>. Based on our discussion in the previous section we can only define free energy functions separately for the north (N) and the south (S) hemispheres as G_N and G_S but not globally:G_N = -J_0m_z^2 + eE|α|/4π ma m_x, G_S=-J_0m_z^2 - eE|α|/4π ma m_x.On each hemisphere, the 2nd term in the free energy implies a magnetization-dependent polarization, which will be interesting to detect experimentally. Moreover, since G_N-G_S∝ m_x, they cannot be connected by a constant energy shift across the equator. The electric field thus shifts the two free energy minima at the north and the south poles in opposite directions, and distorts the equal energy lines in the vertical direction, as shown in Fig. fig:2. In addition, the opposite signs of G_N and G_S very close to the equator make half of the equator dynamically stable, as can be seen from the arrows pointing to the equator from both above and below in Fig. fig:2. Specifically, if we still assume a vanishing intrinsic damping of the ferromagnet, when the magnetization is very close to the equator with ϕ∈ (π,2π), or more generally when it is between the two critical trajectories determined by G_N/S= -eE|α|/4π a, it will follow the equal energy lines and end up on the half equator with ϕ∈ (0,π). Conversely, for a magnetization outside of the region between the two critical trajectories, i.e., G_N/S < -eE|α|/4π a, it will keep precessing around one of the free energy minima. When there is a small damping, the size of the attraction area around the half equator reduces because energy is dissipated during evolution.In the limiting case of strong electric fields eE|α|/4π a > 2J_0 m^2, the critical trajectories disappear on the Bloch sphere and the magnetization will always evolve to the stable half equator. Since without the magnetic field the magnetization has a perpendicular anisotropy due to the topological surface states, electric fields can lead to dynamical switching between easy (out-of-plane) and hard (in-plane) directions. This mechanism is unique to the FM/TI system and is independent of the easy-hard-axes switching due to a negative-definite damping tensor discussed in the last section. Since the electric field enters our formalism only through its modification on momentum [Eq. (<ref>)], our theory can be straightforwardly generalized to other time-varying perturbations that influence wave-packet dynamics in similar ways, which will give both Fermi-surface contributions and Fermi-sea contributions through the Berry curvature Ω_mt. For example, a potential application is the magnetization dynamics driven by sound wave <cit.>.Separately, our formalism can be applied to the slow dynamics of other order parameters in crystalline solids, and to its dependence on electromagnetic fields through the electron degrees of freedom. We acknowledge useful discussions with A. H. MacDonald, R. Cheng, Y. Gao, H. Zhou. This work is supported by National Basic Research Program of China (Grant No. 2013CB921900), DOE (DE-FG03-02ER45958, Division of Materials Science and Engineering), NSF (EFMA-1641101), and the Welch Foundation (F-1255). apsrev4-1 | http://arxiv.org/abs/1709.09513v3 | {
"authors": [
"Bangguo Xiong",
"Hua Chen",
"Xiao Li",
"Qian Niu"
],
"categories": [
"cond-mat.mes-hall"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170927134624",
"title": "Geometric Dynamics of Magnetization: Electronic Contribution"
} |
Properties of Nuclei up to A=16 using Local Chiral Interactions X. B. Wang December 30, 2023 ===============================================================A 1-ended finitely presented group has semistable fundamental group at ∞ if it acts geometrically on some (equivalently any) simply connected and locally finite complex X with the property that any two proper rays in X are properly homotopic. If G has semistable fundamental group at ∞ then one can unambiguously define the fundamental group at ∞ for G. The problem, asking if all finitely presented groups have semistable fundamental group at ∞ has been studied for over 40 years. If G is an ascending HNN extension of a finitely presented group then indeed, G has semistable fundamental group at ∞, but since the early 1980's it has been suggested that the finitely presented groups that are ascending HNN extensions of finitely generated groups may include a group with non-semistable fundamental group at ∞. Ascending HNN extensions naturally break into two classes, those with bounded depth and those with unbounded depth. Our main theorem shows that bounded depth finitely presented ascending HNN extensions of finitely generated groups have semistable fundamental group at ∞. Semistability is equivalent to two weaker asymptotic conditions on the group holding simultaneously. We show one of these conditions holds for all ascending HNN extensions, regardless of depth.We give a technique for constructing ascending HNN extensions with unbounded depth. This work focuses attention on a class of groups that may contain a group with non-semistable fundamental group at ∞. § INTRODUCTION If H is a group, and ϕ:H→ H is a monomorphism, then the notation ⟨ t,H:t^-1ht=ϕ(h)⟩ stands for a presentation of a group G with generators {t}∪ Hand relation set {t^-1 ht=ϕ( h) for allh∈H} union all relations for H. The group G is usually denoted H∗_ϕ and called an ascending HNN extension with base H and stable letter t. By Britton's lemma the obvious map of H into G is an isomorphism onto its image.If F(𝒜) is the free group on the set 𝒜,ϕ:𝒜→ F(𝒜) is a function and ℛ is a set of 𝒜-words, then the group G with presentation 𝒫=⟨ t,𝒜:ℛ,t^-1at=ϕ(a)for alla∈𝒜⟩is an ascending HNN extension of A,the subgroup of G generated by 𝒜. It is important to note that ⟨𝒜:ℛ⟩ need not be a presentation for A. For each integer n>0 and r∈ℛ, ϕ^n(r) may not be in the normal closure of ℛ in F(𝒜), but certainly ϕ^n(r) is a relator of A. In fact, when 𝒜 is finite,one would rarely expect A to be finitely presented. The relationst^-1at=ϕ(a) are called conjugation relations. Semistability of the fundamental group at ∞ for a finitely presented group is a geometric notion defined in <ref>. If a finitely presented 1-ended group G has semistable fundamental group at ∞ then the fundamental group at ∞ of G is independent of base ray. It is unknown if all finitely presented groups are semistable at ∞. To date, the strongest result in the theory of semistability and simple connectivity at∞ for ascending HNN extensions is the following: (M. Mihalik <cit.>) Suppose H is a finitely presented group ϕ:H→ H is a monomorphism and G=⟨ t,H:t^-1ht=ϕ(h)⟩ is the resulting HNN extension. Then G is 1-ended and semistable at ∞. If additionally, H is 1-ended, then G is simply connected at ∞. The line of proof used for this result fails when H is only finitely generated and it has been suggested since the 1980's that a promising place to search for a group with non-semistable fundamental group at ∞ is among the finitely presented ascending HNN extensions with finitely generated base. More specifically, A. Ol'shanskii and M. Sapir <cit.> and <cit.> have constructed a finitely generated infinite torsion group ℋ̅ and finitely presented ascending HNN extension 𝒢 of ℋ̅which has been suggested as a possible group with non-semistable fundamental group at ∞. In <ref>, we show that the collection of finitely presented ascending HNN extensions of finitely generated groups is naturally divided into two classes - those with what is called bounded depth and those of infinite/unbounded depth. If the finitely generated base is finitely presented, then the resulting ascending HNN extension has bounded depth. The Ol'shanskii-Sapir group 𝒢 has bounded depth and is semistable at ∞ by our main theorem.Suppose G is a finitely presented ascending HNN extension of a finitely generated group A and G has bounded depth. Then G has semistable fundamental group at ∞. Semistable fundamental group at ∞ for finitely generated groups was defined in the mid-1980's (<cit.>). While we are not concerned with that notion here, the following result (Theorem 4, <cit.>) is connected to the ideas in this paper. Suppose G is an ascending HNN extension of a finitely generated 1-ended group A. If A is semistable at ∞, then G is semistable at ∞.To prove Theorem <ref> we use the main theorem of <cit.> which implies that a finitely presented group G has semistable fundamental group at ∞ if and only if two (somewhat orthogonal) weaker semistability conditions hold for G.The rest of the paper is organized as follows. In <ref>, we define semistability at ∞ for spaces and groups, and list a number of equivalent formulations of this notion. Two weaker notions, the semistablility of a finitely generated subgroup J in an over group G and, the co-semistability of J in G are defined. In <ref> we prove that if A is an infinite finitely generated base group of a finitely presented ascending HNN extension G and t is the stable letter, then for any N≥ 0, t^NAt^-Nis semistable at ∞ in G (regardless of depth).By the main theorem of <cit.> this reduces the proof of our main theorem to showing that G satisfies the second semistability condition of <cit.>. In <ref> we review the combinatorial group theory of ascending HNN groups and define what it means for such a group to have bounded depth. Examples of Grigorchuk and Ol'shanskii-Sapir of ascending HNN extensions with bounded depth are reviewed and a method for constructing ascending HNN extensions with unbounded depth is given. In <ref> the bulk of the proof of our main theorem is given. We show that if G is an ascending HNN extension of a finitely generated group A, 𝒫 is a finite HNN presentation with bounded depth for G, and X is the Cayley 2-complex for 𝒫, then for each compact subset C of X, there is an integer N(C)≥ 0 such that t^NAt^-N is co-semistable at ∞ in X with respect to C. We also prove a result (Theorem <ref>) that considers the case when A is finitely presented and connects this case to several papers already in the literature. When A is finitely presented and Cis compact in X,we show there is an integer N(C)≥ 0 and compact set Q(C) containing C such that loops in X-(t^NAt^-N) Q are homotopically trivial in X-(t^NAt^-N) Q.§ THE BASICS OF SEMISTABILITY AT ∞ FOR GROUPSSuppose K is a locally finite connected CW complex. A ray in X is a map r:[0,∞)→ K.The space K has semistable fundamental group at ∞ if any two proper rays in K converging to the same end are properly homotopic.SupposeC_0, C_1,… is a collection of compact subsets of a 1-ended locally finite complex K such that C_i is a subset of the interior of C_i+1 and ∪_i=0^∞ C_i=K, and r:[0,∞)→ K is proper, then π_1^∞ (K,r) is the inverse limit of the inverse system of groups:π_1(K-C_0,r)←π_1(K-C_1,r)←⋯This inverse system is pro-isomorphic to an inverse system of groups with epimorphic bonding maps if and only if K has semistable fundamental group at ∞.When K is 1-ended with semistable fundamental group at ∞, π_1^∞ (K,r) is independent of proper base ray r. If for any compact set C in K there is a compact set D in K such that loops in K-D are homotopically trivial in X-C (equivalently the above inverse sequence of groups is pro-trivial), then K is simply connected at ∞.There are a number of equivalent forms of semistability which are collected as Theorem 3.2 of <cit.>. (G. Conner and M. Mihalik <cit.>) Suppose K is a locally finite, connected and 1-ended CW-complex. Then the following are equivalent: * K has semistable fundamental group at ∞.* For any proper ray r:[0,∞ )→ K and compact set C, there is a compact set D such that for any third compact set E and loop α based on r and with image in K-D, α is homotopic rel{r} to a loop in K-E, by a homotopy with image in K-C. * For any compact set C there is a compact set D such that if r and s are proper rays based at v and with image in K-D, then r and s are properly homotopic rel{v}, by a proper homotopy in K-C. If K is simply connected, then a fourth equivalent condition can be added to this list:4. Proper rays r and s based at v are properly homotopic rel{v}.If G is a finitely presented group and Y isa finite complex with π_1(Y)=G then G has semistable (respectively simply connected) fundamental group at ∞ if the universal cover of Y has semistable (respectively simply connected) fundamental group at ∞. This definition only depends on the group G.In <cit.> we consider finitely generated groups acting (perhaps not co-compactly) as covering transformations on 1-ended CW complexes X and we say what it means for such a group to be semistable at ∞ in X with respect to a given compact subset of X. In this paper we only need consider a more simple notion. Suppose A is a finitely generated infinite subgroup of a finitely presented 1-ended group G. Say𝒜∪𝒮 is a finite generating set of G, where 𝒜 generates A. Let X be the Cayley 2-complex for some finite presentation 𝒫 (with generating set 𝒜∪𝒮) of G. So X is the simply connected2-dimensional complex with 1-skeleton equal to the Cayley graph of G with respect to 𝒜∪𝒮. The vertex set of X is G and each edge of X is labeled by an element of 𝒜∪𝒮. For each vertex v of X and relation r of 𝒫 there is a 2-cell with boundary equal to the edge path loop at v with edge labels spelling the word r. Let ∗ be the identity vertex of X. Let Λ(A, 𝒜)⊂ X be the Cayley graph of A with respect to 𝒜. If g∈ G and q is an edge path in gΛ, then q is called an 𝒜-path in X. Note that q is an 𝒜-path if and only if each edge of q is labeled by an element of 𝒜. If g∈ G and C is compact in X then we say gAg^-1 is semistable at ∞ in X (or in G) with respect to C if there is a compact set D(C)⊂ X such that if r and s are two proper edge path rays in gΛ(A,𝒜)-Dbased at the same vertex v∈ gA then r and s are properly homotopic rel{v} by a proper homotopy in X-C. This definition is equivalent to the one of <cit.>. If gAg^-1 is semistable at ∞ with respect to every compact subset of X, then we say gAg^-1 is semistable at ∞ in X (or in G). If A is 1-ended and semistable at ∞, then gAg^-1 is always semistable at ∞ in X (G).In <ref> we prove:If G is a finitely presented ascending HNN extension of a finitely generated infinite group A and t is the stable letter, then for all N≥ 0, t^NAt^-N issemistable at ∞ in G. The main theorem of <cit.> is significantly more general than Theorem <ref>.In <cit.>, the main result does not require an overgroup G acting cocompactly on Y, only that Y be 1-ended and for each compact subset C of Y, the existence a finitely generated group J acting as covering transformations on Y and satisfying conditions 1) and 2) below. The notion of a group J being co-semistable at ∞ in a space is a bit technical and we define this afterwards.(R. Geoghegan, C. Guilbault and M. Mihalik <cit.>) Suppose G is a 1-ended finitely presented group acting cocompactly on a simply connected locally finite CW-complex Y. If for each compact set C⊂ Y there is an infinite finitely generated subgroup J of G such that 1) J issemistable at ∞ in Y with respect to C and 2) J is co-semistable at ∞ in Y with respect to C, then Y (and hence G) has semistable fundamental group at ∞. The converse of Theorem <ref> is rather straightforward.In fact, if Y (equivalently G) has semistable fundamental group at ∞, then suppose C is any compact subset of Y and J is any infinite finitely generated subgroup of G then conditions 1) and 2) hold for J and C. Interestingly, our proof of the main theorem of this paper relies on selecting different groups J for different compact sets C satisfying 1) and 2). We apply Theorem <ref> when G is an ascending HNN extension of a finitely generated group A, and G acts cocompactly on Y the Cayley 2-complex of G with respect to some finite HNN presentation 𝒫 (see <ref>).In our situation, all of the subgroups J of Theorem <ref> will have the form t^NAt^-N for some N≥ 0. Proposition <ref> resolves part 1) of Theorem <ref> for all compacts sets. All that remains to be shown is that for each compact set C in X there is an integer N(C)≥ 0 such that t^NAt^-N is co-semistable at ∞ in Y with respect to C. We now define what that means.Suppose J is an infinite finitely generated group acting as covering transformations on the 1-ended, simply connected and locally finite CW-complex Y. A subset S of Y is bounded in Y if S is contained in a compact subset of Y. Otherwise S is unbounded in Y. Let q:Y→ J\ Y be the quotient map. If K is a subset of Y, and there is a compact subset C_1 of Y such that K⊂ JC_1 (equivalently q(K) has image in a compact set), then K is a J-bounded subset of Y. Otherwise K is a J-unbounded subset of Y. If r:[0,∞)→ Y is a proper edge path ray and qr has image in a compact subset of J\ Y thenr is said to be J-bounded.Equivalently, r is a J-bounded proper edge path ray in S if and only if r has image in J C_1 for some compact set C_1⊂ Y. Let ∗ be a base vertex in Y. When r is J-bounded there is an integer M (depending only on C_1 and fixed terms) such that each vertex of r is (using edge path distance) within M of a vertex of J ∗⊂ Y.We say J is co-semistable at ∞ in Y with respect to the compact subset C of Y if there is a compact subcomplex C_1 of Y such that for each J-unbounded component U of Y-(JC_1), and any J-bounded proper ray r in U “loops in U and based on r can be properly pushed to infinity along r, avoiding C". More specifically: For any loop α:[0,1]→ U with α(0)=α(1)=r(0) there is a proper homotopy H:[0,1]× [0,∞)→ Y-C such that H(t,0)=α(t) for all t∈ [0,1] and H(0,s)=H(1,s)=r(s) for all s∈ [0,∞).§ BASE GROUP SEMISTABILITY IN AN ASCENDING HNN EXTENSION In this section we prove three lemmas that imply Proposition <ref>. This shows that an infinite finitely generated base group is always semistable at ∞ in an ascending HNN extension (regardless of bounded or unbounded depth). Begin with a finite presentation for a group G which is an ascending HNN extension with base group a finitely generated group A with finite set of generators 𝒜:𝒫=⟨ t, 𝒜: ℛ, t^-1at=ϕ(a) for all a∈𝒜⟩ Here ℛ is a finite subset of the free group F(𝒜). Consider the homomorphism P_0:G→ℤ that kills the normal closure of A. If g∈ G and P_0(g)=N, we say g is in level N. Let X be the Cayley 2-complex for the presentation 𝒫 of G. Then P_0 can be extended to P:X→ℝ by taking each 2-cell corresponding to an element of ℛ to P_0(v) for any vertex v of the cell, and if D is2-cell corresponding to a conjugation relation t^-1at=ϕ(a) for a∈𝒜, then P maps D to the interval [N,N+1] (where the edge of D corresponding to a∈𝒜 is mapped by P_0 to N and those corresponding to ϕ(a) are mapped to N+1), in the obvious way.Let e:[0,1]→ X be an edge in X with e(0)=v, e(1)=w and label a∈𝒜. Let r_v and r_w be the edge path rays at v and w (respectively) each of whose edges is labeled t. There is a proper map H_e:[0,1]× [0,∞)→ X such that H_e(t,0)=e(t), H_e(0,t)=r_v(t)), H_e(1,t)=r_w(t) and P(H_e([0,1]× [N,N+1]))⊂ [N,N+1].On [0,1]× [0,1] define H_e to have image the 2-cell at v with boundary label atϕ(a^-1)t^-1. Iterate to define H_e as in Figure 1. Note that if ϕ(a) has length L then the image of H_e on [0,1]× [1,2], consists of L conjugation relation 2-cells (each of which is mapped by P to [1,2]). to 2in < g r a p h i c s >Figure 1To see that H_e is proper, let C be compact in X. Then P(C)⊂ [-N,N] for some integer N≥ 0. But then H_e^-1 (C)⊂ [0,1]× [0,N]. Recall Λ is the Cayley graph of A with respect to 𝒜 and we assume ∗∈Λ⊂ X where ∗ is the identity vertex.Suppose C is compact in X. There are only finitely many 𝒜-edges e in Λ such that the image of H_e (see Lemma <ref>) intersects C. If v∈ A, let r_v be the proper edge path ray at v, each of whose edges is labeled t. If e is an edge of Λ with initial point v, let H_e be the proper homotopy of Lemma <ref>. For any integers S>R≥ 0, P(H_e([0,1]× [R,S]))⊂ [R,S]. Say that P(C)⊂ [-N,N] for N≥ 0. Then for any edge e of Λ,H_e([0,1]× [N+1,∞))∩ C=∅ (since P(C)⊂ [-N,N] and P H_e([0,1]× [N+1,∞))⊂ [N+1,∞)). LetL be the length of the longest word in {ϕ(a_1),…, ϕ(a_n)}. So for any integer K≥ 0, the length of the 𝒜-word H_e([0,1]×{K} is ≤ L^K (if e has label a∈𝒜, then H_e([0,1]×{K} has label ϕ^K(a)).For any edge e of Λ with initial vertex v, H_e([0,1]× [0,N])⊂ St^L^N+N(v). There are only finitely many vertices v of Λ such that St^L^N+L(v)∩ C∅ and so there are only finitely many edges e of Λ such that the image of H_e intersects C. Suppose s=(s_0,s_1,… ) is a proper edge path ray in Λ⊂ X. If v is the initial point of s letr_v be the edge path at v each of whose edges is labeled t, then there is a proper homotopy H_s:[0,∞)× [0,∞)→ X of s to r_v rel{v} defined so that H_s restricted to [N,N+1]× [0,∞) is H_s_N (i.e. H_s(N+x,y)=H_s_N(x,y) for all (x,y)∈ [0,1]× [0,∞)). Since H(0,y)=r_v(y) and H(x,0)=s, H is a homotopy of r_v to s rel{v}. It remains to show that H is proper. If C is compact in X, then by Lemma <ref> there are only finitely many edges e of s such that the image of H_e intersects C. Choose N such that for all n>N, H_s_n avoids C. Then H_s^-1(C)=∪_i=1^NH_s_i^-1(C). This last set is a finite union of compact sets since each H_s_i is proper. (of Proposition <ref>) We show that for any integer N≥ 0, the group t^NAt^-N is semistable at ∞ in X (G). Let C be compact in X. If v∈ A, let r_v be the proper edge path ray at v, each of whose edges is labeled t. If e is an edge of Λ with initial point v, let H_e be the proper homotopy of Lemma <ref>. By Lemma <ref> there are only finitely many edges e of Λ such that the image of H_e intersects t^-NC.Choose D compact such that D contains t^-NC and all of these edges. If s and s' are proper 𝒜-rays at v∈Λ-D then the proper homotopies H_s and H_s' of Lemma <ref> both avoid t^-NC so that both s and s' are properly homotopic rel {v} to r_v by homotopies in X-t^-NC. Combining H_s and H_s' we have s is properly homotopic rel{v} to s' by a homotopy H in X-t^-NC.Now t^NH is a proper homotopy rel{t^Nv} of t^Ns to t^Ns' in X-C and t^NAt^-N is semistable at ∞ in X. § ASCENDING HNN EXTENSION COMBINATORICSSuppose 𝒜 is a finite set, ϕ:F(𝒜)→ F(𝒜) is a homomorphism of the free group, ℛ is a finite set of words in F(𝒜) and G is the (finitely presented) ascending HNN extension with the following HNN presentation:(∗)𝒫=⟨ t, 𝒜: ℛ, t^-1at=ϕ(a) for all a∈𝒜⟩The base group of this HNN extension is A, the subgroup of G generated by 𝒜. In this paper, we are only interested in the case when 𝒜 is finite. In order to define what it means for an ascending HNN extension to have bounded depth, we must first understand ker(p) where p is the homomorphism p:F(𝒜)→ A (defined by p(a)=a for a∈𝒜).Certainly ker(p) contains N_0(ℛ,ϕ)≡ N( ∪ _i=0^∞ϕ^i (ℛ)), where N( ∪ _i=0^∞ϕ^i (ℛ)) is the normal closure of ∪ _i=0^∞ϕ^i (ℛ) in F(𝒜). But it may be that for some wordw∈ F(𝒜) and some integer m, ϕ^m(w)∈ N_0(ℛ,ϕ), and w∉N_0(ℛ,ϕ). Then w∈ ker(p). Consider the normal subgroup of F(𝒜): N^∞(ℛ,ϕ)≡∪ _i=0^∞ϕ^-i(N_0(ℛ,ϕ))◃ F(𝒜).It is well known to experts that ϕ^-i(N_0(ℛ,ϕ))< ϕ^-i-1(N_0(ℛ,ϕ)) (see theorem <ref>) so thatN^∞(ℛ,ϕ) is an ascending union of normal subgroups of F(𝒜) and that N^∞(ℛ,ϕ) is the kernel of p, so A=⟨𝒜:N^∞(ℛ,ϕ)⟩If there is an integer B such that N^∞(ℛ,ϕ)=∪ _i=0^Bϕ^-i(N_0(ℛ,ϕ)) then the presentation 𝒫 of G has bounded depth. Our main theorem shows that if 𝒫 has bounded depth, then G is semistable at ∞ (Theorem <ref>). It is not always the case that such ascending HNN extensions have bounded depth. (See Theorem <ref>.) As in <ref>, P_0:G→ℤ is the homomorphism that kills the normal closure of A. If X is the Cayley 2-complex for the presentation 𝒫 of G given in (∗) (with vertex set G), then P_0 extends to P:X→ℝ. If g∈ G and P_0(g)=N, g is in level N. An edge path loop in level L of X, whose labelingdefines an element of ∪ _i=0^Bϕ^-i(N_0(ℛ,ϕ)), is homotopically trivial by a combinatorial homotopy H such that P(H) has image in (-∞,L+B]. Note that if α is an edge path loop in level L labeled by an element of N(ℛ) (the normal closure of ℛ in F(𝒜)) then α can be killed by a homotopy in level L.If α has initial vertex v in level L and labeling ϕ(r) for r∈ N(ℛ), then using only conjugation relations, α is homotopic to an edge path loop at v with labeling (t^-1,β, t) where β has labeling r and image in level L-1. Since β is homotopically trivial in level L-1, the loop α can be killed by a homotopy H such that P(H) has image in [L-1,L]. This homotopy only uses the homotopy that kills β in level L-1 and theconjugation relation 2-cells connecting α and β. If α has label in ϕ^-1(N(ℛ)) (so ϕ(α)=r∈ N(ℛ)) then α can be killed by a homotopy H such that P(H) has image in [L,L+1]. In the case that A is finitely generated and the image of ϕ:A→ A is of finite index in A, then A is “commensurated" in G and G is semistable at ∞ (see Corollary 4.9 of <cit.>). For 𝒜 finite, the group G=⟨ t, 𝒜: ℛ', t^-1at=ϕ(a)fora∈𝒜⟩ (with ℛ' ⊂ F(𝒜)) is an ascending HNN extension with bounded depth D and root ℛ if the kernel of the homomorphism p:F(𝒜)→ A (defined by p(a)=a for all a∈𝒜) is ϕ^-D(N_0(ℛ,ϕ))≡ϕ^-D(N(∪ _i=0^∞ϕ^i (ℛ))) for some finite set of words ℛ in F(𝒜). In this case, G has finite presentation: ⟨ t, 𝒜:ℛ,t^-1at=ϕ(a)for alla ∈𝒜⟩.R. Grigorchuk (<cit.> and <cit.>) constructed a finitely generated infinite torsion group G of intermediate growth having solvable word problem. He also showed that G was the base group of a finitely presented ascending HNN extension (which is the first example of a finitely presented cyclic extension of an infinite torsion group). I. Lysënok <cit.> produced the following recursive presentation of G:G≡⟨ a,c,d:σ^n(a^2), σ^n((ad)^4), σ^n(adacac)^4), n≥ 0⟩where σ (a)=aca, σ (c)=cd and σ (d)=c. It can be shown that the ascending HNN extension E with presentation: ⟨ a,c,d,t: a^2=(ad)^4=(adacac)^4=1, t^-1at=aca, t^-1ct=dc, t^-1dt=c⟩ has base group G generated by {a,c,d} and E has bounded depth with root {a^2,c^2, d^2, (ad)^4, (adacac)^4}. The group E was the first example of a finitely presented amenable but not elementary amenable group. In 5 of <cit.>, M. Mihalik shows that E is simply connected at ∞. The notion of a finitely generated group being simply connected at ∞ is introduced in <cit.>, and the group G is shown to be simply connected at ∞.A. Ol'shanskii and M. Sapir <cit.> and <cit.> construct a finitely presented ascending HNN extension 𝒢, where the base group ℋ̅ is a finitely generated infinite torsion group. In contrast to Grigorchuk's group (Example <ref>) the base group has finite exponent, and 𝒢 is not amenable (see Theorem 1.1 of <cit.>).The group 𝒢 has been suggested as a possible non-semistable at ∞ group, but it is clear from the equations (5)-(8) in 1.2 of <cit.> that 𝒢 has an ascending HNN presentation with depth one, and so by our main theorem is semistable at ∞.We give a brief summary. A finite set of words ℛ is determined in F_C=⟨ c_1,…, c_m⟩ a free group of rank m.A monomorphism ϕ:F_C→ F_C is defined and ℛ' is defined to be ∪_i=1^∞{ϕ^i(r):r∈ℛ}. The base group of their ascending HNN extension has presentationℋ̅=⟨ c_1,…, c_m: ℛ∪𝒱∪ℛ'⟩where 𝒱 is the set of elements u^nfor all u∈ F_C (and n a fixed large odd number). In particular, ℋ̅ is an infinite torsion group. A finitely presented ascending HNN extensionof ℋ̅ has infinite presentation𝒢=⟨ t, c_1,…, c_m: t^-1c_it=ϕ(c_i), ℛ∪ℛ'∪𝒱⟩(This follows equation (7) of <cit.>.) Clearly the relations ℛ' are a consequence of ℛ and the conjugation relations and so can be removed. It is then argued that each relation v^n of 𝒱 is ϕ^-1 (v') where v' is a consequence of ℛ and the conjugation relations. In particular, the above presentation of 𝒢 can be reduced to the presentation 𝒢=⟨ t, c_1,…, c_m: t^-1c_it=ϕ(c_i), ℛ⟩ and this presentation has depth 1. It seems unlikely that 𝒢 has an ascending HNN presentation with depth 0. One must wonder if for every integer N>0 there are finitely presented ascending HNN groups 𝒢_N with ascending HNN presentations of depth N but 𝒢_N does not have such a presentation of depth N-1.Suppose G is the ascending HNN extension with finite presentation: 𝒫=⟨ t,𝒜: ℛ, t^-1at=ϕ(a)for alla∈𝒜⟩where ϕ:F(𝒜)→ F(𝒜) is a (finite rank) free group homomorphism. Then A, the subgroup of G generated by 𝒜, has presentation:A=⟨𝒜:N^∞(ℛ,ϕ)≡∪_i=0^∞ϕ^-i(N(∪ _j=0^∞ϕ^j(ℛ)))⟩. Furthermore, we have the relations:* ϕ^-i(N(∪ _j=0^∞ϕ^j(ℛ)))⊂ϕ^-(i+1)(N(∪ _j=0^∞ϕ^j(ℛ))) for all i≥ 0, and* ϕ(N^∞ (ℛ,ϕ))⊂ N^∞(ℛ,ϕ)=ϕ^-1(N^∞(ℛ,ϕ))Note that ϕ(N(∪ _j=0^∞ϕ^j(ℛ)))⊂ N(∪ _j=1^∞ϕ^j(ℛ)))⊂ N(∪ _j=0^∞ϕ^j(ℛ)) so thatN(∪ _j=0^∞ϕ^j(ℛ)))⊂ϕ^-1( N(∪ _j=0^∞ϕ^j(ℛ)))and so relation 1) follows. To simplify notation, let N^∞=N^∞(ℛ,ϕ) and N_i=ϕ^-i(N(∪ _j=0^∞ϕ^j(ℛ))) for i≥ 0, so that N^∞=∪_i=0^∞ N_i and by 1), N_i⊂ N_i+1=ϕ^-1(N_i). Suppose a∈ϕ^-1(N^∞). Then ϕ(a)∈ N^∞ and so ϕ(a)∈ N_i for some i≥ 0. Then a∈ϕ^-1(N_i)=N_i+1⊂ N^∞ and we have shown that ϕ^-1(N^∞)⊂ N^∞.Next suppose a∈ N^∞. Then for some i≥ 0, a∈ N_i. By 1), a∈ N_i+1=ϕ^-1(N_i)⊂ϕ^-1(N^∞). We have shown that N^∞(ℛ,ϕ)⊂ϕ^-1(N^∞(ℛ,ϕ)). Combining we have N^∞= ϕ^-1(N^∞) and relation 2) follows. Let A_1 be the group with presentation ⟨𝒜:N^∞(ℛ,ϕ)⟩. To finish the theorem we must show that A=A_1. Let p_1:F(𝒜)→ A_1 (determined by p_1(a)=a for all a∈𝒜)be the quotient homomorphism. By 2), the map ϕ_1:A_1→ A_1 that extends the map ϕ_1(p_1(a))=p_1(ϕ(a)) for all a∈𝒜 is a homomorphism. This gives a commutative diagram: F(𝒜)ϕ⟶F(𝒜) ↓ p_1↓ p_1 A_1 ϕ_1⟶ A_1Next we show that ϕ_1 is a monomorphism. Suppose w_1∈ ker(ϕ_1). Let w∈ F(𝒜) be such that p_1(w)=w_1. Then p_1(ϕ(w))=1 and so ϕ(w)∈ ker(p_1)=N^∞ and w∈ϕ^-1(N^∞) = N^∞.Then w_1=p_1(w)=1∈ A_1 and ϕ_1 is a monomorphism. Consider the ascending HNN extension: A_1∗_ϕ=⟨ t, 𝒜: N^∞(ℛ,ϕ), t^-1at=ϕ(a)⟩ for alla∈𝒜 with base group A_1. Since each relation in N^∞(ℛ,ϕ) is a consequence of ℛ and the conjugation relations, this group also has presentation 𝒫. By Britton's lemma A=A_1. Suppose G has finite presentation ⟨ t, 𝒜: ℛ, t^-1at=ϕ(a) fora∈𝒜⟩. Here ϕ:F(𝒜)→ F(𝒜) is a homomorphism. Let N_0≡ N(∪ _j=0^∞ϕ^j(ℛ))◃ F(𝒜),N_i≡ϕ^-i(N_0) and A be the subgroup of G generated by 𝒜, so that G is the ascending HNN extension, with base A and stable letter t.Let p:F(𝒜)→ A be the homomorphism extending the map taking a to a for all a∈𝒜.It seems that there is some potential to find a finitely presented group that is not semistable at ∞ if one could find a finitely presented ascending HNN extension ⟨ t, 𝒜: ℛ, t^-1at=ϕ(a) fora∈𝒜⟩, such that the ascending chain of normal subgroups N_k of F(A) do not stabilize. The following approach gives a general method of constructing infinite depth ascending HNN presentations. In particular, when A_0 is a non-Hopfian group and ϕ_0:A_0→ A_0 is an epimorphism with non-trivial kernel, then there is a corresponding ascending HNN extension with infinite depth.Suppose the group A_0 has finite presentation ⟨𝒜:ℛ⟩ and ϕ_0:A_0→A_0 is a homomorphism with non-trivial kernel K_0 such that the following diagram (with F(𝒜) the free group on 𝒜 and q(a)=a for a∈𝒜) commutes:F(𝒜)ϕ⟶F(𝒜) ↓ q↓ q A_0ϕ_0⟶ A_0If the ascending sequence {K_i=ϕ_0^-i(K_0)=ker(ϕ_0^i+1)} of normal subgroups of A_0 does not stabilize (in particular when ϕ_0 is an epimorphism), then the group G with ascending HNN presentation 𝒫≡⟨ t,𝒜:ℛ, t^-1at=ϕ(a)for all a∈𝒜⟩has unbounded depth. First observe that if ϕ_0 is an epimorphism, and k∈ K_0-1, then there is k_n such that ϕ_0^n(k_n)=k. In particular, k_n∈ ker(ϕ_0^n+1)-ker(ϕ_0^n). Note that ker(q)=N(ℛ)◃ F(𝒜). If r∈ N(ℛ), then q(ϕ(r))=1 and so ϕ(N(ℛ))⊂ N(ℛ) and (retaining the notation of Theorem <ref>)N_0=N(∪_i=0^∞ϕ^i(ℛ))=N(ℛ)=ker(q).For the subgroup Aof G determined by 𝒜 there is a commutative diagram:F(𝒜)ϕ⟶F(𝒜) ↓ p↓ p A ϕ_1⟶ AObserved that A is a quotient of A_0 where the element q(a) is mapped to p(a) for all a∈𝒜 and the following diagram commutes: A_0ϕ_0⟶A_0 ↓ q_0↓ q_0 A ϕ_1⟶ A (∗) Ifϕ_0 is an epimorphism, then since q_0 is an epimorphism ϕ_1 is also an epimorphism. In any case, G =A∗_ϕ_1 and when ϕ_0 is an epimorphism, ϕ_1 is an isomorphism.Let N_i=ϕ^-i(N_0)◃ F(𝒜). By Theorem <ref>.1 N_i-1≤ N_i. For i>0 we show N_i N_i-1 when K_i K_i-1, so that 𝒫 has unbounded depth when {K_i} does not stabilize. Choose a_n∈ K_n-K_n-1.Choose a̅_n∈ F(𝒜) such q(a̅_n)=a_n. Then q(ϕ^n-1(a̅_n))=ϕ_0^n-1q(a̅_n)=ϕ_0^n-1(a_n) 1so ϕ^n-1(a̅_n)∉N_0=ker(q) and a̅_n∉N_n-1. But, qϕ^n(a̅_n)=ϕ_0(q(ϕ^n-1(a̅_n)))=ϕ_0(a)=1so ϕ^n(a̅_n)∈ ker(q)=N_0 and a̅_n∈ N_n-N_n-1.When A_0 is non-Hopfian and ϕ_0 maps A_0 onto A_0 with non-trivial kernel, Theorem <ref> produces a corresponding ascending HNN extension with unbounded depth. LetA_0=BS(2,3)=⟨ a,b:b^-1a^2b=a^3⟩, and ϕ:F({a,b})→ F({a,b}) by a→ a^2 and b→ b,observe that ϕ^i([b^-iab^i,a])=[b^-ia^2^ib^i,a^2^i]≈ [a^3^i,a^2^i]=1, so that [b^-iab^i,a]∈ N_i. If [b^-iab^i,a]∈ N_i-1 then ϕ^i-1([b^-iab^i,a])∈ N_0 whereN_0=N(b^-1a^2ba^-3)◃ F({a,b})). But ϕ^i-1([b^-iab^i,a])=[b^-ia^2^i-1b^i, a^2^i-1]≈ [b^-1a^3^i-1b,a^2^i-1]a reduced word of syllable length 8 in (the HNN extension) ⟨ a,b:b^-1a^2=a^3⟩. In particular, the following ascending HNN extension presentation with stable letter t and base group generated by {a,b} has infinite depth:⟨ t,a,b:b^-1a^2b=a^3, t^-1at=a^2, t^-1bt=b⟩.Since ϕ_1 is an isomorphism (see (∗)), ⟨ A⟩=⟨ a,b⟩ is normal in G and the main theorem of M. Mihalik's paper <cit.> implies G is semistable at ∞. So this particular approach cannot yield a non-semistable at ∞ ascending HNN extension of unbounded depth when ϕ_0 is an epimorphism.The remainder of this section is of general interest in understanding presentations of ascending HNN extensions, but not important to the proof of our main theorem.Consider a homomorphisms ϕ: F(𝒜)→ F(𝒜) for 𝒜 finite where ϕ has non-trivial kernel. One might wonder if it is possible to have a such a homomorphism so that (even with ℛ=∅), the presentation ⟨ t, 𝒜: t^-1at=ϕ(a)fora∈𝒜⟩does not have finite depth? I.e. is it possible that the ascending collection of normal subgroups of F(𝒜) defined by N_k=⟨∪ _i=1^kker(ϕ^i)⟩ does not stabilize? The answer is no.Consider the sequence F(𝒜)→ϕ(F(𝒜))→ϕ^2(ℱ(𝒜))→⋯ of epimorphisms where each map is ϕ. For i>0,ϕ^i(F(𝒜)) is a free group of rank ≤ rank(ϕ^i-1(F(𝒜))). So, for some integer m≥ 0, rank (ϕ^m(F(𝒜)))=rank (ϕ^m+1(F(𝒜))). As finitely generated free groups are Hopfian, the epimorphism ϕ:ϕ^m(F(𝒜))→ϕ^m+1(F(𝒜)) is an isomorphism and ker(ϕ^m)=ker(ϕ^m+1).Next we show that any homomorphism ϕ :F(𝒜)→ F(𝒜) defining an ascending HNN extension can be replaced by a monomorphism. Suppose 𝒜 is a finite set, ℛ is a finite subset of the free group F(𝒜) and ϕ:F(𝒜)→ F(𝒜) is a homomorphism. Then there is a finite set ℬ, a finite set ℛ'⊂ F(ℬ),a monomorphism ϕ':F(ℬ)→ F(ℬ) and an isomorphism of ascending HNN extensions:⟨ t,𝒜: ℛ, t^-1at=ϕ(a) for a∈𝒜⟩ρ⟶⟨ t,ℬ: ℛ', t^-1bt=ϕ'(b)forb∈ℬ⟩ Furthermore,if q_𝒜:F(𝒜∪{t})→⟨ t,𝒜: ℛ, t^-1at=ϕ(a) for a∈𝒜⟩andq_ℬ:F(ℬ∪{t})→⟨ t,ℬ: ℛ', t^-1bt=ϕ'(b)forb∈ℬ⟩ are the natural projections, thenthere is a epimorphism ρ':F(𝒜∪{t})→ F(ℬ∪{t}) such that: 1) ρ'(t)=t2) ρ' ∘ q_ℬ=q_𝒜∘ρ and 3) ρ'(ℛ)=ℛ',(for N_G(ℛ) the normal closure of ℛ in G) ρ'(N_F(𝒜)(ℛ))= N_F(ℬ)(ℛ') and ρ'(N_F(𝒜∪{t})(ℛ))= N_F(ℬ∪{t})(ℛ')In particular, the following diagram commutes: F(𝒜∪{t}) ρ'⟶F(ℬ∪{t}) ↓ q_𝒜↓ q_ℬ ⟨ t,𝒜: ℛ, t^-1at=ϕ(a)⟩ρ⟶⟨ t,ℬ: ℛ', t^-1bt=ϕ'(b)⟩(Basically ρ is conjugation by t^m for some m≥ 0.) Since free groups are Hopfian, there is an integer m≥ 0 such that ϕ:ϕ^m(F(𝒜))→ϕ^m+1(F(𝒜)) is an isomorphism (see Remark <ref>). Let ℬ be a finite set of free generators for ϕ^m(F(𝒜)) (so F(ℬ)≡ϕ^m(F(𝒜))) and let ϕ':F(ℬ)→ F(ℬ) be defined so that ϕ'(b) is a ℬ-word for ϕ(b) for each b∈ℬ. Note that ϕ'is a monomorphism, since ϕ:ϕ^m(F(𝒜))→ϕ^m+1(F(𝒜))<F(ℬ) is a monomorphism.Define ρ':F(𝒜∪{t})→ F(ℬ∪{t}) such that ρ'(t)=t and ρ'(a)=ϕ^m(a) for all a∈𝒜. Note that ρ' is an epimorphism. Let ℛ'=ϕ^m(ℛ) (written as ℬ-words) and then 3) holds. Since ρ' of each relation of ⟨ t,𝒜: ℛ, t^-1at=ϕ(a)⟩ is a relator of ⟨ t,ℬ: ℛ', t^-1bt=ϕ'(b)⟩, the homomorphismρ can be defined so that 2) holds. Since ρ' is an epimorphism, ρ is an epimorphism. (Basically, ρ is conjugation by t^m.) To show ρ is an isomorphism, it remains to show that if w∈ ker(ρ q_𝒜) then w∈ ker (q_𝒜) (i.e. ρ is a monomorphism). First observe that the exponent sum of t in w is zero. Next observe that, w∈ ker(ρ q_𝒜) (respectively w∈ ker(q_𝒜)) iff t^-jwt^j∈ ker(ρ q_𝒜) (respectively t^-jwt^j∈ ker( q_𝒜)) for every integer j≥ 0. Select a positive integer j such that any initial segment of t^-jwt^j has t-exponent sum ≤ 0. In F(𝒜∪{t}), w=(t^-n_1w_1t^n_1)⋯ (t^-n_sw_st^n_s) where n_i≥ 0 and each w_i∈ F(𝒜). Let w̅≡ϕ^n_1(w_1)⋯ϕ^n_s(w_s)(∈ F(𝒜)). Now, q_𝒜(w)=q_𝒜(w̅) and w̅∈ ker(q_ℬρ'). Note that ρ'(w̅)=ϕ^m(w̅)∈ ker(q_ℬ) (<F(ℬ)). By Theorem <ref>, ϕ^m(w̅)∈ (ϕ')^-k(N(∪_i=0^∞(ϕ')^i(ℛ'))) for some integer k≥ 0.By 3) we have,ϕ^m(w̅)∈ϕ^-k(N(∪_i=0^∞ϕ^i(ϕ^m(ℛ)))) and so w̅∈ϕ^-k-m(N(∪_i=m^∞ϕ^i(ℛ))). By Theorem <ref>, w̅ (and hence w) is an element of ker (q_𝒜).§ BOUNDED DEPTH HNN EXTENSIONSARE SEMISTABILE AT ∞ The group G is an ascending HNN extension of a finitely generated group A and G has bounded depth.We use the notation of <ref>. Let 𝒜={a_1,… , a_n} be a finite generating set for A and 𝒫≡⟨ t,𝒜:ℛ,t^-1at=ϕ(a)for alla∈𝒜⟩a finite presentation for G, where each element of ℛ is an 𝒜-word.Let X be the Cayley 2-complex for this presentation, and Λ be the Cayley graph of A with generating set 𝒜. We assume ∗∈Λ⊂ X where ∗ is the identity vertex for X.We must show condition (2) of Theorem <ref> is satisfied for each compact set C in X.We will show that there is an integer N(C)≥ 0 (defined in Lemma <ref>) such that t^NAt^-N is co-semistable at ∞ in X with respect to C. This requires that we find a compact set D(C) such that loops in X-(t^NAt^-N ) D(C) can be pushed to infinity by proper homotopies in X-C. In every instance D(C) will have the form t^N(C){∗, t^-1,… ,t^-M} for some integer M that depends on C and the depth of the presentation 𝒫 for G.In the case that A is finitely presented, it is interesting to note that our proof will show that for our choice of D(C), each loop in X-(t^-NAt^N)D is homotopically trivial in X-(t^-NAt^N) D (see Theorem <ref>). This sort of behavior is related to the main theorems of <cit.>, <cit.> and <cit.>, and is called strongly coaxial when A is infinite cyclic.Recall that P:X→ℝ is such that for each vertex v∈ G⊂ X, P(v) is the exponent sum of t in v and we say v is in level P(v).The next lemma is a direct consequence of the normal form for elements of G (each element g∈ G has the form t^nat^-m for some n, m≥ 0 and a∈ A).Suppose C is a finite subcomplex of X. For each vertex v∈ C, write v=t^n(v)a_vt^-m(v) for a_v∈ A and n(v), m(v)≥ 0,andN(C)=max{n(v):v∈ C} and M(v,C)=N(C)-n(v)+m(v)(≥ 0). Then vt^M(v,C)∈ t^N(C)A. Note that by definition, N(C)-M(v,C)=n(v)-m(v)=P(v). For v∈ C,v=t^N(C)(t^n(v)-N(C)a_vt^N(C)-n(v))t^-M(v,C).If a'_v=t^n(v)-N(C)a_vt^N(C)-n(v)(∈ϕ^N(C)-n(v)(A)<A) then vt^M(v,C)=t^N(C)a'. Geometrically this say that for each vertex v of C, the edge path at v with each edge labeled t and of length M(v,C) ends in t^N(C)A. Suppose C is a finite subcomplex of X. Let M(C)=max{M(v,C): v is a vertex ofC}.Then for each vertex v∈ C v∈ (t^N(C)At^-N(C))(t^N(C){∗, t^-1,… ,t^-M(C)})and for positive integers M,N and w∈ (t^NAt^-N)(t^N{∗, t^-1,… ,t^-M}) we havewA⊂ (t^NAt^-N)(t^N{∗, t^-1,… ,t^-M}). The first conclusion follows from Lemma <ref>.Note that w=t^Nat^-m for some a∈ A and m∈{0,…, M}.Then wA⊂ t^Na(t^-mAt^m)t^-m and as t^-mAt^m⊂ A:wA⊂t^NAt^-m⊂ (t^NAt^-N)(t^N{∗, t^-1,… ,t^-M}). For integers N,M≥ 0 define D(N,M)≡ t^NA{∗,t^-1,…, t^-M}. If C is compact in X, and B is the bounded depth of our ascending HNN presentation 𝒫 we will use the set D(N(C),M(C)+B+1) to play the roll of the compact set D in X and t^N(C)At^-N(C) to play the roll of J when applying Theorem <ref>. First we must understand the set (t^NAt^-N)D(N,M)=t^NA{∗, t^-1,…, t^-M} and a few geometric definitions will help. If v,w∈ G, we say the coset wA is n levels directly below vA if there is an edge path of length n with each edge labeled t from a vertex of wA to a vertex of vA. Note that if wA is n levels directly below vA then for every vertex u of wA, the edge path at u of length n and with each edge labeled t ends in vA. We say vA is n levels directly above wA. Any coset wA has exactly one coset n(≥ 0) levels directly above it, but the cosets one level directly below vA are in 1-1 correspondence with the cosets of A in G.This means The set D(N,M)=t^NA{∗,t^-1,…, t^-M} is the union of cosets vA that are n levels directly below t^NA for n∈{0,1,…, M}. Note. In order to avoid confusion we may use the notation H· E instead of HE when H is a subgroup of G and E a subset ofX. LetQ(M)={∗, t^-1, t^-2,…, t^-M} (M≥ 0) and notice thatthe next lemma says that it is easy to check if a vertex v of X is in either A · Q(M), K_0 (a special component of X-A· Q(M)) or a component of X-A · Q(M) other than K_0. If v is in a level >0 then v∈ K_0. If v is in level 0 through -M then v is in A· Q if the edge path from v to level 0, with each edge labeled t, ends in A (i.e. vA is -P(v) levels directly below A); and v is in K_0 otherwise. If v is in a level <-M, then v is in K_0 if the edge path from v to level 0, with each edge labeled t, does not end in A; and otherwise, v belongs to a component of X-A· Q other than K_0.Note that t^n∈ K_0 for all n>0, so that under the quotient of X by A, the image of K_0 is not contained in a compact set. If v∈ K where K is acomponent of X-A· Q other than K_0 then vt^n∈ K for all n<0, so under the quotient of X by A, the image of K is not contained in a compact set. Our terminology for this is that K and K_0 are A-unbounded components of X-A· Q.Let Q(M)={∗, t^-1, t^-2,…, t^-M} for M≥ 0. Then 1) A· Q(M) is the set of all vertices v∈ X such that P(v)∈{-M,…, 0} and vt^-P(v)∈ A. Furthermore, if v∈ A· Q(M) then vA⊂ A· Q(M).2) X-A· Q(M) has an A-unbounded component K_0 with stabilizer A and the vertex v of X-A· Q(M) is in K_0 if and only if either P(v)≥ -M or both P(v)<-M and vt^-P(v)∉A, 3) if K is any component of X-A· Q(M) other than K_0, then K is A-unbounded, and if v is a vertex of K, then P(v)<-M and vt^-P(v)∈ A. Part 1): This part follows directly from Lemma <ref> (with N=0). Part 2): Let K_0 be the component of X-A· Q that contains the vertex t. Let v be a vertex of X, then by normal forms, v=t^lat^-m where a∈ A and l,m≥ 0. If P(v)>0, then l>m and the normal form for v defines an edge path from t to v in levels 1 and above, and hence avoiding A· Q. So if P(v)>0, then v∈ K_0. Note thatP(at)=1 for all a∈ A, so that A stabilizes K_0. Suppose v∈ X-A· Q andP(v)∈{-M,…, 0}, then by part 1), vt^-P(v)∉A and no point of the edge path beginning at v with labeling t^-P(v) is a point of A· Q. Since P(vt^-P(v)+1)=1, the edge path at v with labeling t^-P(v)+1 avoids A· Q and ends at a point of K_0. So if v∈ X-A· Q and P(v)∈{-M,…,0} then v∈ K_0. Suppose v∈ X-A· Q and P(v)<-M. Note that P(vt^-P(v))=0. If vt^-P(v)∉A, then we have already shown that vt^-P(v)∈ K_0, and by part 1), no point of the path with labeling t^-P(v) at v intersects A· Q. Hence v∈ K_0. For the converse, suppose v∈ K_0 and P(v)<-M. We must show vt^-P(v)∉A. Let α be an edge path in X-A· Q from t to v. Let β be a tail of α where w, the initial point of β, is the last point of α with P(w)=-M.The first edge of β is labeled t^-1. Note that conjugation relations allow us to move each A-edge of β up to level -M so there is an edge path from w to v labeled (x_1,… x_i, t^-k) where k> 0 and x_i∈{a_1,…, a_n}^± 1. Hence vt^k∈ wA and P(vt^k)=-M. By Part 1), w∉A· Q implies wA ∩ A· Q=∅, so vt^k∉A· Q. Again by Part 1), vt^kt^-P(vt^k)∉A. Then vt^-P(v)=vt^kt^-P(v)-k=vt^kt^-P(vt^k)∉A. This completes part 2). Part 3): If v∈ K K_0 then by Part 2), P(v)<-M and vt^-P(v)∈ A. We need a slightly stronger version of Lemma <ref>.Recall that Q(M)={∗, t^-1, t^-2,…, t^-M}. Then t^NA· Q(M)=t^NAt^-N (t^N(Q(M))).Observe that for any integer m≥ 0 the stabilizer of t^mΛ is t^mAt^-m.LetM,N≥ 0 be integers:1) The set t^NA· Q(M)(=D(N,M))consists of the vertices v∈ X such that P(v)∈{N,N-1,…,N-M} and vt^N-P(v)∈ t^NA. Furthermore, if v∈ t^NA· Q(M) then vA⊂ t^NA· Q(M).2) Let K_0 be the component of X-A· Q(N) described by part 2 of Lemma <ref>. Then t^NK_0 is a (t^NAt^-N)-unbounded component of X-t^NA· Q(N) with stabilizer t^NAt^-N, and the vertex v of X-t^NA· Q is in t^NK_0 if and only if either P(v)≥ N-M or P(v)<N-M and vt^N-P(v)∉t^NA, 3) if K is any component of X-A· Q other than K_0 then t^NK is a (t^NAt^-N)-unbounded component of X-t^NA· Q(M), and if v is a vertex of t^NK, then P(v)<N-M and vt^N-P(v)∈ t^NA.Part 1): If v∈ t^NA· Q(M), then P(v)∈{N,N-1,…, N-M}. Note thatP(t^-Nv)=-N+P(v)∈{0,…, -M}. Lemma <ref> implies, t^-Nv∈ A· Q if and only ift^-Nvt^-P(t^-Nv)∈ A if and only if vt^N-P(v)∈ t^NA. Furthermore if v∈ t^NA· Q(M) then t^-Nv∈ A· Q(M) and by Lemma <ref>, t^-NvA⊂ A· Q(M) so that vA⊂ t^NA· Q(M). Part 2): By Lemma <ref>,t^NK_0 is a component of X-t^N(A· Q(M)). Since t∈ K_0, t^N+1∈ t^NK_0, and so the proper ray at t^N+1 with all edge labels t belongs to t^NK_0. In particular, t^NK_0 is t^NAt^-N-unbounded.Since A stabilizes A· Q(M), t^NAt^-N stabilizes t^NA· Q(M). The vertex vof X belongs to t^NK_0 if and only if t^-Nv∈ K_0, (by Lemma <ref>) if and only if P(t^-Nv)≥ -M or both P(t^-Nv)< -M and t^-Nvt^-P(t^-Nv)∉A, if and only if P(v)≥ N-M or both P(v)<N-M and vt^N-P(v)∉t^NA.Part 3): Suppose v is a vertex of t^NK then t^-Nv∈K.By Lemma <ref>, P(t^-Nv)<-M (so P(v)<N-M) and t^-Nvt^-P(t^-Nv)∈ A (so vt^N-P(v)∈ t^NA). Geometrically, the only difference between Lemma <ref> and Lemma <ref> is that in order to check if a vertex v in a level of X less than N, belongs to either t^NA· Q(M), t^NK_0 or t^NK for K a component of X-A· Q(M) different than K_0, one simply checks if the end point of the path at v with each edge labeled t and ending in level N, ends in t^NA or not. It is also important to observe the following remark.For any integers M,N≥ 0 the set t^NA· Q(M)(=D(N,M)) and any component of X-D(N,M) is a union of cosets vA.Suppose M,N≥ 0 are integers and v is a vertex of thecomponent t^NK_0 of X-t^NA· Q(M). Then for any integer n≥ 0, (vt^nA)∩ A· Q(M)=∅.By 1) ofLemma <ref>, it suffices to show that vt^n∉t^NA· Q(M). But this follows directly from parts 1) and 2) of Lemma <ref>.(∗ ) From this point on we assume the presentation 𝒫 has bounded depth B≥ 0.If α is an edge path loop in X and im(P(α))⊂ (-∞,L], then α is homotopically trivial by a homotopy H such that im(P(H))⊂ (-∞, L+B].Using only conjugation 2-cells, α, is homotopic (by a homotopy H_1) to an edge path loop β, each of whose vertices is in level L. In particular, each edge of β is labeled by an element of 𝒜 and im(P(H_1))⊂ (-∞, L]. The word w determined by the edge labeling of β is in the kernel of the epimorphism p:F(𝒜)→ A. So w∈∪_i=0^Bϕ^-i(N_0(ℛ,ϕ)).By Remark <ref>, the loop β (and hence α) is homotopically trivial by a homotopy H such that im(P(H))⊂ (-∞, L+B].Suppose M, N≥ 0 are integers, α is a loop in X-t^NA· Q(M) and B is the bounded depth of the presentation P. 1) If α has image in a component of X-t^NA· Q(M) other than t^NK_0, then α is homotopically trivial by a homotopy H such that P(H) has image in (-∞, B+N-M],2) if α has image in t^NK_0, v is a vertex of α and r_v is the proper edge path ray at v with each edge labeled t, then there is proper homotopy H:[0,∞)× [0,1]→ t^NK_0 where H(x,0)=H(x,1)=r_v(x), and H(0,y)=α (y). Part 1): By 3) of Lemma <ref>, im(P(α))⊂ (-∞, N-M]. Lemma <ref> finishes part 1). Part 2): Let H be the homotopy that strings together the homotopies H_e of Lemma <ref> for each 𝒜-edge e of α. The image of H avoids t^NA· Q(M) by Lemma <ref> and so is in t^NK_0. The homotopy H is proper since it is a combination of finitely many proper homotopies. By Lemma <ref>, if C is a compact subset of X, there are integers M(C) and N(C) such that C⊂ t^NA· Q(M). Suppose G is an ascending HNN extension of the finitely presented group A and X is the Cayley 2-complex for the HNN presentation with stable letter t and base A (with a finite presentation of A as a sub-presentation). If M, N≥ 0 are integers and α is a loop in X-t^NA· Q(M) (=X-t^NAt^-N· (t^NQ(M))) then α is homotopically trivial in X-t^NA· Q(M).We present the case where N=0 as all others are completely analogous. Let Λ be the Cayley 2-complex for A, determined by the presentation of A within our HNN presentation of G.If K is a component of X-A· Q other than K_0 and α is an edge path loop in K, then each vertex v of α is such that P(v)<-M. Using conjugation relations α is homotopic in K to an A-loop α_1 in level -M-1. Then α_1 lies in a copy of Λ in level -M-1 and so is homotopically trivial in level -M-1.If α is an edge path loop in K_0, then by Lemma <ref>, conjugation relations can be used to show that α is homotopic to a loop α_1 in a single level and this homotopy avoids A· Q. Lemma <ref> also implies that α_1 is in a copy of Λ that avoids A· Q. As α_1 is homotopically trivial in that copy of Λ, α_1 (and hence α) is homotopically trivial in X-A· Q. Suppose M,N≥ 0 are integers and s is a proper edge path ray in X-t^NA· Q(M) with initial vertex v∈ K_0. If q is the quotient of X by the action of t^NAt^-N and qs has image in a compact subset of (t^NAt^-M)\ X (so s is t^NAt^-N-bounded), then each vertex of s is within edge path distance ≤ K of t^NA and Ps has image in theclosed interval [N-K, N+K].Suppose M,N≥ 0 are integers, s is a proper edge path ray in the t^NK_0 component of X-t^NA· Q(M) and s(0) =v. Let r_v be the proper edge path ray at v, each of whose edges is labeled t.If Ps has image in a closed interval then s is properly homotopic to r_v by a homotopy with image in t^NK_0.Assume that the image of Ps is [L,M]. By Lemma <ref>,one can use conjugation relations to slide each A-edge of s along t-edges to level M, by a homotopy with image in t^NK_0. So s is properly homotopic to s', the resulting proper ray which (after removing any backtracking edges (t,t^-1) or (t^-1,t)) is a proper 𝒜-ray. Let r' be the proper edge path ray at the initial point of s' with all edges labeled t (so r' is a sub-ray of r_v).Let H be the proper homotopy of s' to r' defined in Lemma <ref>. By Lemma <ref>, H has image in t^NK_0.(of Theorem <ref>) Let X be the Cayley 2-complex of 𝒫. By Proposition <ref>, t^NAt^-N issemistable at ∞ in X for all N≥ 0 and in <ref> we reduced the proof of Theorem <ref> to showing that for each compact set C in X there is an integer N≥ 0 such that t^NAt^-N is co-semistable at ∞ in X with respect to C. That means:For any finite subcomplex C of X there is an integer N≥ 0, and compact set D such that for any proper t^NAt^-N-bounded ray s in X-t^NAt^-ND andloop α in X-t^NAt^-ND such that α(0)=s(0), there is a proper homotopy H:[0,1]× [0,∞)→ X-C such that H(0,t)=H(1,t)=s(t) and H(t,0)=α.Start with a finite subcomplex C of X.The integer N(C)≥ 0 will play the part of N. Recall that B is the bounded depth of the presentation 𝒫. LetD=t^N(C) Q(M(C)+B+1) Recall Q(M)={∗, t^-1,… ,t^-M}. By Lemma <ref>, for each vertex v∈ C:vA⊂t^N(C)At^-N(C)(t^N(C)Q(M(C)))⊂ t^N(C)At^-N(C)(t^N(C)Q(M(C)+B+1))= t^N(C)At^-N(C)D=t^N(C)A· Q(M(C)+B+1).If v∈ C, then v∈ t^N(C)A Q(M(C)) so that P(v)∈ [N(C)-M(C),N(C)]. Suppose α is a loop in X-t^N(C)At^-N(C)D. Then α is either in t^N(C) K_0 where K_0 is the special component of X-A· Q(M(C)+B+1) (described in part 2) of Lemma <ref>) or α is in t^N(C)K for somecomponent K of X-A· Q(M(C)+B+1) other than K_0. If α belongs to t^N(C)K, then by part 1) of Lemma <ref>, α is homotopically trivial by a homotopy H such that im(P(H))⊂ (-∞, B+N(C)-(M(C)+B+1)]=(-∞, N(C)-M(C)-1]. Since P(C)⊂ [N(C)-M(C),N(C)], the homotopy H kills α in X-C (actually in X-A· C).If α is in t^N(C)K_0, and s is a t^N(C)Dt^-N(C)-bounded proper ray in t^N(C)K_0 such that α(0)=s(0), then by Lemma <ref>, s is properly homotopic (rel{s(0)}) to rthe proper edge path ray at s(0), each of whose edges is labeled t, by a homotopy with image in t^N(C)K_0⊂ X-C. Combining the homotopy of r to s with one given by part 2) of Lemma <ref> (also in t^N(C)K_0) completes the proof.amsalpha Michael MihalikDepartment of Mathematics, Vanderbilt University, Nashville, TN 37240email: [email protected] | http://arxiv.org/abs/1709.09140v1 | {
"authors": [
"Michael Mihalik"
],
"categories": [
"math.GR"
],
"primary_category": "math.GR",
"published": "20170926172406",
"title": "Bounded Depth Ascending HNN Extensions and $π_1$-Semistability at $\\infty$"
} |
firstpage–lastpage Veto Interval Graphs and Variations Breeann Flesch Computer Science Division Western Oregon University 345 North Monmouth Ave. Monmouth, OR 97361 Jessica Kawana^*, Joshua D. Laison^* Mathematics Department Willamette University 900 State St. Salem, OR 97301 Dana Lapides^* Earth and Planetary Science University of California, Berkeley 307 McCone Hall Berkeley, CA 94720-4767 Stephanie Partlow[Funded by NSF Grant DMS 1157105] Mathematics Department Woodburn Wellness, Business And Sports School 1785 N Front St Woodburn, OR 97071 Gregory J. Puleo Department of Mathematics and Statistics College of Sciences and Mathematics Auburn University 221 Parker Hall Auburn, Alabama 36849 Received: Jul 6, 2017 / Accepted: Sep 11, 2017 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== We study particle acceleration and magnetic field amplification in theprimary hotspot in the northwest jet of radiogalaxy Cygnus A. By using theobserved flux density at 43 GHz in a well resolved region of this hotspot, we determine the minimum value of the jet density and constrain the magnitude of the magnetic field.We find that a jet with density greater than 5×10^-5 cm^-3 andhotspot magnetic field in the range 50-400 μG arerequired to explain the synchrotron emission at 43 GHz. The upper-energy cut-off in the hotspot synchrotron spectrum is at a frequency≲ 5×10^14 Hz, indicating that the maximum energy of non-thermal electrons accelerated at the jet reverse shock is E_e, max∼ 0.8 TeV in a magnetic field of100 μG.Based on the condition that the magnetic-turbulence scale length has to belarger than the plasma skin depth, and that the energy density in non-thermal particles cannotviolate the limit imposed by thejet kinetic luminosity, we show that E_e, max cannot beconstrained by synchrotron losses as traditionally assumed.In addition to that, and assuming that the shock is quasi-perpendicular, we show that non-resonant hybrid instabilitiesgenerated by the streaming of cosmic rays with energyE_e, max can grow fast enoughto amplify the jet magnetic field up to 50-400 μG and accelerateparticles up to the maximum energy E_e, max observed in the Cygnus Aprimary hotspot. galaxies: active – galaxies: jets –– acceleration of particles – radiation mechanisms: non-thermal – shock waves § INTRODUCTIONType II Fanaroff-Riley (FR) radiogalaxies exhibit well collimated jets withbright radio synchrotron knots (hotspots) at the termination region.Electrons radiating in the hotspotare locally accelerated in the jet reverseshock, and they reach a maximum energy E_e, maxinferred from theInfrared (IR)/optical cut-off frequency (ν_ c) of the synchrotron spectrum: E_e, max/ TeV∼ 0.8 (ν_ c/5×10^14Hz)^1/2(B/100 μ G)^-1/2, where B is the magnetic field <cit.>.In some cases, X-rays are also detected and modeled assynchrotron self Compton emission and Compton up-scattering of Cosmic Microwave Background photons <cit.>. We note however that in very few cases X-ray synchrotron emission is proposed <cit.>.Ions can also be accelerated in the jet reverse shock. Given that hadroniclosses are very slow in low density plasmas such asthe termination region of FR II radiogalaxy jets,protons might achieveenergies as large as thelimit imposed by the size of the system, usuallycalled"Hillas limit" <cit.>. In particular, mildly relativisticshocks with velocityv_ sh=c/3might accelerate particles withLarmor radius r_ g∼ R_ j, where R_ j is the jet width at the termination region. Particles with such a large r_ ghave energyE_ Hillas/ EeV∼ 100 (v_ sh/c/3) (B/100μ G) (R_ j/ kpc), as expected for Ultra High Energy Cosmic Rays (UHECRs)<cit.>. <cit.> examinethe maximum energy to which Cosmic Rays (CR) can be accelerated by relativistic shocks, showing that acceleration of protons to 100 EeV isunlikely. <cit.>. In <cit.> we have shown that hotspots of FR II radiogalaxies arevery poor accelerators. We have shown that the maximum energy of non-thermal electrons accelerated at the reverse shocks is not determinedby synchrotron losses, unless very extreme conditions in the plasma are assumed[In our previous papers we calledE_ c = E_e, max and E_ uhecr=E_ Hillas.].By equating the acceleration and synchrotron cooling timescales,we show that the mean free path of the most energetic electrons accelerated at thejet termination shocks is greater than the maximum valueimposed by plasma physics for canonical values of the magnetic field and jet density. We demonstrated this byconsidering the sample of 8 hotspots observed with high spatial resolution at optical, IR and radio wavelengths by <cit.>. If synchrotron losses do not balance energy gain, the electrons' maximumenergy E_e, max is ultimately determined by the ability to scatter particles in the shock environment, andthis limit applies to both electrons and protons. Assuming that the jet magnetic field downstream of the shockis quasi-perpendicular,we foundthat non-resonant (Bell) turbulence generated by the streaming ofCRscan grow fast enoughto amplify the jet magnetic field by about two orders of magnitude and accelerate particles up to E_e, max∼ 0.1-1 TeV.In the present paper we study the FR II radiogalaxy Cygnus A, having a redshift z∼ 0.05607 (d ∼227.3 Mpc, where d is the distancefrom Earth) in the Cygnus galaxy cluster <cit.>. The northwest jet terminates at ∼ 60 kpc from the central source where the primary (B) and secondary (A) hotspotsare detected [The northwest primary and secondary hotspots are sometimes called B and A, respectively <cit.>.].<cit.> modelled the radio-to-X-rays non-thermalemission fromthe secondary hotspots in the one-zone approximation and assumed that E_e, max isdetermined by synchrotron cooling. In this work weapply the samemethodology presented in<cit.> to the northwest primary hotspot. We improve our previousmodel by removing the assumption that the jet density (at the terminationregion) is n_ jet=10^-4 cm^-3. Using the 43 GHz high spatial resolution data we constrain the magnetic field and the jet density (Section <ref>). On the other hand, using thecut-off of the synchrotron spectrumdetermined from IR and optical emission we show that E_e, max cannot be determined by synchrotroncooling, unless the jet density is of the order of thedensity in the external medium (Section <ref>).Finally, assuming that the magnetic field downstream of the shock isquasi-perpendicular, we constrain the scale size of magnetic turbulence (Section <ref>) and show that it can be excited throughthe non resonant hybrid (NRH) instability (Section <ref>).We conclude that the primary hotspot in Cygnus A is a clear example whereparticle acceleration is not constrained by synchrotron losses.Throughout this paper we use cgs units and thecosmology H_0 = 71 km s^-1 Mpc^-1, Ω_0 = 1 andΛ_0 =0.73. One arcsecond represents 1.044 kpc on the plane ofthe sky at z = 0.05607. § SYNCHROTRON RADIO EMISSION FROM THE NORTHWEST PRIMARY HOTSPOTThe northwest primary hotspot has been detected with the MERLIN interferometer at 151 MHz <cit.> and with the Very Large Array (VLA) at frequencies from 327 MHz to 87 GHz <cit.>.In addition to that, 230 GHz emission was detected with the BIMA array with 1^'' angular resolution. <cit.> made spectral index maps and found that the 5-230, 5-15, and 15-230 GHz spectral indices areα_5^230 =α_5^15 = α_15^230 =1.13, whereas α_87^230 =1.23. Therefore no spectral break is observed between 5 and 230 GHz.These steep radio spectral indices would indicate that electrons emitting synchrotron radiation at these frequenciesradiate most of their energy in the hotspot.However, recent analysis from the same set of VLA datashows that thespectral index from 5 to 43 GHz is α_5^43∼ 0.72 <cit.>, consistent with standard diffusive shock acceleration in the slow coolingregime. In the following section we will consider thewell resolvedemission at 43 GHz to constrain the value of the magneticfield[The VLA beam-size at 43 GHz is 0.07×0.06 arcsec^2.]. §.§ Constraining the magnetic field with the synchrotron emissionat 43 GHzFigure <ref> shows the hotspot at 43 GHz,where the region considered by <cit.> to calculate the spectral index (α_5^43) is indicated by the grey rectangle of0.7×1.2 arcsec^2. For our study, we select aregion of 0.5× 0.9 arcsec^2 (indicated by thewhite rectangle in Figure <ref>) defined by the half-height points of the emission peak. Considering that the radio emitter is a cylinder of diameter D=0.9^'' and width (projected in the plane of the sky)l_ obs=0.5^'',the emitter volumeat 43 GHz is V = π D^2l_ obs/4 ∼0.32 arcsec^3(i.e. V ∼ 0.36 kpc^3). The background emission corrected flux at 43 GHz is f_43=0.36 Jy <cit.>, and the specific luminosity is L_43 = 43×10^9 f_43 4 π d^2∼ 9×10^41 erg s^-1. We model the synchrotron radio emissionin V as producedby non-thermal electrons followinga power-lawenergy distribution N_e ∝ E_e^-p, withp = 2α_5^43 +1 = 2.44 and E_e ≥ E_e, min = m_ec^2 γ_e, min, where γ_e, min∼ 450 (ν_ min/0.1GHz)^1/2(B/100 μ G)^-1/2 and ν_ min is the frequency of the low-energy turnover[ Using the Low Frequency Array (LOFAR) between 109 and 183 MHz, at an angular resolution of ∼ 3.5^'', <cit.> found that the low energy turnover of the secondary hotspots synchrotron spectra in Cygnus A is at ∼ 150 MHz.] <cit.>.We insertγ_e, min and the numerical values of V, p, ν, and L_43 (see Table <ref>) in equations 20 and 21 in <cit.>.We find thatthe energy densityin non-thermal electrons determined from the synchrotron emission at 43 GHz is U_e/ erg cm^-3∼2×10^-8(ν_ min/0.1GHz)^-0.22(B/100 μ G)^-3/2(V/0.36kpc^3)^-1, and the magnetic field in equipartition with non-thermal electrons andprotons would be B_ eq/μ G∼ 390 (1+a/2)^2/7(ν_ min/0.1 GHz)^-0.06(V/0.36kpc^3)^-2/7, where the energy density in non-thermal protons is U_p = a U_e and therefore the non-thermal energy density is U_ nt = (1+a)U_e.Note the weak dependence of U_e and B_ eq on ν_ min. We keep V fixed in Eqs. (<ref>) and (<ref>).The jets of Cygnus A suggest a precession pattern from which the jet velocitywas estimated as 0.2c < v_ jet< 0.5c in the termination region <cit.>.The jet kinetic energy density is U_ kin/ erg cm^-3∼ 9× 10^-9(n_ jet/10^-4cm^-3) (Γ_ jet -1/0.06), where Γ_ jet = 1.06 is the jet bulk Lorentz factor whenv_ jet = c/3 (see Table <ref>). By setting the extreme condition U_ kin= 2 B_ eq^2/(8π),i.e. all the jet kinetic energy density in the shock upstream region is converted into magnetic (U_ mag) and non-thermal (U_ nt) energydensities in the downstream region (the hotspot) and thatU_ mag=U_ nt=(1+a)U_e (the equipartition condition), the minimum jet matter density (at the termination region) is n_ jet,min= 2(B_ eq^2/8π) (1/m_pc^2(Γ_ jet -1))∼ 1.36×10^-4(1+a/2)^4/7(ν_ min/0.1 GHz)^-0.12×(Γ_ jet -1/0.06)^-1(V/0.36kpc^3)^-4/7cm^-3. In Figure <ref> weplotn_ jet,min for 0≤ a ≤ 50 and considering v_ jet = c/3.Note thatn_ jet,min∝ 1/(Γ_ jet-1) and therefore it is ∼3.7 times larger and 0.41 times smaller than the values plotted inFigure <ref> when v_ jet = c/5 and c/2, respectively. In Table <ref> we list n_ jet,min forv_ jet = c/5, c/3, and c/2, and a=1. In relativistic shocks we do not expect a much larger than 1,andhereafter weconsider a=1. Therefore, the energy density innon-thermal particles is U_ nt = 2 U_e.Given that 2 U_e > B^2/(8π) when n_ jet > n_ jet,min,most of thejet kinetic energy goes to non-thermalparticles when we consider that thereis only magnetic and non-thermal energy in the hotspot. Therefore, we findthe magnetic field minimum value (B_ min) required to explain theemission at 43 GHz by setting the conditionU_ kin = 2 U_e + B_ min^2/(8π).In Figure <ref> we plotB_ min (green-solid line) for v_ jet=c/2, c/3, and c/5, andfrom n_ jet,min to 10^-3 cm^-3∼ 0.1 n_ ext, wheren_ ext∼ 10^-2 cm^-3 is the density in the external medium<cit.>. Valuesof n_ jet larger than 10^-3 cm^-3 would be very unrealistic given that the jet to external medium density ratioin adiabatic flows is expected to be ∼ 10^-2. In fact, <cit.> found that the plasma density in the jet of Cygnus A is smaller than 4×10^-4 cm^-3. In order to provide an analytical expression (B_ min,a) of B_ min we set the unrealisticcondition 2U_e = U_ kin and thereforeB_ min,a/μ G ∼305 (ν_ min/0.1GHz)^-0.15×[ (n_ j/10^-4 cm^-3) (V/0.36kpc^3)(Γ_ jet -1/0.06)]^-2/3. In Figure <ref> we plot B_ min and B_ min,a (green-dashed line),and wesee that B_ min,a is a very good approximation. Finally, the hotspot magnetic field required to explain the synchrotronemission at 43 GHz is 50 ≲ B ≲ 400 μG. We keepV in Eqs. (<ref>) and (<ref>) to show thatB_ eq and B_ min,a, and therefore B_ min, increases when V is smaller than 0.36 kpc^3. This is the case when we take into account that the jet is inclined by anangle∼ 70^∘ with the line of sight<cit.>. In such a case, the real extent ofthe synchrotron emitter is smaller than l_ obs and thereforethe emitter volume is smaller than 0.36 kpc^3 <cit.>.The hotspot magnetic field could also be constrained by modeling the(synchrotron self Compton) X-ray emission<cit.>. However, we need to know theX-ray-emitter volume which is not easy to determine fromthe data in the X-ray domain.§ CUT-OFF OF THE SYNCHROTRON SPECTRUMDiffuse IR (at frequencies 3.798×10^13 and 6.655×10^13 Hz)and optical (ν_ opt = 5.45×10^14 Hz)emission was detected with the Spitzer and Hubble Space Telescopes,respectively <cit.>.The very steep IR-to-optical spectral index, α_ IR-opt∼ 2.16,indicates that the cut-off of the synchrotron spectrum isν_ c < 5×10^14 Hz. <cit.> suggestedthat the optical emissionis the low-energy tail of the synchrotron selfCompton spectrum,as in the case of the Cygnus A northwest secondary hotspot. In such a case, ν_ c < 5×10^14 Hz. The maximum energy ofnon-thermal electrons accelerated at the jet reverse shock isE_e, max∼ 0.8 TeV when ν_ c =5×10^14 Hz and B = 100 μG, as shown in Eq. (<ref>). §.§ Revising the reigning paradigmIt is commonly assumed in the literature that E_e, max is determinedby synchrotron losses <cit.>. In such a case, by equatingthe synchrotron cooling time, t_ synchr∼ 600/(E_e, max B^2) s,with the acceleration timescale t_ acc∼ 20 𝒟/v_ sh^2,where the diffusion coefficient is 𝒟= λ c/3 and λ is the mean-free path, we find that 𝒟/𝒟_ ℬℴ𝒽𝓂 =λ/r_ g∼ 2×10^6 (v_ sh/c/3)^2 (ν_ c/5× 10^14Hz)^-1. In Eq. (<ref>)𝒟_ ℬℴ𝒽𝓂 = r_ gc/3 is the Bohm diffusion coefficient and r_ g = E_e, max/(eB) is the Larmor radiusof E_e, max-electrons (and protons) in a turbulent field B. In the small scale turbulence regime λ = r_ g^2/s, where s isthe plasma-turbulence scale-length <cit.>.Therefore, from Eq. (<ref>), the plasma-turbulence scale-length in the"reigning paradigm" iss∼r_ g^2/λ =r_ g𝒟_ ℬℴ𝒽𝓂/𝒟∼ 8.3×10^6 (ν_ c/5× 10^14Hz)^3/2(B/100 μ G)^-3/2(v_ sh/c/3)^-2cm. Surprisingly, s is smaller than the ion-skin depth c/ω_ pi∼ 10^9 Γ_ jet^0.5(n_ jet/10^-4 cm^-3)^-0.5 cm unless B is smaller thanB_ max,s/μ G∼ 2 (ν_ c/5× 10^14Hz) (n_ jet/10^-4cm^-3)^1/3(v_ sh/c/3)^-4/3. (Note that B_ max,s∝Γ_ jet^-1/3, but we neglect this dependence in Eq. (<ref>) given that1.02≤Γ_ jet≤ 1.15 when c/5 ≤ v_ jet≤ c/2.)In Figure <ref> we plot B_ max,s (blue-solid line) for thecase v_ jet = c/3. We see thatB_ min is larger than B_ max,s for allpossible values ofn_ jet. We mentioned that 5× 10^14 Hz is the upper-limitfor the synchrotron spectrum cut-off. In the case thatν_ c<5× 10^14 Hz, B_ max,s is even smaller than the valueplotted in Figure <ref> whereas B_ min increases.Therefore, ν_ c<5× 10^14 Hz enlarges the gap between B_ minandB_ max,s. Note that B_ min/B_ max,s also increases when weconsider an emission volume (at 43 GHz) smaller than 0.36 kpc^3 as aconsequence of the jet inclination angle <cit.>. In Figure <ref> we plot B_ min/B_ max,s for the cases v_ jet = c/2 (blue-solid line),v_ jet = c/3(green-dot-dashed line), and c/5 (orange-dashed line). We can see that B_ min > B_ max,s for all possible values of v_ jet andn_ jet.Hence weshow that B is larger than B_ max,s for a large range of parametersand therefore E_e, maxcannot be determined by synchrotron cooling in the primary hotspot of Cygnus A, in disagreement with the standard assumption as was pointed out by <cit.>.Note that to reach this conclusion we haveonly used well resolved radio emission at 43 GHz and the requirement s > c/ω_ pi. In the next section we explore a more fundamental limit to constrain E_e, max. § THE CASE OF PERPENDICULAR SHOCKS The maximum energy is ultimately constrained by the ability to scatterparticles back and forth across the shock, and this depends on the geometryof the magnetic field (i.e. the angle between the field vector and the shocknormal). In this section we consider the case of perpendicular shocks, giventhat relativistic shocks are characteristically quasi-perpendicular.Note however that shocks moving at v_ sh∼ c/3 are mildlyrelativistic and therefore they may not be strictly perpendicular. Unfortunately, it is not possible to determine the geometry of the magneticfield in the reverse shock downstream region using thepolarization dataavailable in the literature. §.§ Electrons' maximum energy determined by the diffusion condition To accelerate particlesup to an energy E_e, max in perpendicular shocks,the mean-free path in turbulent magnetic field in the shock downstream region,λ_ d∼ (E_e, max/eB)^2/s has to be smaller than Larmor radius in B_ jd in order to avoid the particles following the B_ jd-helical orbits and cross-field diffusionceasing <cit.>.The condition λ_ d≲ r_ g0, wherer_ g0 = E_e, max/(eB_ jd) is the Larmor radius in the ordered(and compressed)field B_ jd∼ 4 B_ j, where B_ j is the jet magnetic field,is marginally satisfied when the magnetic-turbulencescale-length is s = s_⊥, where s_⊥ =E_e, max/eB(4B_ j/B) ∼ 6.7×10^11(ν_ c/5× 10^14Hz)^1/2(B_ j/μ G) (B/ 100μ G)^-5/2cm. In Figure <ref> we plot s_⊥ for the cases of B = B_ eq(red-dotted line) and B = B_ min (green-dotted lines) and fixingB_ j=1 μG. We plot alsoc/ω_ pi (∝ n_ jet^-0.5). Note thats_⊥ > c/ω_ pi which indicates that the magneticfield is probably not generated by the Weibel instability(that has a characteristic scale length ofc/ω_ pi).§.§ NRH instabilities in perpendicular shocksTurbulence on a scale greater than c/ω_ pi may be excited through the NRH instability, which can grow until s reaches the Larmor radius of the highest energy CR driving the instability <cit.>.Since the scattering rate is proportional to E^-2 in given small scaleturbulence, the distance over which CR currents are anisotropised downstream of the shock is proportional to E^2. Hence the higher energy CR have moretime to drive the NRH instability, and CR with energy E_ e,max are predominantly responsible for generating the turbulence unless the CRspectrum is unusually steep (p>3). As explained above, the maximum CR energyE_ e,max is that of CR whose anistropy decays over a distance equal totheir Larmor radiusin the ordered component of the downstream magneticfield <cit.>We now discuss whether E_ e,max-CRs have sufficient energy density toamplify the magnetic field. To amplify the magnetic field via the NRHinstability in a perpendicular shock, the turbulent field has to grow througharound 10 e-foldings at the maximum growth rate Γ_ max<cit.>.The time available for the instability to grow ist_⊥ = r_ g0/v_ d during which the plasmaflows through adistance r_ g0 in the downstream region at velocity v_ d∼ v_ sh/4.Therefore, the condition formagnetic field amplification by the NRH instability inperpendicular shocks is Γ_ maxt_⊥ > 10. In perpendicular shocks where both the CRcurrent j_ CR andB_ jd are in the plane of the shock and orthogonal to each other, Γ_ max is similar to thelinear growth rate in parallel shocks, as shown by <cit.> and <cit.>,and in agreement with the dispersion relation derived by <cit.>.Therefore,in perpendicular geometry,Γ_ max∼ (j_ CR/c) √(π/ρ_ jet),where ρ_ jet = m_p n_ jet.The current density carried by E_ e,max-CRs isj_ e,max = η_ e,max U_ kin c e/E_ e,max, where η_ e,max =1 notionally represents the condition in which theCR electron number density at energy E _ e,max is equal to U_ kin/E_ e,max and the CR drift alongthe shock surface at velocity c. Allowing for compression of the mass density and magnetic field by a factorof four at the shock, the condition Γ_max t_⊥ >10 leads to alower limit on η _ e,max: η _e, max > η_ min = 80/M_ A where M_ A=v_ sh/v_ A is the Alfven Mach number of the jet at thetermination shock andv_ A=B_ j/√(4 πρ _ jet), giving M_ A= 1400 (v_ sh/c/3)(B_ j/μ G)^-1(n_ jet/10^-4cm^-3)^1/2 and thereforeη_ min = 0.057(v_ sh/c/3)^-1(B_ j/μ G) (n_ jet/10^-4cm^-3)^-1/2. The ordered magnetic field B_ jin the termination region of AGN jets is unknown, but values lower than 1 μ G are reasonableconsidering the lateral expansion of the jet during propagation from itsorigin in the active galactic nucleus. It appears that the CR current is sufficient todrive the NRH instability,but the margins are tight, CR acceleration to energy E _ e,max must beefficient, and the jet magnetic field must be small.In order to check whether these conditions are satisfied in the primary hotspot of Cygnus A, we consider that non-thermal protons are accelerated in the jet reverse shock following a power-law energy distribution with the sameindex as non-thermal electrons (p=2.44). In such a case, the energy densityinE_e, max-protons is U_e, max = K_p E_e, max^2-p, where K_pis the normalization constant of the energy distribution. Considering thatU_e = U_p (see Section <ref>)we findK_p = U_e (p-2)/E_p, min^2-p where E_p, minis the minimum energy of non-thermal protons. By settingE_p, min=1 GeVwe find that the acceleration efficiency of E_e, max-protons is η_e, max ≡U_e, max/U_ kin∼ 0.44(U_e/U_ kin) (E_e, max/ GeV)^-0.44∼ 0.07[(ν_ c/5×10^14Hz) (ν_ min/0.1GHz)]^-0.22(B/ 100 μ G)^-1.28×[(n_ jet/10^-4cm^-3) (Γ_ jet-1/0.06) (V/0.36kpc^3)]^-1 Therefore, to satisfy the condition η_e, max>η_ min(Eq. <ref>) for efficient magnetic field amplification by theNRH-instability in a perpendicular shock, the jet (unperturbed) magneticfield has to be(B_ j/μ G) < 1.2 (n_ jet/10^-4cm^-3)^-1/2(B/ 100 μ G)^-1.28(Γ_ jet-1/0.06)^-1, when V, ν_ c, and ν_ min take the values in Table <ref>, and v_ sh∼ c/3. In such a case, and assuming that the shock is quasi-perpendicular, E_e, max-CRs have sufficient energy density togenerate NRH-turbulence on scale s_⊥ and amplify the magnetic field by a factor B/B_ j∼ 100 in theprimary hotspot of Cygnus A. § CONCLUSIONS We study diffusive shock acceleration and magnetic field amplification inthe northwest primary hostspot inCygnus A. We focus on the well resolved region downstream of the jet reverse shock where most of the synchrotron radiation is emitted.By consideringthe synchrotronflux at43 GHz we determine that the jet density has to be largerthan∼ 5×10^-5 cm^-3 and the hotspot magnetic field is50≲ B ≲ 400 μG (when the energy density in non-thermalprotons is the same as in non-thermal electrons, i.e. a=1, andc/5 < v_ jet < c/2). The cut-off of the synchrotron spectrum is atν_ c≲5×10^14 Hz, implying that the maximum energy ofelectrons accelerated in the hotspots is E_e, max< 1 TeV. By settingthe magnetic-turbulence scale-length s larger than the ion-skin depthc/ω_ pi (in the small-scale turbulence regime) we find that themagnetic field required to be E_e, max determined by synchrotron coolingis smaller than the field required to explain the synchrotron emission at43 GHz. Therefore, we conclude that E_e, max is not constrained bysynchrotron cooling, as traditionally assumed. The maximum energy E_e, max is ultimately determined by the scatteringprocess.By assumingthat the shock is quasi-perpendicular, particles cannot diffuse further than a distance r_ g0downstream of the shock, i.e. λ_ d< r_ g0. To satisfy thiscondition, the magnetic turbulence scale-length has to be larger than ∼ 2×10^10 cm, that is ∼ 10c/ω_ pi(see Fig. <ref>), and therefore B is probably not amplifiedby the Weibelturbulence. On the other hand, the NRH instability amplify the magnetic field on scaleslarger than c/ω_ pi and we show thatNRH-modes generated by CRs with energies E_e, max can grow fast enoughto amplify the jet magnetic fieldfrom ∼1 to 100 μG and accelerate particles up to energies E_e, max∼ 0.8 TeV observed in the primary hotspot of Cygnus Aradiogalaxy. The advantage of magnetic turbulence being generated by CRs current is that B persists over long distances downstream of theshock, and therefore particles accelerated very near the shock can emit synchrotron radiation far downstream. Finally, if E_e, max is determined by the diffusion condition in aperpendicular shock, the same limit applies to protons and therefore themaximum energy of ions is also ∼ 0.8 TeV. As a consequence, relativisticshocks in the termination region of FR II jets are poor cosmic ray accelerators. § ACKNOWLEDGEMENTS The authors thank the anonymous referee for a constructive report. The authors thankS. Pyrzas for providing Figure 3 in <cit.>,and Alexandre Marcowith and Robert Laing for useful comments.The research leading to this article has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013)/ERC grant agreementno. 247039. We acknowledge support from the UK Science and Technology Facilities Council under grantsST/K00106X and ST/N000919/1. mnras | http://arxiv.org/abs/1709.09231v1 | {
"authors": [
"Anabella T. Araudo",
"Anthony R. Bell",
"Katherine M. Blundell",
"James H. Matthews"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170926192633",
"title": "On the maximum energy of non-thermal particles in the primary hotspot of Cygnus A"
} |
IEEEexample:BSTcontrolThe Co-Evolution of Test Maintenance and Code Maintenance through the lens of Fine-Grained Semantic Changes Stanislav Levin The Blavatnik School of Computer Science Tel Aviv UniversityTel-Aviv, [email protected] Amiram Yehudai The Blavatnik School of Computer Science Tel Aviv UniversityTel-Aviv, [email protected] 30, 2023 ================================================================================================================================================================================================================================================Automatic testing is a widely adopted technique for improving software quality. Software developers add, remove and update test methods and test classes as part of the software development process as well as during the evolution phase, following the initial release. In this work we conduct a large scale study of 61 popular open source projects and report the relationships we have established between test maintenance, production code maintenance, and semantic changes(e.g, statement added, method removed, etc.). performed in developers' commits. We build predictive models, and show that the number of tests in a software project can be well predicted by employing code maintenance profiles (i.e., how many commits were performed in each of the maintenance activities: corrective, perfective, adaptive). Our findings also reveal that more often than not, developers perform code fixes without performing complementary test maintenance in the same commit (e.g., update an existing test or add a new one). When developers do perform test maintenance, it is likely to be affected by the semantic changes they perform as part of their commit.Our work is based on studying 61 popular open source projects,comprised of over 240,000 commits consisting of over 16,000,000 semantic change type instances,performed by over 4,000 software engineers. Software Testing; Software Maintenance; Mining Software Repositories; Predictive Models; Software metrics; Human Factors; § INTRODUCTION Automated testing, and automatic unit tests <cit.> in particular, is a popular technique for improving software quality. Our work <cit.> showed that semantic changes [A.K.A, fine-grained source code changes <cit.>.], such as method removed, field added, are statistically significant in the context of software code maintenance. Moreover, semantic changes can be used to effectively classify commits into maintenance activities (as defined by Mockus et al. <cit.>).These studies may indicate that semantic changes can also be useful in the context of test maintenance, and particularly, in exploring the co-evolution of test maintenance and production code maintenance, using the semantic changes as the lowest common denominator.The field of software evolution research can be classified into two groups, the first considers the term evolution as a verb while the second as a noun <cit.>.The verbal view research is concerned with the question of “how”, and focuses on means, processes, activities, languages, methods, tools required to effectively and reliably evolve a software system.The nounal view research is concerned with the question of “what” and investigates the nature of software evolution, i.e., the phenomenon, and focuses on the nature of evolution, its causes, properties, characteristics, consequences, impact, management and control <cit.>.Lehman et. al. <cit.> suggest that both views are mutually supportive. Moreover, it is suggested that the verbal view research will benefit from progress made in studying the nounal view, and both are required if the community is to make progress in mastering software evolution.Our intuition is that exploring the co-evolution of test maintenance and production code maintenance may benefit both the verbal and nounal views. In terms of the verbal view, our study may help managers and practitioners reduce costs by improving the quality of their code artifacts. For example, if certain semantic changes tend to be under-tested, it could be beneficial to design tools that can detect such changes and act upon them, whether by prompting the developers to take measures or by automatically performing scripted mitigation. In terms of the nounal view, in order to be able to design and implement such tools, we must first better understand the nature of the relationship between test evolution and production code evolution in general, and between test maintenance activities and code maintenance activities in particular.This work is also motivated by the rapidly growing number of production grade open source projects hosted on a web based platform such as GitHub <cit.>, BitBucket <cit.> and others <cit.>. All make a great source of publicly available, free, and high quality source code corpora which was not available in the earlier stages of studying software and test evolution. The absence of such corpora, often resulted in the models for growth dynamics being relatively simplistic <cit.>. Moreover, along with the large source code corpora now available, progress has been made in the Big Data ecosystem <cit.>, bringing software tools capable of processing extremely large data volumes to the masses. The combination of the two has created unprecedented opportunities to collect and process an enormous volume of source code <cit.>, and provide insights that were previously exponentially harder, or even impossible to obtain <cit.>.Our study concentrates on the following research questions: * How does (production) code maintenance relate to projects' tests count?* How often is test maintenance performed as part of (production) code maintenance?* How do semantic changes performed in (production) code maintenance relate to test maintenance activities? § DATA COLLECTION We harvest software code repositories from GitHub (<cit.>), a popular repository hosting platform with rich query options. Candidate repositories were selected according to the following criteria, which we designed to target data-rich repositories: * Had over 900 Java commits (i.e., commits where Java files were changed)* Were created before 2015-01-01(i.e., these repositories had been around for a while)* Had size over 2MB (i.e. these repositories are of considerable size)* Had more than 100 stars (i.e. more than 100 users had “liked” these repositories)* Had more than 60 forks (i.e., more than 60 users had “copied” these repositories for their own use)* Had their code updated since 2016-01-01 (i.e., these repositories were active) To perform the data collection and processing tasks we use a designated VCS mining platform we have built on top of Spark <cit.>, a state of the art framework for large data processing. Our final dataset consisted of 61 projects [https://github.com/staslev/paper-resources/blob/icsme-2017/The-Co-Evolution-of-Test-Maintenance-and-Code-Maintenance-through-the-lens-of-Fine-Grained-Semantic-Changes/studied-repos.md] from various domains, such as IDEs, programming languages (that were implemented in Java), distributed databases and storage platforms, integration frameworks and more (see summary statistics in table <ref>). This dataset included a total of 242,567 commits, 4,259 developers and 16,161,680 semantic changes. §.§ Distilling Semantic Changes After downloading (cloning) the repositories from GitHub, for each repository r where 1≤ r ≤ 61 we created a series of patch files {p_i^r}_i=1^N_r, where N_r is the latest revision number for repository r. Each patch file p_i^r was responsible for transforming repository r from revision r_i-1 to revision r_i, where r_0 is the empty repository. By initially setting repository r to revision 1 (i.e. the initial revision) and then applying all patches {p_i^r}_i=2^N_r in a sequential manner, the revision history for that repository was essentially replayed. Conceptually, this was equivalent to the case of all developers performing their commits sequentially one by one according to their chronological order.To distill semantic change types as per the taxonomy defined by Fluri et al., we repeatedly applied a customized version of the ChangeDistiller tool (<cit.>) on every two consecutive revisions of every Java file in every repository we had selected to be part of the dataset. We had to perform some modifications to the original ChangeDistiller tool since we encountered use cases where the distilled change list was incomplete. In particular, the addition and removal of classes did not produce changes corresponding to all the internal methods being added or removed as well. Since this scenario is crucial for the extraction of test maintenance activities (including test method addition and removal as a result of a test class addition or removal), we enhanced the original ChangeDistiller with this feature. §.§ Classifying Commits Into Maintenance Activities Three main classification categories for maintenance activities in software projects were identified by Mockus et al. <cit.>: * Corrective: fixing faults, functional and non-functional.* Perfective: improving the system and its design.* Adaptive: introducing new features into the system.In our work on commit classification into maintenance activities <cit.> we suggested a model for cross-project commit classification that was able to achieve an accuracy of 76%. The suggested technique consists of the following steps: * Manually classify the maintenance activities for a ground truth set* For each maintenance activity (“Corrective”, “Perfective” or “Adaptive”), perform a word frequency analysis using the ground truth set commits' comments in order to obtain the 10 most frequent words for each maintenance activity* Distill the semantic changes from the commits in the ground truth set, and perform a frequency analysis* Use the ground truth set to train a RandomForest <cit.> based classification model, where features are the combination of commits' word frequencies and semantic change type frequencies obtained from in previous stepsFor the purpose of this work, we use the classification model we have devised in <cit.>. §.§ Detecting Test MaintenanceIn the scope of this work, we consider a class to be a test class (a.k.a test suite) either if its name starts with the word “Test”, or, if it ends with the words “Test”, or “Tests”, or “TestCase”. We consider a method to be a test method (a.k.a test case) if it has a “Test” Java annotation, OR starts with the word “test” and resides inside a test class. These heuristics are popular both in the software industry <cit.> and academia <cit.>.We use semantic changes to detect test maintenance by inspecting the following semantic change types: * “ADDITIONAL_FUNCTIONALITY” - the addition of a new method* “REMOVED_FUNCTIONALITY” - the removal of an existing method* “ADDITIONAL_CLASS” - the addition of a new class* “REMOVED_CLASS” - the removal of an existing class* Other change type (see <cit.> for the full list) - if the parent entity qualifies as a test method or a test class, the original entity is classified as a test method update or a test class update, respectively.We detect test method/class activity as part of a commit by inspecting its semantic change types and the corresponding element names.To capture the test maintenance activities, given a commit c we define the following metrics:* Test^Method_A(c), number of test methods added in c.* Test^Method_R(c), number of test methods removed in c.* Test^Method_U(c), number of test methods updated in c.* Test^Class_A(c), number of test classes added in c.* Test^Class_R(c), number of test classes removed in c.* Test^Class_U(c), number of changes inside a test class, but outside any of the test methods in that class, as part of c, e.g., setUp, tearDown <cit.>, and other helper methods.* TestMaintenance, the total number of test activities performed as part of a given commit.TestMaintenance(commit) :=∑_scope∈{Method,Class} activity∈{A,R,U}Test_activity^scope§ STATISTICAL METHODS We use regression models to study the relationships between a set of predictors and an outcome variable. In particular, we use generalized linear modeling(GLM) <cit.> for count and for logistic regressions to explore the effects of maintenance activities and semantic change types on test method and class counts, and test maintenance activities.For count regression models, we report the statistically significant predictors, and use their coefficients to analyse their effect on the outcome variable. We then build predictive models by keeping only the most significant predictors. Finally, we use analysis of variance (ANOVA <cit.>) to establish the magnitude of predictors' effects by observing the reduction in the residual deviance associated with the variable’s effect in the model. We evaluate the predictive models by splitting our dataset into a training and a validation datasets, the former is used for training the models, and the latter for evaluating their predictive powers. We report the p-value as a measure of goodness of fit. A high p-value indicates a lack of evidence to support the hypothesis that the observed counts do not match the expected counts, implying a good fit. In case of logistic regression models, we report both statistically significant predictors, and the odds ratio along with their 95% confidence level intervals. The odds in favor of an outcome A is the ratio of the probability of an outcome A to occur and the probability of the complement of A (i.e., that A will not occur) and is defined as P(A)/P(A^c) <cit.>. The odds ratio represents the odds that an outcome will occur given an exposure to a particular effect, compared to the odds of the outcome occurring in the absence of that exposure. Odds ratios are used to compare the relative odds of the occurrence of the outcome of interest, given exposure to the variable(s) of interest.Odds ratio greater than 1 indicates an increase in the odds in favor of the outcome, while odds ratio less than 1 indicates a decrease in the odds in favor of the outcome. Odds ratio equals 1 indicates no effect on the odds of the outcome given a particular exposure.We use the R statistical environment <cit.> for statistical computations, where we extensively use the R caret package <cit.> for the purpose of model training and evaluation.§ PRODUCTION (CODE) MAINTENANCE AND TEST MAINTENANCE§.§ RQ. 1: How does (production) code maintenance relate to projects' tests count?The statistics for test method and test class counts in the projects we studied can be found in table <ref>. Since these counts vary greatly in absolute numbers and are highly dependent on the size of the project, we report themper 1000 LOC to give a standardized perspective.We also compute the average number of test methods in a test class, which stands at 4.636. To better understand the relations between the number of tests (methods and classes), and maintenance activities we devise GLMs of the form:Test^M(prj) = C^M + ∑_i=1^|Predictors|(coeff^M_i * predictor^M_i(prj))where M is the test metric we model, i.e., M ∈{Methods, Classes}, Predictors is the predictors set, coeff^M_i are the predictor coefficients, predictor^M_i(prj) are predictor values, and C^M is the model constant. The corresponding models for Test^Methods and Test^Classes can be found in table <ref>. All predictors were log transformed to alleviate skewed data, a common practise when dealing with software metrics <cit.>. Statistically significant predictors of interest are highlighted in lime-green, and the standard error is reported in parenthesis below the estimated coefficients.In addition to the variables we are directly interested in, such as the log(corrective), log(perfective) and log(adaptive) we also use log(LOC), log(age) and log(developers) as control variables, in order to reduce the effect of lurking variables which correlate both with the predictors and the predicted (outcome) variable. Control variables are highlighted in light-bisque. The models in table <ref> were devised using all 61 projects in our dataset.§.§ Analysing the count regression modelGreater project size (in LOC) and/or age (in days) are likely to boost the the project's expected number of tests (method and class), as indicated by the positive coefficients of the log(LOC), log(age) predictors. These results align well with our intuition according to which mature projects, that are larger in size (LOC wise) and have been developed over long periods of time, are likely to contain more tests.Higher number of perfective commits, i.e., commits that aim to improve system design and code structure, is also likely to boost the expected number of tests (methods and classes) in a project. Higher numbers of corrective commits on the other hand, i.e., commits that aim to fix faults, is likely to associate with projects with less tests (methods and classes), as indicated by the negative coefficient of the log(corrective) predictor for both Test^Methods and Test^Classes. In spite of the fact that regression models do not provide means to ascertain causality, the negative coefficient of corrective commits on tests (i.e., both methods and classes) is worth considering. Potentially, one could argue that projects with tests may only need little corrective activity due to the high quality of the codebase. The opposite direction, may imply that corrective activity may be required when the test count of a project is low, and the codebase's quality is poor. It is also possible, that test counts and corrective commits do not have a cause and effect relationship at all, in which case they just tend to happen together and are connected via a lurking variable. The effect of adaptive commits on test counts is inconclusive as their coefficients did not demonstrate a statistical significance.In addition to establishing the correlations between maintenance activities and test counts, our models can also be used to predict the latter using the former. In order to improve prediction results we optimise the models by keeping only the most statistically significant predictors (i.e. with p-value < 0.01): log(corrective), log(perfective), log(LOC) and log(age).The resulting models were then trained using 53 randomly selected projects (our of 61) from our dataset, and tested using the remaining 8 (∼13% of the entire dataset). Figure <ref> and figure <ref> chart the predicted vs. the actual values of Test^Methods and Test^Classes, respectively. The x-axis is a running index (1…8, indicating the index of the project in our validation data set), and the y-axis is the number of tests. The red line depicts the actual values, and the turquoise line depicts the predicted ones. Our predictions for projects' test methods and test classes show a substantial accuracy, as indicated by the p-value, which was 0.067 and 0.071, respectively (values greater than 0.05 indicate high goodness of fit).The ANOVA for the predictive models (Test^Methods and Test^Classes), which we use to establish the magnitude of predictors' effects, can be found in table <ref> and <ref>, respectively. Each row indicates the reduction in the residual deviance and degrees of freedom induced by adding a given predictor to the model. By observing the “Deviance” column of tables <ref> and <ref> we learn that both perfective and corrective have a substantial contribution to “explaining” the test counts (tests and methods).Perfective commits have the greatest effect in both the test method and test class models, while the corrective commits's effect is 52% less in the Test^Methods model and only 1% less in the Test^Classes model (see cells highlighted in yellow in table <ref> and table <ref>). This may imply that corrective commits play a greater role in predicting test classes than test methods. Also worth noting is the LOC predictor, which demonstrated high explanatory power in both test method and test class predictive models, indicating that the size of the project does indeed have a considerable effect on the number of test methods and test classes it contains.The predictive models we devise (see table <ref>) show that different maintenance activities have different affect on test counts (method and class). Consequently, it implies that having prior knowledge of a commits class (corrective, perfective or adaptive) may contribute to predicting its affect on the overall test count (method and class) of a project. Moreover, these models demonstrate good prediction powers (see figure <ref> and figure <ref>) which can benefit future applications such as resource allocation and project planning (see section <ref>) §.§ RQ. 2: How often is test maintenance performed as part of (production) code maintenance?We are interested in exploring whether test maintenance (addition, deletion or update of test methods or test classes) are common in the context of software projects. In particular, we wish to investigate whether test maintenance is typically performed as part of production code maintenance. In case the two tend to occur together, can test maintenance activities be attributed to any particular code maintenance activity type (corrective, perfective, adaptive)?First, we inspect how common test maintenance is within a project, without accounting for the various code maintenance activities. That is, we inspect the percentage of the commits that involve test maintenance in the scope of a project (see figure <ref>). The box-plot in figure <ref> shows that proportions of test maintenance is quite different across the projects in our study, while in some projects more than half of the commits involved test maintenance, in others less than 15%. In none of the projects, did the test maintenance occur in more than 68.5% of the commits. Moreover, in a few projects there were no tests at all and therefore no test maintenance took place. This may imply testing practices remain to be determined individually in each project, rather than standardised.We then analyse how common test maintenance is, within each of the code maintenance activities (corrective, perfective, adaptive), see figure <ref>.§.§ Analysing test maintenance portions in commitsThe box-plots in figure <ref> show that for half the projects (i.e., the median) in our dataset, test maintenance was present in less than 24.7% of the corrective commits, less than 30.4% in adaptive commits, and less then 35% in perfective commits. It is visually apparent that the test maintenance tends to be the least common within the corrective commits. However, to gain statistical confidence we perform similar analysis on a developer level, where more data points are available (per-developer data points vs. per-project data points). Figure <ref> complements the per-project perspective with a per-developer-per-project one, where for each developer and a maintenance activity type MA∈{Corrective, Perfective, Adaptive}, we calculate the percentage of commits of type MA that involved test maintenance. The Wilcoxon-Mann-Whitney (<cit.>) test confirms (p-value < 0.01) that the percentage of test maintenance within the corrective commits is the lowest. In addition, we can see that some developers perform test maintenance in all of their corrective commits, some perform test maintenance in all of their adaptive commits, and some perform test maintenance in all of their perfective commits. Perhaps, some perform test maintenance in all of their commits.The phenomena of developers performing test maintenance in 100% of their commits in either of the maintenance categories may indeed indicate such developers are true test maintenance fans, alternatively, such cases may indicate that these particular developers perform only few commits, e.g. 1, which translates into a high relative portion (e.g. 100%) of their commits involving test maintenance.The lack of test maintenance in corrective commits is dominant in both project and developer oriented views. We suggest two hypotheses that may account for this phenomena: * a developer commits a change that breaks one or more tests and thus also breaks the build performed by the Continuous Integration (CI) system <cit.>. Now, she must fix the breakage by performing further changes. In this case, the subsequent commit, which would be considered a corrective one, will not involve a test maintenance activity, since the developer is fixing a bug caught by a test.* a developer performs a fix for a bug she has been made aware of, which is not covered by existing tests. The developer completes the fix, but does not provide an automatically runnable proof-of-fix, i.e., she does not add a new test neither does she update an exiting one, so that it will fail in case of regression.Neither of these scenarios is one that adheres to what may be considered best practices. In case of a broken build, we would have expected developers to first run the tests locally (some CI systems allow running tests on private changes remotely <cit.>), and avoid breaking the build for the rest of the team.In case of a fix aimed at eliminating a bug that is not covered by existing test, we would have expected developers to add coverage, that would protect the system in case of regression. We assume there is a CI system in place, an assumption we feel comfortable with in the context of automated testing.§ SEMANTIC CHANGES AND TEST MAINTENANCE§.§ RQ. 3.: How do semantic changes performed in (production) code maintenance relate to test maintenance activities? In order to explore how test maintenance (see section <ref>) is affected by semantic change types (Fluri et al. <cit.>) in production code, we explore our dataset on an individual commit granularity. Studies show that commits which perform a large rename or move refactorings may involve a great deal of test methods and/or classes moves from one package to another <cit.>. Moreover, commits that add entire packages as part of open sourcing new modules, and other activities of similar nature can yield extreme values for test maintenance.Such extreme values pose a challenge to establishing reliable relationships.We deal with these challenges by removing commits with extreme TestMaintenance values by applying a technique suggested by Hubert at el. <cit.> on positiveTestMaintenance values. This technique operates similarly to the traditional IRQ method where data point greater than Q3 + (1.5 * IQR) are removed, however, different thresholds are employed to accommodate skewed distributions. In addition, since our goal is to establish relations between test maintenance and semantic change types in production code, we remove commits that deal purely with test code and have no changes outside the scope of test classes (files), we also ignore semantic change types made to test files. The clean up stage eliminated roughly 10% of the dataset. To establish relationships between test maintenance and semantic change types in production code, we devise generalised logistic regressions of the form:HasK = Const^K + ∑_i=1^|Predictors|c_i^K * SCT_iwhere HasK is a binary version of the underlying metric K indicated in section <ref>, SCT_i is the semantic change type, c_i^K are the predictor coefficients and Const^K is the model constant. HasK evaluates to TRUE for a commit iff the underlying metric K is greater than 0 for that particular commit. For example, the metric: 1!HasTestMaintenance(commit) := TestMaintenance(commit) > 0is a binary version of the TestMaintenance metric. We use binary metrics to detect the presence of test maintenance, rather then quantifying it. Our goal is to determine whether a developer added any test methods, rather than detecting that a developer added exactly 5 new test methods.We devised a regression model for HasTestMaintenance(commit) (omitted for brevity), which indicated that most semantic change types are statistically significant. We then proceeded to analysing the effect of semantic changes on test maintenance activities by computing the odds ratio for each semantic change type. Semantic change types that increase the odds of the outcome being TRUE, are greater than 1, while semantic change types that decrease the odds of the outcome being TRUE, are less than 1.We visualise the semantic change types with the most dominant odds ratio in figure <ref>. We depict only the odds ratio of the 5 most positively, and 5 most negatively affecting predictors, and only plot predictors that demonstrate sufficient strength which we set to |oddsRatio(predictor)-1| > 0.15. Green bars indicate positive effect, while red bars indicate negative effect. The 95% confidence interval is indicated on top of every bar. §.§ Analysing the odds ratioThe semantic change type “ADDITIONAL_FUNCTIONALITY” has odds ratio of 1.601 which means that an increase of one unit of measure in “ADDITIONAL_FUNCTIONALITY” will increase the odds of HasTestMaintenance being TRUE by a factor of 1.601. In other words, with each addition of a new method to production code, the odds of test maintenance to be present in that commit increase by a factor of 1.601. The odds ratio of REMOVING_METHOD_OVERRIDABILITY is 0.784, which indicates that each introduction of the semantic change type REMOVING_METHOD_OVERRIDABILITY, which stands for adding the “final” keyword to a method's declaration, will increase the odds of HasTestMaintenance being TRUE by a factor of 0.784, and since this factor is less than 1, it actually decreases the odds of test maintenance to be present by a factor of 1/0.784=1.275. In the context of odds ratio, a value Val such that Val < 1 indicates a decrease by a factor of 1/Val in the odds. Figure <ref> visualises the odds ratio forTest^Method_A, Test^Method_R, Test^Method_U, Test^Class_A, Test^Class_R, Test^Class_U. It can be seen that adding new methods and/or classes significantly increases the odds of a test method or a test class to be added as part of a commit, while the removal of a class field tends to decrease them (figures <ref>,<ref>). Removing a class increases the odds of a test method and a test class to be removed by a factor of 2.5 and 4.7 respectively (figures <ref>, <ref>). Changing the return type of a method increases the odds for a test method to be updated by a factor of 1.7 (figure <ref>). Adding a new statement, or updating an existing statement, significantly decreases the odds of a test class or a test method to be removed (figures <ref>, <ref>). Updating an existing statement significantly increases the odds of a test class to be updated (figure <ref>).§ THREATS TO VALIDITYThreats to Statistical Conclusion Validity are the degree to which conclusions about the relationship among variables based on the data are reasonable.* Regression Models. Our dataset consists of 61 projects and over 240,000 commits. Both the model coefficients and the predictions were annotated with statistical significance levels to indicate the strength of the signal. Most of the coefficients were statistically significant (p-value < 0.01). To compare distributions we used the Wilcoxon-Mann-Whitney test and reported its high significance level (p-value < 0.01).We assume commits are independent, however, it may be the case that commits performed by the same developer share common properties. Threats to Construct Validity consider the relationship between theory and observation, in case the measured variables do not measure the actual factors.* Maintenance Activity Classification. We employ a technique we suggested in <cit.>, which demonstrated an accuracy of over 76% and Cohen’s kappa over 0.63 for their test set. It may be the case that some of the commits were misclassified and introduced a bias into our dataset. The classification categories we used are widely accepted and were first put forth by Mockus <cit.>. Hattori et al. <cit.> suggested different classification categories for maintenance activities in the context of open source projects, we did not consider these categories in the scope of this work.* Semantic Change Classification. ChangeDistiller and the VCS mining platform we have built and used are both software components, and as such, are not immune to bugs which could result in inaccurate or incomplete semantic change extraction and data aggregations.* Test Maintenance Classification. We used a widely practiced conventions and heuristics <cit.> for detecting JUnit test methods and test classes. However, the use of heuristics may lead to undetected test maintenance.* Data Cleaning. Prior to devising regression models, we removed extreme data points using a technique suggested in <cit.>.Despite the fact we removed only ∼10% of the data, this process could have introduced bias into the dataset we operated on. Threats to External Validity consider the generalization of our findings. * Programming Language Bias. All analyzed commits were in the Java programming language. It is possible that developers who use other programming languages, have different maintenance activity patterns which have not been explored in the scope of this work.* Open Source Bias. The repositories studied in this paper were all popular open source projects from GitHub (<cit.>). It is possible that developers' maintenance activity profiles are different in an open source environment when compared to other environments.* Popularity Bias. We intentionally selected the popular, data rich repositories. This could limit our results to developers and repositories of high popularity, and potentially skew the perspective on characteristics found only in less popular repositories and their developers.* Mixed Commits. Recent studies <cit.> report that commits may involve more than one type of maintenance activity, e.g. a commit that both fixes a bug, and adds a new feature.Our classification method does not currently account for such cases, but this is definitely an interesting direction to beconsidered for future work (see section <ref>).* Activity Boundary. In this work we assume a commit serves as a logical boundary of an activity. It may be the case, that developers perform test maintenance as part of activities that span multiple commits. Such work patterns were not considered in the scope of this work, but are definitely an interesting direction for future work in this area. § DISCUSSION AND APPLICATIONS A large body of knowledge has formed around two different aspects of defect prediction: the relationship between software defects and product metrics, and the impact of the software process metrics on the defectiveness of software.Existing predictive models usually employ one of the following:* Code/Product metrics - relate to the nature of the code (<cit.>), such as lines of code (LOC), static code complexity (<cit.>), etc.* Process metrics - relate to the process of code change (<cit.>), such as code churn (<cit.>), change size, change complexity (<cit.>), etc.* Combined metrics - mixing process and product metrics (<cit.>).Test maintenance models may complement existing models based on process and product metrics. Integrating our test maintenance models with defect predictive models based on commit classification (e.g., Kim et al. <cit.>) can be beneficial as well. For instance, our models indicate that some semantic changes considerably increase or decrease the odds for testing maintenance activities to be part of a commit. This information may be useful for predicting whether a given commit is buggy.§ RELATED WORK Marsavina et al. <cit.> studied the co-evolution patterns of production and test code by utilizing source code changes (which in this work we referred to as “semantic changes” so as to stress their structural meaning, as opposed to merely being textual edits) on 5 open source projects. They showed that several patterns were apparent, for example, upon the deletion of a production class, its associated test class is also removed, and when a new production class is added, an associated test class is also created, etc.In contrast to Marsavina et al., our work concentrates on exploring the relations between software maintenance activities (see section <ref>) and test maintenance (see section <ref>).Zaidman et al. studied the co-evolution of test and production code <cit.> and observed two main testing strategies: * synchronous, where production and test code are developed simultaneously and* phased, where production and test code are developed in different phases.Pinto et al. <cit.> studied the evolution of tests between subsequent version releases andsuggested a classification for the reasons behind test addition, removal and modification.Moreover, their work suggested that many tests are not really deleted and added, but rather moved or renamed. Also, tests are not only added to check bug fixes or test new functionality, but also to validate changes in the code.Fluri et al. <cit.> analyzed the co-evolution of comments and code, and reported that the type of a source code entity, such as a method declaration or an if-statement, has a significant influence on whether or not it gets commented.Gall et al. <cit.> analyzed which semantic change types frequently appear together, and revealed several patterns in the way developers deal with exception flow handling, apply the single-exit principle (multiple vs. single return statements) and swap conditions in if-else statements. Beller et al. conducted a large-scale field study, where 416 software engineers were closely monitored over the course of five months <cit.>. Their findings indicate that software developers spend a quarter of their work time engineering tests, whereas they think they test half of their time.Our previous work <cit.> explored semantic change types in the context of maintenance activities, and reported predictive models for developers' maintenance activity profiles. Our recent work <cit.> also shows that semantic change types can be effectively used in combination with word frequency analysis in order to obtain high quality predictive models (accuracy and Kappa as high as 76% and 0.63 respectively) for classifying commits into maintenance activities.In contrast to prior work, in addition to exploring test maintenance characteristics of individual commits, we also study developers' test maintenance behaviour by considering all commits performed by the same developer. To the best of our knowledge, our work is the first to study commit and developer level test maintenance on such scale.§ CONCLUSIONS AND FUTURE WORKIn this work we explore the relationships between test maintenance activities, production code maintenance activities, and semantic changes performed as part of developers' commits.Our large scale empirical study provides several observations: * The number of test methods and test classes in a software project can be well predicted using models that employ code maintenance activity profiles (corrective, adaptive, perfective).* Both the number of test methods and test classes has a negative correlation with corrective commits (see discussion in section <ref>).* Test maintenance vary considerably between projects, which may imply that current testing practices are more project specific than standardized.* Empirical evidence show that among the three maintenance activity types, corrective, perfective, and adaptive, the corrective maintenance which deals with fault fixing, involve less test maintenance. We discuss and suggest several hypotheses that may account for such a result (see section <ref>).* Different semantic changes affect test maintenance activities differently. Moreover, certain semantic changes considerably increase or decrease the odds of a particular maintenance activity to take place in a scope of a given commit. The analysis carried out in this work shows what types of changes increase and decrease the odds for test maintenance to take place. For example, with each new method added as part of a commit, the odds of it to involve a test maintenance activity increase by 60%, while each addition of the “final” keyword to a method's declaration decreases such odds by 22% (see section <ref>).We believe these insights can lead to a better understanding of software quality and help practitioners reduce costs and improve software quality. In particular, we believe that the methods used in this work can assist in understanding what kinds of changes are usually tested, in contrast to those that are not.One possible direction would be integrating test maintenance models with defect prediction and test selection models, and exploring whether such a combination could improve existing results.Future directions may also include modeling sets of semantic change in the context of test maintenance. Our intuition is that certain change types are likely to appear together, or better yet, some changes are unlikely to be made on their own, e.g, the addition of the “final” (which decreases the odds for test maintenance to occur by a factor of 1.275) word to a method is likely to be part of a larger change, not a commit on its own.Activity boundary is another direction that may be worth exploring in the context of test maintenance activities. In this work we set the commit as an activity boundary, but it is also reasonable to assume some activities span multiple commits. Therefore, it can be beneficial to explore test maintenance activities in the scope of these multi-commit activities, and explore whether certain developers tend to perform code maintenance and test maintenance in separate commits, but as a single logical activity. Moreover, exploring if an activity boundary may cross the scope of a single developer may also be of interest. Consider the scenario of multiple developers working on a single logical activity, where one or more of them perform test maintenance in designated commits, separated form the non-test code.It could also be beneficial to further investigate the negative correlation of corrective maintenance and the test method and class count. Specifically, the question of whether test methods (and/or classes) and corrective maintenance have a cause-and-effect relationship remains open. IEEEtran | http://arxiv.org/abs/1709.09029v1 | {
"authors": [
"Stanislav Levin",
"Amiram Yehudai"
],
"categories": [
"cs.SE"
],
"primary_category": "cs.SE",
"published": "20170926141333",
"title": "The Co-Evolution of Test Maintenance and Code Maintenance through the lens of Fine-Grained Semantic Changes"
} |
[ [ December 30, 2023 =====================We present stellar-dynamical measurements of the central supermassive black hole (SMBH) in the S0 galaxy NGC 307, using adaptive-optics IFU data from VLT-SINFONI.We investigate the effects of including dark-matter haloes as well as multiple stellar components with different mass-to-light (M/L) ratios in the dynamical modeling.Models with no halo and a single stellar component yield a relatively poor fit with a low value for the SMBH mass (7.0 ± 1.0 × 10^7) and a high stellar M/L ratio (= 1.3 ± 0.1). Adding a halo produces a much better fit, with a significantly larger SMBH mass (2.0 ± 0.5 × 10^8) and a lower M/L ratio (= 1.1 ± 0.1). A model with no halo but with separate bulge and disc components produces a similarly good fit, with a slightly larger SMBH mass (3.0 ± 0.5 × 10^8) and an identical M/L ratio for the bulge component, though the disc M/L ratio is biased high (= 1.9 ± 0.1). Adding a halo to the two-stellar-component model results in a much more plausible disc M/L ratio of 1.0 ± 0.1, but has only a modest effect on the SMBH mass (2.2 ± 0.6 × 10^8) and leaves the bulge M/L ratio unchanged. This suggests that measuring SMBH masses in disc galaxies using just a single stellar component and no halo has the same drawbacks as it does for elliptical galaxies, but also that reasonably accurate SMBH masses and bulge M/L ratios can be recovered (without the added computational expense of modeling haloes) by using separate bulge and disc components. galaxies: structure – galaxies: elliptical and lenticular, cD –galaxies: bulges – galaxies: individual: NGC 307 – galaxies: evolution.§ INTRODUCTIONThe most commonly used technique for measuring the masses of supermassive black holes (SMBH) in galaxy centres is Schwarzschild modeling; fully two-thirds of the SMBH masses in the recent compilations of <cit.> and <cit.> were determined this way. Schwarzschild modeling entails the construction of gravitational potentials based on the combination of a central SMBH and one or more extended stellar components (which are typically based on deprojecting a 2D surface-brightness model of the galaxy in question), with the SMBH mass and stellar mass-to-light (M/L) ratio as variables. A library of stellar orbits is built up by integrating test particles within a given potential defined by particular values of SMBH mass and stellar M/L ratio; these orbits are then individually weighted so as to reproduce the observed light distribution and stellar kinematics of the galaxy. The SMBH mass and stellar M/L ratio are varied until the best match with the data is achieved.Schwarzschild modeling has several advantages over methods based on modeling gas kinematics (the other major approach for measuring SMBH masses): it can be used in any galaxy bright enough for stellar kinematics to be measured, does not require the presence of gas, and does not require simplifying assumptions about the underlying kinematics (e.g., that all orbits are circular and coplanar).Up until recently, the standard approach for Schwarzschild modeling of SMBH masses has been to treat galaxies as having just two components: a central SMBH and a stellar component with a single M/L ratio. This is problematic for several reasons, the principal ones being that galaxies – especially disc galaxies – do not always have uniform M/L ratios, and that galaxies have dark matter as well as stars. Disc galaxies are widely recognized as having spatially varying stellar M/L ratios, something at least partly due to different stellar populations in different subcomponents. <cit.> introduced the idea of using two stellar components with distinct, independent M/L ratios in order to model the combination of an actively star-forming nuclear star cluster within an older bulge in the spiral galaxy NGC 3227. <cit.> modeled the central bulges and main discs as separate stellar components for two spiral galaxies (NGC 3368 and NGC 3489); this was also done by <cit.> for the S0 galaxy NGC 1332. The modeling of separate M/L ratios for bulges is also useful for investigating bulge-SMBH correlations, especially if one wants to determine bulge masses dynamically <cit.>.Elliptical galaxies are in principle simpler to model than disc galaxies,because we can treat ellipticals as having a single stellar component (i.e., they can be approximated as pure “bulge” with no disc). However, they are known – like all galaxies – to possess haloes of dark matter. Recent work has focused on the question of whether the practice of ignoring these haloes in dynamical modeling might bias the resulting SMBH masses and stellar M/L ratios. The key issue is whether the modeling process assigns extra mass to the stellar component in order to account for the (missing) effect of the halo. An increased stellar M/L ratio can then result in a lower SMBH mass, because the stars at small radii will contribute more to the central potential than they would if the M/L ratio were lower; this removes the need for a more massive SMBH.<cit.> found that including a DM halo in their models for M87 resulted in a stellar M/L ratio about half as large – and a SMBH mass about twice as large – as when their models included only a SMBH and the stellar component.Subsequent studies examining the inclusion of DM haloes in elliptical-galaxy models have yielded somewhat conflicting results, with some reporting effects similar to those found by <cit.> – e.g., <cit.> – and some reporting no differences between models with and without DM haloes – e.g., <cit.>.[The <cit.> study is of the bulge-dominated Sa galaxy NGC 4594, not an elliptical.] Studies of larger samples by <cit.> and <cit.> have indicated that DM haloes can be safely ignored in the modeling only if high-spatial-resolution kinematics are available for the centre of the galaxy. Ideally, this means kinematic observations obtained with a point-spread-function whose FWHM is at least 5–10 times smaller than the diameter of the SMBH's sphere of influence <cit.>.What is not clear at this point is whether ignoring the existence of dark matter haloes in dynamical models of disc galaxies has any significant effect on either derived SMBH masses or bulge M/L ratios. In this paper, we investigate this question by measuring the central SMBH mass and stellar M/L ratios for the S0 galaxy NGC 307 using a four different models: first, a simple SMBH + single-stellar-component model; second, a model with a SMBH and two stellar components (bulge and disc) with separate M/L ratios. We then add dark-matter haloes to both the single- and two-stellar-component models.Unless otherwise specified, we adopt a cosmology where Ω_m = 0.7, Ω_Λ = 0.3, and H_0 = 75kpc^-1. § NGC 307 NGC 307 is a poorly-studied early-type galaxy, classified as S0^0 by <cit.>. Although it lies only ∼ 0.5 from the centre of the cluster Abell 119, its much smaller redshift (0.0134 versus 0.044 for the cluster) means there is no physical association. In the group catalog of <cit.>, it is the second-brightest[Based on tabulated values in NED.] member of a small, five-galaxy group (LGG 13, brightest member = NGC 271). We adopt a distance of 52.8 Mpc, based on the (Virgocentric-infall-corrected) redshift of 3959from HyperLEDA. <cit.> reported a central velocity dispersion of 325 ± 15 , but more modern measurements indicate significantly lower values: σ_e = 239has been reported by <cit.>, and <cit.> estimated σ_e = 205 , based on the kinematic and imaging data presented in this paper.[<cit.> used a curve-of-growth analysis of the VLT-FORS1 image to derive a whole-galaxy r_e = 4.76; the light-weighted dispersion within this radius was determined as described in Appendix A of that paper, using the VLT-FORS1 long-slit data.] Using the HyperLeda correctedcolour (0.84) and the colour-based M/L ratios of <cit.> with either the HyperLeda B_ tc magnitude (13.52) or the 2MASS total H magnitude (9.865),[Corrected for Galactic extinction using data from <cit.>, as tabulated in NED.] we find estimated stellar masses of either 5.5 × 10^10 or 6.5 × 10^10, quite close to recent estimates of the Milky Way's stellar mass <cit.>.Figure <ref> shows log-scaled R-band isophotes of NGC 307 using an image from the Wide Field Imager (WFI) on the ESO 2.2m telescope and a higher-resolution image from the FORS1 imager-spectrograph on the VLT; ellipse fits to both images are shown in Figure <ref> (see Section <ref> for more about the images). These fits show a fairly broad peak in isophotal ellipticity of ϵ≈ 0.65 extending from semi-major axis a ∼ 20 to a ∼ 40, with the isophotes becoming significantly rounder (as low as ∼ 0.30) further out. This suggests that we may be seeing a disc embedded within a rounder, luminous halo (we will show in Section <ref> that the latter is unlikely to be just an extension of the central bulge). In addition, unsharp masks suggest the existence of a weak bar or lens within the disc, with semi-major axis ∼ 10; this matches the shoulder in ellipticity seen in the ellipse fits and the corresponding slight twist in the position angle to a local minimum of ∼ 81 at ≈ 9–10.§ OBSERVATIONS§.§ Spectroscopy: SINFONI IFU Data Our primary set of spectroscopic data comes from observations made at the VLT with SINFONI in November of 2008. SINFONI combines the near-IR integral field spectrograph SPIFFI and the adaptive-optical module MACAO <cit.>, using an image-slicer to subdivide the field of view into 32 slitlets, which are subsequently rearranged into a composite pseudo-long-slit image that is passed into the main spectrograph.After dispersion by the grating, the resulting composite spectrum is imaged onto a 2048 × 2048 Hawaii 2RG detector.The pre-optics of SINFONI allow the user to select one of three different spatial resolution modes: 25, 100, or 250 mas, corresponding to fields of view of 0.8× 0.8, 3× 3, or 8× 8.For NGC 307, we used only the middle (100mas) scale, along with the K-band grating, since our primary target was the CO absorption bandheads at 2.3 . A single exposure, when assembled into a datacube, yields rectangular spatial elements with sizes of 50 × 100 mas for the 100 mas mode; when multiple exposures with appropriate dithering are combined, the resulting datacubes have a spatial pixel scale of 50 mas pixel^-1. The resulting K-band velocity resolution is σ = 53 .Since NGC 307 is much larger than the SINFONI field of view, we observed it using a sequence of multiple ten-minute exposures organized into an object-sky-object pattern; the sky exposures were made with an offset of 80 along the galaxy minor axis to avoid contamination by galaxy light. Individual ten-minute exposures were dithered using offsets of a few (spatial) pixels, to reduce the effects of bad pixels in the detector and to allow construction of a final data cube with full spatial resolution. The complete set of observations included 40 minutes of on-target time on each of two night – 2008 November 25 and 26 – for a total of 80 minutes integration time. However, we found the observations from the first night to be of significantly higher quality in terms of AO performance and achieved resolution; since they had sufficient S/N by themselves, we only used that night's data. Observations of telluric-standard B stars, obtained immediately after the galaxy observations and at similar air masses, were used to remove atmospheric absorption (see below).The centre of NGC 307 was not bright enough to serve as an AO guide source by itself, so we used the PARSEC laser guide star (LGS) system at the VLT <cit.>. The LGS mode still requires an extra-atmospheric reference source for “tip-tilt” correction of lower-order atmospheric distortions; we used the galaxy nucleus for this.Data reduction was performed using a custom-built pipeline combining the official ESO SINFONI Pipeline <cit.> with elements from its predecessor, the SPIFFI Data Reduction Software <cit.>. This combined pipeline included the standard bias-correction, dark subtraction, distortion correction, non-linearity correction, flat-fielding, wavelength calibration, and datacube generation stages. Sky subtraction, which used the sky datacube observed closest in time for each galaxy datacube, was augmented using the IDL code of <cit.> to account for variations in night-sky emission-line strengths between the times of the galaxy and sky observations.The galaxy datacubes were then corrected for telluric absorption using the telluric-star datacubes. This involved extracting a single, summed spectrum for the telluric star from its datacube and then dividing it by a blackbody curve with a temperature appropriate for the spectral type of the star (the Paschen γ absorption line in the standard-star spectrum was fit by hand using code written in IDL). The resulting normalized spectrum was then used to correct the individual spectra in the corresponding galaxy datacubes. Finally, the individual datacubes were combined into a single datacube for the night, taking into account the recorded dither positions in the headers. To estimate the resolution obtained by the LGS system, we used “PSF star” observations obtained during or immediately after thegalaxy observations, with exactly the same instrument setup and AO mode (i.e., LGS). To make the match as close as possible, the PSF stars were chosen to have the same R-band magnitude and B - R colour as the galaxy nucleus (measured within a 3-diameter aperture), so that the AO system would respond in a similar fashion. Although it is always possible that the PSF star measurements reflect different observing conditions, <cit.> reported that measurements of PSF stars taken after their (non-AO) VLT-SINFONI observations showed FWHM agreement to within 0.02 for galaxies with bright AGN, where the AGN itself could be used to independently measure the seeing. The combined PSF-star datacube was flattened to produce a K-band image, which was then fit with the sum of two Gaussians using Imfit <cit.>. The inner Gaussian component (37% of the total light), which was mildly elliptical, had FWHM measured along its major and minor axes of 0.20 and 0.16, respectively, for a mean resolution of 0.18. The outer component was nearly circular, with a FWHM of 0.48. This PSF is consistent with previously published SINFONI 100mas K-band values when using the laser guide star; in fact, it is equal to the median value from our previously published observations with the LGS in the same mode <cit.>.§.§ Spectroscopy: VLT/FORS1 and VIRUS-W Observations To obtain measurements of the stellar kinematics outside the central 3× 3 field of view provided by our SINFONI data, we made two sets of optical spectroscopic measurements: long-slit observations along the galaxy major and minor axes with the FORS1 spectrograph in the VLT, and wide-field IFU observations with the VIRUS-Wspectrograph on the McDonald 2.7m telescope. §.§.§ VLT/FORS1We obtained long-slit data along the galaxy major and minor axes with the VLT-FORS1 spectrograph on 2008 October 23 (Programme ID 082.A-0270). We made a total of four 2700s exposures with the slit oriented along the galaxy major axis (PA = 78.1) and two more exposures of the same integration time with the slit along the minor axis (PA = 168.1). The instrument was used with the 1200g grism and a slit width of 1.6 width slit; the instrumental dispersion was σ≈ 79 .The reduction of the FORS1 spectra followed the standard steps of bias subtraction, flat fielding, cosmic-ray rejection, and wavelength calibration to a logarithmic scale using our customized MIDAS scripts <cit.>. We subtracted the sky measured at the ends of the slit and binned the resulting frame radially to obtain a set of spectra with approximatly the same signal to noise ratios.The kinematic analysis of the spectra is discussed in in Section <ref>, and our stellar-population analysis is discussed in Section <ref>. §.§.§ VIRUS-W VIRUS-W is an optical integral-field-unit spectrograph with a 105× 55 field of view, based on the VIRUS IFU design for HETDEX (Hoby-Eberly Telescope Dark Energy Experiment) and adapted to achieve high spectral resolution for deriving stellar kinematics <cit.>. It has 267 fibers with core diameters of 3.14 on the sky, arranged in a rectangular array with a fill factor of one-third. We observed NGC 307 with VIRUS-W mounted on the 2.7m Harlan J. Smith telescope at the McDonald Observatory in Texas on 2010 December 6, as part of commissioning/science-verification time for the instrument. The galaxy was observed using a total of three dither positions (to account for the 1/3 fill factor), each with 1200s exposure time. These were bracketed and interleaved with sky offset exposures, also using 1200s exposure times. The seeing varied in FWHM from 1.2 to 1.9. VIRUS-W has both low- and high-resolution spectral modes; although we observed NGC 307 with both modes, we ended up using just the low-resolution mode data. Since the low-resolution mode has σ_ instr = 39(R = 3300, with a spectral coverage of 4320–6042 Å), it provided more than sufficient spectral resolution for NGC 307.Data reduction used a custom pipeline based on the Cure pipeline for for HETDEX; see <cit.> for details.The result is a datacube with 1.6 × 1.6 spaxels.In order to generate high-S/N spectra for kinematic extraction, we combined spectra from individual spaxels using the Voronoi binning scheme of <cit.>, ending up with a median S/N per bin of 29. The kinematic analysis of the binned spectra is discussed in in Section <ref>.§.§ Imaging DataThe available imaging data for NGC 307 consist of a large-scale R-band image from a 300s exposure at the ESO-MPI 2.2m Wide Field Imager on 2010 July 15 (Programme ID 084.A-9002), with seeing FWHM = 1.62; a 10s-exposure VLT-FORS1 image with smaller field of view (also R-band, with FWHM = 1.00) made during our spectroscopic observations with FORS1 (above); and our VLT-SINFONI combined datacube, collapsed along the wavelength axis to form a ∼ 3× 3 K-band image. These images are, to a degree, complementary: the WFI image is wide enough and deep enough to allow determination of the outer stellar halo and main disc, but has relatively poor resolution; the FORS1 image provides better resolution for the bar/lens and the disc-bulge transition region, but is not as good for characterizing the halo due to its smaller field of view, lower S/N, and the fact that the outer part of the galaxy falls on an inter-chip gap; and the SINFONI image has the best resolution for the inner region of the bulge. Consequently, we construct our final photometric models using a combination of all three images. Since the innermost data are K-band, we calibrated all three images to K-band by a multi-step process similar to that used by <cit.>; the resulting calibration is ultimately based on the publicly available 2MASS K-band image of the galaxy. First, we calibrated the FORS1 image by convolving it to the resolution of the 2MASS image and performing aperture photometry on both images. We then calibrated the SINFONI K-band image to the FORS1 image by iteratively matching surface-brightness profiles from ellipse fits to both images in the region a = 0.6–1.42, including a sky-background term for the SINFONI data.[This is because of variations in the sky background between the times of the galaxy and sky observations with SINFONI, which cannot be completely removed by the data reduction process.] Finally, the WFI image was calibrated to match the K-band-calibrated FORS1 image using a similar ellipse-fit profile-matching technique for the region a = 15–45.§ STELLAR KINEMATICS §.§ SINFONI Kinematics For our SINFONI data, we extracted full, non-parametric line-of-sight velocity distributions (LOSVDs) from the spectra, using a total of 21 bins in velocity space. We used a maximum penalized likelihood (MPL) method originally introduced by <cit.> and a set of stellar template spectra of K and M giants derived from earlier SINFONI observations with the same instrumental setup <cit.>.[The extreme width of the CO bandheads makes the FCQ method we use for our optical spectra (Section <ref>) unusable.] We focused on the spectral region containing the first two CO bandheads ^12CO(2–0) and ^12CO(3–1), which corresponds to a rest-frame spectral range of 2.279–2.340 . In order to minimize template mismatch, we limited our set of template stars to those with equivalent widths for the first CO bandhead which were similar to theequivalent width of the galaxy spectra <cit.>. A trial LOSVD was convolved with a linear combination of template spectra, and the resulting model spectrum was compared with the data. The LOSVD and the weights for the template spectra were adjusted by minimizing a penalizedfunction:χ^2_ P=χ^2+α𝒫,where 𝒫 is the penalty function (the integral of squared second derivative of the LOSVD) and α is a smoothing parameter. The appropriate value of α depends on the S/N of the data and the velocity dispersion of the galaxy; our choice was based on extensive simulations involving MPL fitting of template stellar spectra convolved with different LOSVDs; see <cit.> for more details. An example of one of our fits is shown in the upper panel of Figure <ref>.To increase the S/N of the spectra, we binned individual spaxels into angular and radial bins using luminosity-weighted averaging. This involved dividing the galaxy into four quadrants; the boundaries of the quadrants were set by the major and minor axes of the galaxy. Each quadrant was subdivided into five angular bins and seven radial bins. (See Figure <ref> for the binning, and the first panel for definitions of the quadrants.)Uncertainties for the best-fitting LOSVDs were determined by a Monte Carlo technique, where for each spectrum we created 100 realizations of the best-fitting combined template spectrum, convolved with the best-fitting LOSVD, and then added Gaussian noise based on the measured RMS deviations of the original fit. Each such spectrum was then fit using the same MPL approach, with the final uncertainties based on the distribution of fitted LOSVDs from the Monte Carlo realizations.For presentation purposes, we parameterized the LOSVDs using the Gauss-Hermite moments <cit.> velocity v, velocity dispersion σ, h_3, and h_4. Maps of these four moments are shown in Figure <ref>. Significant rotation can be seen in the velocity field, with an accompanying anti-correlation in the h_3 values. A somewhat noisy trend of increasing velocity dispersion towards the galaxy centre can also be seen. No trends are visible in the h_4 map. §.§ Optical Kinematics Stellar kinematics for both the VLT-FORS1 long-slit spectra and the Voronoi-binned VIRUS-W spectra were derived using the Fourier Correlation Quotient (FCQ) method <cit.>, which models the LOSVD using a Gauss-Hermite decomposition, producing stellar velocity V and velocity dispersion σ values, along with h_3 (skew) and h_4 (kurtosis) deviations from Gaussianity. The FORS1 kinematics were measured as done in <cit.>, chosing the best-fitting template from the simple stellar population model spectra of <cit.>. For the VIRUS-W data, we used a single K2 III template star (HR 2600) spectrum previously observed with VIRUS-W, using a rest-frame spectral range of 4537–5442 Å and removing the continuum using an eight-order polynomial. Error estimates for the V, σ, h_3, and h_4 measurements in both cases were obtained using a Monte Carlo approach <cit.>. Examples of individual fits are shown in the middle and lower panels of Figure <ref>.Figures <ref> and <ref> show the major- and minor-axis stellar kinematics from the FORS1 spectra, and Figure <ref> shows the kinematic maps for the VIRUS-W data. Figure <ref> compares stellar kinematics extracted along the major axis from our three datasets. Given the differences in the resolution for the different observations (the respective FWHM or fiber sizes are indicated by vertical shaded regions in the figures) – and the relative noisiness of the higher-order h_3 and h_4 moments – the overall agreement between the three datasets is good. Both the major- and minor-axis long-slit kinematics show that the velocity dispersion rises quite steeply in the inner r5, suggestive of a kinematically hot central component. This can also be seen, less clearly, in the higher dispersion of the central three bins of the VIRUS-W data. There is, non the less, evidence for significant rotation as in this region as well, as can be seen in the strong inner velocity peak at r ∼ 3 and accompanying V–h_3 anti-correlation in the major-axis kinematics (Figure <ref>). Outside this region, the kinematics are strongly rotation-dominated, with an observed peak velocity of ∼ 200 and a dispersion profile that declines to below 100 for r20 along the major axis.As a whole, then, the stellar kinematics suggest a kinematically hot central region (e.g., a classical bulge, albeit one with significant rotation, or possibly a fast-rotating subcomponent) within the central 5 and a dominant disc component at larger radii. As we will show below, this is consistent with both our stellar population analysis of the FORS1 spectra and with our morphological analysis and 2D decomposition of the galaxy.There is photometric evidence for a weak bar or lens in NGC 307, extending to about 10 in radius (see Sections <ref> and <ref>). Is there any evidence for this bar in the stellar kinematics? We compare the observed kinematics with predictions from N-body models published by <cit.> and <cit.>, paying particular attention to projections where the bar orientation is similar to that in NGC 307 (i.e., with the bar viewed nearly side-on). Although some of the N-body model projections show a “double-hump” major-axis velocity profile, which might seem to agree with the clear double-peak in NGC 307's velocity profile (upper left panel of Figure <ref>), this feature is only visible in the models when the bar is close to end-on, and vanishes when the bar is closer to side-on. The double-hump velocity feature in NGC 307 is thus almost certainly not a bar signature; it is more likely due to a rapidly rotating substructure within the classical bulge region.The models do predict local extrema in h_3 – and maxima in h_4 – near the ends of a strong bar seen side-on and at inclinations of 75 or 80 (e.g., lower right subpanels of Fig. 4 in ). While there are local extrema in NGC 307's h_3 profile at r ∼ 6which might be consistent with this prediction, there are no such features in the h_4 profile (Figure <ref>). We conclude that there is no evidence that the bar/lens strongly affects the stellar kinematics in this galaxy. §.§ Quadrants for Stellar Kinematics As noted above (Section <ref>), the SINFONI kinematics were derived using a radial-angular binning scheme, with the galaxy divided into four quadrants whose boundaries were the major and minor axes of the galaxy. Each quadrant was subdivided into five angular bins of varying width, with seven radial bins spaced logarithmically out to the edge of the SINFONI field of view (Figure <ref>). To include the optical kinematics in the same scheme for our dynamical modeling, we extended the quadrants with additional radial bins and assigned values from the optical kinematics. Since the FORS1 long-slit orientations were along the quadrant boundaries, we assigned their kinematic values to the corresponding bins along the quadrant boundaries – e.g., the major-axis data were assigned to corresponding closest bins along the major-axis boundaries of the quadrants. For the Voronoi-binned VIRUS-W kinematics, we assigned each Voronoi bin's kinematic values to the radial-angular bin containing the center of the Voronoi bin.§ STELLAR POPULATION ANALYSIS To get a preliminary sense of how stellar populations – and thus M/L ratios – might vary within NGC 307, we performed a stellar-population analysis of our FORS1 long-slit spectroscopy.We measured the Lick line strength index profiles from Hβ to Fe5406 as in <cit.>. Following the minimumprocedure described in <cit.>, we determined the age, metallicity, and [α/Fe] overabundance profiles that best reproduced the observed profiles of the Lick indices Hβ, Mgb, Fe5015, Fe5270, Fe5335, and Fe5406 using the simple stellar population (SSP) models of <cit.>, with a <cit.> IMF and the modeling of the Lick indices with α-element overabundance of <cit.>.We are able to reproduce the Mg and Fe indices quite well; however, the measured Hβ is systematically ≈ 0.2 Å smaller than the models. As a consequence of this, the resulting ages hit the maximum allowed value (15 Gyr) of the model grid for most of the cases.The [α/Fe] profile is approximately flat at a level of +0.3 dex, on both the major and minor axes.Figure <ref> show some of the results, including both raw Mgb and Fe5270 index measurements and the overall metallicity ([Z/H]) and K-band stellar M/L ratio estimates.The metallicity is slightly above the solar value in the inner r5 and drops to half-solar outside. The K-band M/L ratio implied by the derived age and metallicity profiles is approximately constant at a value of 1.22 M_⊙/L_⊙ at radii 10, rising to a central peak of ∼ 1.26. Actual radial variations in the M/L ratio are probably underestimated due to the saturated SSP age estimates.Both major- and minor-axis profiles show evidence for a central peak in metallicity, with a correspondingly higher M/L ratio. This is good evidence for a separate, metal-rich population with a higher M/L ratio dominating the inner r5 along the major axis. As noted above, our VLT-FORS1 kinematics (Figure <ref> and <ref>) show that the stellar velocity dispersion increases rapidly towards the centre in this same region, from a nearly constant disc value of ∼ 110–120to values > 200 , suggesting a classical, dispersion-dominated (albeit rapidly rotating) bulge. This is also consistent with the decompositions we perform (below), which argue for a relatively round luminosity component dominating the light at r5, and motivates separating out the bulge component and allowing it to have its own M/L ratio in the modeling process. § PHOTOMETRIC MODELING§.§ General Approaches To properly measure the mass of a galaxy's SMBH, we must construct a dynamical model based on at least two components: the potential of the central SMBH and the potential due to the stellar mass distribution. (In some cases, gas may also form a significant component; however, in , we presented evidence that the molecular gas content in the centres of the disc galaxies we observed with SINFONI – that is, those galaxies where we could detect gas – was much lower than the stellar mass in the same region, and so could reasonably be neglected. In the case of NGC 307, we detected no gas emission at all; an absence of significant gas is consistent with its S0 classification and the lack of visible dust lanes in the optical images.) The stellar-mass potential is the combination of a stellarM/L ratio – something adjusted during the fitting process – and a luminosity-density model for the stellar light. The luminosity-density model, in turn, is derived from the observed stellar light distribution of the galaxy, usually by deprojecting an observed surface-brightness model. In this section, we describe how we devise luminosity-density models for NGC 307.The standard approach for constructing luminosity-density models has been to fit ellipses to the isophotes of a galaxy image, and use the resulting ellipse-fit model – i.e., surface brightness, ellipticity, and possibly symmetric higher-order terms (cos 4 θ, cos 6 θ, etc.) as a function of semimajor axis – as input to the code which then deprojects this to obtain a 3D luminosity-density model. An alternate approach is to model the isophotes as the sum of multiple 2D Gaussians, which can then be deprojected individually and summed to form the luminosity density model <cit.>. In the case of something simple like most elliptical galaxies, this is usually a straightforward operation, since we can assume that the entire galaxy is a single, coherent stellar component.But in constructing photometric models of disc galaxies, we face two problems. The first has to do with questions of stellar M/L ratios. As noted above, most Schwarzschild modeling in the past has assumed a single M/L ratio for the entire stellar component. While this is perhaps reasonable for elliptical galaxies, disc galaxies are known to contain multiple stellar populations which can dominate different regions of a galaxy. In the simplest case, a disc galaxy may have distinct populations belonging to the bulge and to the disc; our spectroscopic analysis suggests this is indeed the case for NGC 307 (Section <ref>).One possible approach is to consider a M/L ratio which varies as a simple function of radius <cit.>. But this may or may not have a plausible physical origin, and there are a potentially unlimited number of possible radial profiles to choose from, with varying numbers of additional free parameters; even a linear function adds two extra free parameters to the modeling process. We choose instead a somewhat more physically motivated approach: we assume that the galaxy can be spatially decomposed into two or more overlapping but distinct stellar components, each with its own M/L ratio <cit.>.The second problem we have when constructing photometric models stems from the fact that our Schwarzschild modeling code assumes an axisymmetric stellar potential, which can be described as a set of coplanar, axisymmetric spheroids with relative thicknesses which can vary as a function of radius (i.e., spheroids with a = b but varying vertical scale heights c). This requires an axisymmetric photometric model as input to the deprojection algorithm: the isophote shapes can vary (in ellipticity and higher-order moments), but their orientations (position angles) cannot. Real disc galaxies, however, are often non-axisymmetric, with bars, spiral arms, and other stellar substructure which show up in ellipse fits as variations in ellipticity and position angle. Since the deprojection process cannot handle position-angle variations, they are ignored, and the result is that changes in isophotal ellipticity due to, e.g., bars or spiral arms are misleadingly converted into changes in vertical thickness in the resulting luminosity-density model.To deal with these issues, we use an approach first described in <cit.> and applied to the galaxies NGC 3368 and NGC 3489 in that paper, and also to NGC 1332 in <cit.>. This consists of first identifying plausible “bulge” and “disc” regions, devising preliminary models corresponding to the bulge and disc, creating separate residual images for the two components (i.e., a “bulge-only” image which has the disc model subtracted off and a “disc-only” image with the bulge model subtracted off), and then treating them in distinct fashions: * The bulge-only residual image is fit with freely varying ellipses in the standard fashion, treating it as though it were the image of a spheroidal, axisymmetric structure with potentially variable c/a axis ratios. * The disc-only residual image is fit with ellipses which are fixed to a common shape and orientation (axis ratio and position angle) corresponding to that of the outer disc. This has the effect of azimuthally averaging whatever non-axisymmetric structure – bars, spiral arms, etc. – may actually exist. Although we generate preliminary models for both bulge and disc based on combinations of simple analytic components (e.g., an elliptical Sérsic component for the bulge), the final surface-brightness models which we pass to the deprojection machinery are based primarily on direct ellipse fits to the residual images as outlined above. This means that the final models – especially for the bulge component – contain as much of the intrinsic galaxy light variation as possible: e.g., our final bulge component is not a pure Sérsic component, but represents the galaxy light after the preliminary disc model has been subtracted.In the specific case of NGC 307, as we will discuss below, there is evidence for a rounder stellar “halo” which dominates the light beyond a certain radius. Thus, we modify the second surface-brightness component described above by allowing the isophotes to have lower ellipticities (as measured by ellipse fits with variable ellipticity) at large radii. §.§ Photometric Modeling of NGC 307As noted above, there is evidence for a central bulge in NGC 307 with a distinct metal-rich stellar population dominating the inner r5, a weak bar or lens contributing to the light at intermediate radii (most strongly at r ∼ 9–10), and a halo dominating the outer light (r50). Therefore, we analysed this galaxy with a 2D decomposition approach, including up to four components: central bulge, bar/lens, disc, and halo. (Note that in this subsection we use “halo” to refer specifically to a stellar component, not to a dark-matter halo.)To start with, we fit the FORS1 image with Imfit <cit.> using several models, including both a simple bulge + disc (B+D = Sérsic + exponential) model and two versions of a bulge + bar/lens + disc (B+b+D) model, which differed in how the bar/lens was modeled. A Moffat PSF based on the median values of fits to stars in the image was convolved with each model during the fitting process.We compared the effectiveness of the models using the Akaike Information Criterion <cit.>, which is automatically computed by Imfit based on the likelihood of the best-fitting model, the number of data points, and the number of parameters.[Imfit actually computes the “corrected” version of AIC (AIC_ c), though given the large number of individual data points involved, the difference between AIC_ c and AIC is minimal.] Lower values of AIC indicated (relatively) better fits. A difference in AIC values between two models of < 2 is considered insignificant, while a difference > 6 is considered strong evidence for the model with lower AIC being better.The best of these models, with the lowest AIC, was the B+b+D model with the bar/lens represented by an elliptical, broken-exponential component <cit.>. Using a Sérsic function for the bar/lens provided a reasonable fit, though not nearly as good (ΔAIC = +675 relative to the broken-exponential model). The baseline B+D model was a much poorer fit, with ΔAIC = +4291 relative to the broken-exponential model.To determine the contribution of the halo component, we then performed a four-component (B+b+D+H) fit to the (larger FOV) WFI image, starting with the best B+b+D model from the FORS1 image fits and adding a Sérsic component with generalized ellipses (i.e., boxy or discy isophote shapes) to represent the halo. Generalized ellipses are described by ( |x|/a)^c_0 +2 +( |y|/b)^c_0 +2= 1,where |x| and |y| are distances from the ellipse centre in the coordinate system aligned with the ellipse major axis, a and b are the semi-major and semi-minor axes, and c_0 describes the shape: c_0 < 0 corresponds to disky isophotes, c_0 > 0 to boxy isophotes, and c_0 = 0 for perfect ellipses. The best-fitting halo component had slightly boxy isophotes (c_0 = 0.57) and a profile essentially indistinguishable from an exponential (Sérsic n = 0.97); this component is slightly misaligned with respect to the disc and bulge (both disc and bulge have PA ≈ 82, while the halo has PA ≈ 77). We then re-fit the (higher-resolution) FORS1 image by including the halo component, keeping most of its structural parameters fixed to the best-fitting values from the WFI fit but allowing the position angle and intensity (I_e) to be free parameters. Figure <ref> compares our final four-component B+b+D+H fit to the FORS1 image (lower panels) with the baseline B+D fit (upper panels); the parameters of the B+b+D+H fit are listed in Table <ref>. In addition to the fact that the second decomposition is a significantly better fit in a statistical sense (e.g., ΔAIC is -5486 relative to the B+D model, and -1491 relative to the best B+b+D model), we can see that the B+D fit has an exceptionally narrow disc (ellipticity = 0.80) and an exceptionally bright bulge component with Sérsic index n = 5.5; the value of n = 2.5 for the bulge in the B+b+D+H fit is much more typical of bulges in S0 galaxies <cit.>. Figure <ref> compares ellipse fits to the data (black) and to the B+D (green) and B+b+D+H (red) model images; the latter does a significantly better (albeit not perfect) job of matching position-angle twists and ellipticity variations in the data.Figure <ref> shows the galaxy's major-axis surface-brightness profile from the FORS1 image, along with major-axis cuts through the PSF-convolved B+b+D+H model (dashed black line) and the individual components of the model. This shows that the inner Sérsic component dominates the light for r5 – making it a very plausible match to the separate stellar population suggested by our spectroscopic analysis (Section <ref>). We note that the ellipticity of this component (0.385) is a good match to the observed outer isophote ellipticity in the SINFONI image (∼ 0.4), where seeing effects are smallest. §.§.§ Generating Final “Bulge” and “Disc” Components for Dynamical ModelingTo generate “bulge-only” images for use in constructing the final bulge model, we constructed model images (using thetool in ) consisting of the bar/lens, disc, and halo components of the best-fitting B+b+D+H model, suitably rescaled and PSF-convolved for the SINFONI, FORS1, and WFI images. These were then subtracted from the data images, and ellipses were fit to the resulting residual images. The final bulge profile consisted of ellipse-fit data from the residual SINFONI image for a < 1.1, FORS1 data for a = 1.1–10, and a Sérsic extrapolation of the inner data (using the bulge parameters in Table <ref>) for largerradii.[The surface brightness of the residual bulge image is too low and noisy to be fit outside a ∼ 10; this is also true if we use the WFI image.] The ellipticity and cos 4 θ values were taken from the residual SINFONI image ellipse fits for a < 1.1 and were set to 0.385 and 0, respectively, for larger radii.The “disc-only” images used for constructing the final disc model (actually the disc + lens + stellar halo) were generated in an analogous fashion: PSF-convolved model-bulge images (using the inner Sérsic parameters from Table <ref>) were subtracted from the FORS1 and WFI images, and the resulting residual images were fit with both fixed and free ellipses. Since, as noted above, NGC 307 has a significant outer halo which is rounder than the disc, the final “disc” model actually incorporates a transition from the azimuthally averaged, constant-ellipticity profileto a profile with declining ellipticity at a = 28. The ellipticity and PA for the fixed-ellipse fits were 0.69 and 82, based on free-ellipse fits to the residual FORS1 image; these values are almost identical to those of the exponential-disc component in the best-fitting B+b+D+H model (Table <ref>). The fixed-ellipse-fit FORS1 data were used for a = 6.4 to 28 in the final profile, with free-ellipse-fit surface-brightness and ellipticity used for a ≥ 28 (using WFI free-ellipse-fit data for a > 47). For a < 6.5, the FORS1 fixed-ellipse-fit surface brightness became extremely noisy and difficult to deproject; thus the surface-brightness data at smaller radii come from a fixed-ellipse fit to an unconvolved model image (built using the disc + bar/lens + halo components from Table <ref>). At these small radii the final luminosity density is dominated by the bulge component, so accuracy in the disc component is less important.Figure <ref> shows the surface-brightness profiles of the final bulge and disc components. The top panel of Figure <ref> compares the surface-brightness profile of our final bulge component (red) with the equivalent (ellipse-fit-derived) surface-brightness profile of the Sérsic function (convolved with the SINFONI PSF) from our 2D decomposition. Although they are very similar, the final bulge component is brighter in the centre than the inward extrapolation of the Sérsic function. If we had simply used the Sérsic function itself as the bulge component for deprojection, we would underestimate the central stellar density and thus potentially overestimate the SMBH mass in our modeling. The bottom panel of the figure makes the same comparison for the disc component.Ideally, one could treat the outer halo as a third stellar component, with its own M/L ratio. However, since our kinematic data are limited to r30 along the major axis, well inside the region where the halo component begins to dominate over the disc (e.g., Figure <ref>), the precise details of the stellar halo do not significantly affect our dynamical modeling.§.§ Deprojection To go from the surface-brightness profiles and the accompanying geometric information (ellipticity, B_4) to actual 3D luminosity density models requires deprojection under certain assumptions. We use an approach based on that of <cit.>. Different realizations of 3D luminosity-density models are projected, assuming an inclination of 76,[Based on the observed maximum ellipticity of ≈ 0.69 in the disc-dominated region, assuming an intrinsic disc thickness of c/a = 0.2.] and compared to the observed 2D surface-brightness model derived from the profiles. A simulated annealing algorithm is used to maximize a penalized log-likelihood function based on the difference between the model and the data in order to determine the best-fitting 3D model. We performed separate deprojections for the bulge and disc components. Since the central regions of the disc component are negligible compared to the bulge component, we ignored the effects of PSF convolution (in fact, as explained in the previous section, the central part of our disc surface-brightness component was derived from an unconvolved model image). For the bulge component, on the other hand, PSF convolution is important, so we used our double-Gaussian model of the SINFONI PSF (Section <ref>) when projecting trial 3D bulge-component models for comparison with the data. § DYNAMICAL MODELING To determine the SMBH mass and stellar M/L ratios for NGC 307, we use Schwarzschild orbit-superposition modeling <cit.> with the three-integral, axisymmetric code of <cit.>, which is based in turn on the code of <cit.> (see also ).The basic outline of our Schwarzschild modeling process is as follows. First, we define general mass models consisting of a SMBH, one or more stellar components, and (optionally) a DM halo. Then, for each such model, we perform fits spanning a grid in the space of free parameters of the model, computing regularizedvalues (see below) for each combination of parameters. Finally, we analyse the resultinglandscape and the corresponding likelihoods to determine best-fit parameter values and corresponding confidence intervals.The fitting process for a given general model is:* Construct a specific mass model and its potential from the general model, based on particular values for the free parameters (SMBH mass, stellar M/L ratios, DM halo parameters). * Integrate test particles within this potential to build a library of orbits. For NGC 307, we used 2 × 14,300 individual orbits, with the duplication achieved by reversing the angular momentum of individual orbits. * Assign weights w_i to the individual orbits so that their weighted sum reproduces the input stellar mass model (this is treated as a boundary condition, so the match is exact to within machine tolerances[This helps ensure self-consistency, so that the generated model reproduces the potential used to compute the orbits.]) and reproduces the observed kinematics. The comparison with the kinematic data is done by simulating kinematic observations of the model using the same spatial and LOSVD bins as the data, convolved with PSFs based on the observations. Avalue is computed based on the comparison between the observed and model kinematics. * Repeat the process with new values of the free parameters. The fit of a given orbit library to the kinematic data is computed by maximizing Ŝ = S - α. This is a regularized version of aminimization, based on a maximum entropy approach, where α is the regularization parameter and S is the Boltzmann entropy:S = -∑_i w_iln( w_i/V_i),with V_i the phase-space volume of orbit i, computed as in <cit.>. Theterm is= ∑_j = 1^N_L∑_k^N_ vel (L_jk, m - L_jk, d)^2/σ_jk^2,which is a sum over the N_L spatial positions j and the N_ vel LOSVD bins k, with L_jk, m and L_jk, d the model and datavalues in each LOSVD bin and σ_jk^2 the corresponding Gaussianuncertainty for the data. Since our modeling code assumes axisymmetry, we treat each quadrant of kinematic data as a separate dataset to which the model is fit. The result is four independent evaluations for each set of model parameters, which can in principle be used as quasi-independent estimates of model uncertainties, as well as a gauge of how well the underlying assumption of axisymmetry is justified <cit.>. Our final analysis is based on combining the results for all four quadrants, as described below.We have four general models. Each features a central SMBH. Model A has single stellar component; Model A+DM adds a DM halo to this. Model B has two stellar components: one for the bulge sub-component and one for the disc;[Where “disc” means the combined disc + bar/lens + stellar halo component, as determined in Section <ref>.] Model B+DM also includes a DM halo. These models are summarized in Table <ref>, and described in more detail in the following subsections.The stellar density components are based on stellar luminosity density components ν (plus a M/L ratio which converts luminosity to mass). The luminosity density components themselves are obtained by deprojecting the surface-brightness components (Section <ref>). For the single-stellar-component model, we simply add the bulge and disc luminosity-density models together and assign the result a single M/L value. To determine best-fit values and confidence intervals for parameters, we use a slightly modified version of the likelihood-based approach of <cit.> and <cit.>.For each value of a given parameter (e.g., ), we compute the relative likelihood (from the ) for a given quadrant by marginalizing over the other parameters; the final relative likelihood is then the product of thelikelihoods for the individual quarters. As an example, the marginalized likelihood value ℒ_n(x) for a model with parameters x, y, and z, evaluated in quadrant n, would be:ℒ_n(x)∝ ∑_y_ min^y_ max∑_z_ min^z_ max e^-1/2χ_n^2(x,y,z)Δ zΔ y ,and the final marginalized likelihood value would beℒ(x) =∏_n = 1^4ℒ_n(x) . To determine the best-fit values and confidence intervals, we use the cumulative of the marginalized likelihood:C(x) =∫_x_ min^xℒ(x^')dx^'/∫_x_ min^x_ maxℒ(x^')dx^'with the best-fit value at the median, where C(x) = 1/2, and the 68% (“1-σ”) confidence interval defined by the values of x for which C(x) = 0.16 and 0.84.The best-fit parameter values and confidence intervals for each model are presented in Table <ref>, along with the total CPU time used for each model.[The code ran in a cluster with approximately 500 Intel Xeon 2.6 GHz E5-2670 CPUs.] The relativeand marginalized likelihood plots for SMBH mass and stellar M/L ratios for all four models are shown in Figure <ref>. The grey shaded areas show the (arbitrarily scaled) marginalized likelihood for the parameter in question, with the best-fit value and confidence intervals indicated by the solid and dashed vertical black lines. The lines show Δ = χ^2(x) - χ^2_0, where χ^2(x) is the minimum for all models with the same value of the parameter in question (marginalized over the other parameters) and χ^2_0 is the minimumover all parameter values. The thin lines show the Δ values for the individual-quadrant fits; the thick lines are the result of summing the individual-quadrantvalues.§.§ Model A: SMBH + Single Stellar ComponentModel A is is the traditional model used for most published dynamical SMBH mass measurements. It consists of a SMBH and a single stellar-density component:ρ= δ(r) + ν_ tot.For NGC 307, the single luminosity-density component ν_ tot is the sum of the bulge and disc luminosity-density components ν_b and ν_d, which are the deprojections (Section <ref>) of the bulge and disc surface-brightness profiles derived in Section <ref>.The relativeand marginalized likelihood plots for this model are shown in the upper left part of Figure <ref>. The best-fit SMBH mass (7 ± 1 × 10^7) is rather low – about a factor of four smaller than what thewould predict (see Section <ref>) – though by itself not obviously implausible. The stellar M/L is apparently quite well-defined.§.§ Model A+DM: SMBH + Single Stellar Component + DM HaloModel A+DM is Model A with the addition of a dark-matter halo, so that the mass model isρ= δ(r) + ν_ tot+ρ_ DM.The DM halo is a standard spherical cored logarithmic model <cit.>, with a density profile given by ρ_ DM(r) =V_h^2/4 π G3 r_h^2 + r^2/(r_h^2 + r^2)^2, where r_h is the core radius (inside of which the density slope is constant) and V_h is the asymptotic circular velocity. Previous studies modeling early-type galaxies with DM haloes have found that similar results are obtained for both cored logarithmic and NFW DM halo models <cit.>. Schwarzschild modeling of SMBH masses including DM haloes have also found that the results do not depend strongly on the specific DM halo model used <cit.>.The lower left part of Figure <ref> shows relativeand marginalized likelihood plots for Model A+DM. The SMBH mass (2.0 ± 0.3 × 10^8) is about three times larger than the Model A value; the stellar M/L ratio is about 15% smaller (= 1.1 ± 0.1).Model A+DM required almost 30 times the computational effort of Model A.§.§ Model B: SMBH + Bulge + DiscModel B is similar to Model A except that there are two stellar-density components in the mass model, each with its own M/L ratio, so the mass model isρ= δ(r) + ν_ b+ ν_ d,where ν_ b and ν_ d are the bulge and disc luminosity-density models, respectively. These two components are deprojections (Section <ref>) of the bulge and disc surface-brightness models derived in Section <ref>.Relativeand marginalized likelihood values for this model are shown in the upper right part of Figure <ref>. The disc-component M/L value () is implausibly high (1.9 ± 0.1); however, the bulge M/L value (1.1 ± 0.1) is lower than the global M/L of Model A, and is in fact identical to the global M/L value of Model A+DM. The SMBH mass (3.0 ± 0.5 × 10^8) is about 50% larger than that from Model A+DM, but still consistent with the latter at the ∼ 2-σ level; it is over four times larger than the Model A value.Model B required about six times the computational effort as Model A, but only one-fifth that of Model A+DM. §.§ Model B+DM: SMBH + Bulge + Disc + DM HaloModel B+DM is the most complex model we consider. It is the same as Model B except that there is also a DM halo, so the that the mass model isρ= δ(r) +ν_ b+ν_ d+ρ_ DM.The DM halo is the same spherical cored logarithmic model as we use in Model A+DM. The combined model thus has a total of five free parameters: , , , r_h, and V_h.The lower right part of Figure <ref> shows relativeand marginalized likelihood values for the SMBH mass and the M/L ratios for the bulge and disc components.The best-fitvalue (2.2 ± 0.6 × 10^8) is in between the best-fit values from Model A+DM and Model B, and is more than three times larger than the best-fit value from Model A. The bulge M/L value is identical to the value in Model B (and the global M/L ratio of Model A+DM). The disc M/L value (1.0 ± 0.1) is only about 60% of the value in Model B, and is thus now lower than the bulge M/L ratio, in qualitative agreement with our spectroscopic analysis (Section <ref>).Since we consider this the best model for NGC 307 (see discussion below), we show details of the fits to the kinematic data in Figures <ref>. This compares the predicted stellar kinematics from the best-fit model with the kinematic data from each of the four quadrants; note that for simplicity we show Gauss-Hermite moments – V, σ, h_3, and h_4 – derived from the full LOSVDs.With a total of five free parameters, Model B+DM required 200,000 CPU hours of computational time – six times that of the other DM-halo model (Model A+DM) and almost 30 times that of Model B. §.§ Comparison and Summary of Modeling The effect of not including a DM halo in the single-stellar-component case (Model A) is easily understood, because it is similar to the effects seen for elliptical galaxies (always modeled as single-stellar-component systems). Without a DM halo, the stellar component needs a higher M/L ratio (= 1.3) in order to match the observed kinematics at large radii, where the (real) DM halo starts to become significant compared to the stars. Since the stellar M/L ratio is the same at all radii, this effect also increases the stellar mass in the inner regions of the galaxy, and so a lower SMBH mass is needed to in order to match the observed kinematics there.When a DM halo is added to Model A (creating Model A+DM), the effect is fairly dramatic: although the stellar M/L ratio decreases only moderately (from 1.3 to 1.1), the SMBH is almost three times larger (= 2.0 ± 0.5 × 10^8). As is true for elliptical galaxies modeled with DM haloes, the halo is able to replace the role of the stellar component in accounting for the outer kinematics. Consequently, the stellar component can acquire a lower value, and the SMBH mass can correspondingly increase.For Model B (the two-stellar-component model without DM halo), the disc component is affected in a fashion similar to (but even stronger than) that of the single stellar component in Model A: in order to explain the observed kinematics at large radii, the disc M/L ratio is biased high (= 1.9) to compensate for the absence of a DM halo. However, the presence of a separate bulge stellar component – which dominates the stellar mass budget at small radii – breaks the direct connection between outer stellar M/L ratio and SMBH mass that bedevils Model A. Instead, the bulge M/L ratio and the SMBH mass can vary as needed to better match the observed central kinematics. The result is a lower M/L ratio for the bulge component and a higher mass for the SMBH. The bulge M/L value (= 1.1 ± 0.1) is identical to the global stellar M/L value in Model A+DM; the SMBH mass is only 50% higher. The only obvious problem with Model B is the unrealistically high M/L ratio for the disc component – almost twice the bulge M/L ratio. This is directly contradicted by the stellar-population analysis in Section <ref>, which indicated that the disc M/L ratio should be lower than the bulge M/L ratio.Adding a DM halo to Model B (Model B+DM) primarily affects the disc M/L ratio: instead of there needing to be excess stellar mass at large radii in order to explain the observed kinematics, mass can be shifted into the DM halo component. The result is a much lower – and much more plausible – M/L ratio for the disc component of = 1.0. Because the outer stellar component remains decoupled from the inner component, the effect on the bulge M/L ratio and thus the SMBH mass is relatively mild. In fact, the bulge M/L ratio is unchanged from the Model B value, and the SMBH mass is in between the values for Model A+DM and Model B.Because our kinematic data do not extend much beyond the baryon-dominated inner regions of the galaxy, they cannot provide strong constraints on the DM halo. In practice, fitting the two models with a DM halo component (A+DM and B+DM) yields only lower limits on the halo radius and somewhat discordant asymptotic velocities (200 ± 20for A+DM, 260 ± 20for B+DM).Do the models with extra components provide significantly better fits to the data in a purely statistical sense? Since we are comparing multiple models which are not simply nested (e.g., while Model A is nested within Model A+DM, Models Band B+DM are not), direct comparisons ofvalues is not valid. Instead, we look at more general comparisons using information-theoretic statistics, which can be used to compare non-nested models that are fit to the same data. Table <ref> compares the best fits of the different models using the Akaike Information Criterion (AIC; see Section <ref>) and also the Bayesian Information Criterion <cit.>. The AIC (actually the “corrected” AIC_ c value) and BIC values are calculated using theterm from Eqn. <ref>. As noted in Section <ref>, lower values of AIC (or BIC) indicate better fits; differences of < 2 are insignificant, while differences of > 6 are considered strong evidence that the model with the lower AIC or BIC is superior.In this context, Model A is clearly the worst model: its AIC values are ∼ 9–55 higher than those of the other models, and its BIC values are 24–70 higher. The other models are practically indistinguishable from each other in terms of AIC and BIC values. For example, only for the Q2 value is Model B+DM clearly superior to Model B. The BIC values actually favor Model B over Model B+DM (ΔBIC ≈ 9) for all datasets except Q2.What this shows is that our kinematic data are insufficient to clearly discriminate between Models A+DM, B, and B+DM. The data, for example, do not allow us to distinguish between the case of a massive disc with no DM halo (Model B) and the case of a low-mass disc with a DM halo (Model B+DM). §.§ Variations: Testing the Sensitivity of Fits to Bulge/Disc Decompositions The method we use for generating the luminosity-density models involves a 2D bulge-disc decomposition (with multiple sub-components for the “disc”). Uncertainties in this process translate into uncertainties in the amount of light assigned to different components. Since for Models B and B+DM we assign potentially different M/L ratios to the bulge and disc components, the decomposition uncertainties could, in principle, affect our derived M/L ratios and SMBH masses.To test how much variations in the bulge/disc decomposition might actually affect the derived model parameters, we ran additional fits of Model B using divergent versions of our bulge/disc decompositions corresponding to 1-σ deviations from the best fit. This is described in more detail in Appendix <ref>. The results can be summarized as effectively no discernable changes in the black hole mass or stellar M/L ratios for fits using ± 1-σ variations on the best-fit decomposition, so we conclude that our results are not significantly affected by uncertainties in the decomposition. §.§ Which Model Is Best? Accuracy Versus Efficiency and Strategies for Modeling We are left with three models – A+DM, B, and B+DM – which are approximately equally good at fitting the kinematic data. How can we choose among them? From a general astrophysical perspective, Model B+DM should be the most correct (or least wrong) model, since it allows for both the possibility of different bulge and disc stellar M/L ratios (something we expect from both our general understanding of disc galaxy evolution and from the spectroscopic evidence for NGC 307 itself) and the existence of a separate DM halo (something we expect for all galaxies). The fact that the derived bulge and disc M/L ratios for Model B+DM qualitatively agree with the spectroscopic results (slightly higher in the bulge-dominated region, lower in the disc outside; Section <ref>) is further reason to prefer it over the other models, although given the uncertainties in M/L ratios, its superiority relative to the Model A+DM is not statistically significant. Although Model B allows for different M/L ratios in the bulge and disc regions, its agreement with the spectroscopic analysis is actually worse, because it has a disc M/L ratio that is higher than the bulge value. Moreover, its disc M/L value (= 1.9) is too high to be physically plausible.While the best model for NGC 307 is thus probably Model B+DM, it does have one practical drawback: the extensive computational time required to evaluate it (200,000 CPU hours in our case).The difficulty posed by computational time for Schwarzschild modeling is illustrated by the fact that recent studies which used the equivalent of our Model A+DM – that is, including two DM halo parameters as part of the fit, for a total of four free parameters – have been devoted to one or at most two galaxies only <cit.>. Studies which included DM haloes for more than two galaxies have avoided the expense of full parameter-space searches by using fixed DM haloes in their models. <cit.> specified fixed halo parameters based on galaxy luminosity, while <cit.> first fit three-parameter stars + DM halo models (excluding the high-spatial-resolution data which probed the SMBH region) to derive halo parameters as a function of , and then fit SMBH + stars + DM halo models – with onlyandas free parameters – to their full kinematic data. Schwarzschild modeling with five free parameters, as in our Model B+DM, has not previously been attempted, and is probably not (yet) a practical approach for more than one or two galaxies at a time.If we are interested in measuring reasonably accurate SMBH masses, and potentially bulge M/L ratios as well, for several galaxies at a time, then Models A+DM and B seem equally apropos: they yield SMBH masses close to the Model B+DM value and the same stellar M/L ratio for the bulge region as in Model B+DM. Model A+DM has a somewhat more accurate SMBH mass, while Model B is clearly the most efficient way to measure these quantities, since it has only three free parameters and requires only ∼ 20% as much computational time as Model A+DM. § DISCUSSION §.§ The SMBH in NGC 307 Our preferred model (Model B+DM) gives a SMBH mass of = 2.2 ± 0.6 × 10^8 for NGC 307. Given the previously published central velocity dispersion of 205<cit.> and our adopted distance of 52.8 Mpc, the diameter of the black hole's sphere of influence would be ≈ 0.18. Our SINFONI observations had a mean FWHM of 0.18, which means that our data (just) resolve the SMBH's sphere of influence.From therelation of <cit.>[Specifically, the CorePowerEClassPC relation, since NGC 307's status as an S0 with a classical bulge places it in that particular sample.] we would derive an estimated SMBH mass of 2.67 × 10^8. Using the Sérsic model from our 2D decomposition in Section <ref>, the bulge of NGC 307 has M_K = -22.65; with the bulge M/L from Model B+DM, this gives = 2.97 × 10^10, so the SMBH is 0.74% of the bulge mass. The predicted SMBH mass from the CorePowerEClassPCrelation in <cit.> would be 1.40 × 10^8. The SMBH in NGC 307 is thus within ∼ 30–40% of what theandrelations would predict,[Differing from the predictions in log by only 0.08 and 0.20 dex, respectively, as compared with the measured RMS of 0.41 and 0.45 for the fits in <cit.>.] and is therefore quite unexceptional.[Note that preliminary SMBH and bulge masses for this galaxy (= 4.0 ± 0.05 × 10^7, = 3.2 ± 0.4 × 10^10) were actually used to construct the relations in <cit.>, but since NGC 307 was only one of 77 galaxies in the CorePowerEClassPC subsample, it did not have a strong effect on the derivation of the relation.]§.§ Implications for SMBH Measurements in Disc GalaxiesOur analysis of NGC 307 suggests that attempts to measure SMBH masses in disc galaxies via stellar-dynamical modeling can suffer from the same limitations that have been found for elliptical galaxies. Specifically, modeling a disc galaxy with just a single stellar component (with a uniform M/L ratio) and a SMBH can lead tounderestimated SMBH masses and overestimated stellar M/L ratios. This can be alleviated by subdividing the stellar model into bulge and disc components (increasing the number of free parameters to three), or by adding a DM halo to the single-stellar-component model (increasing the number of free parameters to four). The best approach is clearly to model multiple stellar components and a DM halo, but this is computationally very expensive, since it involves five free parameters rather than three or four.Schwarzschild modeling of disc galaxies using a single stellar component and no DM halo does not always lead to biased SMBH mass measurements, as the case of NGC 4258 shows. <cit.> obtained a SMBH mass measurement for that galaxy which differed by only ∼ 15% from the very high quality maser measurement. <cit.> showed that biases to SMBH measurements without DM haloes in elliptical galaxies could be avoided if the inner kinematic data used in the modeling had sufficiently high spatial resolution – ideally several times better than the SMBH's sphere of influence. Since the HST STIS kinematic observations used for the Siopis et al. analysis of NGC 4258 (FWHM ≈ 0.1) significantly over-resolved the SMBH sphere of influence (d ≈ 0.7, assuming σ = 115 , D = 7.27 Mpc, and = 3.8 × 10^7 from the compilation in ), Schwarzschild modeling of the SMBH mass would understandably be insensitive to the lack of a DM halo.Based on our findings, and by analogy with the results for elliptical galaxies, it seems plausible that disc galaxies where the SMBH sphere of influence is only just resolved – or is under-resolved – would be the likeliest candidates to have biased SMBH measurements when modeled with only one M/L ratio and no DM halo. From the recent compilation of <cit.>, there are eighteen disc galaxies with SMBH masses from Schwarzschild modeling.[Or seventeen if NGC 524 is considered to be an elliptical galaxy.][Since the details of the measurements for NGC 4736 and NGC 4826 – listed in– have not yet been published, we do not consider them.] Four of these have been modeled with a single stellar component and a DM halo <cit.>, and another four were modeled with two stellar components <cit.>. Of the remainder, we can identify two for which the FWHM of the kinematic observations isthe diameter of the sphere of influence: NGC 1023 (; FHWM = 0.2, = 0.16) and NGC 2549 (; FWHM = 0.17, = 0.10). We suggest that those two galaxies in particular could benefit from remodeling with multiple stellar components or with DM haloes (or both).§.§ Stellar Orbital Structure Schwarzschild modeling produces a distribution of weights for the different pre-calculated orbits in the model potential. From these, it is possible to learn something about the stellar orbital structure in the best-fitting model. As we have done in past studies <cit.>, we examine the radial trend in orbital anisotropy. Specifically, we adopt the approach of Erwin et al. and calculate an anisotropy parameter using the ratio of planar/equatorial velocity dispersion σ_eq to the vertical velocity dispersion σ_z (assuming cylindrical coordinates R, φ, z), where the mean dispersion in the equatorial plane is defined byσ_eq^2 = (σ_R^2 + σ_φ^2)/2 .We compute the averages at each radius from orbits in angular bins that range from θ = -23 to θ = +23 with respect to the equatorial plane. The anisotropy β_ eq = 1 - σ_eq^2 / σ_z^2 is ∼ 0 for isotropic velocity dispersion and < 0 for planar-biased anistropy; values of ∼ -1 are typical for the Galactic disc in the Solar neighborhood<cit.>. Figure <ref> shows that isotropy (β_ eq∼ 0) is the rule for r12. For r5, this is consistent with the evidence from the photometric decomposition and the stellar-population analysis for a classical bulge. The region r ∼ 5–12 is outside the bulge, and so at first glance it is puzzling that the velocity dispersion remains roughly isotropic. However, r ∼ 12 is roughly where our exponential-disc component begins to dominate the light (see Figure <ref>). This suggests that the near-isotropy between ∼ 5 and 12 may be related to the weak bar or lens, which contributes to the light in that radial range. We note that although lenses are in general poorly studied, some previous stellar-kinematic observations and models of barred galaxies have suggested that lenses are kinematically hot, possibly dominated by chaotic orbits or a large fraction of retrograde orbits <cit.>. Thus, is it perhaps not surprising that the lens region in NGC 307 fails to show the rotation-dominated anisotropy of a classical disc. § SUMMARY We have presented 2D photometric decompositions, stellar kinematics from adaptive-optics IFU and large-scale IFU and long-slit spectroscopy, and dynamical modeling of the S0 galaxy NGC 307 with the aim of determining the mass of its central SMBH.We have paid particular attention to the effects of modeling the stellar component as a single entity with one M/L ratio versus modeling it as two sub-components (bulge and disc) with independent M/L ratios, and the effects of including a separate DM halo in the modeling.Our best estimate, from the model with a SMBH, separate bulge and disc components, and a DM halo (Model B+DM), is a black hole mass of 2.2 ± 0.6 × 10^8, K-band bulge and disc M/L = 1.1 ± 0.1 and 1.0 ± 0.1, respectively, and a DM halo (spherical cored logarithmic model) with core radius r_c > 5.6 kpc and circular velocity V_h = 260 ± 30 . The SMBH mass is within ∼ 40% of the predicted value from the–σ relation (assuming σ_0 = 205 ) and is ≈ 0.74% of the bulge stellar mass, making NGC 307 entirely consistent with standard SMBH-bulge relations. The M/L ratios are qualitatively consistent with single-stellar-population modeling of our long-slit spectroscopy, which implies a higher M/L in the bulge region.Modeling the stellar kinematics with both stellar components but without the DM halo (Model B) produces identical results for the bulge M/L ratio (1.1 ± 0.1) and a slightly higher SMBH mass (3.0 ± 0.5 × 10^8). The disc M/L ratio is significantly higher (1.9 ± 0.1), due to the fact that the disc component has to be more massive to account for the effects of the (missing) halo.This approach requires only ∼ 4% of the computational time as Model B+DM.Modeling with a single stellar M/L for both bulge and disc plus a DM halo (Model A+DM) yields a SMBH mass almost identical to that of Model B+DM (2.0 ± 0.5 × 10^8) and a combined stellar M/L = 1.1 ± 0.1; the DM halo then has core radius r_c > 4.5 kpc and circular velocity V_h = 200 ± 20 . The computational time required for this model is ∼ 20% of the time required for model B+DM, but about 4.5 times that for Model B.Finally, the simplest model, with a single stellar M/L ratio and no DM halo, gives a much lower value for the SMBH mass (7.0 ± 0.1 × 10^7) and a higher stellar M/L ratio (1.3 ± 0.1), because the necessity of accounting for the DM halo drives the stellar M/L ratio to high values, increasing the stellar mass everywhere and reducing the amount of mass that can be assigned to the SMBH. This model is also clearly worse than the others in terms of how poorly it fits the kinematic data.This suggests that dynamical modeling of disc galaxies can yield reasonably accurate measurements of SMBH masses and bulge M/L ratios without needing the additional computational time of including a DM halo – if a separate disc component with its own M/L ratio is included, though the disc M/L ratio will then almost certainly be overestimated. Models that treat the entire galaxy as having a single stellar M/L ratio (without a DM halo) can potentially underestimate the SMBH mass by significant amounts, especially if the kinematic data used do not overresolve the SMBH sphere of influence, as has been previously found for elliptical galaxies. We suggest that previous SMBH measurements for the S0 galaxies NGC 1023 and NGC 2549 should be revisited, since they were modeled using single stellar components and no DM haloes, using kinematic data which probably does not fully resolve their SMBH spheres of influence.Our morphological and spectroscopic analysis of NGC 307, including 2D decompositions, suggests that the galaxy has four distinct stellar components: a compact central bulge with a metal-rich stellar population (≈ 33% of the light), a weak bar or lens (≈ 6%), an exponential disc (≈ 36%), and a rounder, luminous stellar halo with slightly boxy isophotes (≈ 25%) which is misaligned by about 5 with respect to the disc and bulge. (In our two-stellar-component dynamical modeling, we treated the disc + bar/lens + stellar halo as one component.) Using our best-fit K-band M/L values, the estimated stellar masses for these components are 3.6 × 10^10 (bulge), 4.3 × 10^9 (bar/lens), 2.7 × 10^10 (disc), and 1.9 × 10^10 (stellar halo), with a total stellar mass of 8.6 × 10^10. The stellar halo is best understood as a separate component rather than being simply the outer part of the bulge; this is consistent with recent 2D decomposition analyses of the Sombrero Galaxy, which indicate a bulge + stellar halo + disc model is a better match to the galaxy than a single bulge component plus the disc <cit.>. § ACKNOWLEDGEMENTS We thank VLT support astronomer Chris Lidman and telescope operator Christian Esparza for their assistance during the VLT-SINFONI observations. 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O.,Gültekin K.,Husemann B.,2016, ApJ, 817, 2[Yıldırım, van den Bosch, van de Ven, Husemann, Lyubenova, Walsh, Gebhardt & GültekinYıldırım et al.2015]yildirim15 Yıldırım A.,van den Bosch R. C. E.,van de Ven G., Husemann B.,Lyubenova M.,Walsh J. L.,Gebhardt K., Gültekin K.,2015, , 452, 1792 § EFFECTS OF VARIATION IN BULGE/DISK DECOMPOSITION ON DYNAMICAL MODELINGOur preferred dynamical models (Models B and B+DM) have separate bulge and “disc” (i.e., disc + bar/lens + stellar halo) stellar components, each with its own M/L ratio. There is the possibility that uncertainties in the bulge/disc decomposition (Section <ref>) – e.g., how much stellar light is assigned to the bulge component – might lead to uncertainties in the two stellar M/L ratios, and thus potentially also to uncertainties in the SMBH mass. To investigate the possible effects of variations in thebulge-disk decomposition, we focused on the fits to the VLT-FORS1 image (Section <ref>). Using the bootstrapping facility in Imfit (see Section 5 of ), we generated 1000 resampled versions of the FORS1 image and fit each with the same B+b+D+H model as we used for the main decomposition (Table <ref>). We then computed the B/T values for each best-fitting model. The standard deviation of the 1000 B/T values was σ_B/T = 0.0046 or ∼ 1.4% of the original best-fit model's B/T of 0.3265.We selected two of the bootstrap-resampled fits, with B/T values equal to the best-fit value ±σ_B/T. We then generated bulge and disc model surface-brightness profiles and deprojected these to form bulge and disc luminosity-density components, as in Section <ref>. Finally, we ran our dynamical modeling process using these new stellar components. For the underlying general dynamical model we used Model B, which has SMBH, bulge, and disc components. (We chose this general model because it contains separate bulge and disc M/L ratios but requires considerably less time to run than Model B+DM.)The results of the dynamical fits to these two decompositions are shown in Figure <ref>. The SMBH mass is, within in our admittedly somewhat coarse sampling, identical to our standard best-fit results (= 3.0 ± 0.5) for Model B (see Table <ref>). The bulge and disc M/L ratios are also identical (= 1.1, = 1.9). We conclude that the nominal uncertainties of our bulge-disc decomposition have minimal effect on the results of our dynamical modeling, and in particular have negligible effect on the SMBH mass determination.§ LONG-SLIT STELLAR KINEMATICS FOR NGC 307 The stellar kinematics (both major- and minor-axis) from our long-slit spectraof NGC 307 are presented in Table <ref>. § VIRUS-W IFU KINEMATICS FOR NGC 307 The Voronoi-binned stellar kinematics from our VIRUS-W observations of NGC 307 are presented in Table <ref>. The definitions of the bins in terms of individual fibers, and the positions of the latter on the sky, are presented in Table <ref>. | http://arxiv.org/abs/1709.08956v1 | {
"authors": [
"Peter Erwin",
"Jens Thomas",
"Roberto P. Saglia",
"Maximilian Fabricius",
"Stephanie P. Rusli",
"Stellar Seitz",
"Ralf Bender"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170926120303",
"title": "NGC 307 and the Effects of Dark-Matter Haloes on Measuring Supermassive Black Holes in Disc Galaxies"
} |
AIP/[email protected] of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino, ItalyMedacta International SA, Castel San Pietro, SwitzerlandDepartment of Environmental, Land and Infrastructure Engineering, Politecnico di Torino, Torino, Italy A network-based approach is presented to investigate the cerebrovascular flow patterns during atrial fibrillation (AF) with respect to normal sinus rhythm (NSR). AF, the most common cardiac arrhythmia with faster and irregular beating, has been recently and independently associated with the increased risk of dementia. However, the underlying hemodynamic mechanisms relating the two pathologies remain mainly undetermined so far; thus the contribution of modeling and refined statistical tools is valuable. Pressure and flow rate temporal series in NSR and AF are here evaluated along representative cerebral sites (from carotid arteries to capillary brain circulation), exploiting reliable artificially built signals recently obtained from an in silico approach. The complex network analysis evidences, in a synthetic and original way, a dramatic signal variation towards the distal/capillary cerebral regions during AF, which has no counterpart in NSR conditions. At the large artery level, networks obtained from both AF and NSR hemodynamic signals exhibit elongated and chained features, which are typical of pseudo-periodic series. These aspects are almost completely lost towards the microcirculation during AF, where the networks are topologically more circular and present random-like characteristics. As a consequence, all the physiological phenomena at microcerebral level ruled by periodicity - such as regular perfusion, mean pressure per beat, and average nutrient supply at cellular level - can be strongly compromised, since the AF hemodynamic signals assume irregular behaviour and random-like features. Through a powerful approach which is complementary to the classical statistical tools, the present findings further strengthen the potential link between AF hemodynamic and cognitive decline. From time-series to complex networks: application to the cerebrovascular flow patterns in atrial fibrillation Luca Ridolfi December 30, 2023 ============================================================================================================= The paper presents a network-based perspective to investigate the cerebrovascular flow patterns during atrial fibrillation (AF) with respect to normal sinus rhythm (NSR). There has been recently growing evidence that AF, the most common cardiac arrhythmia with faster and irregular beating, is independently associated to the increased risk of dementia. The topic has a high social impact given the number of individuals involved and the expected increasing AF incidence in the next forty years. Although several mechanisms try to explain the relation between the two pathologies, causality implications are far from being clear. In particular, little is known about the distal and capillary cerebral circulation. Thus, the contribution of modeling and statistical tools is valuable. Here, we exploit the powerful techniques of complex network theory to better characterize the cerebrovascular patterns (in terms of in silico pressure and flow rate time-series) during AF with respect to NSR. Each cerebral region (from large carotid arteries to capillary districts) is represented through a network, whose topological features evidence the differences between arrhythmic and normal beating, thus highlighting in the micro-circulation plausible mechanisms for cognitive decline during AF. § INTRODUCTION The growing size of numerical data coming from multiscale simulations, highly-resolved imaging and computational fluid dynamics approaches <cit.> requires refined quantitative tools to appropriately analyze biomedical signals. Complex network theory, by combining elements from the graph theory and statistical physics, offers an innovative and synthetic framework to handle and interpret complex systems with a huge number of interacting elements <cit.>. In the last decades, beside the well-established applications to Internet, World Wide Web, economy and social dynamics <cit.>, complex networks have been employed in a variety of physical and engineering systems, e.g. from hydrology and climate dynamics (e.g., <cit.>), turbulent and fluid flows (e.g., <cit.>), to biomedical applications (e.g., <cit.>). The network-based approach has been extensively proposed for the characterization of time series which - by means of algorithms based on recurrence plot <cit.>, visibility graph <cit.>, correlation matrix <cit.> and pseudo-periodicity <cit.> - are converted into complex networks.We here present a network-based approach, which relies on the correlation matrix and exploits the pseudoperiodic framework proposed for the electrophysiological signals <cit.>, to study the cerebrovascular flow patterns during atrial fibrillation (AF). This cardiac pathology, characterized by irregular and accelerated heart-beating, is the most common arrhythmia with an estimated number of 33.5 million individuals affected worldwide in 2010 <cit.>. Beside the well-known disabling symptoms (such as palpitations, chest discomfort, anxiety, decrease of blood pressure, limited exercise tolerance, pulmonary congestion) which deteriorate the quality of life, AF is related to thromboembolic transient ischemic attacks <cit.> and increases the risk of suffering an ischemic stroke by five times <cit.>. Through a collection of hemodynamic mechanisms, such as silent cerebral infarctions <cit.>, altered cerebral blood flow <cit.>, hypoperfusion <cit.> and microbleeds, an independent association between AF and cognitive decline has been recently observed.Although an increasing number of very different observational studies suggests this possible association (e.g., among the most recent <cit.>), to the best of our knowledge none of them definitely states a causal relation between AF hemodynamics and cognitive impairment. Recent works highlight AF consequences on the cerebral circulation (e.g. lower diastolic cerebral perfusion and decreased blood flow in the intracranial arteries), but the causal connections still remain mostly undetermined <cit.>. Moreover, current techniques - such as transcranial doppler ultrasonography and intracranialpressure measures - are difficult to be obtained and usually fail in capturing the cerebral micro-vasculature fluid dynamics. For all these reasons, little is known about AF effects on altered pressure levels and irregular cerebral blood flow. However, an accurate cerebral hemodynamic mapping would be helpful in revealing AF feedbacks on brain circulation. For example, anomalous pressure levels could be symptomatic to predict haemorrhagic events such as microbleeds, while impair blood flow repartition can locally alter brain oxygenation.Awaiting direct clinical evidences which are nowadays lacking, the efficiency of the computational hemodynamics is a promising branch <cit.>, as it can be extremely helpful in isolating single cause-effect relations and understanding which AF features lead to specific hemodynamic changes. In recent works, AF and normal sinus rhythm (NSR) cerebral hemodynamic signals were in silico simulated and compared <cit.>, revealing a dramatically altered scenario during AF, where critical events - such as hypoperfusions and hypertensions - are more likely to occur in the peripheral brain circulation.The aim of the present work is to exploit a validated modeling algorithm to analyze, through some suitable advanced metrics adopted in the complex network theory, how the peripheral cerebral circulation is altered by AF, in particular, in those regions where in vivo measures are still missing. Hemodynamic signals of pressure and flow rate over 1000 heartbeats in NSR and AF are evaluated along four representative cerebral sites (internal carotid artery, middle cerebral artery, distal region, capillary-venous district), exploiting the in silico data recently obtained <cit.>. A total number of sixteen time series is thus analyzed. Each time series is then transformed into a network, associating with each temporal segment a node and identifying possible links among nodes through the linear correlation coefficient. If the statistical interdependence between two nodes is above a sufficiently high threshold, a link between the two nodes exists. The resulting sixteen networks are then studied with the network metrics, revealing in the peripheral circulation different topological features between NSR and AF.§ METHODS A stochastic modeling approach is adopted to reproduce pressure and flow rate signals in the cerebral circulation during NSR and AF. Two lumped-parameter models describing the cardiovascular and cerebral circulations are exploited in series and stochastically forced through artificially-built RR series, where RR (s) is the temporal range between two consecutive heart beats. The cardiovascular and cerebral models have been previously calibrated and extensively validated as long as literature data are available <cit.> to mimic realistic NSR and AF conditions for the cerebrovascular circulation. Thus, the validated in silico outputs are the object of the present work and used to understand how the cerebral fluid dynamics and the temporal structure of the hemodynamic signals modify during AF, especially in the peripheral regions where clinical measures are still lacking. This goal is achieved through a refined and innovative method for signal analysis, based on the complex network theory, by broadening and strengthening what recently observed with more classical statistical tools <cit.>. §.§ Cerebral time series: pressures and flow rates in NSR and AF The NSR and AF signals have been acquired from a recent computational hemodynamic study <cit.>, which consists of an algorithm made of three subsequential steps. In Figure 1, the modeling approach adopted (panels 1, 2, 3) is described and representative pressure time series obtained as outputs (panel 4) are displayed. The present computational framework combines two different lumped models in sequence: the cardiovascular model is run to obtain the systemic arterial pressure, P_a, which is next exploited as forcing input for the cerebral model.To obtain the cerebral time series, the first step is to artificially generate the RR intervals, in NSR and AF conditions. We used artificially-built RR-intervals to avoid the patient-specific details (e.g., sex, age, weight, cardiovascular diseases, ...) inherited by real RR beating. To purely catch the overall impact of a fibrillated cardiac status, we consider the same healthy young adult configuration, first forced through NSR and then by AF rhythm. Being normal beating a remarkable example of pink noise <cit.>, RR beats during NSR have been extracted from a pink-correlated Gaussian distribution (mean μ=0.8 s, standard deviation, σ=0.06 s), which is the typical distribution observed during sinus rhythm <cit.>. Differently to the white noise which is uncorrelated, the adoption of a pink noise introduces a temporal correlation, which is a common feature of the normal beating <cit.>. The AF distribution is commonly unimodal (60-65% of the cases) and is fully described <cit.> by thesuperpositionoftwo statisticallyindependenttimes,RR=φ+η.φisextractedfromacorrelatedpinkGaussiandistribution, ηis insteaddrawnfroman uncorrelated exponential distribution (rate parameter γ). The resulting AF beats are thus obtained from an uncorrelated Exponentially Gaussian Modified distribution (mean μ=0.8 s, standard deviation σ=0.19 s, rate parameter γ=7.24 Hz). The AF beating results less correlated and with a higher variability than NSR, as clinically observed <cit.>. The RR parameters of both configurations aresuggestedbythe available data <cit.>, and by considering that the coefficient of variation, cv, is around 0.24 during AF <cit.>. Since these RR time sequences have been validated and tested over clinically measured beating <cit.>, we adopt them as the most suitable and reliable RR time-series to model NSR and AF conditions. Moreover, both RR series have been chosen with the same mean heart rate (75 bpm) to facilitate comparison between the two conditions. More details on RR extraction are reported elsewhere <cit.>.1000 cardiac cycles, shown in Fig. 1 (first panel), are considered for each configuration. This allows us to achieve statistically stable results. Namely, we tested that other extractions of the same number of beats give results analogous to those described in the following sections III and IV.As second step, a lumped cardiovascular model was run using the RR signals as input, to obtain the signals of systemic arterial pressure, P_a, in NSR and AF conditions (Fig. 1, first panel), which will be the forcing inputs of the cerebral model. The cardiovascular modeling, first proposed by Korakianitis and Shi <cit.> and then widely adopted in the computational hemodynamics <cit.>, consists of a network of electrical components - such as compliances, resistances and inductances - and describes the whole systemic and pulmonary circulation, with an active representation of the four heart chambers. Parameters were fitted to reproduce the physiological hemodynamics of a young healthy man, by providing results which are in agreement with the expected real behavior <cit.>. Moreover, by suitably changing the parameters, the computational approach is able to capture the main features of different cardiovascular pathologies, such as hypertension and valvular diseases <cit.>. During AF, the model was first tuned and extensively validated in resting conditions over more than thirty clinical studies <cit.>. Then, it has been successfully adopted to evaluate different AF aspects, such as the concomitant presence of left valvular diseases <cit.> and the role of increased heart rate at rest <cit.> and under effort <cit.>. To mimic the absence of the atrial kick, which is observed in AF, left and right atria are imposed as passive.In the end, the P_a signals are introduced in a lumped parameter model which simulates the entire cerebral circulation <cit.> (Fig.1, second panel). The cerebral model is based on electrical counterparts and accurately describes the cerebral circulation up to the peripheral and capillary regions. It is able to reproduce several different pathological conditions characterized by heterogeneity in cerebrovascular hemodynamics and can be divided into three main sections: large arteries (Fig. 1, light blue box), distal arterial circulation (Fig. 1, green box), and capillary/venous circulation (Fig. 1, yellow box). The computational approach has been validated mainly over mean flow rates up to the middle cerebral circulation <cit.>, since current clinical techniques for pressure and flow rate measures lack the resolving power to give any insights on the micro-vasculature. For this reason, the cerebral modeling can be a useful predictive tool to understand how the hemodynamic signals change towards the micro-circulation in presence of AF <cit.>. The left vascular pathway ICA-MCA (i.e., internal carotid artery - middle cerebral artery) evidenced in Fig. 1 (Panel 3, red path) is here analyzed as representative of the blood flow and pressure distributions from large arteries to the capillary-venous circulation: left internal carotid artery (P_a and Q_ICA,left), middle cerebral artery (P_MCA,left and Q_MCA,left), middle distal district (P_dm,left and Q_dm,left), capillary-venous circulation (P_c and Q_pv). Pressure time series of the left ICA-MCA pathway are reported as exemplificative of NSR and AF behaviors, in the fourth panel of Fig. 1. More details on the cerebral model are offered elsewhere <cit.>. All the cerebral pressure and flow rate time series selected to build the corresponding networks have a frequency of 250 Hz and are composed, as previously mentioned, by 1000 heartbeats. §.§ Complex network metrics Some fundamental concepts of the complex network theory are here summarized <cit.>, by recalling only the measures and definitions which are relevant to the present analysis.A network is defined by a set V = 1,...,N of nodes and a set E of links {i,j}. We assume that a single link exists between a pair of nodes and no self-loops can occur. The adjacency matrix, A: A_ij =0,{i,j}∉ E1,{i,j}∈ E, accounts whether a link is active or not between nodes i and j. The network is considered as undirected, thus A is symmetric, moreover A_ii=0 since self-loops are not allowed.The normalized degree centrality of the i-th node is defined as k_i = ∑_j=1^N A_ij/N-1, and represents the number of neighbors of node i, normalized over the total number of possible neighbors (N-1).The eigenvector centrality, measuring the influence of the node i in the network, is given by x_i = 1/λ∑_k A_ki x_k, where A_ki is the adjacency matrix and λ is its largest eigenvalue (in modulus) <cit.>. In matrix notation, we have: λ x = x A, where the centrality vector x is the left-hand eigenvector of the adjacency matrix A related to the eigenvalue λ.The assortativity coefficient, r, <cit.> is defined as r = 1/σ_q^2∑_jkjk (e_jk - q_j q_k), where q_j is the distribution of the remaining degree, that is the number of edges leaving the node other than the one we arrived along. e_jk is the joint probability distribution of the remaining degrees of the two nodes, j and k (for undirected networks, e_jk=e_kj). This quantity follows the rules: ∑_jke_jk=1 and ∑_j e_jk=q_k. The assortativity coefficient is the Pearson correlation coefficient of the degree between pairs of linked nodes, thus r ∈ [-1,1]. When r = 1, the network has perfect assortative mixing patterns, meaning that high-degree nodes tend to connect each other (e.g., rich-club effect). When r = 0 the network is non-assortative or uncorrelated, which is typical of random graphs, while at r = -1 the network is completely disassortative, that is high-degree nodes are linked to low-degree nodes.The local clustering coefficient of node i is lc_i = e(Γ_i)/k_i(k_i-1)/2, where Γ_i is the set of first neighbors of i, e(Γ_i) is the number of edges connecting the vertices within the neighborhood Γ_i, and k_i(k_i-1)/2 is the maximum number of edges in Γ_i, 0 ≤ lc_i ≤ 1. The local clustering coefficient gives the probability that two randomly chosen neighbors of i are also neighbors. The global clustering coefficient is the mean value of lc_i, lc = ∑_i=1^N lc_i /N.The betweenness centrality of node k is bc_k = ∑_i,j ≠ kσ_ij(k)/σ_ij, where σ_ij are the number of shortest paths connecting nodes i and j, while σ_ij(k) represents the number of shortest paths from i to j, across node k. If node k is crossed by a large number of all existing shortest paths (i.e., high bc_k values), then it can be considered an important mediator for the information transport in the network.The closeness centrality of node i is cc_i = N-1/∑_j=1^N d_ij where the shortest path length, d_ij, is the minimum number of edges that have to be crossed from node i to node j, with i, j ∈ V (d_ii=0). If i and j are not connected, the maximum topological distance in the graph d_ij = N-1 is used in the sum. Closeness centrality is normalized as follows, 0 ≤ cc_i ≤ 1. According to this definition, node i has a high closeness centrality value when it is topologically close to the rest of the network. §.§ Building the networks: from time-series to complex networks To transform the time series into complex networks, the approach proposed for pseudo-periodic series <cit.> has been adopted. The complete temporal signal is divided into sequential cycles according to the RR intervals, which represent the portions of series corresponding to each beat. Fig. 2a shows an example of pressure temporal series divided into 5 time segments, according to the RR beating. Every temporal segment is associated to a node of the network, with the convention that node i corresponds to the i-th beat (i∈ [1,1000]). Therefore, nodes are ordered in agreement with the beating sequence. In the example reported in Fig. 2, the network is thus composed by 5 nodes.For each hemodynamic signal the cross-correlation matrix R is built, where the element R_ij represents the maximum value of the linear correlation coefficient between the i-th and j-th temporal segments. The maximum correlation value is taken when the two segments have different temporal lengths, by shifting the shortest signaling segment all along the length of the longest segment. Each matrix R is symmetric (R_ii=1 by definition) and has dimension n x n, where here n=1000 is the number of the heartbeats (i.e., nodes) analyzed. Exploiting the symmetry of the R matrix, the correlation coefficients to be evaluated are the matrix elements above the diagonal and correspond to the number of possible links, n(n-1)/2. In the example of Fig. 2b, the correlation matrix is (5x5), thus we have 10 correlation coefficients.To compare pairs of cycles, the maximum value of the correlation coefficients has been selected among other distance criteria. Two other distance measures have been checked. The first analyzed is the phase space distance <cit.>, defined as M_ij = min_l1/min(l_i,l_j)∑_k=1^min(l_i,l_j) ||X_k - Y_k+1||, where l_i and l_j are the lengths of the cycle i and j, while X_k and Y_k are the k-th elements (e.g., hemodynamic signals) of the cycles i and j, respectively. M represents the minimum value of the sum of the modules of the differences between the samples of all the possible pairs of cycles. The second measure is based on the mean value distance. For each cycle, i, a mean value of the hemodynamic signal is computed, x_i. Then, the distance matrix, D, is built considering the distance, d_i, between the mean value of the cycle i and the average value of the complete signal. The element D_ij is defined as |d_i - d_j|.The three distance measures here introduced lead to similar results, however the most significative to evidence the difference between NSR and AF conditions turned out to be the correlation matrix, R. Therefore, results in Sections III and IV are presented only by using the linear correlation.To define the adjacency matrix, A, and the corresponding network, a link between nodes i and j exists whether R_ij≥τ, where τ is the ninth decile of the matrix R (computed excluding the diagonal). In so doing, each resulting network is undirected (A_ij=A_ji) and unweighted, since A_ij=1 (unitary weight) as long as R_ij≥τ. In the example of Fig. 2b, the ninth decile of the corresponding R matrix is 0.995, thus two nodes are connected by a link if the corresponding correlation coefficient is equal or above this value. The active link (1-5) is highlighted through a thick blue line.The choice of the threshold, τ, has been long discussed in correlation-based networks <cit.> and is a non-trivial aspect of building the network. It should represent a good compromise between a very high degree of correlation and a suitable network dimension. Our goal is to compare the networks here built, which represent different cerebral regions (from the internal carotid to the capillary regions) in physiological (NSR) and pathological (AF) conditions. Therefore, we preferred using a threshold which, case by case, through the ninth decile accounts for the maximum correlation of the local dynamics considered, rather than a unique threshold (e.g., τ=0.8) fixed for all the configurations, which could be not equally meaningful in all the districts and conditions. Since τ is the only arbitrary parameter involved in the network building, a sensitivity analysis with different τ percentile values (namely, 85^th and 95^th percentiles) is reported in the Appendix A.The described mapping of time series into networks has been achieved for pressure and flow rate series during NSR and AF in the 4 aforementioned cerebral regions. Thus, 16 networks are built and analyzed starting from the corresponding 16 hemodynamic series.§ RESULTS Outcomes for the metrics introduced in Section IIB are here presented for the 16 networks. Apart from the assortativity which is a global network parameter, each node of a network has a value for the analyzed metrics (degree centrality, eigenvector centrality, local clustering coefficient, betweenness centrality and closeness centrality). To synthesize this large amount of information, we build the probability density function (PDF) and the cumulative function (CDF) distributions for all the metrics of each network. Then, at the same district and for the same hemodynamic signal (pressure and flow rate) we compare NSR and AF conditions. We recall that, given a beating condition (NSR or AF), results shown in the following are insensitive to the specific RR values composing the sequence of 1000 cardiac cycles, since this number of beats allows the statistical stationary to be reached. In fact, we checked that other extractions with the same number of beats give analogous trends, with negligible differences with respect to the differences observed between NSR and AF.A first concise outcome is revealed by the matrix, P, defined asP_ij = ∫_D |p_i(x) - p_j(x)| dx,i,j=1,...,4,where p_i and p_j are two PDFs to be compared, while D = D_i ∪ D_j is the union of the domains D_i and D_j, where the PDFs are defined. The values of the subscripts i and j vary from 1 to 4 according to the cerebral districts considered (1: large arteries, 2: middle cerebral region, 3: distal region, 4: capillary/venous circulation). P, for each metric and hemodynamic signal, accounts for the area of the difference (in module) between pairs of PDF distributions along the different districts in NSR and AF conditions. P has the same dimension [4x4] as the cerebral regions studied. In particular, each element of the diagonal of P represents the area of the difference between two NSR and AF PDFs in the same region (from P_11 for the large arteries to P_44 for the capillary/venous circulation). The upper triangular part of P takes into account the difference between two PDFs at different districts in NSR (e.g., P_13 represents the area of the difference between the PDF at large arteries and the PDF at the distal region during NSR). The lower triangular part of P expresses the difference between two PDFs at different districts in AF (e.g., P_24 represents the area of the difference between the PDF at the middle cerebral region and the PDF at the capillary/venous region during AF). Since the PDF has unitary area, the area of the difference between the two compared PDFs can vary between 0 (when the distributions coincide) and 2 (when the distributions have completely different domains). An example of P matrix is computed over the betweenness centrality in Fig. 3a for the pressure signal.By considering the diagonal values (from top to bottom) of all the P matrices, one can infer how much each metric is significant to capture AF-induced variations (with respect to NSR) along the ICA-MCA pathway, as reported in Fig. 3b. All the metrics studied are displayed for the pressure signal (similar results are found for the flow rate), where on the y-axis lie the P_ii values, while on the x-axis the cerebral stations are localized. Fig. 3b shows that the differences between healthy and fibrillated conditions are minimal in large arteries and increase in the distal and capillary circulation. Both the degree and the eigenvector centrality indicators, however, are not optimal metrics in this context, since they do not significantly inherit structural signaling variations induced by AF. The local clustering coefficient, lc, together with the closeness, cc, and the betweenness, bc, centralities are the best metrics to highlight the differences between NSR and AF.The analysis of P matrices is exploratory since allows us to discern the most useful metrics from those which here are not so meaningful. To this end, in the following we only focus on lc, bc and cc distributions, as well as on the assortative mixing. Moreover, the preliminary examination of P matrices reveals the presence of important variations, in absolute terms, between NSR and AF when going towards the microcirculation. However, evaluating P matrices is not sufficient, as we are not able to observe whether the metrics increase or decrease during AF with respect to NSR along the ICA-MCA pathway. The bc and cc distributions are thus analyzed in a more extensive way through their CDFs. CDFs are shown instead of PDFs because they are less sensitive to oscillations of high-tail values and can be more easily interpreted.Figure 4 shows the CDFs of the closeness centrality (left) and betweenness centrality (right), starting from the pressures signals. Top and bottom panels refer to NSR and AF conditions, respectively. In each panel the distributions at the four cerebral districts are reported. Figure 5 is organized similarly to Fig. 4, but the results are obtained with the flow rate signals.For both pressure and flow rate, during NSR the distributions of cc and bc (top panels of Fig. 4 and 5) assume similar values along the ICA-MCA pathway. During AF, closeness centrality dramatically increases towards the distal and capillary/venous regions (Fig. 4c and 5c), with significantly higher values reached especially for the pressure (Fig. 4c). On the contrary, betweenness centrality in AF tends to meaningfully decrease when entering the microcirculation (Fig. 4d and 5d), for both pressure and flow rate. The two hemodynamic signals (pressure and flow rate) confirm that in normal conditions no relevant variation occurs along the ICA-MCA pathway, while during AF closeness centrality increases and betweenness centrality decreases.Apart from local clustering and assortativity coefficients, the other metrics (k, x, bc, cc) here discussed are measures of a node prominence in a network. However, as observed so far, these measures behave differently and their trends along the cerebral pathway reveal non-trivial behaviors <cit.>. Degree and eigengvector centrality parameters remain almost constant when entering the cerebral regions. As the two metrics represent a similar degree of node centrality, it is reasonable they are quite correlated. On the contrary, closeness and betweenness measures of centrality present opposite trends along the ICA-MCA pathway.At the large arteries level in AF conditions, the network has lower cc values and higher bc values with respect to the microcirculation region. Low cc and high bc values mean that the network is topologically elongated and chained. Each node (i.e., beat) is generally linked to the previous and the next nodes (beats), as well as to other far nodes. On average, every node has a low closeness centrality since it has direct links (i.e., low shortest path) only with its neighborhood, while the shortest paths with the rest of the network are in general high. Moreover, each node is equally important with respect to the others for the information transmission. In fact, given two non-directly connected nodes of the network, information has to necessarily pass through the intermediate nodes (beats) between them. This last aspect implies a high bc value for almost all the nodes.In the microcirculation region during AF, the network has higher cc values and lower bc values with respect to the cerebral input. This configuration means that a node is more incline to connect with nodes which are not not its precedent and subsequent nodes. On average, each node needs a limited shortest path to reach all the other nodes of the network. As a consequence, cc value is high. On the contrary, since information has not to cross all the intermediate nodes between pairs of nodes, the betweenness centrality is averagely decreased. Being the shortest paths in general shorter than at large arteries, the stream of information now excludes several nodes which were instead crossed at the carotid entrance. The network is topologically more circular, with non-consecutive distant beats (nodes) which often share a link.Since during AF degree and eigenvector centrality distributions remain basically constant along the ICA-MCA pathway, this entails that on average each node maintains the same number of links with the other nodes. What makes the difference is the link topology. At the carotid entrance, the links of a node are created with the surrounding nodes as well as with long-range links (beats). Going towards the capillary/venous region, the links connecting consecutive nodes are almost all broken and substituted with long-range links (i.e., links between temporally distant beats).Evidence of this different link distribution during AF along the cerebral circulation is given by the assortativity coefficient, r, along the ICA-MCA pathway (Fig. 6). We recall that r measures the tendency of a network to present link between similar (r=1) or dissimilar (r=-1) nodes. An assortativity value close to zero means that none of the above trends is evidenced, links emerge with no preference between similar or dissimilar nodes, thus the network is uncorrelated. During NSR, both pressure and flow rate reveal a quite high assortativity (0.65-0.7) which is maintained constant along the ICA-MCA pathway (blue curves, panels a and b). In AF condition, the assortativity has analogous values as in NSR at the carotid entrance, with a significant drop towards the distal and capillary/venous circulation (red curves, panels a and b). In the peripheral regions, links are no more between nodes sharing the same properties (i.e., degree centrality), but spurious long-range links are predominant. The network here resembles the features of a random uncorrelated graph.We conclude the Results Section with a comment on the local clustering coefficient. In Fig. 3b, diagonal elements of the P matrix showed a slight increase of this metric towards the capillary region. In Fig. 7, the mean values, lc, are reported for both pressure and flow rate signals during NSR and AF, together with the dispersion due to the standard deviation values, lc±σ_lc, where σ_lc is the standard deviation value of the lc distribution. It can be noted that, the global local clustering values are quite constant along the ICA-MCA pathway in NSR as well as in AF. However, in the distal and capillary regions, the data dispersion increases (for both NSR and AF), being higher during AF than NSR. Thus, going towards the microcirculation during AF, the neighborhood of each node can be either almost fully connected or poorly connected. This is a further symptom of the increased variability and unpredictability induced by AF on the peripheral hemodynamic signals.§ DISCUSSION As evidenced in Section III, NSR and AF networks in the cerebral microcirculation present significantly different structures. To better highlight the topological features, we graphically represent the pressure (Fig. 8) and flow rate (Fig. 9) networks at the beginning and towards the end of the ICA-MCA pathway in both NSR and AF conditions. Networks are visualized through the open-source software package Gephi <cit.>, with respect to the closeness centrality values. Red color corresponds to low cc values, while blue-colored nodes correspond to high cc values. In addition, the size of a node is proportional to its cc value.Let us first consider the networks obtained by the pressure signals, in Fig. 8. At the large arteries level (Fig. 8, bottom panels), the NSR and AF networks present an elongated and almost planar shape, which is the typical feature of networks based on pseudo-periodic series, with similar mean closeness centrality values, cc (cc: 0.257 (NSR), 0.247 (AF)). In the capillary region, the NSR network is slightly less elongated (Fig. 8, top left panel), but the mean closeness centrality value (cc= 0.282) is not far from the corresponding value at the entrance (cc=0.257). During AF the scenario is instead completely altered (Fig. 8, top right panel). The network assumes a markedly circular and three-dimensional shape, which is usually encountered in random networks, with an average closeness centrality value (cc=0.458) which almost doubles the corresponding one at the cerebral entrance (cc=0.247).The networks obtained from flow rate signals have similar average closeness centrality values at the large arteries level (Fig. 9, bottom panels), for both NSR (cc= 0.333) and AF (cc= 0.337). The NSR network (Fig. 9, bottom left panel) has an elongated shape, although less pronounced than the corresponding pressure NSR network (Fig. 8, bottom left panel). During AF, the flow rate network assumes a triangular shape, preserving its bi-dimensional form (Fig. 9, bottom right panel). In the distal region, the NSR network maintains its mainly-planar shape (Fig. 9, top left panel) and accentuates its elongation with respect to the carotid entrance. The average closeness centrality value (cc=0.284) does not change significantly with respect to the Q_ICA,left network (cc=0.333). On the contrary, in AF the network becomes circular and fully three-dimensional (Fig. 9, top right panel), with an increased closeness centrality mean value (cc=0.412). In this configuration, the network features resemble those of random networks.The network-based approach is able to fully catch the structural differences between NSR and AF hemodynamic signals, which, for some metrics, are striking. The proposed scenario evidences a dramatic signal variation towards the distal/capillary cerebral regions during AF, which has no counterpart in NSR conditions. All the most significant network parameters (bc, cc, lc, r) agree in locating this alteration. During AF, the input signals (P_a and Q_ICA,left) exhibit pseudo-periodic features, which are almost completely lost towards the microcirculation, where instead P_c and Q_dm,left signals reveal random-like characteristics.The hemodynamic consequences of this substantial alteration can be of high biomedical impact. All the physiological phenomena at microcerebral level ruled by periodicity - such as regular perfusion, mean pressure per beat, average nutrient supply at cellular level - are not guaranteed and can be strongly compromised, since the AF hemodynamic signals assume irregular behaviour and random-like features. The cardiovascular implications of the highlighted alteration surely deserve to be further quantified through clinical evidences, although invasive and accurate measurements are still difficult to be accomplished, also because of the signal complexity induced by the heart rate variability. Awaiting necessary in vivo validation, the network-based hints here emerged can plausibly explain the hemodynamic mechanisms leading to cognitive impairment in presence of persistent AF.§ CONCLUSIONS The network analysis performed over cerebral hemodynamic signals highlighted that the degree centrality, which is usually the most intuitive and firstly analyzed metric, is here not much informative. Other local (i.e., eigenvector centrality) and mesoscopic (i.e., local clustering coefficient) measures are not crucial in discerning NSR and AF hemodynamic features. A deeper examination of the adjacency matrix was necessary, especially in terms of global metrics. In particular, the markers of betweenness and closeness centrality as well as the assortativity coefficient turned out to be meaningful. From the combined analysis of the network metrics through their probability distributions, during AF it emerges that towards the peripheral cerebral circulation the closeness centrality increases, while the betweeness centrality is reduced. The AF pressure and flow rate networks change from an elongated shape (which is characteristic of pseudo-periodic series) in the large artery region to a circular-like shape (which is a feature of random series) in the capillary-venous districts.The complex network analysis evidences in a synthetic and innovative way how hemodynamic signals in the cerebral microcirculation are deeply altered by AF. This result, on one hand, confirms that the complex network theory can be successfully extended to explore other pathological biomedical signals in complex geometries, such as stenotic flows across aortic valve and flow dynamics in brain aneurysms. On the other hand, the present findings further strengthen the always more evident link between AF hemodynamic and cognitive decline, through a powerful approach which is complementary to the classical statistical tools.§ SENSITIVITY ANALYSIS A sensitivity analysis on the threshold value τ is here performed, recalling that τ is the only arbitrary parameter involved for building the network. Beside the 90^th percentile of the R matrix, other two values have been chosen to set the threshold τ, namely the 85^th and 95^th percentiles. In Fig. A1, the diagonal of P matrix is reported for the degree centrality, closeness centrality and betweenness centrality for the three percentile values, for both pressure (top panel) and flow rate (bottom panel) signals. It can be noted that, apart from the specific values, the trend of the P_ii values remain unaltered by changing the threshold τ. In fact, for all the percentile values considered, the degree centrality, k, remains almost constant towards the peripheral circulation, while betweenness and closeness centrality values experience significative variations with respect to large arteries. The proposed percentile (90^th) is therefore a good choice to catch the network behavior which, despite the specific values assumed at each cerebral district, turns out to be substantially insensitive to the threshold adopted.* | http://arxiv.org/abs/1709.09087v1 | {
"authors": [
"Stefania Scarsoglio",
"Fabio Cazzato",
"Luca Ridolfi"
],
"categories": [
"physics.data-an",
"physics.flu-dyn",
"physics.med-ph",
"q-bio.TO"
],
"primary_category": "physics.data-an",
"published": "20170926152436",
"title": "From time-series to complex networks: Application to the cerebrovascular flow patterns in atrial fibrillation"
} |
=1 arrows,positioning,shapes.geometric#1#1 #1#1#1#1 | http://arxiv.org/abs/1709.09712v2 | {
"authors": [
"Arindam Das",
"Partha Konar",
"Arun Thalapillil"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170927194133",
"title": "Jet substructure shedding light on heavy Majorana neutrinos at the LHC"
} |
firstpage–lastpage 2013 Precise Reduction of Solar Spectra Observed by the 1-meter New Vacuum Solar Telescope [=====================================================================================A number of groups have recently been active in searching for gradients in the observed Faraday rotation measure (RM) across jets of Active Galactic Nuclei (AGNs) on various scales and estimating their reliability. Such RM structures provide direct evidence for the presence of an azimuthal magnetic fieldcomponent, which may be associated with a helical jet magnetic field, as is expected based on the results of many theoretical studies. We present new parsec-scale RM maps of 4 AGNs here, and analyze their transverse RM structures together with those for 5 previously published RM maps. All these jets display transverse RM gradients with significances of at least 3σ. This ispart of an ongoing effort to establish how common transverse RM gradientsthat may be associated with helical or toroidal magnetic fields are in AGNson parsec scales.§ INTRODUCTION The relativistic jets of Active Galactic Nuclei (AGNs) emit radio synchrotron emission, which can be linearly polarized up to about 75% in optically thinregions with uniform magnetic fields, with the polarization angle χorthogonal to the projection of the magnetic field B onto the plane ofthe sky (Pacholczyk 1970). The degree of linear polarization is considerably lower in optically thick regions,up to 10–15%, with χ parallel to the projected B (Pacholczyk 1970). In the standard theoretical picture of AGN jets offered by Blandford &Königl (1979), emission is observed only near and beyond the locationin the jet outflow where the optical depth is near unity, τ≈ 1,representing the transition between optically thick regions closer to thecentral engine and optically thin regions farther along the jet. Thisτ≈ 1 surface is a theoretical construction that is locatedsomewhere within the “core” observed in Very Long Baseline Interferometry(VLBI) images. Although there is a tendency to think broadly in terms ofan optically thick core and optically thin jet, there is abundant evidencethat the VLBI cores observed at centimeter wavelengths are in fact mixedregions of partially optically thick emission corresponding to the vicinity of the theoretical τ≈ 1 surface and optically thin emission from theinner jet (see, e.g., Gabuzda 2015). Multi-frequency VLBI polarization observations provideinformation about the wavelength dependence of the parsec-scale polarization, in particular, Faraday rotation occurringat various locations between the emitting region and observer.When Faraday rotation occurs in regions of thermal (non-relativistic or only mildly relativistic) plasma outsidethe emitting region the rotation is given byχ_obs - χ_o =e^3λ^2/8π^2ϵ_om^2c^3∫ n_e𝐁· d𝐥≡ RMλ^2where χ_obs and χ_o are the observed and intrinsicpolarization angles, respectively, -e and m are the charge andmass of the particles giving rise to the Faraday rotation, usuallytaken to be electrons, c is the speed of light, ϵ_o the permittivity constant, n_e thedensity of the Faraday-rotating electrons, 𝐁 the magneticfield, d𝐥 an element along the line of sight, λthe observing wavelength, and RM (the coefficient of λ^2) is theRotation Measure (e.g., Burn 1966).The action of external Faraday rotationcan be identified using simultaneous multifrequency observations, throughthe linear λ^2 dependence, allowing the determination of both theRM (which reflects the electron density and line-of-sight B fieldin the region of Faraday rotation) and χ_o (the intrinsic direction of the source's linear polarization, and hence the synchrotron B field,projected onto the plane of the sky).Many theoretical studies and simulations of the relativistic jets of AGNs have predicted the development of a helical jet B field, which comes about due to the combination of the rotation of the central black hole and its accretion disk and the jet outflow (e.g. Nakamura, Uchida & Hirose 2001,Lovelace et al. 2002; see Tchekhovskoy and Bromberg 2016 for a recent example). Researchers have long been aware that the presence of a helical jet B field could give rise to a regular gradient in the observedRM across the jet, due to the systematic change in the line-of-sightcomponent of the helical field (Perley et al. 1984, Blandford 1993).Statistically significant transverse RM gradients acrossthe parsec-scale jets of more than 25 AGN have been reported in the refereed literature (e.g. Gabuzda et al. 2015 andreferences therein), interpreted as reflecting the systematic change in theline-of-sight component of a toroidal or helical jet B field acrossthe jets.The Monte Carlo simulations of Hovatta et al. (2012), Mahmud et al. (2013) and Murphy & Gabuzda (2013) clearly indicate that the key factors in determining the trustworthiness of an RM gradient (i.e., the probability that it is not spurious) are (i) monotonicity, (ii) the range of valuesencompassed by the gradient relative to the uncertainties in the RMmeasurements and (iii) steadiness of the change in the RM values acrossthe jet (ensuring a seeming “gradient” is not due only to values in afew edge pixels), rather than the width spanned by the gradient. The second criterion reflects the result of the Monte Carlo simulations that, when an RM gradient encompasses values differing by at least 3σ and spanseven a small distance comparable to one beamwidth, the probability that it is spurious is very low – less than 1%. Mahmud et al. (2013), Gabuzda et al. (2014a, 2014b) and Motter and Gabuzda (2017) have carried out new Faraday-rotation analyses employingthe empirical error formula of Hovatta et al. (2012), focusing on monotonicity, steadiness of thegradient across the jet and a significance of at least 3σ as the key criteria for reliability of observed transverse RM gradients. Results published earlier by Gabuzda et al.(2004, 2008) were reanalyzed using this same approach by Gabuzda etal. (2015), who also reported 8 new cases of monotonic, statisticallysignificant transverse RM gradients across AGN jets based onpreviously published and unpublished maps.In the current study, we have applied this approach to analyze 4 RM imagespreviously published by Zavala & Taylor (2002, 2003), based on VLBA data at 7 frequencies between 8.1 and 15.2 GHz, 1 RM image published by Kharb et al. (2009) based on 3 frequencies between 5.0 and 15.3 GHz and 4 RM images published here for the first time, based onVLBA data at 6 frequencies between 4.6 and 15.4 GHz. All of these AGNs display statistically significant transverse RM gradients across their jets.This is part of a larger study aiming to build up statistics for AGN jets displaying transverse RM gradients with the ultimate goal of analyzing the collectedproperties of the RM gradients detected.§ OBSERVATIONSLike Gabuzda et al. (2015), we present some new analyses of previouslypublished Faraday RM maps together with RM images published here for the firsttime.In all cases,the observations were obtained on the NRAO Very Long Baseline Array. We carried out the imaging and analysis for the data considered here in the same way as is described by Gabuzda et al. (2015), including matching the resolutions at the different frequencies, aligning the images at the different frequencies when significant misalignments were present, and correction for Faraday rotation occurring in our Galaxy when significant. The integrated RM measurements of Taylor et al. (2009) for all the sourcesconsidered here, based on the VLA Sky Survey (NVSS) observations at two bands near 1.4 GHz, are given in Table 1.The integrated RMs for 0212+735,0300+470, 0305+039, 0415+379, 0945+408, 1502+106 and 1611+343 are small,no higher than about 24 rad m^-2, which is smaller than the typicaluncertainties in the parsec-scale RM values. We did notremove the effect of these small integrated RM values, since this will not have a bearing on the interpretation of our results. On the other hand, the integrated RMs for 2005+403 (-171 rad m^-2) and 2200+420(-199 rad m^-2) are substantial, and we accordingly removed these RMsfrom all values in the RM maps shown for these two sources.§.§ 4.6–15.4 GHz, September 2007 The data considered here were obtained as part of the same project as the maps published by Gabuzda et al. (2014b), and were obtained on 26th September 2007. The observations and calibration procedures are described by Gabuzda et al. (2014b).§.§ 5.0–15.3 GHz, September 2005The data considered here were obtained on 10th September 2005 and werepreviously analyzed by Kharb et al. (2009), who describe the observationsand calibration procedures. Their results were based on observations atthree frequencies: 5.0, 8.4 and 15.3 GHz. We used final, fully calibrated UV data files kindly provided by P. Kharb to construct Stokes Q, Stokes U, PANG, and PANGN maps. The effect of the integrated (Galactic) Faraday rotation was not removed, as the integrated RM (10 rad m^-2)was appreciably smaller than the typical RM uncertainties (≃ 50 rad m^-2). These essentially reproduced the RM map of Kharb et al. (2009), but using the more accurate error estimates given by the formula of Hovatta et al. (2012). §.§ 8.1–15.2 GHz, June 2000 Zavala & Taylor (2002, 2003) present 15.2-GHz Faraday RM maps for 20 AGNs based on VLBA observations obtained on 27th June 2000 at 8.1, 8.2, 8.4, 8.6, 12.1, 12.6 and 15.2 GHz.We retrieved these data from the VLBA archive and calibrated them using the same procedures asthose described by Zavala & Taylor (2002, 2003).As in Gabuzda et al (2015), we used the human eye as an initial gradient detector; this indicated four candidates for AGNs with transverse RMgradients across their jets: 0212+735, 0415+379 (3C111), 1611+343 and2005+403.Our RM maps for these four sources basically reproduced theRM maps published by Zavala & Taylor (2002, 2003), but applying the more accurate error-estimation formula of Hovatta et al. (2012), and withthe Galactic RM value removed for 2005+403.§ RESULTS The source names, redshifts, pc/mas values and integrated rotation measures are summarized in Table 1.The pc/mas were determined assuming a cosmology with H_o =71 km s^-1Mpc^-1, Ω_Λ = 0.73 and Ω_m =0.27; the redshifts and pc/mas vales were taken from the MOJAVE projectwebsite (http://www.physics.purdue.edu/MOJAVE/).Polarization maps for all of these sources at 15 GHz can befound in Lister & Homan (2005) and on the MOJAVE website(http://www.physics.purdue.edu/MOJAVE/).Like Motter and Gabuzda (2017), in all cases, we present totalintensity (Stokes I) and RM maps made using the naturally weightedelliptical convolving beams as well as circular convolving beams havingequal area; in other words, the circular beam used has a full widthat half maximum equal to √((BMAJ)(BMIN)), where BMAJ andBMIN are the full widths at half maximum for the major and minoraxes of the nominal elliptical restoring beam.As is explained by Motter and Gabuzda (2017), the maps made usingcircular convolving beams were used to test the robustness of RM structuresvisible in the maps made using the elliptical beams — for example, an RM gradient that seemed to be present in the original RM map butdisappeared upon convolution with the equal-area circular beam would notbe considered reliable (see, e.g., the case of 2155–152 presented by Gabuzda et al. 2015). Such comparisons are especially helpful when theelliptical beam is very elongated. In addition, in some cases, the mapsmade using equal-area circular beams helped clarify the relationship between structure in the RM map and the local direction of the jet. Note that this does not bias the resulting circular beams to maximizethe resolution in any particular direction of interest, e.g., thedirection across the jet. Total intensity and RM maps made using the naturally weighted elliptical and equivalent-area circular convolving beams are shown inFigs. <ref>–<ref>, together withexample slices in regions of visible transverse RM gradients.The frequency, peak and bottom contour of the intensity maps shown in these figures are given in Table 2; the contours increase in steps of a factor of two, and the ranges of the RM maps areindicated by the colour wedges shown with the maps. The panels show the intensity maps made using the nominal elliptical beams with the corresponding RM distributions superposed (left), the corresponding maps made using equal-area circular beams (middle), and slices taken along the lines drawn across the RM distributions in the middle panels (right); theletter “S” at one end of these lines marks the side corresponding to the starting point for the slice (a slice distance of 0). In all cases, we aimed to take the RM slices shown inFigs. <ref>–<ref> perpendicular to thelocal jet direction.We did not formally fit a ridge line to the jet. When RM gradients weredetected relatively far from the core in relatively straight jets (0305+039, 0415+379, 2200+420), the jet direction was estimated directly from the intensity images.When RM gradients were detected in the region of the VLBI core and/or inner jet (0212+735, 0300+470, 0945+408, 1502+106,1611+343, 2005+403), we took the slices perpendicular to the direction ofthe innermost VLBI jet visible in the 15-GHz images for the corresponding observing epochs, taking into account the visual appearance of these maps and the distribution of CLEAN components, as is also described by Motter & Gabuzda (2017). Slices in the core region weretaken at locations near the center of the region where the gradients arevisible, which was sometimes slightly upstream or downstream of the intensitypeak; these locations were not particularly chosen to maximize the significanceof these transverse RM gradients. RM gradients are visible across the core regions of a number of thesources considered here.As is discussed by Motter & Gabuzda (2017)and Wardle (2017), the polarized emission in the core region in practicearises in optically thin regions in the innermost jets that are blendedwith partially optically thick regions in the observed VLBI “core.” In fact, as is shown by the calculations of Cobb (1993) [see also Wardle(2017)], the 90^∘ rotation of the polarization position angleassociated with the optically thin–thick transition occurs near opticaldepth τ≃ 6, far upstream of the most optically thick regions inthe observed VLBI core, located near τ≃ 1. We have accordingly analyzed and interpreted transverse RM gradients in the core region in the same way as those observed farther out across the jet structures.The statistical significances of the transverse gradients detected in our RM maps are summarized in Table 3.We took the significance of an RMgradient to be the magnitude of the difference between the RMs at the twoends of the gradient, divided by the uncertainty in this difference,taken to be the sum of the individual RM uncertainties added in quadrature: Significance=|RM_1 - RM_2|/√(σ_RM1^2 +σ_RM2^2)Note that this approach is more conservative that the procedure used by Hovatta et al. (2012), who comparedthe magnitude of the RM difference and the the maximum error along theslice; typically, our significance estimates will be about a factorof √(2) lower, helping to ensure that we do not overestimate the significance of the gradients. When plotting the slices inFigs. <ref>–<ref> and finding thedifference between the RM values at two ends of a gradient, we did not include uncertaintyin the polarization angles due to EVPA calibration uncertainty, since EVPA calibration uncertainty cannot introduce spurious RM gradients (see discussions by Mahmud et al. (2009) and Hovatta et al.(2012)).In all cases, in both the jet and core regions, the dependence ofχ_obs as a function of λ^2 is consistent with the linearbehaviour expected for external Faraday rotation, to within the uncertaintiesin χ_obs.Results for each of the datasets and each of the AGNs considered hereare summarized below. §.§ 4.6–15.4 GHz data 0300+470.The RM maps for this source are shown in Fig. <ref> (toprow).The RM map made using the elliptical convolving beam (left panel) showsan RM gradient across the core region, whose significance is about 3σ. The RM map made using an equal-area circular beam (middle panel) likewise shows a clear, monotonic transverse RM gradient across the core region,with a significance of 4.9σ (right panel).0945+408.The RM maps for this source are shown in Fig. <ref> (secondrow). Although the shift between the intensity images at the different frequencies due to the change in the position of the VLBI core with frequency was essentially negligible, there was a large shift between the maps at 4.6 and 5.0 GHz and the maps at the higher frequencies, due to thefact that a bright knot in the inner jet was the brightest feature at 4.6 and5.0 GHz, while the core was the brightest feature at the higher frequencies.The polarization angle images at 4.6 and 5.0 GHz were shifted to correct forthis large misalignment before the RM maps were made.The RM map made using the elliptical convolving beam (left panel) shows a transverse RM gradient, which has a significance of about 3.5σ. The RM map made using an equal-areacircular beam (middle panel) shows a clear region of transverse RM gradientsacross the core, with significances of 4-5σ (right panel).1503+106.The RM maps for this source are shown in Fig. <ref> (thirdrow).The RM map made using the elliptical convolving beam (left panel) shows a transverse RM gradient across the core region, which has a highsignificance of 7.4σ, but is nevertheless somewhat uncertain due to thehighly elliptical beam.There is also a transverse gradient in the opposite direction in the jet, with a significance of about 2.5σ. Both of these gradients become more clearly visible inthe RM map made using an equal-area circular beam (middle panel), withsignificances reaching 8σ for the core region and 4-5σ for the jet.2200+420.The RM maps for this source are shown in Fig. <ref>.TheRM map made using the elliptical 4.6-GHz beam (upper row, left map) showshints of a transverse RM gradient in the jet (higher RM values on the eastern side of the jet), but it is not fully monotonic. A region with a monotonic transverse RM gradient with a significance of about 4σ appears at the end of the detected RM distribution in the RM map made withthe equal-area circuar beam (upper row, right map and slice), but weconsidered this gradient to be uncertain due to the small region where itis present at the end of the detected RM distribution.To test the reliability of this transverse gradient,we made the 4.6-GHz intensity map and the RM map using the slightly smaller 7.9-GHz elliptical beam (lower row, left map)and a corresponding equal-area circular beam (lower row, right map).This beam is about a factor of 0.6 smaller than the naturally weighted 4.6-GHz beam; use of such a beam is justified by the analysis of Coughlan& Gabuzda (2016), who used Monte Carlo simulations to demonstrate thatboth Maximum-Entropy and CLEAN deconvolutions yielded reliable results for intensity, polarization and RM images when resolving beams down to halfthe size of the full naturally weighted CLEAN beam were used.Transverse RM gradients in the same direction are visible across the jetin these slightlyhigher-resolution maps. The gradient in the region of the RM slice shown in the top right panel of Fig. <ref> has fallen to about2.3σ, but a region closer to the core has RM gradients reaching3-4σ. Although we believe this gradient is likely real, it would be valuable to verify this result using other multi-frequency data withsimilar resolution.§.§ 5.0–15.3 GHz data 0305+039 (3C78).The RM maps for this source are shown in Fig. <ref>.The RMmap made using the elliptical beam (left panel) shows a roughly transverse RM gradient with a significance of about 2.7σ.The RM map made withan equal-area circular beam (middle panel) shows a clear, monotonictransverse RM gradient across this region, with a significance of3.6σ, whose direction is very close to orthogonal to the jet. §.§ 8.1–15.2 GHz data RM maps of these sources based on the same visibility data but with somewhat different weighting were originally published by Zavala & Taylor (2002,2003).In all cases, our RM maps are very similar to the originallypublished maps.0212+735.The RM maps for this source are shown in Fig. <ref> (top row). Both the RM maps made using the elliptical (left panel) and equal-area circular (middle panel) beams show a very clear, monotonic transverseRM gradient across the core and inner jet region, with significances of7-8σ.0415+379 (3C111).The RM maps for this source are shown in Fig. <ref> (second row). The polarization in the core and innermost jet is weak, leading to the measurement of RM values only in the jet, well separated from the core region. Both the RM maps made using the elliptical (left panel) andequal-area circular(middle panel) beams show a clear, monotonic transverse RM gradient acrossthe jet approximately 6 mas from the core. The gradient has significances of 3.7σ and about 6σ in the elliptical-beam and circular-beam RM maps, respectively. 1611+343.The RM maps for this source are shown in Fig. <ref> (third row). Both the RM maps made using the elliptical (left panel) andequal-area circular (middle panel) beams show a clear, monotonic transverse RM gradient across the core region. The significance of this gradient is 2.5σ in the elliptical-beam RM map, but increases to 3.3σ in the equal-areacircular-beam RM map. 2005+403.The RM maps for this source are shown in Fig. <ref> (fourth row). Bearing in mind that the direction of the inner jet is slightly south of east (as can be seen, for example, in the 15-GHz maps of Zavala & Taylor (2003) and Lister & Homan 2005), the RM map made with theelliptical convolving beam shows a possible transverse gradient in the vicnity of the core with a significance of about 5σ.The RM mapmade using anequal-area circular beam shows this gradient and its direction relative tothe jet much more clearly; its significance in the circular-beam RM map is 8-9σ. The reduced clarify of this gradient in the elliptical-beam RM map is due to the fact that,although the elliptical beam is not extremely elongated, its orientation isclose to orthogonal to the jet direction. § DISCUSSION§.§ Significance of the Transverse RM Gradients Table 3 gives a summary of the transverse Faraday rotation measure gradientsdetected in the images presented here. The statistical significances of these gradients typically lie in the range 3-5σ, although four of the transverse RM gradients we have detected exceed 6σ. In the case of 0305+039 and 1611+343, the transverse gradients in the RM mapsmade with their naturally weighted elliptical beams have significancesdetermined using our conservative approach of 2.7σ and 2.5σ,respectively, but these increase to 3.6σ and 3.3σ, respectively,when the RM maps are made with a circular beam of equal area. Recall that,as we pointed out above, our significances will typically be about a factorof √(2) lower than would be obtained using the approach of Hovattaet al. (2012) [using the maximum error rather than the two RM errors addedin quadrature when determining the significance];these two elliptical-beam significances would increase to about 3.8σ and 3.5σ, respectively, using this latter approach. In reality, it is likely that our significances are slightly underestimated, while those of Hovatta et al. (2012) are somewhat overestimated; the true significances probably lie somewhere between the two, andwe accordingly believe both of these gradients to be significant.Thus, the Monte Carlo simulations of Hovatta et al.(2012) andMurphy & Gabuzda (2013) indicate that the probability that any of thetransverse RM gradients in Table 3 are spurious is less than 1%; thisprobability willbe even lower for clear, monotonic gradients encompassing differences ofappreciably greater than 3σ (0212+735, 0415+379, 0945+408, 1502+106, 2005+403).§.§ Sign Changes in the Transverse RM Profiles In general, Faraday-rotation gradients can arise due to gradients in the electron density and/or line-of-sight magnetic field. However, since gradients in the electron density cannot bring about changes in the sign of the Faraday rotation, a monotonicRM gradient encompassing an RM sign change unambiguously indicates a changein the direction of the line-of-sight magnetic field, such as that due to the presence of a toroidal B fieldcomponent in the region of Faraday rotation. Signficant sign changes are observed in the transverse RM gradients detectedin 5 of the 9 AGNs considered here: 0300+470, 0415+479,1502+106, 2005+403and 2200+420.This supports the idea that these gradients aredue to toroidal, possibly helical, jet B field. As has been noted previously (e.g. Motter & Gabuza 2017), the absence of a sign change in an transverse RM gradient does not rule out the possibility that the origin of this gradientis toroidal field component, since RM gradients encompassinga single sign can sometimes be observed, depending on the helical pitch angle and jet viewing angle. §.§ Core-Region Transverse RM Gradients As is noted in the Introduction, in the standard theoretical picture,the VLBI intensity “core” represents the intensity “photosphere” ofthe jet, where the optical depth is roughly unity.The observed VLBI core encompasses this partially opticallythick region together with much more highly polarized optically thinregions in the innermost jet, with the latter dominating overall observed“core” polarization.Various theoretical models and other high-resolution observations support this picture. For example, Marscher et al. (2008) use the modelof Vlahakis (2006) to explain rapid, smooth rotations of the opticalpolarization position angles as reflecting the motion of a distinct region of polarized emission along a helical stream-line located upstream of the observed VLBI core at millimeter wavelengths; since correlatedpolarization-angle rotations in the optical and radio have also been observed (d'Arcangelo et al. 2009), this implies the presence of opticallythin emission regions upstream of the 7-mm VLBI core, which Marscher et al. (2008) suggest may actually represent the region of a recollimationshock, rather than a τ=1 surface. Similarly, Gómez et al. (2016) have reported the detection of polarized emission upstream of the VLBI core in their high-resolution observations of 2200+420.Since the observed “core” polarization at centimeter and long millimeterwavelengths is thus actually dominated by the contributions of effectively optically thin regions, the simplest interpretation of transverse RM gradientsobserved across a core region is that, like transverse gradients fartherout in the jets, they reflect the presence of a toroidal B fieldcomponent. Monotonic transverse RM gradients with significances of 3σ or more are observed across the core regions of 0212+735, 0300+470, 0945+408, 1502+106, and 1611+343. The simulated RM maps of Broderick & McKinney (2010) and Porth et al. (2011) explicitly show the presence of clear, monotonic transverse RM gradientsin core regions containing helical magnetic fields, with relativistic andoptical depth effects only occasionally giving rise to non-monotonic behaviourfor some azimuthal viewing angles.In addition, the slightly non-monotonicbehaviour displayed by some of these calculated RM profiles will be smoothed byconvolution with a typical centimeter-wavelength VLBA beam, giving rise to monotonic gradients of the sort reported here [see, e.g., the lower rightpanel in Fig. 8 of Broderick & McKinney (2010)]. Therefore, whena smooth, monotonic, statistically significant transverse RM gradient isobserved across the core region, it is justified to interpret thisas evidence for helical/toroidal jet B fields on the corresponding scales; this is particularly so given that the 90^∘ rotation in the polarization angle associated with the optically thin/thick transition does not occur until optical depths τ≃ 6 (Cobb 1993, Wardle 2017).§.§ RM-Gradient Reversals We have detected evidence for distinct regions with transverse Faradayrotation gradients oriented in opposite directions in 1502+106. Similar reversalsin the directions of the RM gradients in the core region and inner jethave been reported for 0716+714, 0923+392, 1749+701 2037+511 (Mahmud etal. 2013, Gabuzda et al. 2014b). Our results for 2200+420 are also of interest here. Motter & Gabuzda (2017) presented a 1.4–1.7-GHz VLBA RM map of this same object for epoch August 2010, about three years after the 4.6–15.4 GHz observations considered here. Their map also shows an RM gradient across the core of their image, but in the opposite direction to the one presented here. Further, Gómez et al. (2016) have produced a high-resolution RM map based on joint analysis of22-GHz RadioAstron space-VLBI data and ground-based VLBI data obtained at15 and 43 GHz in November 2013, which likewise shows a transverse RM gradientin the vicinity of the high-frequency VLBI core, in the same direction asthat observed by Motter et al. (2016). This raises the possibility that the predominant direction of the transverse RM gradients in 2200+420 may varywith time, as has also been observed for 1803+784 (Mahmud et al. 2009) and 0836+710 (Gabuzda et al. 2014).As is discussed by Mahmud et al. (2009, 2013), one reasonable interpretation of both of these phenomena (RM-gradient reversals along the jet and in time) is a picture with a nested helical field structure, with opposite directions for the azimuthal field componentsin the inner and outer regions of helical field. The total observed Faraday rotation in the vicinity of the AGN jet includes contributions from both these regions, and a change in the directionof the net observed RM gradient could be due to a change in dominance fromthe inner to the outer region of helical field, in terms of their overall contribution to the observed Faraday rotation. One possible physical picture giving rise to such a nested helical-field structure is described by Christodoulou et al. (2016). Lico et al. (2017) have used this type of model to explain changes in the sign of the 15–43 GHz VLBA core RM of Mrk 421 in a similar way. §.§ Transverse RM Gradients and AGNs with “Spine–Sheath” Magnetic-Field StructuresThe objects observed in September 2007 at 4.6–15.4 GHz (0300+470, 0945+408, 1502+106 and 2200+420) were part of the same sample of sourcesdisplaying “spine–sheath”polarization structures considered by Gabuzda et al. (2014b).One possible origin for this polarization structure is a helical jet B field, with the sky projection of the helical fieldpredominantly orthogonal to the jet near the jet axis andpredominantly longitudinal near the jet edges. This suggests that transverse RM gradients may also be common across these jets.In all, 22 AGNs with spine–sheath polarization structure were observed as part of this experiment (BG173): 12 sources on 26th September 2007 and 10sources on 27th September 2007.This is to our knowledge the only set ofVLBI observations aimed at Faraday rotation studies in which the target AGNswere selected based on the hypothesis that they were good candidates forAGNs with RM distributions showing transverse RM gradients (with jetscarrying helical magnetic fields).Taking the results presented here together with those of Gabuzda et al.(2014b), 12 of these 22 AGNs (i.e., about 55%) displayed statistically significant transverse RM gradients across their jets,while the remaining 10 AGNs did not show significant transverse RM gradients. In contrast, the re-analysis of the multi-frequency data of Zavala & Taylor(2002, 2003, 2004) carried out by Gabuzda et al. (2015) and in the current paper indicates the presence of statistically significanttransverse RM gradients in 6 out of 40 AGNs (i.e., about 15%).Thus, we have found a considerably higher fraction of AGNs displaying partialor full “spine–sheath” transverse polarization structures to display firm evidence for transverse RM gradients, compared to the AGN sample of Taylor (2000), which was selected to have 15-GHz flux greater than 2 Jy and declinations greater than -10^∘.This supports the hypothesis thatboth the “spine–sheath” polarization structure and the relatively high incidence of transverse RM gradients among the 22 AGNs considered byGabuzda et al. (2014b) and in the current paper is due to the fact that these AGN jets carry helical B fields.Another possibility is that our criterion that the sources display“spine–sheath” polarization structure has essentially selected a setof sources with transversely resolved linear polarization structure, whosetransverse RM structures are likewise relatively well resolved, making iteasier to detect transverse RM gradients in these sources; in this case, this would suggest that a relatively large fraction of allAGN might display transverse RM gradients.The high fraction of transverse RM gradients observed in the “spine–sheath” sources thatdisplay sign changes in the RM (9 of 12) also supports the idea that thesegradients are due to toroidal or helical jet B fields, since an RMsign change can only be explained by a change in the direction of theline-of-sight B field, not a change in the electron density.§ CONCLUSION We have presented new polarization and Faraday RM measurements of 4 AGNs based on 4.6–15.4 GHz observations with the VLBA,together with a reanalysis of RM maps for 4 AGNs published previously by Zavala & Taylor (2002, 2003) and an RM map for the radio galaxy 0305+039 (3C78) published previously by Kharb et al. (2009).All 9 of these AGNs display Faraday rotation measure (RM)gradients across their core regions and/or jets with conservatively estimated statistical significances of at least 3σ. One of the AGNs considered here— 1502+106 — shows evidence for distinct regions with transverse Faraday rotation gradients oriented in opposite directions, and another — 2200+420 — for changes in the direction of its transverse RM gradients with time.Similar reversals in the directions of the RM gradients in the core regions and inner jets have been observed for 4 other AGNs (Mahmud et al. 2013, Gabuzda et al. 2014), while reversals in the directions of the RMgradients with time have been observed for 2 other AGNs (Mahmud et al. 2009, Gabuzda et al. 2014b). These have beeninterpreted as evidence for a nested helical field structure, withthe inner and outer regions of helical field having oppositely directed azimuthal components.Our results for 0300+470, 0945+408 and 1503+106 support the conclusion of Gabuzda et al. (2014b) that statistically signficant transverseRM gradients are common across the VLBI jets of AGNs displaying transverse polarization structures with a “spine” of transverse magnetic field and a “sheath” of longitudinal magnetic field.This is natural if both the transverse polarization structure and the transverse RM gradients in these AGNs have their origin in a helical B field located in the jets and in their immediate vicinity, as would come about through the winding up of an initiallongitudinal field component by the rotation of the central black hole and its accretion disk.§ ACKNOWLEDGEMENTS Partial funding for this research was provided by the Irish ResearchCouncil (IRC). We thank A. Reichstein and S. 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"authors": [
"Denise C. Gabuzda",
"Naomi Roche",
"Amy Kirwan",
"Sebastian Knuettel",
"Matt Nagle",
"Caolan Houston"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170926144521",
"title": "Parsec Scale Faraday-Rotation Structure Across the Jets of 9 Active Galactic Nuclei"
} |
Tensor Product Generation Networks for Deep NLP Modeling Qiuyuan Huang, Paul Smolensky, Xiaodong He, Li Deng, Dapeng Wu{qihua,psmo,xiaohe}@microsoft.com, [email protected], [email protected] Research AIRedmond, WA This work was carried out while PS was on leave from Johns Hopkins University. LD is currently at Citadel.DW is with University of Florida, Gainesville, FL 32611.December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================== We present a new approach to the design of deep networks for natural language processing (NLP), based on the general technique of Tensor Product Representations (TPRs) for encoding and processing symbol structures in distributed neural networks. A network architecture — the Tensor Product Generation Network (TPGN) — is proposed which is capable in principle of carrying out TPR computation, but which uses unconstrained deep learning to design its internal representations. Instantiated in a model for image-caption generation, TPGN outperforms LSTM baselines when evaluated on the COCO dataset. The TPR-capable structure enables interpretation of internal representations and operations, which prove to contain considerable grammatical content. Our caption-generation model can be interpreted as generating sequences of grammatical categories and retrieving words by their categories from a plan encoded as a distributed representation. § INTRODUCTIONIn this paper we introduce a new architecture for natural language processing (NLP). On what type of principles can a computational architecture be founded? It would seem a sound principle to require that the hypothesis space for learning which an architecture provides include network hypotheses that are independently known to be suitable for performing the target task. Our proposed architecture makes available to deep learning network configurations that perform natural language generation by use of Tensor Product Representations (TPRs) <cit.>. Whether learning will create TPRs is unknown in advance, but what we can say with certainty is that the hypothesis space being searched during learning includes TPRs as one appropriate solution to the problem.TPRs are a general method for generating vector-space embeddings of complex symbol structures. Prior work has proved that TPRs enable powerful symbol processing to be carried out using neural network computation <cit.>. This includes generating parse trees that conform to a grammar <cit.>, although incorporating such capabilities into deep learning networks such as those developed here remains for future work. The architecture presented here relies on simpler use of TPRs to generate sentences; grammars are not explicitly encoded here.We test the proposed architecture by applying it to image-caption generation (on the MS-COCO dataset, <cit.>). The results improve upon a baseline deploying a state-of-the-art LSTM architecture <cit.>, and the TPR foundations of the architecture provide greater interpretability.Section <ref> of the paper reviews TPR. Section <ref> presents the proposed architecture, the Tensor Product Generation Network (TPGN). Section <ref> describes the particular model we study for image captioning, and Section <ref> presents the experimental results. Importantly, what the model has learned is interpreted in Section <ref>. Section <ref> discusses the relation of the new model to previous work and Section <ref> concludes.§ REVIEW OF TENSOR PRODUCT REPRESENTATIONThe central idea of TPRs <cit.> can be appreciated by contrasting the TPR for a word string with a bag-of-words (BoW) vector-space embedding. In a BoW embedding, the vector that encodesis the same as the one that encodes : J + K + s where J, K, s are respectively the vector embeddings of the words , , .A TPR embedding that avoids this confusion starts by analyzingas the set {/subj, /obj, /verb}. (Other analyses are possible: see Section <ref>.) Next we choose an embedding in a vector space V_F for , ,as in the BoW case: J, K, s. Then comes the step unique to TPRs: we choose an embedding in a vector space V_R for the roles subj, obj, verb: r_ subj, r_ obj, r_ verb. Crucially, r_ subj≠r_ obj. Finally, the TPR foris the following vector in V_F⊗ V_R: v_ = J⊗r_ subj + K⊗r_ obj + s⊗r_ verb Each word is tagged with the role it fills in the sentence;andfill different roles.This TPR avoids the BoW confusion: v_≠v_ because J⊗r_ subj + K⊗r_ obj≠J ⊗ r_ obj + K⊗r_ subj. In the terminology of TPRs, in ,is the filler of the role subj, and J⊗r_ subj is the vector embedding of the filler/role binding /subj. In the vector space embedding, the binding operation is the tensor — or generalized outer — product ⊗; i.e., J⊗r_ subj is a tensor with 2 indices defined by: [J⊗r_ subj]_φρ≡ [J]_φ [r_ subj]_ρ.The tensor product can be used recursively, which is essential for the TPR embedding of recursive structures such as trees and for the computation of recursive functions over TPRs. However, in the present context, recursion will not be required, in which case the tensor product can be regarded as simply the matrix outer product (which cannot be used recursively); we can regard J⊗r_ subj as the matrix product Jr_ subj^⊤. Then Equation <ref> becomes v_ = Jr_ subj^⊤ + Kr_ obj^⊤ + sr_ verb^⊤ Note that the set of matrices (or the set of tensors with any fixed number of indices) is a vector space; thus↦v_ is a vector-space embedding of the symbol structures constituting sentences. Whether we regard v_ as a 2-index tensor or as a matrix, we can call it simply a `vector' since it is an element of a vector space: in the context of TPRs, `vector' is used in a general sense and should not be taken to imply a single-indexed array.Crucial to the computational power of TPRs and to the architecture we propose here is the notion of unbinding. Just as an outer product — the tensor product — can be used to bind the vector embedding a fillerto the vector embedding a role subj, J⊗r_ subj or Jr_ subj^⊤, so an inner product can be used to take the vector embedding a structure and unbind a role contained within that structure, yielding the symbol that fills the role.In the simplest case of orthonormal role vectors r_i, to unbind role subj inwe can compute the matrix-vector product: v_r_ subj = J (because r_i^⊤r_j = δ_ij when the role vectors are orthonormal). A similar situation obtains when the role vectors are not orthonormal, provided they are not linearly dependent: for each role such as subj there is an unbinding vector u_ subj such that r_i^⊤u_j = δ_ij so we get: v_u_ subj = J. A role vector such as r_ subj and its unbinding vector u_ subj are said to be duals of each other. (If R is the matrix in which each column is a role vector r_j, then R is invertible when the role vectors are linearly independent; then the unbinding vectors u_i are the rows of R^-1. When the r_j are orthonormal, u_i = r_i. Replacing the matrix inverse with the pseudo-inverse allows approximate unbinding if the role vectors are linearly dependent.)We can now see how TPRs can be used to generate a sentence one word at a time. We start with the TPR for the sentence, e.g., v_. From this vector we unbind the role of the first word, which is subj: the embedding of the first word is thus v_u_ subj = J, the embedding of . Next we take the TPR for the sentence and unbind the role of the second word, which is verb: the embedding of the second word is then v_u_ verb = s, the embedding of . And so on.To accomplish this, we need two representations to generate the t^th word: (i) the TPR of the sentence, S (or of the string of not-yet-produced words, S_t) and (ii) the unbinding vector for the t^th word, u_t. The architecture we propose will therefore be a recurrent network containing two subnetworks: (i) a subnet S hosting the representation S_t, and a (ii) a subnet U hosting the unbinding vector u_t. This is shown in Fig. <ref>.§ A TPR-CAPABLE GENERATION ARCHITECTUREAs Fig. <ref> shows, the proposed Tensor Product Generation Network architecture (the dashed box labeled N) is designed to support the technique for generation just described: the architecture is TPR-capable. There is a sentence-encoding subnetwork S which could host a TPR of the sentence to be generated, and an unbinding subnetwork U which could output a sequence of unbinding vectors u_t; at time t, the embedding f_t of the word produced, _t, could then be extracted from S_t via the matrix-vector product (shown in the figure by “×”): f_t = S_tu_t. The lexical-decoding subnetwork L converts the embedding vector f_t to the 1-hot vector x_t corresponding to the word _t.Unlike some other work <cit.>, TPGN is not constrained to literally learn TPRs. The representations that will actually be housed in S and U are determined by end-to-end deep learning on a task: the bubbles in Fig. <ref> show what would be the meanings of S_t, u_t and f_t if an actual TPR scheme were instantiated in the architecture. The learned representations _t will not be proven to literally be TPRs, but by analyzing the unbinding vectors u_t the network learns, we will gain insight into the process by which the learned matrices _t give rise to the generated sentence.The task studied here is image captioning; Fig. <ref> shows that the input to this TPGN model is an image, preprocessed by a CNN which produces the initial representation in S, S_0. This vector S_0 drives the entire caption-generation process: it contains all the image-specific information for producing the caption. (We will call a caption a “sentence”even though it may in fact be just a noun phrase.)The two subnets S and U are mutually-connected LSTMs <cit.>: see Fig. <ref>. The internal hidden state of U, p_t, is sent as input to S; U also produces output, the unbinding vector u_t. The internal hidden state of S, S_t, is sent as input to U, and also produced as output. As stated above, these two outputs are multiplied together to produce the embedding vector f_t = S_tu_t of the output word _t. Furthermore, the 1-hot encoding x_t of _t is fed back at the next time step to serve as input to both S and U.What type of roles might the unbinding vectors be unbinding? A TPR for a caption could in principle be built upon positional roles, syntactic/semantic roles, or some combination of the two. In the caption , the initialandmight respectively occupy the positional roles of pos(ition)_1 andpos_2;might occupy the syntactic role of verb;the role of Spatial-P(reposition); whilemight fill a 5-role schema Det(erminer)_1 N(oun)_1 P Det_2 N_2. In fact we will provide evidence in Sec. <ref> that our network learns just this kind of hybrid role decomposition; further evidence for these particular roles is presented elsewhere. What form of information does the sentence-encoding subnetwork S need to encode in ? Continuing with the example of the previous paragraph,needs to be some approximation to the TPR summing several filler/role binding matrices. In one of these bindings, a filler vector f_ — which the lexical subnetwork L will map to the article— is bound (via the outer product) to a role vector r_Pos_1 which is the dual of the first unbinding vector produced by the unbinding subnetwork U: u_Pos_1. In the first iteration of generation the model computes _1u_Pos_1 = f_, which L then maps to . Analogously, another binding approximately contained in _2 is f_r_Pos_2^⊤. There are corresponding approximate bindings for the remaining words of the caption; these employ syntactic/semantic roles. One example is f_r_V^⊤. At iteration 3, U decides the next word should be a verb, so it generates the unbinding vector u_V which when multiplied by the current output of S, the matrix _3, yields a filler vector f_ which L maps to the output . S decided the caption should deployas a verb and included inan approximation to the binding f_r_V^⊤. It similarly decided the caption should deployas a spatial preposition, approximately including inthe binding f_r_Spatial-P^⊤; and so on for the other words in their respective roles in the caption.§ SYSTEM DESCRIPTIONAs stated above, the unbinding subnetwork U and the sentence-encoding subnetwork S of Fig. <ref> are each implemented as (1-layer, 1-directional) LSTMs (see Fig. <ref>); the lexical subnetwork L is implemented as a linear transformation followed by a softmax operation.In the equations below, the LSTM variables internal to the S subnet are indexed by 1 (e.g., the forget-, input-, and output-gates are respectively f̂_1, î_1, ô_1) while those of the unbinding subnet U are indexed by 2.Thus the state updating equations for S are, for t=1,⋯,T = caption length:𝐟̂_1,t=σ_g(𝐖_1,f𝐩_t-1-𝐃_1,f𝐖_e𝐱_t-1+𝐔_1,fŜ_t-1) î_1,t=σ_g(𝐖_1,i𝐩_t-1-𝐃_1,i𝐖_e 𝐱_t-1+𝐔_1,iŜ_t-1) ô_1,t=σ_g(𝐖_1,o𝐩_t-1-𝐃_1,o𝐖_e 𝐱_t-1+𝐔_1,oŜ_t-1) 𝐠_1,t=σ_h(𝐖_1,c𝐩_t-1-𝐃_1,c𝐖_e 𝐱_t-1+𝐔_1,cŜ_t-1) 𝐜_1,t= 𝐟̂_1,t⊙𝐜_1,t-1+ î_1,t⊙𝐠_1,t Ŝ_t= ô_1,t⊙σ_h(𝐜_1,t)Here 𝐟̂_1,t, î_1,t, ô_1,t, 𝐠_1,t, 𝐜_1,t, Ŝ_t∈ℝ^d× d, 𝐩_t∈ℝ^d; σ_g(·) is the (element-wise) logistic sigmoid function; σ_h(·) is the hyperbolic tangent function; the operator ⊙ denotes the Hadamard (element-wise) product; 𝐖_1,f,𝐖_1,i,𝐖_1,o,𝐖_1,c∈ℝ^d× d × d, 𝐃_1,f, 𝐃_1,i, 𝐃_1,o, 𝐃_1,c ∈ ℝ^d× d × d, 𝐔_1,f, 𝐔_1,i, 𝐔_1,o, 𝐔_1,c ∈ ℝ^d× d × d× d. For clarity, biases — included throughout the model — are omitted from all equations in this paper. The initial state Ŝ_0 is initialized by:Ŝ_0=𝐂_s (𝐯-𝐯̅)where 𝐯∈ℝ^2048 is the vector of visual features extracted from the current image by ResNet <cit.> and 𝐯̅ is the mean of all such vectors;𝐂_s ∈ℝ^d× d × 2048. On the output side, 𝐱_t ∈ℝ^V is a 1-hot vector with dimension equal to the size of the caption vocabulary, V, and 𝐖_e ∈ℝ^d× V is a word embedding matrix, the i-th column of which is the embedding vector of the i-th word in the vocabulary; it is obtained by the Stanford GLoVe algorithm with zero mean <cit.>. 𝐱_0 is initialized as the one-hot vector corresponding to a“start-of-sentence” symbol.For U in Fig. <ref>, the state updating equations are: 𝐟̂_2,t=σ_g(Ŝ_t-1𝐰_2,f-𝐃_2,f𝐖_e 𝐱_t-1+𝐔_2,f𝐩_t-1) î_2,t=σ_g(Ŝ_t-1𝐰_2,i-𝐃_2,i𝐖_e 𝐱_t-1+𝐔_2,i𝐩_t-1) ô_2,t=σ_g(Ŝ_t-1𝐰_2,o-𝐃_2,o𝐖_e 𝐱_t-1+𝐔_2,o𝐩_t-1) 𝐠_2,t=σ_h(Ŝ_t-1𝐰_2,c-𝐃_2,c𝐖_e 𝐱_t-1+𝐔_2,c𝐩_t-1) 𝐜_2,t= 𝐟̂_2,t⊙𝐜_2,t-1+ î_2,t⊙𝐠_2,t 𝐩_t=ô_2,t⊙σ_h(𝐜_2,t) Here 𝐰_2,f, 𝐰_2,i, 𝐰_2,o, 𝐰_2,c∈ℝ^d, 𝐃_2,f, 𝐃_2,i, 𝐃_2,o, 𝐃_2,c ∈ ℝ^d × d, and 𝐔_2,f, 𝐔_2,i, 𝐔_2,o, 𝐔_2,c ∈ ℝ^d× d. The initial state 𝐩_0 is the zero vector.The dimensionality of the crucial vectors shown in Fig. <ref>, u_t and f_t, is increased from d × 1 to d^2× 1 as follows. A block-diagonal d^2× d^2 matrix S_t is created by placing d copies of the d × d matrix Ŝ_t as blocks along the principal diagonal. This matrix is the output of the sentence-encoding subnetwork S. Now the`filler vector' 𝐟_t∈ℝ^d^2 — `unbound' from the sentence representation _t with the `unbinding vector' u_t — is obtained by Eq. (<ref>).𝐟_t= 𝐒_t𝐮_tHere 𝐮_t∈ℝ^d^2, the output of the unbinding subnetwork U, is computed as in Eq. (<ref>), where 𝐖_u ∈ℝ^d^2× d is U's output weight matrix. 𝐮_t=σ_h(𝐖_u𝐩_t) Finally, the lexical subnetwork L produces a decoded word 𝐱_t∈ℝ^V by𝐱_t=σ_s(𝐖_x𝐟_t)where σ_s(·) is the softmax function and 𝐖_x ∈ℝ^V× d^2 is the overall output weight matrix. Since 𝐖_x plays the role of a word de-embedding matrix, we can set𝐖_x=(𝐖_e)^⊤where 𝐖_e is the word-embedding matrix. Since 𝐖_e is pre-defined, we directly set 𝐖_x by Eq. (<ref>) without training L through Eq. (<ref>). Note that S and U are learned jointly through the end-to-end training as shown in Algorithm <ref>.§ EXPERIMENTAL RESULTS §.§ DatasetTo evaluate the performance of our proposed model, we use the COCO dataset <cit.>. The COCO dataset contains 123,287 images, each of which is annotated with at least 5 captions. We use the same pre-defined splits as in <cit.>: 113,287 images for training, 5,000 images for validation, and 5,000 images for testing. We use the same vocabulary as that employed in <cit.>, which consists of 8,791 words. §.§ EvaluationFor the CNN of Fig. <ref>, we used ResNet-152 <cit.>, pretrained on the ImageNet dataset. The feature vector 𝐯 has 2048 dimensions. Word embedding vectors in 𝐖_e are downloaded from the web <cit.>. The model is implemented in TensorFlow <cit.> with the default settings for random initialization and optimization by backpropagation. In our experiments, we choose d=25 (where d is the dimension of vector 𝐩_t).The dimension of 𝐒_t is625 × 625 (while Ŝ_t is 25 × 25);the vocabulary size V=8,791; the dimension of 𝐮_t and 𝐟_t isd^2=625. The main evaluation results on the MS COCO dataset are reported in Table <ref>. The widely-used BLEU <cit.>, METEOR<cit.>, and CIDEr<cit.> metrics are reported in our quantitative evaluation of the performance of the proposed model. In evaluation, our baseline is the widely used CNN-LSTM captioning method originally proposed in <cit.>. For comparison, we include results in that paper in the first line of Table <ref>. We also re-implemented the model using the latest ResNet features and report the results in the second line of Table <ref>. Our re-implementation of the CNN-LSTM method matches the performance reported in <cit.>, showing that the baseline is a state-of-the-art implementation. As shown in Table <ref>, compared to the CNN-LSTM baseline, the proposed TPGN significantly outperforms the benchmark schemes in all metrics across the board. The improvement in BLEU-n is greater for greater n; TPGN particularly improves generation of longer subsequences. The results attest to the effectiveness of the TPGN architecture. §.§ Interpretation of learned unbinding vectorsTo get a sense of how the sentence encodings _tlearned by TPGN approximate TPRs, we now investigate the meaning of the role-unbinding vector u_t the model uses to unbind from _t — via Eq. (<ref>) — the filler vector f_t thatproduces — via Eq. (<ref>) — the one-hot vectorx_t of the t^th generated caption word. The meaning of an unbinding vector is the meaning of the role it unbinds.Interpreting the unbinding vectors reveals the meaning of the roles in a TPR thatapproximates.§.§.§ Visualization of 𝐮_tWe run the TPGN model with 5,000 test images as input, and obtain the unbinding vector 𝐮_t used to generate each word 𝐱_t in the caption of a test image. We plot 1,000 unbinding vectors 𝐮_t, which correspond to the first 1,000 words in the resulting captions of these 5,000 test images. There are 17 parts of speech (POS) in these 1,000 words. The POS tags are obtained by the Stanford Parser <cit.>.We use the Embedding Projector in TensorBoard <cit.> to plot 1,000 unbinding vectors 𝐮_t with a custom linear projection in TensorBoard to reduce 625 dimensions of 𝐮_t to 2 dimensions shown in Fig. <ref> through Fig. <ref>.Fig. <ref> shows the unbinding vectors of 1000 words; different POS tags of words are represented by different colors. In fact, we can partition the 625-dim space of 𝐮_t into 17 regions, each of which contains 76.3% words of the same type of POS on average; i.e., each region is dominated by words of one POS type. This clearly indicates that each unbinding vector contains important grammatical information about the word it generates. As examples, Fig. <ref> to Fig. <ref> show the distribution of the unbinding vectors of nouns, verbs, adjectives, and prepositions, respectively.§.§.§ Clustering of 𝐮_tSince the previous section indicates that there is a clustering structure for 𝐮_t, in this section we partition 𝐮_t into N_u clusters and examine the grammar roles played by 𝐮_t .First, we run the trained TPGN model on the 113,287 training images, obtaining the role-unbinding vector 𝐮_t used to generate each word 𝐱_t in the caption sentence. There are approximately 1.2 million 𝐮_t vectors over all the training images. We apply the K-means clustering algorithm to these vectors to obtain N_u clusters and the centroid μ_i of each cluster i (i=0,⋯,N_u-1).Then, we run the TPGN model with 5,000 test images as input, and obtain the role vector 𝐮_t of each word 𝐱_t in the caption sentence of a test image. Using the nearest neighbor rule, we obtain the index i of the cluster that each 𝐮_t is assigned to. The partitioning of the unbinding vectors 𝐮_t into N_u = 2 clusters exposes the most fundamental distinction made by the roles. We find that the vectors assigned to Cluster 1 generate words which are nouns, pronouns, indefinite and definite articles, and adjectives, while the vectors assigned to Cluster 0 generate verbs, prepositions, conjunctions, and adverbs. Thus Cluster 1 contains the noun-related words, Cluster 0 the verb-like words (verbs, prepositions and conjunctions are all potentially followed by noun-phrase complements, for example). Cross-cutting this distinction is another dimension, however: the initial word in a caption (always a determiner) is sometimes generated with a Cluster 1 unbinding vector, sometimes with a Cluster 0 vector. Outside the caption-initial position, exceptions to the nominal/verbal ∼ Cluster 1/0 generalization are rare, as attested by the high rates of conformity to the generalization shown in Table <ref> .Table <ref> shows the likelihood of correctness of this `N/V' generalization for the words in 5,000 sentences captioned for the 5,000 test images; N_w is the number of words in the category, N_r is the number of words conforming to the generalization, and P_c=N_r/N_w is the proportion conforming. We use the Natural Language Toolkit <cit.> to identify the part of speech of each word in the captions.A similar analysis with N_u = 10 clusters reveals the results shown in Table <ref>; these results concern the first 100 captions, which were inspected manually to identify interpretable patterns. (More comprehensive results will be discussed elsewhere.)The clusters can be interpreted as falling into 3 groups (see Table <ref>). Clusters 2 and 3 are clearly positional roles: every initial word is generated by a role-unbinding vector from Cluster 2, and such vectors are not used elsewhere in the string. The same holds for Cluster 3 and the second caption word.For caption words after the second word, position is replaced by syntactic/semantic properties for interpretation purposes. Thevector clusters aside from 2 and 3 generate words with a dominant grammatical category: for example, unbinding vectors assigned to the cluster 4 generate words that are 91% likely to be prepositions, and 72% likely to be spatial prepositions. Cluster 7 generates 88% nouns and 9% adjectives, with the remaining 3% scattered across other categories. As Table <ref> shows, clusters 1, 5, 7, 9 are primarily nominal, and 0, 4, 6, and 8 primarily verbal. (Only cluster 5 spans the N/V divide.)§ RELATED WORKThis work follows a great deal of recent caption-generation literature in exploiting end-to-end deep learning with a CNN image-analysis front end producing a distributed representation that is then used to drive a natural-language generation process, typically using RNNs <cit.>. Our grammatical interpretation of the structural roles of words in sentences makes contact with other work that incorporates deep learning into grammatically-structured networks <cit.>. Here, the network is not itself structured to match the grammatical structure of sentences being processed; the structure is fixed, but is designed to support the learning of distributed representations that incorporate structure internal to the representations themselves — filler/role structure.TPRs are also used in NLP in <cit.> but there the representation of each individual input word is constrained to be a literal TPR filler/role binding. (The idea of using the outer product to construct internal representations was also explored in <cit.>.) Here, by contrast, the learned representations are not themselves constrained, but the global structure of the network is designed to display the somewhat abstract property of being TPR-capable: the architecture uses the TPR unbinding operation of the matrix-vector product to extract individual words for sequential output.§ CONCLUSION Tensor Product Representation (TPR) <cit.> is a general technique for constructing vector embeddings of complex symbol structures in such a way that powerful symbolic functions can be computed using hand-designed neural network computation. Integrating TPR with deep learning is a largely open problem for which the work presented here proposes a general approach: design deep architectures that are TPR-capable — TPR computation is within the scope of the capabilities of the architecture in principle. For natural language generation, we proposed such an architecture, the Tensor Product Generation Network (TPGN): it embodies the TPR operation of unbinding which is used to extract particular symbols (e.g., words) from complex structures (e.g., sentences). The architecture can be interpreted as containing a part that encodes a sentence and a part that selects one structural role at a time to extract from the sentence. We applied the approach to image-caption generation, developing a TPGN model that was evaluated on the COCO dataset, on which it outperformed LSTM baselines on a range of standard metrics. Unlike standard LSTMs, however, the TPGN model admits a level of interpretability: we can see which roles are being unbound by the unbinding vectors generated internally within the model. 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In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3156–3164, 2015.yogatama2016learning Dani Yogatama, Phil Blunsom, Chris Dyer, Edward Grefenstette, and Wang Ling. Learning to compose words into sentences with reinforcement learning. arXiv preprint arXiv:1611.09100, 2016. | http://arxiv.org/abs/1709.09118v5 | {
"authors": [
"Qiuyuan Huang",
"Paul Smolensky",
"Xiaodong He",
"Li Deng",
"Dapeng Wu"
],
"categories": [
"cs.CV",
"cs.CL"
],
"primary_category": "cs.CV",
"published": "20170926163220",
"title": "Tensor Product Generation Networks for Deep NLP Modeling"
} |
Departamento de Física, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile Centro Científico Tecnológico de Valparaíso, Casilla 110-V, Valparaíso, Chile [email protected] de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, Ciudad de México, 04510, México [email protected] of Nuclear Physics, Polish Academy of Sciences, Poland [email protected] Laboratory of Nuclear Problems, Joint Institute for Nuclear Research, Dubna, Russia [email protected] Up to date, there is no consensus regarding the origin of ultra high energy cosmic rays (UHECR) beyond the Greisen–Zatsepin–Kuzmin (GZK) limit. In order for these UHECR to reach the Earth, an extremely suppressed interaction between them and the cosmic microwave background (CMB) is required, which is impossible for standard model (SM) particles, except neutrinos. In this letter, we present constraints on the parameter space of models involving axion-like particles and dark photons, as candidates for UHECRs, by assuming that these particles can traverse the visible Universe without decaying. In the case of axion-like particles, the constraints are tighter than those given in previous works. We present for the first time constraints on the parameter space of dark photon models, by considering their kinetic term mixing with SM photons. Limits on beyond standard model messengers as ultra high energy cosmic rays Jilberto Zamora-Saa December 30, 2023 ===========================================================================§ INTRODUCTION According to the standard model of particle physics, if cosmic rays are primarily composed of protons, there should be a bound on the maximum energy of the cosmic rays coming from distances greater than 50. This bound was first established in Refs. <cit.>, and a more accurate estimation was reported later in <cit.> giving 100, and it is known as the GZK limit. In spite of this, UHECR with energies above the GZK limit have been detected from places where apparently there are no nearby sources <cit.>. It seems therefore, that there is a missing piece in our understanding of the sourcing, nature, and/or propagation of the cosmic rays. If UHECR events present a small-scale clustering, their sources could be considered as point-like at cosmological scales <cit.>, and it has been suggested—based on coincidence of arrival direction—that certain astrophysical object could act as sources of some of the highest energy events <cit.>. Nonetheless, these sources are at red-shift z>0.1,[Equivalent to a comoving radial distance of 421.3.] exceeding the GZK horizon (R_gzk≈100). Indeed, the Pierre Auger Collaborationhas recently reported an anisotropy in the arrival directions of the UHECRs with morethan 5.2σ of significance <cit.>, supporting the hypothesisof their extragalactic origin. This would mean that it is difficult for primary ultra high energy particles to be protons <cit.>, since for energies around e20, the attenuation length is approximately R_gzk due to the interaction with extragalactic radio background. On the other hand, the radio background can be simulated by means of numerical propagation codes <cit.>, and they show that it is very unlikely for UHECR to be photons <cit.>.Within the SM, the only particle that can reach our galaxy without (significant) loss of energy are neutrinos. Therefore,different scenarios modeling UHECR by neutrinos have been proposed. In one of them, neutrinos produce nucleons and photons via resonant Z-production with relic neutrinos clustered within approximately 50 from the Earth, giving rise to angular correlations with high redshift sources <cit.>. However, in order for this model to be compatible with experimental data,a huge neutrino flux should be produced at the source, along with aclustering of relic neutrinos <cit.>. Other models consider extradimensional scenarios, where an enhancementof high-energy neutrino-nucleon cross sections can be produced by the exchange of Kaluza–Klein graviton modes <cit.>, or by an exponential increase of the number of degrees of freedom in string theory models <cit.>. Furthermore, if Kaluza–Klein axions areconsidered, their oscillation into photons allows the latter to travellarge distances without interacting with the CMB, producing in principle the UHECRs events above the GZK limit <cit.>. A thirdscenario allowing to circumvent this limit is the Lorentz invariance violation <cit.>, which is already constrained by astrophysical experiments (see e.g. <cit.>) and can be further tested with global cosmic-ray analyses, as proposed by Cosmic Rays Extremely Distributed Observatory (CREDO) Collaboration <cit.>, e.g. by considering cascades produced by primary ultra high energy photons in the photon decay scenario (see <cit.> and references therein). It is worthwhile to note that the latter model implies non-observation of single UHE photons on Earth, as their lifetimes would be extremely short, of the order of 1 second (see e.g. <cit.> for a review]). Another interesting way around to the GZK cutoff, is the possibility thatUHECR are composed mainly of particles beyond the standard model. In order for these particles to be acceptable candidates of UHECR, they should: (i) to be stable enough to reach the Earth from cosmological distances; (ii) interact very weakly with the CMB and extra galactic magnetic fields, so that they do not lose much energy; (iii) be produced with a significant flux at the source; and (iv) interact sufficiently strong in the near galaxy, with the Sun or Earth magnetic field,or with the Earth atmosphere.Axions, for instance, could be considered as candidates to avoid the GZK cutoff <cit.>. Nevertheless, it has been shown that it is unlikely that the axion production, together with their convertion to photons by the galactic magnetic field, accounts for UHECR within the present exclusion limits <cit.>. Another possibility is to consider particles with similar features of axions, such as axion-like particles. This case has been studied in Refs. <cit.>, constraining their parameter space according to current experimental data.The aim of this paper is to consider two beyond the standard model particles, such as axion-like particles and dark photons, as candidates for UHECRs by consideringthat they can traverse the visible Universe without decaying, producing the detected events on Earth. We present an update on the parameter space of axion-like particles in light of the new experimentalresults <cit.> and,for the first time, we present constraints on the parameter space of dark photons characterized by their mass and mixing with ordinary photons.The paper is structured as follows: In Sec. <ref>, we review a model of dark photons acting as source of UHECR. In Sec. <ref>, we review a few models of axions and axion-like particles. Next, in Sec. <ref> we analyse the astrophysical implications of considering the mentioned particles as sources of UHECR, constraining the values of the coupling constants. Finally, we present our concluding remarks in Sec. <ref>. § DARK PHOTONS The astronomical evidence of matter distribution in galactic rotation curves, mass-to-light ratios due to gravitational lensing, measurements of the CMB anisotropy, among others, suggest the existence of either non-standard matter fields, dubbed dark matter <cit.>, modified gravitational dynamics <cit.>,or both. Due to the lack of direct evidence, it have not been possible todiscriminate between the responsibles for these phenomena. In this section, we shall consider the case where dark matter might explain the aforementionedexperimental data.A way to understand the origin of dark matter, is provided by theoretical models which introduce the concept of dark sector, consisting of singlet fields under the standard model gauge group, but transforming nontrivially under a dark gauge group. In general,there is no constraint on the size of this dark sector in comparison with the visible one. Then, the experimentalsensitivity depends on the coupling and mass scale. In order to explain the absence of these ultra high energy particles within the standard model, we focus on a model including an additional U(1) gauge group <cit.>, whose corresponding gauge boson can aquire its mass through either Stückelberg or Higgs mechanism.It is assumed that the interaction between standard model particles and dark matter will be mediated solely by a new Abelian U'(1) gauge boson, A^', dubbed dark photon, mixed with ordinary photons γ. The interaction between γ and A^' is given by the kinetic mixing <cit.>ℒ_int=-1/2 g_dp F_μν A^'μν,where F^μν and A^'μν are field strength of the ordinary and dark photon, respectively, and g_dp is their mixing parameter. It is important to note that g_dp could be energy-scale dependent (see Ref. <cit.> and references therein). The kinetic mixing (<ref>) induce a γ - A^' oscillation, similar to the case of massive neutrinos <cit.>. Although dark photon interactions have not been detected in experimental searches <cit.>, it does not exclude that—provided their existence—they could be relevant in higher energy scales. Thus, any source of photons could produce a kinematically allowed massive A^' state, in accordance with the mixings. Within the heavy dark matter frameworks, processes as shown in Fig. <ref> (dark matter annihilation) can produce UHECR which energies reaching the scale of heavy dark matter masses M_dm, in the range between e2e19, depending on the dark matter model (see for example Refs. <cit.>). Assuming the existence of dark photons, these oscillate into photons (and vice-versa) with an efficiency driven by their mass m_A^' and the mixing coupling g_dp. Therefore, dark photons can decay into secondary lepton pairs through A^'→ℓ^+ +ℓ^-,providing a way to compare with the collected data of future experiments.Thus, it would be interesting to analyse the correlations between anisotropy ofUHECR sources and regions with high dark matter densities. If such correlationis found, it would support the model of dark photons.The partial decay width of the dark photon (with mass m_ A^'≥ 2 m_ℓ) into a secondary lepton pair is given by <cit.>Γ(A^'→ℓ^+ℓ^-) = α_em3m_ A^' g_dp^2 √(1-4 m_ℓ^2m_A^'^2)(1 + 2 m_ℓ^2m_A^'^2)where m_ℓ is the lepton mass and α_em is the fine-structure constant of quantum electrodynamics. Assuming the dark matter mass to be much larger than dark photon one, implies the latter to be highly boosted after the dark matter annihilation. Thus, thedecay angle ϕ, schematically represented in Fig. <ref>, is givenby <cit.>ϕ≈√(m_A'^2 - 4m_ℓ^2)/m_dm,where m_dm is the dark matter mass, the assumption , and from now on we use that m_ℓ = m_e ≈ 0.511 MeV. § AXIONS AND AXION-LIKE PARTICLES The quark sector of the standard model possesses two different sources of charge-parity (CP) symmetry violation: (i) the Cabibbo–Kobayashi–Maskawa matrix M, arising from the electroweak symmetry breaking; and (ii) the nontrivial structure of the quantum chromodynamics (QCD) vacuum, known as the θ-vacuum <cit.>. The Cabibbo–Kobayashi–Maskawa matrix is generically endowed with a complex phase, and their diagonalization induces a redefined parameter, θ̅ = θ +M, which encodes these two contributions of CP violation within the quark sector. The effective Lagrangian which accounts for these phenomena is given by_eff = _QCD + θ̅α_s/4π[G_μνG̃^μν],where G̃^μν≡12ϵ^μνρσG_ρσ, α_s = g^2_s/4π is the coupling constant of QCD, and the trace is taken over group indices. The last term in Eq. (<ref>) is the Pontryagin density for the SU(3) group, and it is known as the θ̅-term of QCD. This term is a topological invariant which can be locally written as a boundary term, adding no dynamics to the field equations. At the quantum level, however, it contributes to CP-odd observables such as the neutron electric dipole moment <cit.>. Its experimental value has been highly constrained, giving an upper limit of |θ̅|<10^-10 <cit.>. The strong CP problem is known as the lack of explanation for the smallness of θ̅, in order to fit its experimental value.One of the most popular solutions to this problem was proposed by Peccei and Quinn, by introducing an additional global axial symmetry to the standard model Lagrangian <cit.>. If this symmetry were exact, one would be able to rotate the θ̅-parameter away. However, it is clear that this symmetry cannot remain unbroken. Peccei and Quinn showed that one is still able to rotate the θ̅-term away if such a symmetry is spontaneously broken, while the (pseudo-)Nambu–Goldstone boson associated with the breakdown of such a symmetry, the axion <cit.>, replaces the θ̅-parameter by a dynamical field, i.e., θ̅→a(x)f_a, where f_a is the scale of the axial symmetry breaking. Nonperturbative effects of QCD generate a potential for the axion, which selects a vacuum expectation value that cancels the θ̅-angle exactly, solving the strong CP problem dynamically.[This potential also generates a mass for the axion, determined solely by the scale of symmetry breaking <cit.>.] Although the original proposal has been ruled out by experimental data <cit.>, extensions to this model have been proposed by considering a higher scale of symmetry breaking, causing that the mass of these axions is rather small <cit.>. These generalizations belong to the so-called invisible axion models.There exist models which predict pseudoscalar particles with similar features as the QCD axion, and they have been collectively called axion-like particles (ALPs).[The difference between models involving axions and ALPs is their number of free parameters: while the former is determined only by the scale of symmetry breaking, the latter is determined by their mass and characteristic energy scale as independent quantities.] Among these models, we can mention: models with extra dimensions <cit.>, two-Higgs-doublet models <cit.>, majorons <cit.>, relaxions <cit.>, familons <cit.>, gravitationally-induced axions <cit.>, etc. Although either axions and ALPs couple to the electromagnetic Pontryagin density, a crucial difference between them arises by considering their coupling to the gluon Pontryagin density: while axions do couple to the latter, ALPs do not.The characteristic Lagrangian of models involving ALPs has the following form_alp = 1/2∂_μ a ∂^μ a - 1/2m_a^2 a^2 - g_alp/4 aF_μνF̃^μν,where m_a is the mass of the ALP, g_alp is a model-dependent ALP-photon coupling, F_μν and F̃_μν are the electromagnetic field strength and its dual, respectively. The interaction between ALPs and photons provides a decay channel for , which plays a key role in experimental searches. To leading order, the decay width for this process is given by <cit.>Γ(a→γγ) = g_alp^2 m_a^3/64 π. Axions and ALPs can be converted into photons (and vice-versa) through the Primakoff effect <cit.>, when a strong external electromagnetic field is present (see Fig. <ref>). This process might induce an ALP-photon oscillation as well, similar to the case of massive neutrinos <cit.>. This interactionchanges the polarization of photons traveling in external magnetic fields, providing an additional mechanism in order to detect these pseudoscalar particles <cit.>. The ALP-photon conversioncould cause an apparent dimming of distant sources as well, affecting theluminosity-redshift relation of Ia supernovae, the dispersion of quasar spectra, and the spectrum of the CMB <cit.>. Constraints on these models have been collected in Fig. <ref>. § ASTROPHYSICAL IMPLICATIONS As we aforementioned, current experimental data suggests an extragalactic origin for UHECRs with energies above the GZK limit <cit.>. Here, we consider particles beyond the stantard model, such as dark photons and ALPs, as candidates for producingthese ultra high energy events. These particles oscillate into ordinary photons and vice-versa, allowing them to travel across the Universe essentially without decaying. The decay length of a particleis given byL = E_p/Γ_p m_p,where E_p, Γ_p, and m_p are the energy, decay width and mass of the particle p, respectively.If we require that the decay length hasto be at least of the order of magnitude of the observed universe R_U as considered in Ref. <cit.>, one finds the following conditionR_U ≲ L ≡E_p/Γ_p m_p⇒Γ_p≲E_p/R_Um_p.By means of Eq. (<ref>), Eq. (<ref>) and Eq. (<ref>) it is possible to establish a restriction on the parameter space in order for the ALPs and DPs to reach the Earth from distances beyond the R_gzk radius. These restrictions are: * For dark photons:g_dp≲(3E_A^'/α_emR_Um_A^'^2( 1+2 m_ℓ^2/m_A^'^2) √(1-4 m_ℓ^2/m_A^'^2))^1/2.* For ALPs:g_alp≲(64 π E_a/R_U m_a^4)^1/2.In addition to the limits imposed by Eq. (<ref>) for models involving dark photons, their coupling and mass are constrained by Big Bang Nucleosynthesis and the physics of the CMB, as shown in Fig. <ref>. It is manifest that these limits do not alter the cosmological evolution of the Universe. Experimental bounds have been found by considering direct detection <cit.>, colliders and fixed-target experiments <cit.>, indirect detection <cit.>, and Supernovae data <cit.>. These bounds are consistent with the ones found in the present analysis. Furthermore, dark photons could also be produced through dark matter annihilation at the center of the Earth and Sun, and may be detected by IceCube, or Alpha Magnetic Spectrometer (AMS-02) <cit.>. As shown in Ref. <cit.>, the sensitivity of AMS-02 allows to search for dark photons with m_A'∼100 and . This region in the parameter space is excluded in Fig. <ref>. Thus, dark photons as candidates for UHECRs cannot be detected by AMS-02 within its present sensitivity. In the case of ALPs, the limits imposed by Eq. (<ref>) are tighter than those of Ref. <cit.>, allowing to constrain a region in the parameter space that was not excluded in previous analysis <cit.>. In addition to the present exclusion limits imposed on ALPs models, their parameter space have been constrained by experimental data coming from “light shining through a wall” technique used in the GammeV experiment <cit.>, gamma rays data from H.E.S.S observations <cit.>, helioscope technique used by the CAST collaboration <cit.>, cosmological data <cit.>, among others, which have been collected in Fig. <ref>. It is worth mentioning that, although the bounds presented here do not reach the sensitivity of current experiments, such as CAST, nor future ones like ALPS-II and IAXO, they still allow for detecting ALPs with lower energies by means of gamma ray telescopes as proposed in Ref. <cit.>.By considering the arclength described by the ultra high energy lepton pair produced through the dark photon decay, to be less than the Earth diameter, i.e., D_E ϕ≤ 2 R_⊕,where R_⊕≃6370 is the radius of the Earth, we find D_E ≤2 R_⊕ m_dm/√(m_A'^2 - 4m_ℓ^2).It is worth noticing that if the DP decay occurs inside of a region with radius given by Eq. (<ref>), the two final leptons can reach the Earth and may produce two super pre-showers correlated in time and space. These signals can be tested in future cosmic rays experiments, as we will discuss in Sec. <ref>. § DISCUSSIONIn this paper, we have discussed scenarios where UHECRs are produced by beyond the standard model particles, namely, dark photons andaxion-like particles. It is known that these particles must satisfy some conditions in order to produce the UHECRs events: * long lived, in order to travel cosmological distances,* weakly interacting with radiation,* produced significantly at the source,and, in addition, they have to interact strongly enough near the Earth. Under these assumptions and criteria explainedin Sec. <ref>, we constrained the parameterspace of the proposed scenarios, enhancing the present exclusion limitsby considering these particles as responsibles for the UHECRs.Although the constraints imposed by observations in the ALP models are much tighter than former estimations (see Ref. <cit.>), it can be seen in Fig. <ref> that our constraints cannot match the sensitivity of neither CAST 2017. However, the limits presentedhere exclude a new parameter region in which both m_a and g_alp are larger in comparison with the previous analysis of Ref. <cit.>.On the other hand, the assumption that UHECRs are produced by dark photons decays, excludes a large region of the parameter space, providing limits as satisfactory as those imposed by Big Bang Nucleosynthesis. Furthermore, the region below g_dp< 10^-15can only be constrained using data from the CMB. It is worth mentioning thatthe imposed constraints are compatible with the sensitivity of several current experiments <cit.>.It is possible to find place for improvement in the UHECR research during the next decade, by considering the proposal of organizing the existing professional detectors together with smart devices—such as mobiles or tablets—as a network capable of global monitoring and analysis of muons coming from showers produced by primary cosmic rays in the atmosphere. A wide spatial distribution of the devices contributing to such a network will help to detect and study possibly correlated cosmic ray events, known as ensembles of cosmic rays <cit.>. These ensembles might be composed of widely distributed events spanning even the whole cosmic-ray energy spectrum which might be observable only by widely spread and possibly dense network of detectors, while even the largest individual observatories might fail to give a trigger. The involvement of the users of smart devices will increase the collective surface of the whole network only mildly, but their geographical spread will significantly increase the capability of the network to observe and analyze possibly existing large scale correlations in the cosmic ray data, as we may expect for dark photons decaying inside the region showed in Fig. <ref>. The proposal of a global cosmic-ray network is being implemented by the Cosmic Rays Extremely Distributed Observatory (CREDO) Collaboration <cit.>. It should be also noted that there are already three smartphone applications enabling the cosmic-ray detection mode (i) Distributed Electronic Cosmic-ray Observatory (DECO) <cit.>; (ii) Cosmic RAYs Found In Smartphones (CRAYFIS) <cit.>; and (iii) CREDO Detector <cit.>. The cascade approach proposed by CREDO will help to probe both classical and exotic scenarios, whenever cascades of particles/photons are initiated above the atmosphere, it will also be an instrument prepared to detect theoretically unexpected manifestations of New Physics which can be observed as zero-background events of correlated excesses of cosmic-ray rates recorded by distant detectors <cit.>.Recently, the DES collaboration has released their results from the first year of data, including an analysis of the weak lensing mass map with sources at redshift .2 < z < 1.3 <cit.>, providing for the first time an astronomical survey of possible dark matter clusters. This could be usefulto analyse the correlation between the location of UHECR sources and the anisotropies in the dark matter distribution, which might used to test the hypothesis that primary UHECRs are composed by dark matter and/or that they are generated by the decay of super-heavy dark matter. The authors thanks to W. Bietenholz, Y. Bonder and S. Troitsky for critical remarks. O.C-F. would like to thank to the Physics Department ofUniversidad de La Serena for the hospitality during the (partial) development of this paper. The work of O.C-F is supportedby the projects PAI-79140040 and FS0821 (CONICYT–Chile), C.C. by UNAM-DGAPA-PAPIIT Grant IA101116 and UNAM-DGAPA postdoctoral fellowship (México), J.Z-S by theGrant Becas Chile No. 74160012, CONICYT, and P.H. by the National Science Centre (Poland) grant No. 2016/23/B/ST9/01635. | http://arxiv.org/abs/1709.09144v2 | {
"authors": [
"Oscar Castillo-Felisola",
"Cristobal Corral",
"Piotr Homola",
"Jilberto Zamora-Saa"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170926173017",
"title": "Limits on beyond standard model messengers as ultra high energy cosmic rays"
} |
[email protected]^1Institute of Nuclear Sciences, Hacettepe University, 06800, Ankara, Turkey ^2Department of Physics, Middle East Technical University, 06800, Ankara, Turkey ^3Department of Applied Physics, Atilim University, 06836, Ankara, Turkey^4The Center for Solar Energy Research and Applications (GUNAM), Middle East Technical University, 06800, Ankara, Turkey^5Micro and Nanotechnology Program of Graduate School of Natural and Applied Sciences, Middle East Technical University, 06800, Ankara, Turkey Plasmonic nanostructures enhance nonlinear response, such as surface enhanced Raman scattering (SERS), by localizing the incident field into hot spots. The localized hot spot field can be enhanced even further when linear Fano resonances (FR) take place in a double resonance scheme. However, hot spot enhancement is limited with the modification of the vibrational modes, the break-down of the molecule and the tunnelling regime. Here, we present a method which can circumvent these limitations. Our analytical model and solutions of 3D Maxwell equations show that: enhancement due to the localized field can be multiplied by a factor of 10^2 to 10^3. Moreover, this can be performed without increasing the hot spot intensity which also avoids the modification of the Raman modes. Unlike linear Fano resonances, we create a path interference in the nonlinear response. We demonstrate on a single equation that enhancement takes place due to cancellation of the contributing terms in the denominator of theSERS response. Fano enhancement of SERS signal without increasing the hot spot intensity Mehmet Emre Tasgin^1 December 30, 2023 =========================================================================§ INTRODUCTION Metal nanoparticles (MNPs) confine incident electromagnetic field into nm-size hot spots as plasmonic oscillations. Field intensity at the hot spots can be 5 orders of magnitude larger compared to the incident one <cit.>. It is also reported that self-repeating cascaded materials can confine light even better compared to the gaps between MNPs <cit.>. Intense fields give rise to appearance of nonlinear processes such as second harmonic generation (SHG), four wave mixing (FWM), and surface enhanced Raman scattering (SERS) <cit.>. Actually, enhancement is squared since the field of the converted frequency is also localized <cit.>. In an efficient conversion, nonlinear process takes place between plasmonic excitations of different frequencies due to the localization <cit.>. Recent studies show that MNPs with plasmon resonances at both excitation and Stokes frequencies (double resonance) provide better enhancement factors for Raman intensities <cit.>.Hot spots also provide enhanced light-matter interaction. When a quantum emitter (QE) is placed into a hot spot, localized plasmon field interacts strongly with the QE. Small decay rate of the QE creates Fano resonances, a dip in the plasmonic spectrum <cit.>. In this process, the localized plasmon field provides the weak hybridization. Fano resonance also appears when excited plasmon mode couples to a long-live dark plasmon mode <cit.>.Fano resonances can extend the lifetime of plasmon excitations <cit.> which makes the operation of coherent plasmon emission (spaser) possible <cit.>. They also lead to further enhancement of the localized hot spot field <cit.>. This extra enhancement in the hot spot field is cleverly adopted for the enhancement of the nonlinear response in FWM <cit.> and SERS <cit.>. Similar to double resonance scheme <cit.>, both the excited and Stokes shifted frequencies are aligned with two Fano resonances <cit.>. The double Fano resonance scheme provides much stronger enhancement in the SERS signal. Fano resonances are also shown to provide control over other nonlinear processes such as SHG <cit.>, third harmonic generation <cit.>, and FWM <cit.>.SERS is a very useful imaging technique. It provides information about the chemical composition of newly synthesized molecules by determining the existing bond types. Single-molecule detection via SERS is studied in many fields of science, including chemistry <cit.>, nanobiology <cit.>, tumor targeting and cancer applications <cit.>. Even more, mapping of inner structure and surface configuration of a single molecule is achieved recently <cit.> using a double resonance scheme <cit.>. Such an imaging requires very intense fields at the nm-size hot spots. When the imaging tip gets closer to the metal surface, the intensity at the hot spot –where the molecule lies– increases. If the hot spot intensity is increased further, e.g. via a double Fano resonance scheme <cit.>, fragile molecules can be damaged <cit.>. It is also experimented that vibrational modes of a Raman-imaged nanostructure (i.e. a carbon nanotube) can be modified due to the close spacing of the tip <cit.>. Additionally, electron tunnelling can limit the intensity enhancement in the gaps <cit.>.In this manuscript, we study the SERS signal from a double resonance system. A Raman reporter molecule is placed close to the gap of a MNP dimer, see Fig. <ref>. We additionally place an auxiliary QE (e.g. a molecule or a nitrogen vacancy centre) to the other side of the gap. 3D solutions of Maxwell equations show that SERS can be enhanced by a factor of 10^3 without increasing the field intensities at the excited and the Stokes-shifted hot spots. This enhancement multiplies the enhancement due to localization. On a basic analytical model, we demonstrate the underlying reason for the enhancement. Coupling of the auxiliary QE with the Stokes-shifted plasmon mode modifies frequency conversion paths dramatically. It yields a cancellation in the denominator of the SERS response, i.e. Eq. (<ref>). 3D simulations show that enhancement predicted by the analytical model, also appears in the presence of retardation effects. The presented phenomenon can be adopted to further increase the efficiency of SERS imaging for systems which are already operating in the break-down or tunnelling regimes. Better signal intensities with larger tip-surface spacing or with smaller laser intensities can be achieved by avoiding modifications in the Raman vibrational modes.Our problem setting, which involves configuration of a MNP dimer coupled to a Raman reporter molecule and an auxiliary QE, can be implemented controllably using several nanotechnological methods such as e-beam lithography <cit.> or DNA based biomolecular recognition <cit.> that provide ultimate nanoscale spatial control <cit.>. One can also conduct an experiment based on the stochastic distributions of many molecules <cit.>. A practical implementation would be the following. A gold coated AFM tip decorated (can also be considered as contamination) with carefully chosen auxiliary molecules (QEs) as shown in Fig. <ref>, using a technique reminiscent of dip-pen lithography, will produce more intense SERS signal without increasing near-field intensity. In spasers <cit.>, where MNPs are surrounded by molecules, linear Fano resonance increases the plasmon lifetime and fluorescence intensity of the molecules <cit.>. Fano resonances can also be adopted in an all-plasmonic setting <cit.>. In the following, we first present the basic analytical model from which we anticipate the presence of the enhancement. We introduce the effective Hamiltonian for a double resonance SERS system coupled with an auxiliary QE. We obtain the equations of motion (EOM) using Heisenberg equations. We manage to obtain a simple expression for the steady-state of the Stokes field amplitude, Eq. (<ref>). On this expression we explain why such an enhancement should emerge. Next, we perform simulations of the exact solutions of the 3D Maxwell equations in order to test the retardation effects. In this case, the spectrum for which enhancement appears narrows down compared to the analytical result. Even so, an enhancement of 3 orders of magnitude on top of the localization can be observed.§ HAMILTONIAN AND EQUATION OF MOTION We consider a double resonance scheme with two plasmon bands, â and â_ R, with resonances Λ=c/Ω=532 nmand Λ_ R=c/Ω_ R=780 nm respectively, see Fig. <ref>. A strong incident laser field, λ_ L=c/ω=593 nm, excites the plasmon polaritons in the â-mode. The substantial overlap between the hot spots of the two modes, â and â_ R, and the Raman reporter molecule yields a significant overlap integral χ for the Stokes Raman process. Hence, a plasmon in the excited â mode generates a Stokes shifted plasmon polariton with c/ω_ R=λ_ R=700 nm in the lower energy mode â_ R <cit.>.When an auxiliary QE is inserted in the system, it also interacts strongly with the hot spot of â_ R-mode, into which the nonlinear conversion takes place. Level spacing of the QE, ω_eg, is chosen about Ω_ R. Hamiltonian for such a system, including the Raman conversion, can be written as the sum of the terms Ĥ_0+Ĥ_ QE+Ĥ_ L+Ĥ_ int+Ĥ_ R, withĤ_0 = ħΩâ^†â+ħΩ_ Râ_ R^†â_ R+ħΩ_ phâ_ ph^†â_ ph Ĥ_ QE = ħω_eg|e⟩⟨e| Ĥ_ L = iħ(â^†ε e^-iω t-âε^*e^iω t), Ĥ_ int = ħ(fâ_ R|e⟩⟨g|+f^*â_ R^†|g⟩⟨e|),Ĥ_ R = ħχ(â_ R^†â_ ph^†â+â^†â_ phâ_ R),where Ĥ_0 includes the energies for the driven â, and Raman shifted â_ R plasmon modes as well as the molecular vibrations, â_ ph. Ĥ_ QE is the energy of the auxiliary QE. Ĥ_ L is the laser pump, Ĥ_ R denotes the Raman process and Ĥ_ int is the interaction of the Stokes-shifted plasmon polaritons of â_ R-mode with the auxiliary QE. χ determines the strength of the Raman process, while ε denotes the power of the incident laser source. Here, we do not consider the anti-Stokes shift in the Hamiltonian to simplify our results, however, we have verified that the enhancement values and the spectroscopic behaviour of the system remain similar in such a case. The interaction strength between the auxiliary QE and â_ R-mode is denoted by f. Coupling of the auxiliary QE to â-mode is not considered due to far-off-resonance and simplicity. |g⟩ and |e⟩ represents the ground and excited states for the auxiliary QE. Ĥ_ R is a standard Hamiltonian for a Raman process, described, for instance, in the Refs. <cit.>. A similar form of Ĥ_ R could have also been derived <cit.> from a radiation pressure like interaction <cit.>. We obtain the dynamics via Heisenberg equations, iħâ̇=[â, Ĥ]. We note that, since we do not consider the quantum optical effects, we are able to replace the operators with complex numbers <cit.>; â→α, â_ R→α_ R, â_ ph→α_ ph, ρ̂_eg=|e⟩⟨g|→ ρ_eg. We find the EOM as α̇_ R = (-iΩ_ R-γ_ R)α_ R-iχα_ ph^*α-if^*ρ_ge,α̇ = (-iΩ-γ)α-iχα_ phα_ R+ε e^-iω t,α̇_ ph = (-iΩ_ ph-γ_ ph)α_ ph-iχα_ R^*α+ε_ phe^-iω_ pht,ρ̇_eg = (-iω_eg-γ_eg)ρ_eg+ifα_ R(ρ_ee-ρ_gg),ρ̇_ee = -γ_eeρ_ee+if^*α^*_ Rρ_eg-ifα_ Rρ^*_eg, where we introduce the damping rates γ, γ_ R, γ_ ph, γ_eg, and γ_ee. We also have the constraint ρ_ee+ρ_gg=1. ε_ ph is introduced for the vibrations, due to the finite ambient temperature <cit.>. Its actual value has no influence in the relative enhancement/suppression factors.In the steady-state, solutions are in the form α_ R(t) = α̃_ Re^-iω_ R t, α(t) = α̃e^-iω t, α_ ph(t) = α̃_ phe^-iω_ ph t, ρ_ eg(t) = ρ̃_ ege^-iω_ R t, ρ_ ee(t) = ρ̃_ ee, where exponentials cancel in each equation, Eqs. (<ref>)-(<ref>). In other words, this is the energy conservation in the long term limit. Eqs. (<ref>)-(<ref>) become[i(Ω_ R-ω_ R)+γ_ R]α̃_ R = -iχα̃_ ph^*α̃-if^*ρ̃_ eg, [i(Ω-ω)+γ]α̃ = -iχα̃_ phα̃_ R+ε, [i(Ω_ ph-ω_ ph)+γ_ ph]α̃_ ph = -iχα̃_ R^*α̃+ε_ ph, [i(ω_eg-ω_ R)+γ_eg]ρ̃_ eg =ifα̃_ R(ρ̃_ ee-ρ̃_ gg),γ_ eeρ̃_ ee =-ifα̃_ Rρ̃^*_ eg+if^*α̃_ R^*ρ̃_ eg. We can obtain a simple expression for the Stokes-shifted plasmon amplitude (SERS signal) by using Eqs. (<ref>) and (<ref>) 0.88!α̃_ R=-iχε_ ph^*/β_ ph^*([i(Ω_ R-ω_ R)+γ_ R]-|f|^2y/[i(ω_eg-ω_ R)+γ_eg])-|χ|^2|α̃|^2α̃, where β_ ph=[i(Ω_ ph-ω_ ph)+γ_ ph]. Here y=ρ_ee-ρ_gg is the population inversion for the auxiliary QE. |χ|^2|α̃|^2 term is small compared to other ones in the denominator and hence, can be neglected.We use Eq. (<ref>) merely to anticipate the enhancement/suppression effects. All the presented results are obtained by numerical time evolution of Eqs. (<ref>)-(<ref>).Enhancement. A quick examination of the denominator of Eq. (<ref>) reveals that for the proper choice of ω_eg, nonresonant term (Ω_ R-ω_ R) in the denominator can be cancelled with the term containing f, the MNP-QE coupling. This condition isω_eg^*=ω_ R+|f|^2y/2(Ω_ R-ω_ R)-√(|f|^4|y|^2/4(Ω_ R-ω_ R)^2-γ_ eg^2). This choice for the level spacing enables us to minimize the denominator, consequently enhancing the Raman signal amplitude. This type of enhancement does not necessitate an arrangement in the inner structure of plasmon modes.To examine the dependence of the enhancement with respect to the level spacing (λ_eg=c/ω_eg), we time evolve the EOM (<ref>)-(<ref>). The parameters are chosen as γ=0.01ω, γ_ R=0.005ω, γ_ ph=0.001ω. Nevertheless, one can realize that Ω_ph and γ_ph play no role in the cancellation of the denominator in Eq. (<ref>). The damping rate (spectral width) of the auxiliary QE is taken to be γ_eg=10^-5ω. Here, the frequency of the incident light (ω) is related to λ_ L as ω=c/λ_ L=593 nm. χ is assumed a small value 10^-5ω, where it is verified that the value of χ does not affect the enhancement factors, and ε=0.1ω. f is also varied in order to explore the effect of the coupling in the MNP-QE system. The enhancement factor is calculated with respect to the |α_ R|^2 intensity for f=0.The results are depicted in Fig. <ref>(b), where enhancement factors of ≈300 are observed. As suggested by Eq. (<ref>), Ω_ R<ω_ R, cancellation (enhancement) takes place for longer wavelengths as MNP-QE coupling, f, increases. The spectral position of λ_eg^*=c/ω_eg^* also justifies our assumption for off-resonant â-QE coupling. If Eq. (<ref>) is examined, it can be realized that the amount of enhancement can be increased by introducing more interference paths via additional QEs <cit.> or additional plasmon conversion modes. Eq. (<ref>) is a single and simple equation which enables us to predict possible interference effects without including the complications emerging in 3D simulations. Before moving forward, we underline that our aim is to present a simple understanding for the enhancement process, without getting lost in details.Linear Fano resonances, commonly referred in the literature, appear if one of the two coupled oscillators has longer lifetime <cit.>. Here, interference of the nonlinear frequency conversion paths demonstrates us an interesting incident. Even when the spectral width (damping rate) of the auxiliary object is equal to the damping rate of the MNP hot spot, 25 times enhancement can emerge due to cancellation in the denominator of Eq. (<ref>). The presented enhancement factor is obtained for a plasmon mode of fair quality γ=0.01ω. When a higher quality MNP <cit.> is used available enhancement factor grows up.§ 3-DIMENSIONAL SIMULATIONSWe also perform simulations with the exact solutions of 3D Maxwell equations and use the setup in Fig. <ref>. We note in advance that we do not aim a one to one comparison between the analytical solutions and the 3D simulations. We aim to observe if the retardation effects wipe-out the enhancement phenomenon predicted by our basic analytical model. Making a one to one comparison between the theoretical findings and the 3D simulations, which is a very sophisticated process, is out of the scope of this work. In Fig. <ref>(a), we present a nano dimer with two gold spheres of radii 90 nm and 55 nm, whose linear response is depicted in Fig. <ref>(b). The dimer supports two plasmon modes at Λ=c/Ω=530 nm and Λ_ R=c/Ω_ R=780 nm. â-mode is driven by a strong laser of wavelength λ_ L=593 nm. We place a Raman reporter molecule with radius of 4 nm <cit.> (blue) close to the hot spot of the MNP dimer. For a “proof-of-principle” demonstration, we consider a single vibrational mode, ν=2600 cm^-1 <cit.>, for the Raman reporter molecule. The Stokes signal appears at λ_ R=700 nm and couples to a_ R mode of the double-resonance scheme <cit.>. We model the auxiliary QE by a Lorentzian dielectric function ϵ(ω) <cit.> of resonance λ_eg=c/ω_eg and damping rate γ_eg. We compare the Raman intensities with the results where no auxiliary QE is present, i.e. enhancement factor. In Fig. <ref>(a), we also change the position of the auxiliary QE along the z-axis in order to alter the interaction with the MNP. When distance to the hot spot centre (z), increases, the interaction of the plasmon mode with the auxiliary QE, [f in Fig. <ref>(b)] decreases. We observe that enhancement occurs at larger wavelengths for stronger MNP-QE coupling as suggested by the basic model. Furthermore, maximum enhancement in Raman signal takes place around λ_eg^*≃834 nm, which is farther apart from the Stokes line λ_ R=700 nm. A linear Fano resonance would yield the strongest hot spot enhancement when λ_eg≃λ_ R <cit.>. λ_eg^* is in this regime both for our simple model and for 3D simulations. On the other hand, when auxiliary QE is positioned much closer to the dimer-centre, enhancement decreases 2 orders of magnitude <cit.>. In this regime, excitations cannot be modelled with the presented treatment: strong hybridization arises <cit.>. This is also observed in SHG process <cit.>. Even though many complications may arise in 3D solutions, e.g. coupling to dark modes in the MNP dimer, the results match “qualitatively” with our basic model as a “proof-of-principle” demonstration. Retardation effects allow Fano resonances to appear in a narrower band compared to our model, similar to Ref. <cit.>. We also note that our analytical model does not account for the change of density of states, i.e., Purcell factor. It can be seen even in the simple setting of Fig. <ref>, a setting which can be optimized by further elaboration, an average of a factor of 10^2 to 10^3 further enhancement factor is achieved at a QE distance of ±1 nm from the MNPs. It can be said that the hot spot field intensity within the 4 nm gap between MNPs is not a particularly strong one with respect to achievable hot spot enhancement factors of 10^5-10^6 at which an even larger distance between the QE and the MNPs could produce similar orders of further enhancement. § SUPPRESSIONOur model also predicts that SERS can be suppressed several orders of magnitude, Fig. <ref>, for the choice of the auxiliary QE, ω_eg=ω_ R. Simply, for this case, Fano resonance (transparency) prohibits the plasmon oscillations of the converted frequency ω_ R from emerging into the â_ R-mode. One can realize that path interference in the nonlinear response is actually not so different from the one taking place in the linear response <cit.>. That is, modification of the denominator both in the nonlinear <cit.> and the linear response <cit.> have a common form <cit.>.Similar silencing phenomenon is observed in the SHG experiments and in 3D simulations <cit.> and can be demonstrated with a simple analytical model <cit.>. Denominator of Eq. (<ref>) also shows why a suppression effect can take place similar to the one observed in SHG <cit.>. If one chooses ω_eg=ω_ R, the extra term becomes γ_eg^-1| f|^2 y. This term is very large since γ_eg^-1∼ 10^5 and f=0.1 in units scaled with the laser frequency ω (≈PHz). We stress that Figs. <ref>(b) and <ref> are generated through the exact time evolutions of Eqs. (<ref>)-(<ref>). That is, no approximation is used to obtain the results. On the other hand, the suppression phenomenon –neither in the SHG <cit.> nor in the Raman cases– cannot be demonstrated with the 3D simulations of Ref. <cit.>. This is simply because 3D simulation method <cit.> is only a first-order approach. Demonstration of the suppression phenomenon necessitates the self-consistent solution of Maxwell equations, as in Eqs. (<ref>)-(<ref>). Self-consistent 3D simulation of a Raman process is a numerical art on its own. § SUMMARY AND DISCUSSIONSWe introduce a new method which can increase the SERS signal without increasing the hot spot intensities. In other words: SERS signal can be further multiplied by a factor of 10^2-10^3, on top of the hot spot formation by plasmon mediated field enhancement, without heating the Raman reporter molecule further. This is different than linear Fano resonances which enhance the hot spot field <cit.>. The phenomenon takes place due to the modification of the Raman conversion paths, in the presence of an auxiliary QE. Both the 10^2-10^3 enhancement and the unvarying hot spot intensities are confirmed with 3D simulations. This phenomenon can be used not only to increase the Raman signal in materials already operating in the break-down or tunnelling regimes and to avoid the modifications of vibrational modes. But it can also be adopted for high spatial resolution imaging of molecules. Raman signal emerges from the region where the two plasmon modes overlap spatially. When this overlap area is kept small, better spatial resolution can be obtained. However, SERS process also weakens with reduced overlap integral. The suggested method can help in increasing the SERS signal to observable values again.The presented method is not physically intriguing only, but the model provides simple implementations, new phenomena and utilization of new enhancement tools. 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"authors": [
"Selen Postaci",
"Bilge Can Yildiz",
"Alpan Bek",
"Mehmet Emre Tasgin"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170926192340",
"title": "Fano enhancement of SERS signal without increasing the hot spot intensity"
} |
[table]capposition=top [figure]capposition=top§ STARTSECTIONSECTION1 @-2.5EX PLUS -0.5EX MINUS -0.1EX0.5EX PLUS 0.1EX §.§ startsectionsubsection2 @-2.25ex plus -0.3ex minus -0.2ex0.05ex plus 0.05ex§.§.§ startsectionsubsubsection3 @-2.25ex plus -0.3ex minus -0.2ex0.05ex plus 0.05ex assumptionAssumption lemmaLemma | http://arxiv.org/abs/1709.09115v1 | {
"authors": [
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"Xiaoxia Shi",
"Matthew Shum"
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"published": "20170926162452",
"title": "Inference on Estimators defined by Mathematical Programming"
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Image of the close companion in front of the primary VLTI-PIONIER imaging of V766 Cen and its close companion European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching bei München, Germany, [email protected] Dpt. Astronomia i Astrofísica, Universitat de València, C/Dr. Moliner 50, 46100, Burjassot, Spain Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía S/N, 18008, Granada, Spain Centro Astronómico Hispano Alemán, Calar Alto, (CSIC-MPG), Sierra de los Filabres, E-04550 Gergal, Spain Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,Lagrange, CS 34229, 06304 Nice Cedex 4, France Observatori Astronòmic, Universidad de València, 46980 Paterna, Spain European Southern Observatory, Casilla 19001, Santiago 19, Chile Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn,Germany South African Astronomical Observatory, PO Box 9, Observatory 7935,South Africa Astronomy Department, University of Cape Town, 7701, Rondebosch, South Africa National Institute for Theoretical Physics, Private Bag X1, Matieland, 7602,South Africa Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg,Germany The star V766 Cen (=HR 5171A) was originally classified as ayellow hypergiant but lately found to more likely be a 27–36 M_⊙red supergiant (RSG). Recent observations indicated a close eclipsing companionin the contact or common-envelope phase. Here, we aim at imaging observations of V766 Cento confirm the presence of the close companion.We used near-infrared H-band aperture synthesis imaging at threeepochs in 2014, 2016, and 2017, employing the PIONIER instrument at the Very Large Telescope Interferometer (VLTI). The visibility data indicate a mean Rosseland angular diameter of 4.1±0.8 mas, corresponding to a radius of 1575±400 R_⊙.The data show an extended shell (MOLsphere) of about 2.5 timesthe Rosseland diameter, which contributes about 30% of the H-band flux. The reconstructed images at the 2014 epoch show a complex elongated structure within the photospheric disk with a contrast of about 10%. The second and third epochs show qualitatively andquantitatively different structures with a single very brightand narrow feature and high contrasts of 20–30%. This feature is located toward the south-western limb of the photosphericstellar disk. We estimate an angular size of the featureof 1.7±0.3 mas, corresponding to a radius of650±150 R_⊙, and giving a radius ratio of 0.42^+0.35_-0.10compared to the primary stellar disk. We interpret the images at the 2016 and 2017 epochs as showing theclose companion, or a common envelope toward the companion,in front of the primary. At the 2014 epoch, the close companion is behindthe primary and not visible. Instead, the structure andcontrast at the 2014 epoch are typical of a single RSG harboring giantphotospheric convection cells. The companion is most likely acool giant or supergiant star with a mass of 5^+15_-3M_⊙. Multi-epoch VLTI-PIONIER imaging of the supergiant V766 Cen Based on observations made with the VLT Interferometer at Paranal Observatory under programme IDs 092.D-0096, 092.C-0312,and 097.D-0286. Olivier Chesneau was PI of the program 092.D-0096.He unfortunately passed away before seeing the results coming out of it. This letter may serve as a posthumous tribute to hisinspiring work on this source. M. Wittkowski1F. J. Abellán2B. Arroyo-Torres3,4A. Chiavassa5J. C. Guirado2,6J. M. Marcaide2A. Alberdi3W. J. de Wit7K.-H. Hofmann8A. Meilland5F. Millour5S. Mohamed9,10,11J. Sanchez-Bermudez12Received …; accepted … ========================================================================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTIONRed supergiants (RSGs) are cool evolved massive stars before their transition toward core-collapse supernovae (SNe). Their characterization and location in the Hertzsprung-Russell(HR) diagram are important to calibrate stellar evolutionary models of massive stars and to understand their further evolution towardSNe <cit.>.The majority of massive stars are members of binary systems with a preference for close pairs <cit.>. Binary interactions have profound implications forthe late stellar evolution of massive stars toward the different types of SNe and gamma-ray bursts (GRBs). For example, <cit.> and <cit.> argued that the progenitor of SN1987A was likely a blue supergiant that was a member of a close binary system, where the companion dissolved completely during a common-envelope phase when the primary was a RSG.The massive evolved star V766 Cen (=HR 5171 A) was originally classified as ayellow hypergiant <cit.>.It is known to have a wide B0 Ib companion at a separationof 9.7. The distance to V766 Cen is well establishedat 3.6±0.5 kpc <cit.>. <cit.> found evidence that the primary component itself (HR 5171 A) has an eclipsing close companion, most likely in a contact or common-envelope phase. Based on VLTI-AMBER spectro-interferometry,<cit.> reported that V766 Cen is a high-luminosity (log L/L_⊙= 5.8 ± 0.4) source of effectivetemperature 4290 ± 760 K and radius 1490 ± 540 R_⊙, located in the HR diagram close to both the Hayashi and Eddington limits.With this location and radius, it is more likelya RSG before evolving to a YHG, and consistent with a 40 M_⊙track of current mass 27–36 M_⊙. This mass is consistent with a system mass of 39^+40_-22M_⊙ and mass ratioq ≤ 10 by C14. Here, we present near-infrared H-band aperture synthesis images ofV766 Cen with the VLTI-PIONIER instrument at multiple epochs to detect the close companion by imaging, and to investigate the surface structure of the primary.§ OBSERVATIONS AND DATA REDUCTIONWe obtained observations of V766 Cenwith the PIONIER instrument <cit.> ofthe Very Large Telescope Interferometer (VLTI) and its four auxiliarytelescopes (ATs).We took data at three epochs with mean Julian Day2456719 (Feb–Mar 2016, duration 11 d), 2457528 (May–Jul 2016,55 d), and 2457839 (Feb-Apr 2017, 64 d). The durations correspond to 0.8%, 4.2%, and 4.9%, respectively, of theestimated 1304 d period of the close companion (C14). In 2014, the data were dispersed over three spectral channels with central wavelengths 1.59 μm, 1.68 μm, 1.77 μm and channel widths of ∼ 0.09 μm. In 2016 and 2017, the data were dispersed over six spectral channels withcentral wavelengths 1.53 μm, 1.58 μm, 1.63 μm, 1.68 μm,1.72 μm, 1.77 μm, and widths of ∼ 0.05 μm. Observations of V766 Cen were interleaved with observationsof interferometric calibrators.The calibrators were HD 122438 (spectral type K2 III, angular uniform disk diameter Θ_UD^H=1.23 ± 0.08 mas,used in 2014), HD 114837 (F6 V, 0.78 ± 0.06 mas, 2014), and HR 5241 (K0 III, 1.68 ± 0.11 mas, 2016 and 2017). The angular diameters are from the catalog by <cit.>. Table <ref> provides the log of our observations. Figure <ref> shows the uv coverages at each epoch, where u and v are the spatial coordinates of the baselines projected on sky. We reduced and calibrated the data with the pndrs package <cit.>. Figure <ref> shows all resulting visibility data of the three epochs together with a model curve as described in Sect. <ref>, and synthetic visibility valuesbased on aperture synthesis imaging as described in Sect. <ref>.§ DATA ANALYSISThe visibility data in Fig. <ref> indicate an overall spherical stellar disk. However, deviations from a continuously decreasing visibility in the first lobe and closure phases different from 0/180at higher spatial frequencies indicate sub-structure within the stellar disk.Changes in the closure phase dataamong the three epochs indicate variability of the structure with time.Previous observations <cit.> indicated the presence of an extended molecular layer, also calledMOLsphere <cit.>. We used a two-component model for the overall stellar disk, consisting of a PHOENIX model atmosphere <cit.> describing the stellar photosphere and auniform disk (UD) describing the MOLsphere.We chose a PHOENIX model from the grid of <cit.> with parameters close to the values from <cit.>: mass 20 M_⊙, effective temperature 3900 K, surface gravity log g=-0.5,and solar metallicity.The fit was performed in the same way as in <cit.>. We treated the flux fractionsf_Ross and f_UD both as free parameters to allow for an additional over-resolved background component. Figure <ref> shows our best-fit models compared to the measured data, showing that the model is successful in describing the visibility data in thefirst lobe. The contribution of the PHOENIX model alone is plotted toillustrate that a single-component model cannot reproduce themeasured shape of the visibility function. Table <ref> shows the best-fit parameters for each epoch. Differences among the three epochs may be caused by avariability of the overall source structure, or by systematic effects such as the sparse coverage of visibility points at low spatial frequencies at epoch I. We used the averaged values and their standard deviations as final fit results as listed in Tab. <ref>. Our value of the Rosseland angular diameter Θ_Rossof 4.1±0.8 mas is consistent with the estimates ofΘ_UD=3.4±0.2 mas by C14 and Θ_Ross=3.9±1.3 mas by <cit.>.§ APERTURE SYNTHESIS IMAGINGWe used the IRBis image reconstruction package by <cit.> to obtain aperture synthesis images at each of our three epochs and at each of the PIONIER spectral channels. The reconstructions were performed in a similar way as for the carbon AGB starR Scl by <cit.>.We usedthe best-fit models from Sect. <ref>as start images. We used a flat prior, and the six available regularization functions of IRBis. We chose a pixel size of 0.3 mas, and we convolved the resulting images with a point spread function (PSF) of twice the nominal array angular resolution (λ/2B_max∼1.2 mas). We used a field of view of 128×128 pixels, corresponding to38.4×38.4 mas, chosen to correspond to twice the best-fit size of the MOLsphere. As final images, we adoptedan average of the images obtained with regularization functions 1 (compactness), 3 (smoothness), 4 (edge preservation), 5 (smoothness), and 6 (quadratic Tikhonov), which resulted in very similar images. Function 2 (maximum entropy) resulted in poorer reconstructions.We performed a number of image reconstruction tests including the use of the reconstruction packages SQUEEZE <cit.> and MiRA <cit.>, further regularization functions, and different start images. All reconstructions were very similarto those obtained with IRBis.Figure <ref> shows our reconstructed imagesfor the three epochs at three spectral channels with central wavelengths 1.58/ 1.59 μm, 1.68 μm, and 1.77 μm. In Fig. <ref>, we over-plot the synthetic squared visibility amplitudes and closure phases based on the reconstructed imagesto the measured values. The residuals between both of them are also displayed. The synthetic visibility values based on the reconstructions are in good agreement with the measured values. There are discrepancies at small spatial frequencies, which increase with wavelength. This is a known systematic calibration effect of PIONIER data caused by different magnitudes orairmass between science and calibrator measurements. Values of χ^2 for the squared visibility amplitudes range between0.4 and 5.3, and for the closure phases between 0.22 and 2.5 for the different epochs and spectral channels. The achieved dynamic range varies between about 10 and 20.The reconstructed images at epoch I show the stellar disk with elongatedsurface features approximately oriented along the East-West direction. The images at epoch II and epoch III are qualitatively different to those at epoch I. They show a dominating narrower single bright feature. The feature is located on top of the stellar disk toward its south-western limb at epoch II and oriented slightly farther toward thesouthern limb at epoch III. The extended molecular layer or MOLsphere as present in our model fits from Sect. <ref> is not well visible in thereconstructed images because it lies just below our achieved dynamic range.We estimated the contrastδ I_rms/<I> <cit.> of our reconstructed images after dividing them by the best-fit modelimage to correct for the limb-darkening effect, and obtained values– averaged over the spectral channels and regularization functions–of 10%±4% for epoch I, 21%±6% for epoch II, and 31%±6% forepoch III. The contrasts at epochs II and III are significantly higher than those at epoch I.We estimated the angular diameter of the feature at epochs II and III to 1.7±0.3ṁas, averaged over the epochs and spectral channels. This gives a ratio of 0.42^+0.35_-0.10 compared to the Rosseland photospheric radius of the primary component. § DISCUSSION AND CONCLUSIONS The images at epoch I are consistent with predictions by three-dimensional (3D) radiative hydrodynamic (RHD) simulations of RSGs, such as those shown by <cit.>. As an example, the contrast of this RHD H-band snapshot is 9%, after convolution to our spatial resolution and correction for the limb-darkening. This value is consistent with our observed value of 10%±4%. We interpret the observed surface features at epoch I by giant convection cells within the stellar photosphere.The images at epochs II and III are significantly different to those at epoch I in terms of their appearance, that is the dominant narrower feature and their significantly higher contrast. While it may be possible that the morphology of the convection features has changed fromepoch I to epochs II and III in this way, the significantly increasedcontrast by a factor of 3 is not consistent with current 3D simulations such as those mentioned above. In the following we explore a scenario in which the images at epochs II and III are dominated by the presence of the close companion (as suggested by C14) located in front of the primary and where at epoch I this companion is located behind the primary and not visible.For reasons of consistency, we derived the positions at our epochs with the same fit procedure as in C14[There was an error in the sign convention of the script used for C14, which has now been corrected.]. Table <ref> lists the resulting positions. The best-fit positions agree with our reconstructed images. We adopt errors of the positions of half the array angular resolution, that is 0.9 mas for the 2012 AMBER data and 0.6 mas for the 2016 and 2017 PIONIER data.Figure <ref> (left) shows a sketch of V766 Cen with the photospheric disk, the MOLsphere, the Nai shell <cit.>, and these companion positions including that of C14.C14 analyzed available V band light curve data and available radial velocity data. They could not find conclusive orbital parameters of the system, meaning that we are not able to compare our positions to a given orbit. However, they were able to constrain the orbital period to 1304±6 d based on the light curve and radial velocity data. In order to test whether our companion positions are consistent with this orbital period and thus with the light curve and radial velocity data, we explored Keplerian orbits of V766 Cen and its close companion using the fixed period of 1304 d. With the limitations of the available data, we were not able to derive any conclusive determination of the orbital parameters. However, we found an indication that semi-major axes between 2 and 5 mas with eccentricities as large as 0.5 may produce plausible orbits that are consistent with our observed angular sizes, our companion positions, and with the orbital period by C14. For illustration, Fig. <ref> (left) includes an example of a plausible NE-SW orbit with a semi-major axis of 3 mas. This example orbit has a total mass of 108 M_⊙, which is above the estimates by C14 and <cit.>. The mass is sensitive to the semi-major axis and goes down to 32 M_⊙ at a semi-major axis of 2 mas. This may point to smaller angular radii of the components and a smaller semi-major axis within our error ranges. Nevertheless, this example illustrates that indeed our positions are consistent with that of C14 and with their orbital period based on the light curve and radial velocity data.In conclusion, we interpret our images in the most likely scenario of the close eclipsing companion that is located behind the stellar disk at epoch I and in front of the stellar disk at epochs II and III. The lower contrast surface features at epoch I as well as the residual features at epochs II and III are caused by giant convection cells on the surface of the primary.Assuming that the system is in contact or is in the common envelope phase, that is, both stars are filling their Roche lobes, we can derive the mass ratio of the components directly from the radius ratio by solving the Roche potential (Fig. <ref>). For a system in contact, we found that our radius ratio R_Comp/R_Prim of 0.42^+0.35_-0.10 corresponds to a mass ratio of 0.16^+0.40_-0.07. This result would only be marginally affected if the system was in the common envelope phase as it mostly affects the stellar extension along the orbital plane. Figure <ref> (right) illustrates an in-contact system with such a radius and mass ratio.Our imaging observations confirm the presence of a close companion to V766 Cen observed in front of the stellar disk at two epochs. With an angular diameter of 1.7±0.3 mas, corresponding to a radius of 650±150 R_⊙ and a mass of 2–20 M_⊙, it is most likely a cool giant or supergiant. We may be witnessing a system similar to the progenitor system of SN1987 A, where a low-mass companion was dissolved during the common envelope phase when the massive progenitor was a RSG.aaThis research has made use of the SIMBAD database, operated at CDS, France, and of NASA's Astrophysics Data System. AA acknowledges support from the Spanish MINECO through grant AYA2015-63939-CO2-1-P, cofunded with FEDER funds.§ ADDITIONAL MATERIAL | http://arxiv.org/abs/1709.09430v1 | {
"authors": [
"M. Wittkowski",
"F. J. Abellan",
"B. Arroyo-Torres",
"A. Chiavassa",
"J. C. Guirado",
"J. M. Marcaide",
"A. Alberdi",
"W. J. de Wit",
"K. -H. Hofmann",
"A. Meilland",
"F. Millour",
"S. Mohamed",
"J. Sanchez-Bermudez"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170927101347",
"title": "Multi-epoch VLTI-PIONIER imaging of the supergiant V766 Cen: Image of the close companion in front of the primary"
} |
0000-0002-1200-0820]Yao-Yuan Mao Department of Physics and Astronomy and the Pittsburgh Particle Physics, Astrophysics and Cosmology Center (PITT PACC), University of Pittsburgh, Pittsburgh, PA 15260, USAArgonne National Laboratory, Lemont, IL 60439, USA 0000-0003-1468-8232]Katrin Heitmann Argonne National Laboratory, Lemont, IL 60439, USAArgonne National Laboratory, Lemont, IL 60439, USA0000-0001-5501-6008]Andrew J. Benson Carnegie Observatories, Pasadena, CA 91101, USAMcWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USAInstituto de Astrofísica de La Plata (CCT La Plata, CONICET, UNLP), Paseo del Bosque s/n, B1900FWA, La Plata, Argentina Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Observatorio Astronómico, Paseo del Bosque, B1900FWA La Plata, ArgentinaKavli Institute for Particle Astrophysics and Cosmology & Department of Physics, Stanford University, Stanford, CA 94305, USAMcWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA 0000-0002-7832-0771]Salman Habib Argonne National Laboratory, Lemont, IL 60439, USA 0000-0003-2219-6852]Andrew P. Hearin Argonne National Laboratory, Lemont, IL 60439, USA 0000-0002-6825-5283]J. Bryce Kalmbach Department of Physics, University of Washington, Seattle, WA 98105, USA 0000-0002-4410-7868]K. Simon Krughoff Large Synoptic Survey Telescope, 950 N Cherry Avenue, Tucson, AZ 85719, USA 0000-0001-7956-0542]François Lanusse McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USALawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 0000-0003-2271-1527]Rachel Mandelbaum McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA 0000-0001-8684-2222]Jeffrey A. Newman Department of Physics and Astronomy and the Pittsburgh Particle Physics, Astrophysics and Cosmology Center (PITT PACC), University of Pittsburgh, Pittsburgh, PA 15260, USA 0000-0001-9850-9419]Nelson Padilla Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile 0000-0002-4637-2868]Enrique Paillas Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile 0000-0003-2265-5262]Adrian Pope Argonne National Laboratory, Lemont, IL 60439, USA 0000-0002-5294-0630]Paul M. Ricker Department of Astronomy, University of Illinois, Urbana, IL 61801, USA 0000-0001-5035-4913]Andrés N. Ruiz Instituto de Astronomía Teórica y Experimental (CONICET-UNC) and Observatorio Astronómico (UNC), Laprida 854, X5000BGR, Córdoba, ArgentinaMcWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA 0000-0002-4998-7606]Cristian A. Vega-Martínez Instituto de Astrofísica de La Plata (CCT La Plata, CONICET, UNLP), Paseo del Bosque s/n, B1900FWA, La Plata, Argentina 0000-0003-2229-011X]Risa H. Wechsler Kavli Institute for Particle Astrophysics and Cosmology & Department of Physics, Stanford University, Stanford, CA 94305, USA SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USADepartment of Physics and Astronomy and the Pittsburgh Particle Physics, Astrophysics and Cosmology Center (PITT PACC), University of Pittsburgh, Pittsburgh, PA 15260, USA 0000-0001-6966-6925]Ying Zu Department of Astronomy, Shanghai Jiao Tong University, 955 Jianchuan Road, Shanghai 200240, People's Republic of China Center for Cosmology and AstroParticle Physics (CCAPP), Ohio State University, Columbus, OH 43210, USA (The LSST Dark Energy Science Collaboration) The use of high-quality simulated sky catalogs is essential for the success of cosmological surveys. The catalogs have diverse applications, such as investigating signatures of fundamental physics in cosmological observables, understanding the effect of systematic uncertainties on measured signals and testing mitigation strategies for reducing these uncertainties, aiding analysis pipeline development and testing, and survey strategy optimization. The list of applications is growing with improvements in the quality of the catalogs and the details that they can provide. Given the importance of simulated catalogs, it is critical to provide rigorous validation protocols that enable both catalog providers and users to assess the quality of the catalogs in a straightforward and comprehensive way. For this purpose, we have developed the DESCQA framework for the Large Synoptic Survey Telescope Dark Energy Science Collaboration as well as for the broader community. The goal of DESCQA is to enable the inspection, validation, and comparison of an inhomogeneous set of synthetic catalogs via the provision of a common interface within an automated framework. In this paper, we present the design concept and first implementation of DESCQA. In order to establish and demonstrate its full functionality we use a set of interim catalogs and validation tests. We highlight several important aspects, both technical and scientific, that require thoughtful consideration when designing a validation framework, including validation metrics and how these metrics impose requirements on the synthetic sky catalogs. January 8, 2018 the Astrophysical Journal Supplement § INTRODUCTIONThe Large Synoptic Survey Telescope (LSST) will conduct the most comprehensive optical imaging survey of the sky to date, yielding a wealth of data for astronomical and cosmological studies. LSST data will offer many exciting scientific opportunities, including the creation of very detailed maps of the distribution of galaxies, studies of transient objects in new regimes, investigations of the inner and outer solar system, observations of stellar populations in the Milky Way and nearby galaxies, and studies of the structure of the Milky Way disk and halo and other objects in the Local Volume. A broad description of the LSST scientific goals is provided in the LSST Science Book <cit.>. Several science collaborations have been formed in order to prepare for the arrival of the rich and complex LSST data set. One of these collaborations, the LSST Dark Energy Science Collaboration (LSST DESC), focuses on the major dark energy investigations that can be carried out with LSST, including weak- and strong-lensing measurements, baryon acoustic oscillations, large-scale structure (LSS) measurements, supernova distances, and galaxy cluster abundance. An overview of LSST DESC goals is provided in the White Paper authored by the <cit.>; a detailed Science Road Map can be found at the Collaboration Web site[lsstdesc.org].Science opportunities relevant to LSST DESC will pose many new analysis challenges on different fronts, including controlling systematic errors, extracting subtle signals from large data sets, combining different dark energy probes, cross-correlations with other observations, etc. The best tools for these tasks must extract the science of interest while simultaneously enabling control of systematic contaminants, whether observational or modeling induced. Before the data set arrives, robust synthetic sky catalogs that are validated against a range of observational data are essential to the development of the required analysis tools. The comprehensive and systematic validation of synthetic sky catalogs presents major challenges. Observational data sets used for validation must be carefully curated and frequently updated with the best available measurements. Tests comparing observations with synthetic data sets have to be designed to address the wide range of tasks for which the catalogs will be used, e.g., tests of photometric pipelines, extraction of cosmological parameters, mass estimates for clusters, etc. The list is essentially as long as the set of analysis tasks to be covered by the survey. For each of these tasks, a set of requirements, such as accurate clustering statistics, best possible match to observed colors, detailed galaxy properties, or results over a range of different redshift epochs, needs to be defined and implemented as part of the validation tests. The synthetic catalogs will be revised, enhanced, and improved over time; a controlled and easy-to-use mechanism to expose new synthetic catalogs to a full battery of observational tests is essential to validating the catalogs properly. In addition, for the users of the catalogs, it is very desirable to have a convenient method to check the catalog quality for their specific needs. In order to provide an environment that can address all of these challenges in a streamlined way, we present DESCQA, a validation framework that has been configured to compare and validate multiple different synthetic sky catalogs in an automated fashion. There are a number of requirements that must be met in designing a comprehensive and flexible framework intended to assess the performance of multiple synthetic sky catalogs. The synthetic catalogs that the system has to handle will have a range of very different characteristics and features depending on how the catalogs are constructed and their ultimate purpose. There are many different ways to build synthetic sky catalogs. Because of the large survey volumes required for cosmological studies, most current methods are based on gravity-only simulations rather than on the significantly more expensive hydrodynamic methods. A range of approaches is used to assign galaxies to dark matter halos in post-processing. These include halo occupation distribution modeling <cit.>, subhalo abundance matching <cit.>, and semi-analytic modeling <cit.>. The choice of method used depends on the base simulation (e.g., resolution, available information with regard to time evolution) and the observational data sets that the catalog is expected to match. All known methods are, at best, only partially predictive, and each individual choice has its own advantages and disadvantages, such as resolution requirements, predictive power, ease of implementation, time to run, etc. In addition, different science cases impose different requirements on the catalogs. Roughly speaking, resolving smaller physical scales increases modeling difficulty, while including broader classes of galaxies adds to the complexity. The galaxy properties required will also influence the choice of method employed.Given the current uncertainties in galaxy modeling, it is not possible to address the full range of science issues with only one catalog construction method (or a single base simulation). Instead, catalog providers choose the methods that are best suited to address specific questions (or classes of such questions) of interest. The heterogeneity among the catalogs manifests itself in both the implementation details, such as file formats, units, and quantity labels, and scientific details, such as the choice of halo-finder algorithms, mass definitions, and filter-band definitions. This heterogeneity presents a significant barrier for users who wish to use several of these catalogs. The framework therefore needs to be capable of ingesting a wide range of synthetic sky catalogs that have very different intrinsic characteristics.The framework's success hinges on a set of well-thought-out validation tests that can act seamlessly on the catalogs. The tests, as well as the criteria for how well a catalog performs in a given test, will be provided by domain experts (e.g., members of analysis working groups). They will have the best understanding of the requirements and often also of the validation data. The difficulty of adding a new test to the framework therefore has to be minimal so that most contributions can be made without significant assistance from the framework developers. In addition, the execution of the test, once it is implemented, needs to be carried out automatically on all catalogs that provide the necessary information for the test. Tests and catalogs will be improved over time and new ones will be added. The framework needs to provide straightforward methods to accommodate these updates and additions.Finally, since the validation tests will be run on a wide range of synthetic sky catalogs, it is very desirable to have a convenient method to check which catalogs meet the specific needs for certain tasks. The results must be presented in a way that gives the catalog users an easy-to-use interface to peruse the different tests and catalogs. At the same time, the interface should provide a useful and easily interpretable overview. For this reason, sets of summary statistics and simple assessment criteria of the catalog quality need to be provided.To summarize, a validation framework used by a survey collaboration such as LSST DESC should be able to (1) process a wide range of heterogeneous catalogs, (2) automate the validation tests, (3) provide straightforward methods to update and add catalogs and tests, and (4) provide easy access to catalog feature summaries and quality assessments for catalog users. DESCQA addresses the above requirements and provides interfaces to access synthetic catalogs and to run a set of pre-specified validation tests. The results of the tests are graphically displayed and evaluated via well-defined error metrics. DESCQA is accessed via a Web-based portal, which is currently set up at the National Energy Research Scientific Computing Center (NERSC). The portal itself can, in principle, be set up anywhere, but collocating the portal with the storage and analysis resources is convenient.In order to demonstrate the full functionality of the framework and to have a sufficiently complex environment for testing and development, it is vital to use realistic, scientifically interesting synthetic sky catalogs and validation tests.LSST DESC is still in the process of defining the requirements for each of its analyses and producing new synthetic sky catalogs (including comprehensive full-coverage light-cone catalogs). Therefore, we have chosen to implement a set of interim validation tests and requirements to assist us in the development of the current version of DESCQA; we present these tests as our case studies of the DESCQA framework. Also, since one major requirement of this framework is to process a wide range of heterogeneous catalogs, we also select a set of interim catalogs that cover the major synthetic methods to be used in our case studies. The interim catalogs and tests presented in this work fulfill functions beyond the provision of demonstration examples. Although these catalogs and tests are not the final versions that will be used for LSST DESC science, their realistically complicated features provide a unique opportunity to delve into the conceptual challenges of building a validation framework. These challenges originate from the different choices made by the creators of the catalogs and tests, such as the definitions of physical quantities. These intrinsic differences cannot be easily homogenized by the framework; however, the framework can highlight them for the scientists who use the framework. Working with this set of interim catalogs and tests, we have identified several such conceptual challenges. Furthermore, our implementation of the validation framework also provides a concrete platform for publicly and quantitatively defining the requirements for a particular scientific analysis.The paper is organized as follows. We first describe the design and implementation ofthe DESCQA framework in <ref>. We explain in detail our method for adding synthetic catalogs to the framework and show how the method enables the automated testing and validation of these catalogs. We also discuss how the Web interface helps the user to navigate the catalogs and validation tests.Then, in <ref>, we present our case studies of five interim validation tests to demonstrate the features of the framework. The description of the different methods employed to build a range of interim synthetic sky catalogs can be found in <ref>. We conclude in <ref> with a summary and discussion of DESCQA and future development paths.§ DESCQA FRAMEWORKIn this section, we describe the DESCQA framework, a unified environment for comparing different synthetic catalogs and data sets using a number of validation tests.The DESCQA framework is based on DESQA, which was originally developed for validating catalogs for the Dark Energy Survey. DESQA, in turn, originated from the FlashTest framework, which was developed for standard regression testing (software testing) of the Flash code. Since regression testing is a considerably simpler task than the validation of catalogs, we had to make multiple changes to the framework to accommodate the design goals discussed in <ref>.The basic structure common to all of the variants of the framework is a set of scripts that execute the tests and a Web interface that displays the results. For DESCQA, although much of the original framework has been revised or replaced, the use of the Python programming language is retained (but revised to be compatible with Python 3), along with some portion of the original Web interface, and several of the original concepts used in FlashTest.<ref> presents the organization of the framework, which possesses four main components: (1) the reader interface, (2) the validation test interface, (3) the automated execution process, and (4) the Web interface.Together, they enable an expandable, automatable, and trackable validation process. The code of the DESCQA framework is publicly available in a GitHub repository[github.com/LSSTDESC/descqa]. A frozen version of the code can be found in <cit.>. §.§ Design GuidelinesIn designing the framework for DESCQA, our priorities were to provide a unified environment for a set of validation tests that examine and compare different synthetic catalogs, while ensuring that new catalogs and validation tests can be added or updated with minimal effort required from the framework developers, the test developers, and the catalog providers. At the same time, we also want to ensure that the validation results generated by the framework and delivered to the user are easy to understand and compare.The above set of considerations all aim at minimizing the overhead in meeting the requirements imposed by the framework. To help achieve this goal, we separate the framework into three major components, which are as independent as possible. These components are detailed in the following sections: the reader interface, the validation test interface, and the Web interface. In fact, the only requirement for the catalog providers and test developers is to conform to these common application programming interfaces (APIs) for accessing the catalogs (with the reader interface) and for executing the validation tests (with the validation test interface). There is no formal requirement regarding the underlying implementation. The flexibility of Python enables providers and users to implement their readers and validation tests using methods ranging from reading existing files to running an external executable, as long as they provide a Python interface that is consistent with the API specification.In practice, most catalog providers (test developers) need similar high-level functionality in their catalog readers (validation tests). From the point of view of code development, it is desirable to reduce code duplication in order to maintain consistency and reduce human errors. We have designed base classes for both catalog readers and validation tests. Catalog providers and test developers write subclasses that inherit from the base classes, but overwrite the core functions with their own, if needed.All of the above features reflect our design philosophy of providing an efficient, flexible, automated framework that is capable of including a diverse set of synthetic catalogs and validation tests with the fewest possible requirements imposed on the contributors and code developers.§.§ Reader InterfaceGiven the heterogeneity among different synthetic catalogs, it is impractical to access all the different catalogs directly in their native formats within the framework. The standardized reader interface solves this problem by associating with each catalog a corresponding reader that loads the catalog in its native format, provides metadata and a list of galaxy properties available in that catalog, and processes any necessary unit or definition conversions. The catalog provider implements the corresponding reader that conforms to the specification of the standardized reader interface. In this fashion, all of the catalogs can be stored in their native formats, and no change to the catalog's generation pipeline is required. Similarly, the reader can also be used to fix minor errors in the native catalogs (e.g., incorrect definitions or units) during the conversion between native quantities and user-facing quantities, thereby reducing the number of catalog files while still propagating updates to the users.We implemented a base class that contains some basic catalog process methods. To implement a new reader for a specific catalog, one would first subclass this base class, and then supply or overwrite the methods (e.g., data loading routine) to accommodate the catalog under consideration. Since we expect that different versions of a specific kind of catalog would be accessed using the same code, we allow the same reader to be used with different configuration parameters that specify, for example, the paths of the catalog files and versions.These configuration parameters are passed as arguments when the subclass is initialized. The reader can also check the catalog's version against an online catalog repository and warn the user if the catalog in use is out of date.Although some catalog variations such as units and quantity labels can be homogenized by the reader interface, others such as mass definitions and cosmology cannot. We do not ask the catalog providers to conform to a specific list of standards, but instead ask that they specify their choices as metadata available in the readers. Similarly, we do not require the catalog providers to include all quantities that are needed for all validation tests. During execution, a catalog that does not have some requested quantities for a specific test will be skipped and noted.In this particular study, the reader interface is used to enforce consistent units across different catalogs. We use comoving Mpc (not scaled to h=1) for all distance units, physical km s^-1 for all velocity units, and(not scaled to h=1) for all masses (including stellar mass and halo mass).Note that the reader interface itself can actually do more than serving data to the DESCQA framework. It can, in fact, be used as a standalone catalog data server or as a converter to convert catalogs from their native format into a database with common schema. We package the reader interface as a standalone Python module,[github.com/LSSTDESC/gcr-catalogs] which allows people to access the homogenized synthetic galaxy catalogs conveniently outside the validation framework. Under this new structure, the DESCQA framework itself becomes a user of the reader interface. §.§ Validation Test InterfaceAn important part of our framework is the quality assurance (QA) implementation, which allows test developers to design validation tests and provides a convenient interface for users to assess the quality of the synthetic catalogs. In DESCQA, a validation test is carried out to establish whether a synthetic catalog meets some particular requirements that have been set by the test developer. The validation test consists of two parts. First, the catalog is checked to see if it provides the quantities required for the specific test. Next, the catalog is tested to see if it can reproduce relevant observational data over a specified range at the required accuracy.We have designed a standardized validation test interface which is similar in concept to the reader interface.We implemented a base class with abstract methods for the validation test interface. Each individual test is a subclass of this base class and contains the non-abstract methods to conduct the test.The test interface also separates the configuration from the code that carries out the actual computation to allow convenient changes to the specific settings for each test.Each test uses specified catalog quantities (already wrapped by the reader interface) as input, carries out necessary calculations, creates figures and reduced data, and finally returns a summary statistic. As mentioned in <ref>, if a catalog does not provide all of the required quantities for a particular test, the test will automatically skip the catalog and proceed with the remaining catalogs.Each test must provide a summary statistic (score) for each catalog on which it runs and also provide a score threshold to determine if a catalog “passes” the test. The score and the passing threshold are both up to the test developer to set in the most useful way for that particular test. The notion of “passing” and “failing” a test is intended to give the user a quick method to inspect the summarized results using the Web interface, as we detail below. The notion is not to judge the quality of a catalog, as each catalog has its own features. Furthermore, many users are only interested in a subset of validation tests, so a catalog does not need to pass every test in order to be scientifically useful. In addition to the score, the framework allows the validation tests to generate figures, tables, and other products. These supplementary products are saved on the filesystem so that they can be accessed by the web-interface component of the framework as described below. Many users also find it helpful to have a summary figure that displays the relevant statistics for all available catalogs. Although the validation test interface does not formally require all tests to generate plots, we do provide a convenient plotting module to produce basic summary figures which test developers can utilize. Alternatively, developers may supply their own plotting modules.We will show examples of these summary figures that are generated by our common plotting module in <ref> for each of our currently implemented tests. We should note that currently all validation tests implemented in this framework are for demonstration purposes. Although this set of tests represents the major tests that are relevant to LSST DESC science, the choice of summary statistics and the passing criteria presented here are preliminary and introduced only as interim values. In future, test developers and catalog users will set more realistic criteria by which to evaluate the catalogs according to the LSST DESC science goals.§.§ Automated Execution and Web Interface Since our design separates the configurations and the actual reader and test code (which are implemented as classes), a master execution script is required to run the tests.For both catalogs and tests, the master execution script reads in the configuration files (specified in the YAML format), identifies the corresponding reader or test classes to load, and passes the configurations to the class and executes the class methods. The master execution script has access to all available catalogs and tests, and can execute all desired combinations. By default, the master execution script is set to run periodically, thereby automating the full validation process.The master execution script is also equipped to handle failures, relying on Python's context manager. The execution script captures all exceptions and traceback information, together with any content printed to standard output or standard error during runtime, and stores them in a log file. This is done for each of the combinations of all tests and catalogs, and when one of them fails, the master execution script writes out the log file, and continues to the next combination without being interrupted. This design makes our framework easily expandable. Including new catalogs or validation tests requires no changes to existing code. Once the new reader or test class and its corresponding configuration file are placed in the pre-specified location, the master script will automatically include the new catalog or test in future runs.In this fashion, catalog readers and validation tests can remain agnostic about what catalogs and tests are available, and hence do not require updates when new catalogs and tests are added.For the seven tests and eight catalogs presented in <ref>, since the catalogs are fairly small (made out of the 100 Mpch^-1 box), it takes only about 10 minutes on a single CPU core to run all of the combinations. The eight catalogs take less than 25 GB of disk space (excluding the underlying DMO simulation). If we were to run a test whose computational cost scales linearly with the number of galaxies on a mock catalog that corresponds to about 10,000 square degrees, it would take tens of minutes on a single CPU core, and this catalog would take about 1TB of disk space (depending on how many galaxy properties are stored). These numbers are still manageable by modern standards; however, in the future when facing very large catalogs and more computationally involved tests (such as two-point statistics), we will need distributed computation. Although we have not yet explored this direction, we believe our framework is flexible enough to accommodate distributed computation. In particular, our framework can be configured to run different tests on different catalogs using separate cores, without the need for cross-node communication (i.e., embarrassingly parallelizable).All of the results (including the plots and summary statistics generated by the validation tests) can be archived periodically. Although the user can certainly inspect individual output files to access validation results, this process will rapidly become tedious with the increasing numbers of catalogs and tests. To avoid this difficulty, we have built a Web interface[portal.nersc.gov/project/lsst/descqa] at NERSC to assist users in quickly inspecting validation results. When users visit the Web interface, the Common Gateway Interface scripts will read in the output files available on the filesystem, and present users with a visual summary of the results.The Web interface also allows the user to browse through different runs. When each run is executed, a copy of the code used is recorded so that the results can be easily tracked. <ref> shows an example of the summary page of the Web interface, which is presented in the form of a validation matrix. This matrix provides a quick summary of all validation tests (rows) and all available catalogs (columns). Each colored cell shows the corresponding test result.Each of the current set of validation tests provides a score that is between 0 and 1. A higher score indicates a larger discrepancy between the catalog under consideration and the validation data set. When the score is higher than a certain predefined value, it is noted as “failed.” As already noted, the specific values and passing criteria are for demonstration purposes and do not reflect the actual LSST DESC requirements. In this matrix view, users can further click on the header or the cell to see the associated plots and output files.This interface helps the users to quickly find the catalogs that satisfy their desired requirements.The DESCQA framework utilizes the filesystem to serve the Web interface and avoids direct communication between the test scripts and the Web interface. Hence, the framework can be easily adapted and applied more broadly to many other comparison tasks, such as a comparison of different code implementations.§.§ Documentation and Maintenance The DESCQA framework faces many different types of users, and hence requires many different forms of documentation. The framework is highly modularized, enabling different types of users to contribute without the overhead of understanding the full framework.Here we describe the different requirements on and our implementation of the documentation as well as related issues regarding maintenance.* For Web interface users who want to browse the results and plots of validation tests: the Web interface is self-explanatory and requires little documentation. On the front page, we provide basic instructions of how to navigate the interface, along with links to our papers, code repositories, and internal documentation pages, where users can easily find further information. * For catalog users who want to access the synthetic catalogs through the reader interface: we provide both API documentation[yymao.github.io/generic-catalog-reader/] to the reader interface and also an example code[github.com/LSSTDESC/gcr-catalogs/tree/master/examples] (many of which use easily browsable Jupyter notebooks) to help users understand quickly how to access the synthetic catalogs using the reader interface.* For users who want to implement a validation test to be included in the DESCQA framework: as discussed above, since the validation tests themselves are very much independent of the rest of the framework, the knowledge required for a test writer to be able to contribute is much reduced. In particular, test developers need only to implement a subclass that inherits the base class of validation tests and to follow the instructions[github.com/LSSTDESC/descqa/tree/master/descqa] on how to implement a few specific member methods. We also provide step-by-step instructions on how to manually trigger the validation framework to test newly implemented validation tests[github.com/LSSTDESC/descqa/blob/master/README.md and github.com/LSSTDESC/descqa/blob/master/CONTRIBUTING.md].* For catalog providers who want to contribute their catalogs: similar to the case for new test developers, new catalog providers need only to implement a subclass that inherits the reader base class and to follow the instructions^<ref> on how to implement a few specific member methods. The catalog providers can test their newly implemented readers either with the full DESCQA framework or through the importable reader module.* For users who maintain the Web interface and execution scripts, which includes DESCQA framework maintainers but not most regular users: we have made these components of our framework highly modularized and self-documenting such that future maintainers can navigate the code easily. We are working on other visual aids such as flowcharts to help future maintainers understand the code structure better.While we continue to improve these various aspects of documentation, recent feedback from users both within the DESC Collaboration and elsewhere suggest that we already have adequate documentation for the different types of users to utilize or to contribute to this framework. In addition, we work closely with the computing infrastructure working group of the Collaboration to ensure that the code base of this framework is kept up to date with the development environment (e.g., Python and Python packages), so as to reduce potential dependencies on deprecated packages and to benefit from better performance and new features.The Collaboration intends to continue to use, support, and develop the DESCQA framework, and will help to ensure that a period of overlap and effective communication is enabled between the current and future maintainers.§ CASE STUDIESTo demonstrate the design and features of the DESCQA framework, we present five validation tests as case studies. <ref> provides a summary of these five tests and the corresponding validation data sets and criteria.These criteria can be defined to satisfy specific science goals; however, as LSST DESC is still finalizing science requirements, the criteria used here are for demonstration purposes.As mentioned earlier, one important requirement of this framework is that it needs to be suitable for a wide range of heterogeneous synthetic catalogs.Hence, we select eight realistic synthetic catalogs that encompass the major classes of methods that are generally used to create synthetic galaxies (HOD, SHAM, SAM, and hydrodynamical simulations) to use in our case studies.<ref> summarizes the eight catalogs used here. In particular, the hydrodynamical galaxy catalog is extracted from the MassiveBlack-II (MBII) simulation, further described in <ref>. All other catalogs are built upon the dark structures of the same gravity-only simulation, MBII DMO, which is a companion run of MBII using the same initial conditions, resolution, and box length as its hydrodynamical counterpart. The description of MBII DMO can also be found in<ref>. The details of how each catalog is implemented can be found in the rest of the subsections of <ref>.In order to keep the creation of the catalogs as simple as possible, we choose to compare them only at a single snapshot (fixed redshift). This approach is sufficient to establish the functionality of the DESCQA framework. In the near future, we will expand the framework to include light-cone catalogs.For each test, we present a summary comparison plot to compare the results from different catalogs and the validation data set, and discuss how these case studies have in turn influenced the design of the framework. We summarize these findings in <ref>. All plots presented here are directly taken from the output of the DESCQA framework, without any further editing.We encourage the reader to further investigate all results directly using the DESCQA Web interface,[ is where this particular run locates] which includes some tests that we did not display here and also includes plots that compare the results for each catalog with validation data separately.§.§ Stellar Mass Function For each of the synthetic catalogs, we calculate the stellar mass density as a function of the total stellar mass for each galaxy. The densities are derived from the number counts of galaxies in each stellar mass bin, divided by the simulation volume. These densities are compared with the stellar mass functions from the MBII hydrodynamic simulation and from <cit.>. <ref> shows a comparison between these two stellar mass functions. Stellar masses are most commonly defined as the mass locked up in long-lived stars and stellar remnants. However, synthetic catalogs and the validation data sets may all have defined stellar mass differently, and the discrepancy cannot be homogenized by the reader interface. For the SAM models, the total stellar mass is the sum of the disk and spheroid components. For the SHAM-based models, the stellar masses correspond to the galaxy catalogs to which they were matched. On the other hand, the stellar masses used to construct the <cit.> stellar mass function are taken from the New York University Value-Added Galaxy Catalog (NYU-VAGC; ), and were derived from the kcorrect code <cit.>. As such, they do not include the mass in stellar remnants (white dwarfs, neutron stars, etc.), which is more commonly included in the definition of stellar mass. For a <cit.> initial mass function and using stellar data from <cit.> and <cit.>, we find that the fraction of the original population mass in stellar remnants for a single-age population of age 10 Gyr is 14.6%, with 39.7% of the original population's mass remaining in stars after this time. Therefore, we shift the <cit.> mass function masses by +0.136 (i.e., log [39.7%/(39.7%+14.6%)]) in order to include the mass in stellar remnants. Estimates of the stellar masses of galaxies also suffer from other sources of systematic error. For example, <cit.> show that uncertainties arise from the template-fitting procedures used to estimate stellar masses from multiband photometry. Although they considered other surveys, they demonstrated that systematics at the 0.1 dex level can arise from these error sources. In the specific case of the stellar mass function from the Sloan Digital Sky Survey (SDSS), <cit.> reanalyzed the SDSS photometry and concluded that there are significant systematic biases in the inferred stellar masses arising from the choice of surface brightness profile fit to the data. In their latest work, <cit.> estimate that the photometric systematic errors are at the level of 0.1 dex. Here we have chosen the <cit.> measurement as an example, which is significantly different from the measurement in MBII.As mentioned in <ref>, each test must provide a summary score. For this test, the summary score is the probability for a χ^2 distribution, given the number of bins, to have a value less than the one calculated from the comparison of the catalog result to the validation data: χ^2 = ∑_i,j(ϕ_i - ϕ̂_i) [(C + Ĉ)^-1]_i,j(ϕ_j - ϕ̂_j),where ϕ_i and ϕ̂_i are the differential stellar mass number density for mass bin i calculated from the catalogs and from the validation data, respectively, C is the covariance matrix calculated from the catalog using the jackknife resampling method, and Ĉ is the covariance matrix calculated from the validation data, in which case we include only the diagonal terms.To evaluate C, we implement the jackknife resampling method by dividing the simulated box into 5^3=125 smaller cubic boxes.The criterion to pass this test is set to be a score less than 0.95 (equivalent to having a right-tail p-value larger than 0.05).When designing this test, we also notice that in most cases, the passing criterion may not apply to the full range of stellar masses. For example, the low-mass end is bound to be affected by resolution, and, depending on the user's application, this may or may not be an issue.Hence, we design the test to have a configurable validation range. We demonstrate this feature here (as the white band in <ref>) and require the synthetic catalogs to reproduce stellar masses above 10^9. We also exclude the most massive bin when calculating the score as the last bin is dominated by cosmic variance. <ref> shows the results for the stellar mass function compared to the observational measurements from <cit.>. The reader is encouraged to inspect the results in more detail with the help of the DESCQA Web interface.The validation data is shown in black solid circles with error bars, and the synthetic catalogs are represented by the colored lines with shaded error bands.By construction, CAM_LiWhite and SHAM_LiWhite are almost identical to the <cit.> validation data set as they are based on the abundance-matching technique, which guarantees an exact match to the input stellar mass function.For the same reason, CAM_MBII and SHAM_MBII are very close to the MBII stellar mass function.was originally tuned to fit <cit.> as well, leading to good agreement in this test.More interesting are the results from the two SAM approaches. Both of them overpredict the stellar mass function at low masses compared to the <cit.> measurement, SAG somewhat more than Galacticus, similar to MBII. The shape of the stellar mass function for the SAMs does show a hint of a knee, similar to what is seen in the validation data set and unlike the MBII catalog, but the shape of the measurement is still not captured very well.This test demonstrates that, if a catalog user wants to impose a stringent requirement such as the one we use here, currently only SHAM-based models would pass the test due to their construction method. Hence, careful consideration is advised when designing the requirement for SMF tests. §.§ Halo Mass Function (HMF) The mass distribution of halos is one of the essential components of precision cosmology and occupies a central place in the paradigm of structure formation. There are two common ways to define halos in a simulation. One of these, the spherical overdensity definition, is based on identifying overdense regions above a certain threshold. The threshold can be set with respect to the (time-varying) critical density ρ_c = 3 H^2 / 8 π G or the background density ρ_b = Ω_m ρ_c. The mass M of a halo identified this way is defined as the mass enclosed in a sphere of radius r_Δ whose mean density is Δρ_c, with common values ranging from 100 to 500. The other method, the friends-of-friends (FOF) algorithm, is based on finding neighbors of particles and neighbors of neighbors as defined by a given separation distance <cit.>. The FOF algorithm is essentially an isodensity estimation method (mass enclosed within a given isodensity contour). FOF halos can have arbitrary shapes, since no prior symmetry assumptions have been made; the halo mass is simply the sum of the masses of particles that are halo members.Here, we calculate the HMF from each catalog for the distinct halos and provide a comparison to the well-established analytic fit by <cit.> for spherical overdensity-defined halos (M_100c), which is accurate at the 5–10% level at z=0 for a ΛCDM cosmology. We have also implemented (not shown) the <cit.> and <cit.> fits for the FOF halos, in addition to many other analytic mass function fits; for details, see <cit.>. The original code was written in Fortran, and we provide a simple Python interface to include the code in DESCQA; we have made the code publicly available[github.com/zarija/HaloMassFunction]. This test uses the same summary statistic as the SMF test as described in <ref>, except that the covariance of the validation data Ĉ is set to Poisson errors for the diagonal terms and zeros for the off-diagonal terms for this particular test, as the validation data is an analytic fit.To compare the mass function for only distinct (host) halos from each catalog, this test requires two important pieces of information: (1) the halo mass associated with each galaxy and (2) whether or not the galaxy is a central galaxy. We then select only halos that are associated with central galaxies. Hence, although all the interim catalogs presented here use the same base simulations, catalogs that assign central galaxies differently may have different HMFs.We note that the HMF we evaluate here is defined slightly differently from the usual HMF in the sense that we require the halos to host central galaxies. Distinct halos that do not contain any central galaxy are not included in these catalogs. As a result, different catalogs include very different abundances of low-mass halos, depending on their halo occupation function at the low-mass end.This effect can be clearly seen in <ref>: above ∼10^11, all mass functions agree extremely well and follow the Tinker fit at the expected level of accuracy. Below this mass, the catalogs start to disagree. These issues should be taken into account when designing the requirement for the HMF test. Other possible discrepancies that cannot be homogenized by the reader interface include cosmology, halo mass definitions, and halo finders. Different research groups often use different halo mass definitions and halo finders, and hence flexibility in this regard is important. Our test routine provides different fitting functions that can be chosen to match the underlying cosmology and halo mass definition used to create the synthetic catalog.Differences in halo finders, on the other hand, are more difficult to deal with. For example, the MBII hydrodynamic simulation-based catalog was generated using an FOF halo finder with linking length of b=0.2, while all other synthetic catalogs that we used in this study are based on the same N-body simulation (MBII DMO) analyzed with the Rockstar halo finder and the same spherical overdensity definition. In <ref> we see that the MBII HMF is overall lower relative to results from the populated catalogs, in particular for the low-mass halos. This result is not only due to different halo mass definitions, but also due to the presence of baryons as indicated by the findings by <cit.>, where the MBII HMF was compared to the DMO mass function (for the same linking length b=0.2) and a difference at the 15% level was found for halos with masses of ∼ 10^11. Similar results were found in the OverWhelmingly Large Simulations <cit.>, and in the Magneticum simulations <cit.>. This shows that baryonic effects and halo definitions are potentially important for designing the requirements for an HMF test.§.§ Stellar Mass–Halo Mass (SMHM) Relation The SMHM relation, defined as the stellar mass of central galaxies within a distinct halo of total spherical overdensity mass, is a now well-established estimator of the efficiency of gas cooling and star formation over a wide range of halo masses. Both observational and theoretical works have shown that the efficiency of the stellar mass assembly (M_*/M_halo) peaks at the scale roughly corresponding to the knee of the stellar mass function and declines at smaller and larger masses. This is generally interpreted in simple terms as the efficiency of the stellar mass assembly being suppressed at the low-mass end by supernova feedback and at the high-mass end by active galactic nucleus (AGN) feedback <cit.>. Similar to the HMF test, the SMHM relation is not directly comparable to observational data. Not all validation tests, however, need to be tests that compare synthetic catalogs with actual observations, as some tests are designed to aid catalog users in understanding the features and characteristics of each catalog, or to provide comparisons with other results in the literature, or from other models. For example, when a user validates the SMHM relation, they might actually be verifying if the catalogs match a specific SMHM relation that is derived from a hydrodynamical simulation or inferred from a theoretical model. Moreover, the user can also validate only a certain regime of the SMHM relation, for example, to verify if the effect of AGN feedback is present in the catalog under consideration.In the current DESCQA test implementation, we use the results from the MBII hydrodynamical simulation as an interim validation data set. As mentioned in <ref> for MBII, the halos were identified with an FOF halo finderwith linking length b=0.2. The halo definition is therefore different for the validation data set than for the synthetic catalogs, where the mass definition is the virial mass (M_vir). In addition, due to baryonic effects, the halo masses from the MBII hydrodynamic simulation are lower than those from the DMO simulation (seeand <ref>).The SMHM relation uses only distinct halos (host halos) and excludes subhalos. As such, most of the caveats that apply in the HMF test would also apply here. This test also uses the same summary statistic as the stellar mass function test as described in <ref>, and the same validation range as the HMF test (<ref>). <ref> shows the results from MBII and the catalog results with error bands. The MBII result is trivially perfect by construction because it is a self-comparison. SHAM_MBII and CAM_MBII perform quite well over most of the mass range as these models were tuned to the MBII stellar mass function. However, at the high-mass end, the SMHM relation from SHAM_MBII flattens out due to the constant scatter used in the abundance-matching technique, while MBII's SMHM relation exhibits a much smaller scatter in stellar mass at high halo mass. SHAM_LiWhite and CAM_LiWhite perform reasonably well over the intermediate-mass range where the MBII and Li & White stellar mass function are closest. The overprediction of MBII at the low- and high-ass end compared to Li & White is reflected in the discrepancy seen in SHAM_LiWhite and CAM_LiWhite. Galacticus is overall lower than MBII but has the correct rise at the high-mass end. The SAG catalog underpredicts the SMHM relation compared to the MBII test for the low-mass halos.overall fits reasonably well though the results are worse at extreme mass values.Note that, except for MBII (which is a self-comparison), none of the catalogs passes the current validation criterion. This indicates that more thoughtful criteria and more realistic validation data sets (e.g., ones derived from empirical models) should be adopted if catalog users view the SMHM relation as an essential measurement that catalogs must reproduce.§.§ Projected Two-point Correlation Function The projected galaxy two-point auto-correlation function, w_p(r_p), is one of the most-used clustering statistics for testing both cosmology and galaxy formation models. Here we describe our test to compare w_p(r_p) among different synthetic catalogs and against observational and simulated data. Since our interim synthetic catalogs are given at a single epoch, we calculate w_p(r_p) using the thin-plane approximation.We use the catalogs at one epoch and then add redshift space distortions along one spatial axis (z-axis). We then calculate the projected pair counts, with a projection depth of ±40 h^-1Mpc.We assume periodic boundary conditions for all three spatial axes. This test also uses the same summary statistic as the stellar mass function test described in <ref>, though the evaluation of the covariance for the catalog, C, is slightly different, and we include the full covariance of the validation data, Ĉ.We estimate the sample variance of w_p(r_p) using the jackknife technique. We divide the projected 2D plane (x-y plane) into 10 × 10 smaller regions, with each region having an area of (10 h^-1Mpc)^2. We then re-evaluate the w_p(r_p) when removing one region at a time. The code that calculates w_p(r_p) and its jackknife variance is publicly available[Module `CorrelationFunction' in bitbucket.org/yymao/helpers].In this paper, we compare the w_p(r_p) from each catalog for all galaxies that have a stellar mass larger than 10^9.8h^-2. We use two interim validation data sets for this test: the w_p(r_p) calculated from the MBII hydrodynamical simulation and the w_p(r_p) from SDSS as presented in <cit.>. This measurement was made on the volume-limited samples from the NYU-VAGC catalog <cit.>, based on Data Release 7 from the SDSS <cit.>.In <ref> we show the results of comparisons with the SDSS data being used for validation; the reader is encouraged to inspect the comparisons with MBII as the validation data directly on our Web interface. Overall, the agreement between the catalogs and SDSS data is rather good, and most catalogs pass our preliminary validation metric. Most synthetic catalogs underpredict the small-scale clustering when compared with SDSS data, though they are still within the 2σ errors. Due to the small volume of our simulation box, the jackknife sample variance dominates the error budget. The data are likely to better distinguish between the same models if they are applied to a larger cosmological volume. Hence, one should consider including the volume of the catalogs as part of the evaluation for a w_p(r_p) test.§.§ Galaxy Color Distribution We also include a test of how well the synthetic galaxy color distributions compare to the observed colors of galaxies. In principle, this test should be done with light-cone catalogs that cover the same redshift range as the observed data set, with the same set of cuts on observed properties. However, for proof of concept, we present it using our current catalogs at a single epoch, z=0.0625 (except for the SAG catalog, which is at z=0, and the Galacticus catalog, which is at z=0.05).We determine the color distributions in these catalogs for comparison with our validation data set — measurements of the ugriz colors of a volume-limited sample of 0.06<z<0.09 galaxies from SDSS DR13 <cit.>. In the future, when light-cone synthetic catalogs are included in the framework, we will incorporate a broader range of SDSS galaxies, as well as objects with deeper imaging, e.g., from CFHTLS <cit.> or DES <cit.>, and spectroscopy, e.g., from GAMA <cit.>, DEEP2 <cit.>, and DESI <cit.>.In order to compare SDSS colors with synthetic galaxy colors, we use the SDSS apparent magnitudes to construct K-corrected absolute magnitudes. First, we select SDSS galaxies in the redshift range 0.06<z<0.09, where the variation of the distribution of K-corrected colors with redshift is small, and correct for Galactic extinction (we implicitly assume that the color evolution for 0<z<0.09 is negligible when comparing to the single-epoch catalogs). We then use the kcorrect code of <cit.> to find the rest-frame spectral energy distributions (SEDs) and obtain K-corrected absolute magnitudes for each SDSS galaxy (e.g., M_i for i-band absolute magnitude). Since different catalogs include galaxy colors that are K-corrected to different redshifts and this difference cannot be eliminated by the reader interface, this test K-corrects the SDSS data to the same redshift that each of the catalogs uses for its passbands.To minimize the effects of incompleteness in the validation data set, we construct a volume-limited sample by applying a cut in r-band absolute magnitude M_r<M_r,max, where M_r,max is chosen to be the value of the 85th percentile of the SDSS M_r distribution in a narrow redshift bin at 0.089<z<0.09. The same M_r<M_r,max cut is also applied to the synthetic catalogs. We then compare the K-corrected colors of the volume-limited samples from SDSS and from the synthetic galaxy sample. To obtain a quantitative estimate of the level of difference between the SDSS and synthetic color distributions, we calculate the two-sample Cramér-von Mises (CvM) statistic <cit.>. The CvM test is a nonparametric test for whether multiple data sets are drawn from the same probability distribution, similar to the Kolmogorov–Smirnov (K–S) test. However, instead of only looking at the maximal difference in the cumulative distribution function (CDF), as is done in the K–S test, the CvM test statistic ω (defined below), calculates the average L^2 distance across the entire CDF.As a result, it is more sensitive to differences in the tails of the distribution than the K–S statistic, and constrains the closeness of two CDFs at every point along them.The CvM statistic is calculated from the formulaω^2 = ∫_-∞^+∞(F_1(x) - F_2(x))^2d H(x)where F_1(x) and F_2(x) are the CDFs of each sample estimated from the data, H(x) = (n_1F_1(x) + n_2F_2(x))/N, and n_1 and n_2 are the numbers of objects in the two samples, with N=n_1+n_2. In our case, F_1(x) and F_2(x) are the CDFs for one SDSS color (e.g., g-r) and the equivalent synthetic color; ω provides a measure of the RMS difference between these two CDFs.As an example, <ref> shows the CDFs of the color distribution from SDSS and one of the synthetic catalogs. To remove potential zero-point offsets between SDSS and the synthetic catalogs (whether due to photometric zero-point uncertainties or issues with K-corrections), we apply a constant shift to the synthetic galaxy colors so that their median matches the median SDSS color;we calculate ω for both unshifted and shifted colors. The current criterion for a synthetic catalog to pass the color test is that the ω calculated from the shifted colors must be smaller than 0.05 for all four SDSS colors (u-g, g-r, r-i, and i-z), i.e., the RMS difference between the color CDFs must each be smaller than this threshold.This criterion is very stringent and may not reflect the final actual requirements, but we do expect that LSST will have stringent requirements on the distribution and evolution of galaxy colors to mitigate systematic errors in photometric redshifts.A set of summary plots for all catalogs with available colors is shown in <ref>. The SHAM catalogs agree well with SDSS in i-z but are redder in u-g, g-r and r-i. The CAM catalog agrees well with SDSS in g-r and r-i, but not as well in u-g and i-z, although the difference in i-z might be due to a zero-point offset. Different releases of SDSS, and moderately different redshift ranges, were used in the production of the abundance-matching-based catalogs and the validation catalog, and these two factors likely contribute to the differences. The two stellar mass functions (from MBII and from ) yield only a very small difference in the color distributions. The two catalogs from semi-analytic models do not produce very realistic color distributions, with SAG color distributions exhibiting a stronger bimodality than SDSS and Galacticus having long blue tails.We see that some of the features in the color distributions are not captured by the summary statistics. Although the framework requires each test to report a summary statistic for the ease of quick overall comparison, for tests like the color distribution test, the figures are essential components as they provide more information for the catalog users.§.§ How These Case Studies Influence Our DesignAs mentioned above, our case studies have helped us identify several features that are particularly useful for a validation framework like DESCQA. Here we summarize these features. * Uniform interfaces for both reading catalogs and executing tests. To carry out the validation tests presented in our case studies, we can easily see the necessity of the two main components of the DESCQA framework: the reader and test interfaces. By providing a uniform interface, it enables the users to access different catalogs in a uniform way and also standardizes the necessary elements of a validation test.* Allowing absent quantities. Since each validation test only accesses a subset of quantities, the framework should not impose a global set of required quantities for all catalogs. Each catalog will be validated by the tests for which it contains the quantities needed. Thus, the reader interface should provide a method for checking available quantities. * Documenting the intrinsic differences in quantity definitions. Some quantities may be defined differently (e.g., halo mass, magnitudes) in different catalogs. In the cases where these differences cannot be homogenized by the reader interface, the difference should be recorded in the metadata, and be exposed to the tests through the reader interface. It is up to each test developer to decide how to deal with these intrinsic differences. * Adaptive tests. In the cases where intrinsic differences that cannot be homogenized exist among the catalogs, sometimes it is computationally more efficient for the test to change its configuration on the fly to adapt to each catalog. For example, in our case studies, the HMF validation data set is computed analytically and hence can be tuned to different redshifts to match the catalogs. Similarly, the color distribution test also applies K-corrections to the validation data set to match the specification in the catalogs. * Configurable tests. For a given test, it may be desirable to have some variants that use different validation data sets, passing criteria, or validation ranges. Hence, the validation test interface should provide a convenient method to allow quick changes of these settings. In practice, these settings can be specified in a configuration file that is read in by the test interface.* Providing both numerical and visual results. A potential concern with a highly automated validation system is that some important issue is buried under a simple pass-or-fail result. Hence, we encourage the validation test developers to create plots in their test implementation, and we have designed an automated framework to manage such plots and display them on the Web interface. The framework also collects a numerical score from the tests to present in the summary view (<ref>) so that users can spot any potential issues more quickly.§ CONCLUSION AND OUTLOOKIn this paper, we have presented DESCQA, a framework that enables the automated validation of synthetic sky catalogs.The major aims of the framework are (1) to provide simulators an easy interface to test the quality of their catalogs and (2) to provide the LSST DESC and larger community a platform that enables them to easily choose synthetic catalogs that best fulfill their needs to test their analysis pipelines. The necessity of this framework arises from the fact that, with the wide variety of cosmological investigations possible with LSST—weak to strong lensing, cluster cosmology, LSS measurements, and supernova distances—no one single synthetic catalog will be optimal for every task. For example,obtaining large volumes for LSS investigations will clearly only be possible with limited mass resolution, while on the other hand, photo-z tests do not necessarily require large volumes, but rather excellent modeling of the color distribution.Hence, a framework that can test a wide range of synthetic sky catalogs against a large set of different target requirements is essential in systematically preparing for LSST science. The goal of the DESCQA framework is to minimize the burden on both the catalog creators and users when they deal with an inhomogeneous set of catalogs and tests.We have designed common APIs for both accessing and testing the synthetic sky catalogs with the Python Programming Language, and have also built a Web interface for ease of comparison. Nevertheless, the fundamental challenge here is to design a framework that can actually respond to the scientific needs of catalog validation.To meet this challenge during the development of the framework, we have selected a set of realistic synthetic sky catalogs and validation tests to test and improve our framework.Although these interim synthetic sky catalogs and validation tests are not necessarily the final product or requirements that LSST DESC will eventually adopt, they have provided useful insights into questions such as how to homogenize a diverse set of synthetic sky catalogs and how to design meaningful validation tests, as we have summarized in this paper.Thanks to the use of these realistic trial catalogs and tests during our development process, we have already identified several needed improvements for upcoming LSST DESC Data Challenges. For example, although the code itself is maintained in a GitHub repository^<ref> and all the outputs are stored on the NERSC filesystem, a more rigorous catalog and test version control system for the framework is still needed. We also need to improve the ability of the framework to process even larger sky catalogs efficiently and to enable a convenient way to download catalogs of interest (currently only available to LSST DESC members in our NERSC LSST project space).Another major step is to include light-cone catalogs, which is essential for realistic comparison with photometric data. In addition, the set of validation tests will also be considerably extended to cover a large range of possible LSST DESC projects. During our development process, it became clear that more consideration is needed when designing the catalog requirements. In particular, the validation tests need to carefully handle the intrinsic differences between catalogs that cannot be homogenized by the framework; we have highlighted many issues of this kind here, as summarized in <ref>.With this study and the implementation of the DESCQA framework, we have made an important step toward the full utilization of the wide variety of synthetic sky catalogs. This paper has undergone internal review by the LSST Dark Energy Science Collaboration. The internal reviewers were Douglas Applegate, Deborah Bard, and Dominique Boutigny. The contributions from the authors are listed below. Y.-Y.M. is one of the main developers of the DESCQA framework, led the transition from the original FlashTest framework to the current framework, generated the MBII DMO halo catalogs and merger trees, provided the SHAM-ADDSEDS galaxy catalogs, developed the validation tests, and contributed to the manuscript. E.K. is one of the main developers of the DESCQA framework, provided the Galacticus catalogs, developed the validation tests, and contributed to the manuscript. K.H. initiated this project, has overseen the design and development as a co-convener of the Cosmological Simulation Working Group, and contributed substantially to the manuscript. T.D.U. contributed to the current framework and also the transition from the original FlashTest framework to the current framework, and assisted in integrating catalogs and tests to reach the current validation capability. A.J.B. is the main developer of the Galacticus model, assisted in generating the Galacticus galaxy catalog, and contributed to the manuscript. D.C. provided the CAM catalogs and contributed to the DESCQA framework and the manuscript. S.A.C. worked on the SAG catalog development. J.D. was the main developer of the SHAM-ADDSEDS catalogs. T.D.M. helped initiate the project and contributed to the MBII simulations and catalogs. S.H. contributed to the manuscript and provided advice on the overall framework functionality and error metrics. A.P.H. contributed code for generating the CAM catalogs and oversaw D.C.'s implementation of the method into the DESCQA framework. J.B.K. contributed to the initial implementation of the framework and initial color tests. K.S.K. participated in this project as a co-convener of the Cosmological Simulation Working Group. F.L. provided the reader interface to the MBII hydrodynamic galaxy catalog. Z.L. produced the halo mass function test, and contributed to the project discussions and manuscript. R.M. helped initiate the project and contributed to the iHOD catalog and the manuscript. J.A.N. provided advice on metrics and methods, especially for the galaxy color distribution test. N.P. and E.P. provided the SAG galaxy catalog and contributed to the manuscript. A.P. helped coordinate infrastructure to produce the Galacticus catalogs. P.M.R. created the original prototype for the FlashTest framework, established the mapping from the FlashTest concepts to the DESCQA ones, and contributed to several utilities in the framework. A.N.R. contributed to the SAG catalog. A.T. assisted in generating the MBII hydrodynamic galaxy catalog, provided validation data sets used in several tests, and contributed to the manuscript. C.V.-M. contributed to the SAG catalog generation. R.H.W. helped initiate the project and contributed to the manuscript, overall project guidance, and to the SHAM-ADDSEDS catalogs. R.Z. developed the galaxy color distribution test and contributed to the manuscript. Y.Z. provided the iHOD galaxy catalog and contributed to the manuscript. The MBII hydrodynamical simulation was run on the BlueWaters facility at the National Center for Supercomputing Applications. Argonne National Laboratory's work was supported under the DOE contract DE-AC02-06CH11357.Part of this work uses the computational resources at the SLAC National Accelerator Laboratory, a U.S. DOE Office; we thank the support of the SLAC computational team. The framework used for this project is derived from a framework that was developed for FLASH and applied to the Dark Energy Survey by the DOE-supported ASC/Alliances Center for Astrophysical Thermonuclear Flashes at the University of Chicago; we thank Gus Evrard, Michael Busha, and Andrey Kravtsov for their contributions. We thank Joanne Cohn for discussions on validation tests. This research has made use of NASA's Astrophysics Data System. Y.-Y.M. is supported by the Samuel P. Langley PITT PACC Postdoctoral Fellowship and was supported by the Weiland Family Stanford Graduate Fellowship. S.A.C. acknowledges grants from Consejo Nacional de Investigaciones Científicas y Técnicas (PIP-112-201301-00387), Agencia Nacional de Promoción Científica y Tecnológica (PICT-2013-0317), and Universidad Nacional de La Plata (UNLP 11-G124), Argentina. T.D.M. acknowledges funding from NSF ACI-1614853, NSF AST-1517593, NSF AST-1009781, NSF AST-1616168, and the BlueWaters PAID program. F.L. and R.M. acknowledge the support of the Department of Energy Early Career Award program. Z.L.'s work was partially funded by the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. DOE Office of Advanced Scientific Computing Research and the Office of High Energy Physics. J.A.N. and R.Z. acknowledge support from DOE under grant DE-SC0007914.N.P. acknowledges support from Fondecyt 1150300, BASAL PFB-06 Centro de Astrofísica y Tecnologías Afines. P.M.R. acknowledges the support and hospitality of the University of Michigan Department of Physics, where initial framework development was done during a sabbatical visit. C.V.-M. was supported by a fellowship from CONICET, Argentina.Y.Z. acknowledges the support of the CCAPP Fellowship. The DESC acknowledges ongoing support from the Institut National de Physique Nucléaire et de Physique des Particules in France; the Science & Technology Facilities Council in the United Kingdom; and the Department of Energy, the National Science Foundation, and the LSST Corporation in the United States.DESC uses the resources of the IN2P3 Computing Center (CC-IN2P3–Lyon/Villeurbanne - France) funded by the Centre National de la Recherche Scientifique; the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under contract No. DE-AC02-05CH11231; STFC DiRAC HPC Facilities, funded by UK BIS National E-infrastructure capital grants; and the UK particle physics grid, supported by the GridPP Collaboration.This work was performed in part under DOE contract DE-AC02-76SF00515.Python 2.7, Python 3.6, Astropy <cit.>, h5py (http://www.h5py.orgh5py.org), Matplotlib <cit.>, NumPy <cit.>, SciPy <cit.>§ SIMULATIONS AND SYNTHETIC SKY CATALOGSHere we describe the set of synthetic catalogs used in our case studies (<ref>). We first provide a description of the MassiveBlack-II simulations (<ref>), for both the hydrodynamical and gravity-only runs. In the following subsections, we discuss the six different methods (one hydrodynamical simulation, one HOD-based model, two SHAM-based models, and two SAMs) to generate the eight synthetic catalogs used in <ref>. All of the synthetic methods used to generate the catalogs are applied to the same dark matter structures (i.e., halos and merger trees) of the MBII DMO simulation. A brief summary of these catalogs is listed in <ref>. §.§ The MassiveBlack-II (MBII) SimulationsMBII is a state-of-the-art, high-resolution cosmological hydrodynamic simulation <cit.> of structure formation with subgrid model physics described in detail below. A companion simulation, MBII DMO, uses the same volume, resolution, cosmological parameters, and initial conditions but only takes gravitational forces into account <cit.>. Both of these simulations have been performed in a cubic periodic box of size 142.45Mpc on a side using the cosmological TreePM Smooth Particle Hydrodynamics (SPH) code p-gadget, which is a hybrid version of the parallel code, gadget2 <cit.>, that has been upgraded to run on petascale supercomputers. The total number of dark matter particles in both simulations is 1792^3 with an equal (initial) number of gas particles in the hydrodynamical simulation. The cosmological parameters are chosen for consistency with WMAP7 <cit.>, with amplitude of matter fluctuations set by σ_8 = 0.816, the scalar spectral index n_s = 0.96, matter density parameter Ω_m = 0.275, cosmological constant density parameter Ω_Λ = 0.725, baryon density parameter Ω_b = 0.046 (in MBII), and Hubble parameter h = 0.702.<ref> lists the box size (L_box), force softening length (ϵ), total number of particles including dark matter and gas (N_part), mass of dark matter particles (m_DM), and mass of gas particles (m_gas) for the two simulations. The major results from the hydrodynamical simulation, MBII, are available in <cit.>. In addition to gravity and SPH, MBII also includes the physics of a multiphase interstellar medium model with star formation <cit.>, and black hole accretion and feedback <cit.>. Radiative cooling and heating processes are included <cit.>, as is photoheating due to an imposed homogeneous ionizing ultraviolet background.For the analysis of the MBII DMO (gravity-only) simulation, we use Rockstar[bitbucket.org/gfcstanford/rockstar (commit #ca79e51)], a six-dimension phase-space halo finder, to identify halos and subhalos <cit.>, and use Consistent Trees[bitbucket.org/pbehroozi/consistent-trees (commit #2ddc70a)] to build the halo merger trees <cit.>. The halo catalogs and merger trees are available on the NERSC filesystem and will be made publicly available.The halos are defined with spherical overdensity at virialization <cit.>. At z=0, this virial overdensity (Δ_vir) is approximately 97.7 for this cosmology. Subhalos are defined as halos whose centers are within the virial radius of any other larger halo. When building the merger trees, we skip some very close-timed snapshots and use in total 177 snapshots from the simulation, with the earliest snapshot at z=20. In the halo catalogs that we provide for the synthetic catalog creators, each halo or subhalo in the catalog must have at least 20 particles associated with it. The catalog creators may apply more stringent halo mass cuts if required. §.§ MBII Galaxy CatalogThe MBII hydrodynamical simulation was analyzed by applying an FOF procedure to dark matter particles, with a dimensionless linking length of b=0.2.Gas, star, and black hole particles were then associated to their nearest dark matter particles. Subhalos were identified with the subhalo finder SUBFIND <cit.>. The galaxy stellar mass is the total mass of all the star particles bound to the subhalo. The SED of star particles in MBII are generated using the Pegase.2 stellar population synthesis code <cit.>, based on the ages, masses, and metallicities of the stars, with the assumption of a Salpeter initial mass function. Nebula (continuum and line) emissions are also added to each star particle SED, along with a correction for absorption in the intergalactic medium using the standard <cit.> prescription. The SED of a galaxy is then obtained by summing the SEDs of all the star particles in the galaxy, from which SDSS-band luminosities are calculated, based on the respective filter.More details can be found in <cit.>. Here we use the SUBFIND halo catalogs for the MBII hydrodynamical simulation as those match the published version, but we use the Rockstar–Consistent Trees catalogs for the MBII DMO run as they provide more robust merger histories <cit.>.Therefore, we caution the reader that when comparing results in <ref>, it should be kept in mind that some differences in the tests that rely on halo masses are to be expected because the MBII hydrodynamic and DMO runs use different halo finders and different mass definitions. §.§ Improved Halo Occupation Distribution () ModelThemodel <cit.> aims to provide a probabilistic mapping between halos and galaxies, assuming that the enormous diversity in the individual galaxy assembly histories inside similar halos would reduce to a stochastic scatter about the mean galaxy-to-halo connection by virtue of the central limit theorem. Therefore, the key is to derive the conditional probability distribution of host halos at fixed galaxy properties, P(𝐡̃|𝐠̃),where 𝐠̃ and 𝐡̃ are the corresponding vectors that describe the most important sets of properties. For 𝐠̃, those properties can be the stellar mass, central/satellite dichotomy, color, velocity, and alignment, and for 𝐡̃ the dark matter mass, concentration, and tidal environment.Building on the global HOD parameterization of <cit.>, <cit.> developed theformalism to solve the mapping between galaxy stellar mass and halo mass, i.e., P(M_h | M_*), using the spatial clustering and the galaxy–galaxy lensing of galaxies in SDSS. Compared to the traditional HOD methods,can include ∼84% more galaxies while taking into account the stellar mass incompleteness of galaxy samples in a self-consistent fashion.In order to link galaxy colors to the underlying dark matter halos and constrain galaxy quenching, <cit.> extended themodel to describe galaxies of different g-r colors, i.e., 𝐠̃≡{M_*, g-r}, by considering two popular quenching scenarios: (1) a “halo” quenching model in which halo mass is the sole driver for turning off star formation in both central and satellite galaxies and (2) a “hybrid” quenching model in which the quenched fraction of galaxies depends on their stellar mass while the satellite quenching has an extra dependence on halo mass. <cit.> found that the halo quenching model provides significantly better fits to the clustering and galaxy–galaxy lensing of blue galaxies above stellar masses of 10^11. The best-fittingquenching model of P(M_h | M_*, g-r) also correctly predicts the average halo mass of the red and blue centrals, showing excellent agreement with the direct weak-lensing measurements of central galaxies <cit.>.Themodeling of galaxy colors provides strong evidence that the physical mechanism that quenches star formation in galaxies above stellar masses of 10^10 is tied principally to the masses of their dark matter halos rather than the properties of their stellar components or halo age.<cit.> further demonstrated that themodel provides an excellent description of the environmental dependence and conformity of galaxy colors observed in SDSS. The currentmodel, as constrained by the clustering and galaxy–galaxy lensing of red and blue galaxies in SDSS, also correctly reproduces the stellar mass functions within each color observed by SDSS. For the purpose of this paper, we populate the halo catalog using the best-fit parameters from <cit.>. §.§ SHAM-ADDSEDS Model This synthetic catalog is a combination of the SHAM <cit.>and the ADDSEDS algorithm (R. H. Wechsler et al. 2018, in preparation; J. DeRose et al. 2018, in preparation) which is explained in more detail below. The SHAM technique is a generic scheme to connect one galaxy property (e.g., stellar mass or luminosity) with one (sub)halo property (e.g., virial mass) by assuming an approximately monotonic relation between these two properties. The two properties are matched at the same cumulative number density, and the resulting galaxy catalog, by explicit construction, preserves the input stellar mass (or luminosity) function. Common choices of the matching (sub)halo properties include M_peak and V_peak, which are the mass and the maximal circular velocity, respectively, at their peak values along the accretion history of the (sub)halo.Scatter between the galaxy and (sub)halo properties can be introduced into the matching procedure. For a constant log-normal scatter in stellar mass or luminosity, one can follow the procedure in <cit.>: first deconvolve the scatter from the stellar mass (or luminosity) function, match the two properties, and finally add a random log-normal scatter in the catalog.To generate the synthetic catalogs used in this work, we use a publicly available SHAM code,[bitbucket.org/yymao/abundancematching] which follows the procedure we outlined above, to match the (sub)halo V_peak function to the stellar mass functions from <cit.> and MassiveBlack-II, respectively.In both cases, we adopt a constant log-normal scatter of 0.15 dex in stellar mass <cit.>. Once we obtain the stellar mass for each synthetic galaxy, we also assign an absolute r-band magnitude by simply matching the stellar mass function to the luminosity function of <cit.>. We then further generate multiband magnitudes using the ADDSEDS algorithm. For each synthetic galaxy, we measure the projected distance to its fifth nearest neighbor. We then bin galaxies in absolute r-band magnitude and rank-order them in terms of this projected distance. We compile a training set consisting of the magnitude-limited spectroscopic SDSS DR6 VAGC cut to z<0.2 and local density measurements from <cit.>. This training set is rank-ordered the same way as the simulation. Each simulated galaxy is assigned the SED from the galaxy in the training set with the closest density rank in the same absolute magnitude bin. The SED is represented as a sum of templates from <cit.>, which can then be used to shift the SED to the correct reference frame and generate magnitudes in SDSS bandpasses. In our case, we assume that all of our galaxies are at redshift z=0, and the magnitudes are K-corrected to z=0.§.§ Conditional Abundance Matching (CAM) Model This synthetic galaxy catalog is created using the CAM technique <cit.>. We provide a brief description of the catalog and technique here, and point the reader to <cit.> for details.The CAM technique is similar to the SHAM technique (see <ref>), but further assigns a secondary galaxy property (e.g., specific star formation rate) according to a secondary (sub)halo property (e.g., mass accretion history). In this work, the primary galaxy property is the stellar mass, which is assigned to the V_peak of (sub)halos using a simple SHAM technique. We include a fixed log-normal scatter σ_SMHM in stellar mass at fixed V_peak. We deconvolve the scatter from the stellar mass function before matching.As for the secondary properties, specific star formation rates (sSFRs) are assigned to galaxies such that there is a correlation between sSFR and a_1/2 at fixed stellar mass, where a_1/2 is the scale factor at which the (sub)halo reached half its peak mass.This step requires drawing from a P(sSFR|M_*) distribution, here based on the Main Galaxy Sample from the SDSS Data Release 7 <cit.>, specifically a re-reduction of DR7 in the form of the NYU-VAGC LSS sample <cit.>. sSFRs are taken from the MPA-JHU DR7 catalog based on the method of <cit.>.The strength of this correlation is encoded in a rank-order correlation statistic, ρ_sSFR, which can generally take values in the range [-1,1], perfect anticorrelation to perfect correlation.The result is that our model has two explicit parameters that can be tuned:σ_SMHM and ρ_sSFR.For this study, we use fiducial values of 0.15 and 1.0, respectively. Absolute magnitudes in five bands (ugriz), K-corrected to z=0, are associated with each galaxy in the synthetic catalog by selecting a galaxy in the NYU-VAGC LSS catalog with similar stellar mass and sSFR, and carrying over each of the magnitudes.In this way, the conditional stellar mass, sSFR, and colors are preserved in the synthetic catalog.§.§ Semi-analytic Galaxies (SAG) ModelThe Semi-analytic Galaxies (SAG) approach is based on the model developed by <cit.>, which, as is usual with semi-analytic models, combines merger trees extracted from a gravity-only cosmological simulation with a set of coupled differential equations for the baryonic processes taking place within these merger trees as time evolves. The most up-to-date version of the SAG model has been further improved from the model described in <cit.>; however, those improvements are not used in the current study in order to achieve good performance.The model used here assumes that the hot gas in dark matter halos is isothermally distributed, with an initial mass calculated using the cosmic baryon fraction. This hot gas cools to form an exponential disk where stars form quiescently. Gas cooling takes place only in central galaxies (i.e., the galaxy residing in the main subhalo of a given dark matter halo); the hot gas atmosphere is stripped instantly when a galaxy becomes a satellite (strangulation scheme). Stars also form through starbursts, which can be triggered by mergers and disk instabilities contributing to bulge formation. In that case, the gas is transferred to a reservoir that is continuously consumed by star formation in a given timescale. This reservoir can be modified by successive mergers and instabilities <cit.>. Bursts are the main channel for supermassive black hole growth. Gas accretion onto these objects produces AGN feedback <cit.>. The stars formed in each star formation event produce a number of supernovae depending on the selected initial mass function. These supernovae reheat the cold gas transferring it back to the hot phase (supernovae feedback). Chemical elements produced by stellar winds and supernova explosions (both core-collapse and Type Ia supernovae) are tracked in different baryonic components, taking into account the lifetime of stars <cit.>. The current chemical implementation has been updated with new stellar yields <cit.>. Stellar luminosities and colors are modeled using the <cit.> stellar synthesis models for the stellar populations generated in model galaxies (at each integration time step).SAG depends on a number of parameters. These are tuned using the Particle Swarm Optimization technique <cit.>. For this particular run, we consider a set of best-fitting parameters obtained from the application of SAG to one of the MultiDark gravity-only cosmological simulations <cit.> with a cubic volume of (1475.6 Mpc)^3 and Planck cosmological parameters <cit.>. The observational constraints used for this calibration are the stellar mass function and the black hole–bulge mass relation, both at z=0. For the former, we adopt the compilation of data used by <cit.>, while for the latter we combine the data sets from <cit.> and <cit.>.§.§ Galacticus Galacticus <cit.> is another semi-analytic model of galaxy formation that we employ in this paper. Galacticus models the baryonic physics of galaxy growth within an evolving, merging hierarchy of dark matter halos.Baryonic processes (including gas cooling and inflow, star formation, stellar and AGN feedback, and galaxy merging) are described by a collection of differential equations that are integrated to a specified tolerance along each branch of the merger tree. Also included are instantaneous transformations, such as starbursts, that are associated with merger events.Galacticus is designed to be fully modular, allowing the physical components and processes in galaxies and halos to be interchanged easily. This permits the possibility of running everything from simplistic models based on empirical fitting functions for the rates of key processes through to fully physical models incorporating detailed treatments of chemical enrichment, galaxy and halo dynamics, black hole accretion disks, and feedback.The output of Galacticus is a catalog of galaxies at all redshifts that includes both physical properties, such as stellar masses, sizes, and morphologies, and observational properties, such as luminosities in any specified bandpass filter. The luminosities are computed by convolving the star-formation history for each galaxy with spectra obtained from stellar population synthesis models. For this paper, we computed rest-frame luminosities in SDSS ugriz filters.Note that, for this paper, in order to ensure consistency of the input halo catalogs with the other synthetic methods, we disable a standard convergence-testing feature in Galacticus, which ensures that they reach sufficient temporal and mass resolution.The parameters of the model are constrained through either particle swarm optimization or Markov Chain Monte Carlo techniques to match a wide variety of data on the galaxy population, including the stellar mass function from z=0 to z=5, the z=0 Hi mass function, the galaxy size–mass relation, and the two-point correlation function of galaxies. The resulting models can, in principle, accurately reproduce key observables of the galaxy population across a wide range of redshifts. However, the parameters also depend on simulation details such as the mass resolution, and so, in practice, the parameters need to be tuned for each simulation. For this paper, we used a default parameter set obtained by tuning on Press–Schechter trees, and so we do not expect to find good agreement between the Galacticus catalog and the validation data. yahapj | http://arxiv.org/abs/1709.09665v2 | {
"authors": [
"Yao-Yuan Mao",
"Eve Kovacs",
"Katrin Heitmann",
"Thomas D. Uram",
"Andrew J. Benson",
"Duncan Campbell",
"Sofía A. Cora",
"Joseph DeRose",
"Tiziana Di Matteo",
"Salman Habib",
"Andrew P. Hearin",
"J. Bryce Kalmbach",
"K. Simon Krughoff",
"François Lanusse",
"Zarija Lukić",
"Rachel Mandelbaum",
"Jeffrey A. Newman",
"Nelson Padilla",
"Enrique Paillas",
"Adrian Pope",
"Paul M. Ricker",
"Andrés N. Ruiz",
"Ananth Tenneti",
"Cristian Vega-Martínez",
"Risa H. Wechsler",
"Rongpu Zhou",
"Ying Zu"
],
"categories": [
"astro-ph.IM",
"astro-ph.CO"
],
"primary_category": "astro-ph.IM",
"published": "20170927180000",
"title": "DESCQA: An Automated Validation Framework for Synthetic Sky Catalogs"
} |
Department of Mathematics and Statistics, Texas Tech University Department of Mathematics and Statistics, Texas Tech UniversityDepartment of Mathematics and Statistics, Texas Tech University Department of Mathematics and Statistics, Texas Tech University [giacomoCorresponding]Corresponding Author [email protected][giacomoFootnote]1108 Memorial Circle, Department of Mathematics and Statistics, Texas Tech University, Lubbock TX 79409, USAIn this paper we study convergence estimates for a multigrid algorithm with smoothers of successive subspace correction (SSC) type, applied to symmetric elliptic PDEs. First, we revisit a general convergence analysis on a class of multigrid algorithms in a fairly general setting, where no regularity assumptions are made on the solution. In this framework, we are able to explicitly highlight the dependence of the multigrid error bound on the number of smoothing steps. For the case of no regularity assumptions, this represents a new addition to the existing theory. Then, we analyze successive subspace correction smoothing schemes for a set of uniform and local refinement applications with either nested or non-nested overlapping subdomains. For these applications, we explicitly derive bounds for the multigrid error, and identify sufficient conditionsfor these bounds to be independent of the number of multigrid levels. For the local refinement applications, finite element grids with arbitrary hanging nodes configurations are considered. The analysis of these smoothing schemes is cast within the far-reaching multiplicative Schwarz framework. Multigrid; SSC algorithm; Domain Decomposition; Hanging nodes; Local refinement; V-cycle.§ INTRODUCTION Multigrid algorithms have been introduced in the literature since the 1960's with pioneering works such as <cit.>. A wide literature of both theoretical and computational works has been developed ever since, driven by appealing features such as optimal computational complexity <cit.>. Multigrid methods have been studied for the approximate solution of partial differential equations in various discretization schemes,starting with finite differences <cit.>and then moving to finite elements <cit.> and other settings. The first results were obtained for elliptic operators of either symmetric <cit.> or non-symmetric type <cit.>. Convergence proofs of multigrid algorithms usually rely on two propertiesreferred to as the smoothing and the approximation property<cit.>. The former is related to the definition of the smoothing operator involved in the algorithm, while the latter is usually proved assuming full elliptic regularityfor the solution of the partial differential equation.A breakthrough in the convergence analysis took place with <cit.>. In this work the elliptic regularity assumption has been dropped. The error bound obtained is not optimal in the sense that it becomes worse as the number of multigrid levels increases; moreover, no dependence of the bound on the number of smoothing iterations is shown. Further work has been done in this direction by Bramble and Pasciak <cit.>,where they showed that optimal convergence can be obtained provided that partial regularity assumptions are made. However, no dependence on the number of smoothing iterations was reported yet. In <cit.>, convergence estimates were obtained by the same authors for the case of a multigrid algorithm with non-symmetric subspace correction smoothers under no regularity assumptions. The error bound obtained for applications to both uniform and local refinementshowed quadratic dependence on the total number of multigrid spacesbut not on the number of smoothing steps. An improvement in addressing this matter was made in <cit.>,where the author showed that the multigrid error bound is optimaland can be improved when increasing the number of smoothing iterations, under partial regularity assumptions and using a Richardson relaxation scheme.More recently, further work on multigrid methods that rely on minimal regularity assumptions has been done in <cit.>, where graded meshes obtained by a variant of the newest vertex bisection method are considered.This work aims at first to further contribute to the description of multigrid methodsby carrying out a general convergence analysis that does not require any regularity assumption.Only three clear assumptions on the smoothing error operator are identified which produce a multigrid error bound that shows dependence on the number of smoothing steps and on the continuity constants of the smoothing error operators in the topology of the energy norm. As mentioned in the abstract, the explicit dependence of the multigrid error bound on the number of smoothing iterations is a new result under no-regularity assumptions. The setup of this framework is then used to analyzesmoothing schemes of successive subspace correction (SSC) type. A unifying scheme encompassing successive subspace correction algorithmsis given by Xu in <cit.>. See also <cit.> for an analysis of smoothers in a general frameworkto which subspace correction smoothers of either additive or multiplicative type belong. We will study both uniform and local refinement applications with arbitrary hanging nodes configurations, to show under what conditions on the subdomain solvers multigrid convergence is achieved, and when it is possible to obtain optimal multigrid error bounds, i.e., independent of the total number of levels. For the uniform refinement case, such results upgrade the ones in <cit.>. For the two local refinement applications,we derive ad-hoc decompositions of finite element spacesand set suitable choices of approximate subdomain solvers. These are needed when dealing with hanging nodes that are introduced by the local refinement procedure. In the first local refinement case, we construct a decomposition that has the advantage of being easy to implement and suitable forstandard finite element codes, but it does not allow freedom in the choice of the subdomains on which the subspaces are built. A second decomposition requires additional work to ensure continuity of the finite element solutionand so it requires a non-standard finite element implementation.However, it enables a choice of the subdomains that does not depend on the multigrid level. Numerical results for a similar choice of decomposition, together with a complexity analysis of the resulting algorithm, have been provided in <cit.>. In such a work, the smoothing procedure is carried out only locally,rather than at all nodes of a given multigrid level, as we do in this theory. A convergence analysis for this local smoothing approach is available in <cit.>. In both the aforementioned local refinement applications studied in this paper,the multigrid error bound shows a quadratic dependence on the multigrid leveland this agrees with what was found in <cit.>.Furthermore, we explicitly show a dependence on the number of smoothing steps. This allows us to identify conditions on the number of smoothing iterationsthat guarantee convergence as well as optimality of the error bound.Basically, these conditions establish a balance between the action of the smoothing errorand the number of smoothing steps. In the applications, we obtain smoothing error boundsthat are either constant or increase tending to one with increasing level,thus corresponding to a poorer smoothing action with increasing level. In order to have convergence, only one smoothing iteration at each level is sufficient. Nevertheless, we find that optimality of the multigrid error bound may be obtained only with a quadratically increasing number of smoothing steps. Thus, the convergence deterioration of the smoother with an increasing number of levelsmay be compensated by an ad-hoc number of smoothing stepsin order to obtain optimality. We remark that when a local smoothing procedure is conducted, increasing the number of smoothing iterationsat a given multigrid level would only improve the multigrid error bound up to a given saturation value. As a consequence, a deterioration of the error bound that goes with the total number of levels could not be balanced by increasing the number of smoothing steps.The outline of the paper is as follows.In Section <ref> the multigrid algorithm isdescribed together with a general convergence theory. Such theory is based on three assumptions on the smoothing error operator and needs no regularity. Section <ref> illustrates the algorithm used for the smoothing iterationand shows how it can be related to the multigrid convergence theory. Uniform and local refinement applications of the analysis described in the previous sections are presented in Section <ref>, where convergence bounds are obtained for the specific cases. Finally, we draw our conclusions. § THE MULTIGRID ALGORITHM In this section we describe the multigrid algorithm subject to our analysis. Throughout the paper, the total number of levels will be denoted as J. For k = 0,…,J, let V_k be a finite-dimensional vector space such thatV_0 ⊂ V_1 ⊂…⊂ V_J, and let (·,·) anda(·,·) be two symmetric positive definite (SPD) bilinear forms on V_k.Hence, both bilinear forms are inner products on V_k.Let ||·|| = √(( · , · ))and ||·||_E = √(a ( · , · )) be the corresponding induced norms.Associated with these inner products, let us also define the operators Q_k : V_J → V_k and P_k : V_J → V_kas the orthogonal projections with respect to (· , · ) and a(· , · ) respectively, namely, for all v ∈ V_J and allw ∈ V_k (Q_k v , w)= (v , w ), a(P_k v , w)= a(v , w ).Note that from this definition it follows thata((I - P_k) v , w) = 0w ∈ V_k. The multigrid algorithm seeks solutions of the following problem: given f ∈ V_J, find u ∈ V_J such that a(u,v) = (f, v)v ∈ V_J. Before we can present the multigrid algorithm studied in this paper,we need to introduce a few operators that will be used in the description of the method.For k = 0,…, J, define the operators A_k : V_k → V_k as(A_k u, v) = a(u , v)u,v ∈ V_k. The operator A_k is SPD with respect to (·,·)as a consequence of the symmetry and positive definiteness of a(·,·).If we set f_k = Q_k f, then at level k the problem we want to solve consists in finding u_k ∈ V_k such that A_k u_k = f_k. The prolongation I_k-1^k : V_k-1→ V_k andrestriction I^k-1_k : V_k→ V_k-1 operators are defined for all v ∈ V_k-1 and all w ∈ V_k byI_k-1^k v= v ,(I^k-1_k w , v)= (w , I_k-1^k v).We are now ready to present the multigrid algorithm considered in this paper.Let B_k: V_k→ V_k denote a smoothing operator. Associated to B_k we can define a smoothing error operator S_k : V_k→ V_kas S_k = I - B_k A_k,whose properties will be discussed later in detail.For k=0, …, J, letMG_k: V_k× V_k→ V_k be the multigrid operators. The purpose of the operators MG_k is to yield an approximate solution to(<ref>).They are defined here in a recursive manner. Note that the total number of pre-smoothing iterations m_k is assumed to be dependent on the level k. Also, we are assuming the same number m_k of pre-smoothing and post-smoothing steps at each level.We also remark that we consider a symmetric version of the multigrid algorithm as in <cit.>,in the sense that both pre-smoothing and post-smoothing are performed.Since we have only one iteration for the error correction step, this algorithm is referred to as V-cycle <cit.>. §.§ Convergence analysis Here we present a general convergence analysis of the multigrid algorithm <ref>. We do so by introducing sufficient assumptions on the smoothing error operator S_kfor the derivation of the convergence results. It is then clear that the convergence properties of the multigrid algorithmare intimately dependent on the smoothing procedures. Before listing the assumptions, we recall the expressions of the error operatorsassociated to B_k and MG_k. Let z_k^(i) be the output of a pre- or post-smoothing iteration at level k. If we denote the associated error as e_k^(i) = u_k - z_k^(i), then substituting for z_k^(i) we havee_k^(i) = u_k - z_k^(i-1) - B_k (f_k - A_k z_k^(i-1)) = S_ke_k^(i-1) ,so that the effect of the smoothing step can be described ase_k^(m_k) = S_k^m_ke_k^(0).The multigrid error operator E_k : V_k → V_k associated to MG_k is defined recursively asE_0= 0,E_k= S_k^m_k [I - (I - E_k-1) P_k-1] S_k^m_k .Note that E_0 is assumed to be zero since we are using a direct solver at level k=0. This means that B_0 = A_0^-1 and S_0 = 0. Here we summarize the properties of the E_k operators.For a proof see <cit.> or <cit.> for the special case where z_k^(0) = 0.Let z_k^(0)∈ V_k, and let u_k be the exact solution to A_k u_k = f_k. Then u_k - MG_k( z_k^(0),f_k) = E_k ( u_k- z_k^(0) ), k≥0. Moreover, the E_k's are symmetric positive semidefinite with respect to a(· , · ) for k ≥ 0.We now state sufficient hypotheses for multigrid convergence.As the expression of the multigrid error operator (<ref>) suggests,once the operators P_k are given by the differential problem at hand, multigrid convergence is affected by the properties of S_k and by the number of smoothing steps m_k.These features are reflected in the following assumptions. For all k = 1,…, J ,S_k is a symmetric positive semidefinite operator on V_k with respect to a(· , · ). This means that for all v, w in V_k we have a(S_k v , w) = a(v , S_k w)a(S_kv , v ) ≥ 0.For all k = 1,…, J there exists a number δ_kwith 0 < δ_k < 1 such thata(S_k v ,v) ≤δ_k a(v ,v)v ∈ V_k.For all k = 1,…, J, the finite sequence ψ_k = m_k (1-δ_k) is non-increasing,where m_k is the number of smoothing steps per leveland δ_k is the quantity in Assumption <ref>. §.§.§ Smoothing and approximation properties Assumptions <ref> and <ref> guarantee that the operators S_ksatisfy certain monotonicity propertiesgiven by Lemmas <ref> and <ref> below. These properties will lead to the smoothing property of Lemma <ref>.Let Assumptions <ref> and <ref> hold. Let α and β be two integers such that 0 ≤α≤β. Then,a(S_k^β v , v ) ≤ a(S_k^α v , v)v ∈ V_k.We will prove it for β=α+1 and the result will then follow by induction. By (<ref>), S_k is positive semidefinite with respect to a(·,·), therefore also S_k^α is.Then there is a unique positive semidefinite square root operator S_k^α/2, see <cit.>. By the symmetry of S_k it follows that S_k^α/2 is symmetric as well. Considering also (<ref>), we have that for v ∈ V_ka(S_k^α+1v ,v)= a(S_k^αS_k v , v) = a(S_k^α/2S_k^α/2S_k v , v)= a(S_k^α/2S_k v , S_k^α/2v) = a(S_k S_k^α/2v , S_k^α/2v)≤ a(S_k^α/2v , S_k^α/2v) =a(S_k^α v , v) .Using the previous lemma, we can prove the next result.Let Assumptions <ref> and <ref> hold.Let α and β be two integers such that 0 ≤α≤β. Then,a((I-S_k)S_k^β v , v ) ≤ a((I-S_k)S_k^α v , v)v ∈ V_k. As we did before, we will prove it for β=α+1 and the result will follow by induction.a((I-S_k)S_k^α+1 v , v )= a((I-S_k)S_k^α+1 v , (I -S_k + S_k)v )= a((I-S_k)S_k^α+1 v , (I -S_k)v) + a((I-S_k)S_k^α+1 v , S_kv)= a(S_k^α+1(I-S_k) v , (I -S_k)v) + a((I-S_k)S_k^α+1 v , S_kv) ≤ a(S_k^α(I-S_k) v , (I -S_k)v) + a((I-S_k)S_k^α+1 v , S_kv)=a(S_k^α(I-S_k) v , v)- a(S_k^α(I-S_k) v , S_k v)+ a((I-S_k)S_k^α+1 v , S_kv) =a(S_k^α(I-S_k) v , v)- a(S_k^α+1(I-S_k) v , v)+ a(S_k^α+1(I-S_k) v , S_kv) = a(S_k^α(I-S_k) v , v) - a(S_k^α+1(I-S_k) v , (I-S_k)v) ≤ a(S_k^α(I-S_k) v , v) = a((I-S_k) S_k^αv , v).Now we are ready to prove the smoothing property of the operator S_k. Let Assumptions <ref> and <ref> hold. Let v ∈ V_k. Then,a((I - S_k)S_k^2m_kv ,v ) ≤12m_ka((I - S_k^2m_k)v , v). By Lemma <ref> we get(2m_k)a((I-S_k)S_k^2m_kv , v) =a((I-S_k)S_k^2m_kv , v)+ … + a((I-S_k)S_k^2m_kv , v)_2 m_k times≤ a((I-S_k)v , v) + a((I-S_k)S_kv , v) + … + a((I-S_k)S_k^2m_k - 1v , v)= a( (I - S_k + S_k - S_k^2 + … + S_k^2m_k-1 - S_k^2m_k) v , v) = a((I-S_k^2m_k)v , v).Before we can show a bound on the error E_k, we need to establish the approximation property. Let Assumptions <ref> and <ref> hold, and let w ∈ V_k.Then,a((I - P_k-1)w , w) ≤(11-δ_k )a((I-S_k)w, w).Let y = (I - P_k-1) w. Note that from the definition of P_k-1 and from the nestedness of the spaces (<ref>)we have that a(y , P_k-1w) = 0 .By Assumption <ref> we havea((I-S_k)w , w)≥ (1 - δ_k) a(w , w)= (1 - δ_k)a(y + P_k-1w , y + P_k-1w) = (1 - δ_k)( a(y , y) + a(P_k-1w , P_k-1w)) (by (<ref>))≥ (1 - δ_k)a((I-P_k-1)w , (I-P_k-1)w) = (1 - δ_k)a((I-P_k-1)w , w).Notice that in this setting the approximation property given by Lemma <ref>is dependent on the smoothing property. This is in contrast with other analyses of multigrid methodsin which the smoothing and the approximation properties are derived independently <cit.>. As a consequence of the approximation property, we have the following result. Let Assumptions <ref> and <ref> hold.Let v ∈ V_k. Then,a((I-P_k-1)S_k^m_kv , (I-P_k-1)S_k^m_kv)≤(11-δ_k )12m_k a((I -S_k^2m_k)v , v).We havea((I-P_k-1)S_k^m_kv , (I-P_k-1)S_k^m_kv)= a((I-P_k-1)S_k^m_kv , S_k^m_kv) ≤(11-δ_k ) a((I-S_k)S_k^m_kv , S_k^m_kv)= (11-δ_k ) a((I-S_k)S_k^2m_kv , v)≤(11-δ_k ) 12 m_k a ((I - S_k ^2 m_k)v , v).§.§.§ Error bound We are now in a position to obtain a bound on the multigrid error operator E_J that gives convergence.Let Assumptions <ref>, <ref> and <ref> hold.For k = 0, 1, …, J letγ_k = 11 + 2 m_k (1 - δ_k). Then, if v ∈ V_J,a(E_J v , v) ≤ γ_Ja(v ,v). By Assumption <ref>, we haveγ_k-1≤γ_k.The proof will be done by induction as in <cit.>. For k=0, E_0 = 0 so the result is obvious. By induction assume thata(E_J-1 v , v) ≤ γ_J-1a(v ,v) ∀ v ∈ V_J-1 .Now consider v ∈ V_J, thena(E_J v , v)= a(S_J^m_J v , S_J^m_J v) - a(P_J-1S_J^m_Jv , P_J-1S_J^m_Jv) + a(E_J-1P_J-1S_J^m_J v, P_J-1S_J^m_Jv)= a((I - P_J-1)S_J^m_Jv , (I - P_J-1)S_J^m_Jv) + a(E_J-1P_J-1S_J^m_J v, P_J-1S_J^m_Jv) ≤a((I - P_J-1)S_J^m_Jv , (I - P_J-1)S_J^m_Jv)+ γ_J-1a(P_J-1S_J^m_J v, P_J-1S_J^m_Jv) ≤ a((I - P_J-1)S_J^m_Jv , (I - P_J-1)S_J^m_Jv)+ γ_Ja(P_J-1S_J^m_J v, P_J-1S_J^m_Jv) =(1-γ_J) a((I - P_J-1)S_J^m_Jv , (I - P_J-1)S_J^m_Jv) +γ_J a((I - P_J-1)S_J^m_Jv , (I - P_J-1)S_J^m_Jv)+ γ_Ja(P_J-1S_J^m_J v, P_J-1S_J^m_Jv)=(1-γ_J) a((I - P_J-1)S_J^m_Jv , (I - P_J-1)S_J^m_Jv) + γ_Ja(S_J^m_J v, S_J^m_Jv)≤(12 m_J (1-δ_J)) (1-γ_J) a((I - S_J^2m_J)v ,v) + γ_Ja(S_J^m_J v, S_J^m_Jv) = γ_J a((I - S_J^2m_J)v ,v) + γ_Ja(S_J^m_J v, S_J^m_Jv)= γ_J a(v,v) - γ_J a(S_J^m_Jv , S_J^m_Jv) + γ_J a(S_J^m_Jv, S_J^m_Jv)= γ_J a(v,v)It is evident that the convergence of the multigrid algorithm is dependent on Assumption <ref>, which liesboth on the number of smoothing steps m_k and on the constants δ_k of the smoothing error operator, jointly.No other parameters affect the convergence rate.Since the behavior of δ_k is determined by the choice of S_k,different choices on m_k can be taken subsequently so that (<ref>) holds.For instance, if the S_k are such that δ_k is non-decreasing, then it is sufficient to take m_k to be non-increasing. In fact, if δ_k ≥δ_k-1 and m_k ≤ m_k-1,Assumption <ref> holds sincem_k (1-δ_k) ≤ m_k-1 (1-δ_k-1) . Although sufficient,a non-increasing m_k is not necessary. Also, observe that a particular case of non-increasing m_k is m_1 = m_2 = … = m_J = m. In this case it is directly visible how an increasing m can lead to better convergence rates, as well as an increasing number of multilevel spaces J can lead to worse convergence. While this last situation was shown in <cit.>, to the best of our knowledge the first featurewas not shown in similar multigrid frameworks without, or with minimal regularity assumptions(see, e.g., <cit.>).In Section <ref> a characterization of δ_k will be given when the smoothing process is chosen to be a successive subspace correction algorithm.From this characterization, proper choices of the number of smoothing steps m_kcan lead to convergence and, in addition, to optimal (i.e., with a value of γ_J that is independent of J)multigrid error bounds.Since m_k ∈ℕ and δ_k ∈ℝ,our analysis suggests that the determination of an optimal multigrid error bound may take place by a proper choice of m_kif and only if the quantity (1-δ_k) is inversely proportionalto an integer-valued function of k.The achievement of an optimal convergence bound seems otherwise impossible, unless a radically new setup of the multigrid algorithm is formulatedthat is oriented to that purpose.§ SUCCESSIVE SUBSPACE CORRECTION (SSC) ALGORITHMSNow we describe the Successive Subspace Correction (SSC) algorithm. The SSC algorithm is an iterative method to approximate the solution of SPD linear systems <cit.>. We link the multigrid convergence theory of the previous section with the SSC theory by using smoothers of subspace correction type for the multigrid algorithm <ref>.The SSC algorithm yields an approximate solution to (<ref>)and is based on a decomposition of the finite-dimensional space as an algebraic sum of subspaces. In the multigrid algorithm presented above, smoothing is performed at each level k = 1, ... , J,therefore we decompose each V_k using subspaces V_k^i ⊂ V_k such thatV_k = ∑_i=0^p_k V_k^i = { v| v = ∑_i=0^p_k v_k^i , v_k^i ∈ V_k^i } .Notice that the number of subspaces p_k is in general different for each level. In order to present the algorithm, we first definefor all i,with u ∈ V_k and u_k^i , v_k^i ∈ V_k^i, the operatorsQ_k^i : V_k → V_k^i ,P_k^i: V_k → V_k^i,A_k^i : V_k^i → V_k^i , (Q_k^i u , v_k^i) = (u , v_k^i),a(P_k^i u , v_k^i) = a(u , v_k^i),(A_k^i u_k^i , v_k^i) = (A_k u_k^i , v_k^i).It follows from its definition that A_k^i is an SPD operator. Moreover, as a consequence of the above definitions we have that A_k^i P_k^i = Q_k^i A_k .Hence if u_k is the exact solution of (<ref>), then P_k^i u_k = u_k^i will be the solution of A_k^i u_k^i = f_k^i,where f_k^i = Q_k^i f_k. Equation (<ref>) is in general solved approximately, therefore we introduce for all i the operatorsR_k^i : V_k^i → V_k^i ,T_k^i : V_k → V^i_k,T_k^i := R_k^i Q_k^i A_k = R_k^i A_k^i P_k^i. The operators R_k^i act as approximate inverses of A_k^i.If R_k^i is taken to be an exact solver then T_k^i = P_k^i.When no confusion arises, we drop the subscript k for p_kas well as for the operators Q_k^i, P_k^i, A_k^i, R_k^i and T^i_k. We now define the SSC algorithm.The error operator associated to this algorithm is denoted as E_p. If u_k is the exact solution of (<ref>), then for i = p , … , 0 we have(u_k - z^α + p+1-i/p+1) = (I - T^i) (u_k - z^α+ p-i/p+1).This yields(z_k - z^α + 1) = E_p (z_k - z^α) ,E_p= (I - T^0) ( I - T^1) ⋯ (I - T^p). The symmetric version of the SSC algorithm is given here.The error operator of this algorithm is denoted as E_p^s and is given byE^s_p = (I - T^p) ⋯ ( I - T^1)(I - T^0)^2( I - T^1) ⋯ (I - T^p).Because of the symmetry requirements for the smoother in Assumption <ref>, we will fit the smoother within the symmetric SSC framework.§.§ Convergence analysis We recall the main convergence result about the SSC algorithm,whose proof can be found in <cit.>. First, we introduce sufficient assumptions. [Bound on w_1] Theoperators R^i are SPD with respect to (·,·) and satisfy w_1 < 2,where w_1 = max_i=0, … ,pρ(R^i A^i), ρ(R^i A^i) being the spectral radius of R^i A^i and p being the number of subspaces in the decomposition (<ref>). [Existence of K_0] There existsK_0 such that for any v ∈ V_k there exists a decomposition v = ∑_i=0^p v_i, with the property∑_i=0^p ( (R^i)^-1 v_i , v_i) ≤ K_0 (A_k v , v).[Existence of K_1] Given the same p as in Assumption <ref>, there existsK_1 such that for any S ⊂{0,1, … , p} × {0,1, … , p} and u_i , v_i ∈ V_k fori = 0,1, … , p we have∑_(i,j)∈ S | a(T^i u_i , T^j v_j ) |≤ K_1( ∑_i=0^p a(T^i u_i , u_i ) )^1/2 ( ∑_j=0^p a(T^j v_j , v_j ) )^1/2 . Notice that all assumptions are related to the choice of the operators R^i. Assumption <ref> involves only functions in V_k, without using the decomposition in Assumption <ref>. We remark that the absolute value in Equation (<ref>) is sufficient but not necessary for convergence, see <cit.>. Due to Assumption <ref>, R^i is invertible so that (<ref>) is well-defined. Also, recall the following property. Let Assumption <ref> hold. The operator T^i is symmetric and positive semi-definite with respect to a(· , · ). Let u and v be in V_k.Using the fact that A_k is symmetric with respect to (·,·) in V_k, R_k^i is symmetric with respect to (·,·) in V_k^i, together with the definition of Q_k^i, we havea(T_k^i u, v) = a(R_k^i Q_k^i A_k u, v) = (R_k^i Q_k^i A_k u ,A_k v)= (R_k^i Q_k^i A_k u, Q_k^i A_k v) = (Q_k^i A_k u, R_k^iQ_k^i A_k v)= (Q_k^i A_k u, T_k^i v) = (A_k u, T_k^i v) = a(u, T_k^i v).This shows T_k^i is symmetric with respect to a(·,·). To see that it is also positive semi-definite, we use the same properties as above and geta(T_k^i u, u) = (R_k^i Q_k^i A_k u , A_k u)= (R_k^i Q_k^i A_k u,Q_k^i A_k u).Since R_k^i is SPD with respect to (·,·) by Assumption <ref>, the result follows.By the symmetry of T^i with respect to a(·,·), I - T^i is symmetric with respect toa(·,·). Hence E_p^* = (I - T^p) ⋯ ( I - T^1)(I - T^0) is the adjoint of E_p with respect to a(·,·), so that E_p^s = E_p^* E_p.Thus, we have a(E_p^s v , v) = || E_p v||_E^2 ∀v ∈ V_k.We now state the convergence result.Let Assumptions <ref>, <ref> and <ref>hold. Then, we have|| E_p ||_E^2 ≤ 1 - 2 - w_1K_0 (1 + K_1)^2 ,where w_1 was defined in Assumption <ref> and K_0 and K_1 are constants related to the ones in Assumptions <ref> and <ref>.See <cit.> for a proof.Let us point out that the quantity in the right-hand side of (<ref>) has a nonzero denominator, is larger than 0 and less than 1.In fact, notice that Assumption <ref> implies that (R^i)^-1 is also SPD with respect to (·,·),which, together with the fact that A is SPD with respect to (·,·),implies K_0 > 0. Then,2 - w_1K_0 (1 + K_1)^2 is well-defined and by Assumption <ref> we have2 - w_1K_0 (1 + K_1)^2 > 0.Finally, the constant K_0 can be majorized by another constant such that (<ref>) still holds and 2 - w_1K_0 (1 + K_1)^2 < 1. The convergence of the symmetric version of the SSC algorithm is then a direct consequence of (<ref>).As we dropped the subscript k to make the notation more readable, we point outthat the quantities w_1, K_0 and K_1 in general depend on k.Moreover, also the quantity p in Algorithms <ref> or <ref> can be chosen differently for different multigrid levels.§.§ Sufficient conditions for multigrid convergence with smoothers of SSC type Our intent is to fit the properties of the SSC smoother to the sufficient conditions neededfor the convergence of the multigrid algorithm <ref>. Choosing the symmetric SSC iteration as the smoother for our multigrid algorithm, we then haveS_k = E_p_k^s.Our purpose is to consider a set of choices of V_k in the multigrid algorithmand of smoothers S_k = E_p_k^sfor which Assumptions <ref>, <ref> and <ref> are satisfied.First, the existence of a suitable error operator E_p_k^s with norm less than one is given by the fulfillment of Assumptions <ref>, <ref> and <ref>. Once the existence of this operator is granted, Assumptions <ref> and <ref> are true.In fact, Assumption<ref> is a consequence of the fact that E_p_k^s is SPD with respect to a(·,·). Concerning Assumption <ref>,we set δ_k asδ_k = 1 - 2 - w_1K_0 (1 + K_1)^2 .It then follows by (<ref>) and (<ref>) thatAssumption <ref> holds.The only assumption that remains to be checked is Assumption <ref>, which again depends on all the Assumptions <ref>, <ref> and <ref>. This is due to the fact that δ_k in (<ref>) is determined by w_1, K_0 and K_1,all of which in general depend on k. Different definitions of V_k and S_k correspond to multigrid algorithmson different spaces with different subspace correction smoothing schemes. Some examples will be described in Section <ref>. In these, a verification of Assumptions <ref>, <ref>, <ref> is provided,along with a characterization of the constants w_1, K_0 and K_1 in terms of k.This characterization leads to the identification of the conditions for Assumption <ref> to hold. The conditions for the optimality of the multigrid error bound are also determined. § REFINEMENT APPLICATIONS WITH SUBSPACE CORRECTION SMOOTHING SCHEMESIn this section, we apply the multigrid algorithm with SSC-type smoother described earlier, to applications involving uniform and local refinement and also domain decomposition smoothing. Multiplicative domain decomposition algorithms can in fact be seen as instances of SSC algorithms <cit.>. In the following, we introduce the model problem and its finite element discretization. The applications of the theory that we consider here differin the definition of V_k,in its decomposition into appropriate subspaces V^i_kand in the choice of the operators R^i_k. We first address a case of uniform refinement with exact subsolvers. Then, we present two local refinement applicationsthat deal with two possible ways of enforcing continuity,corresponding to appropriate choices of the subspace decomposition of the multigrid spaces V_k.§.§ Model problem and finite element discretization To fix the ideas, let Ω be a polygonal subset of ℝ^n, let Θ = (θ_ij) be a symmetric matrix and consider the elliptic boundary value problem- ∑_i=1^n∑_j=1^n ∂∂ x_i( θ_ij∂ u∂ x_j) = f Ω u = 0∂Ω,then u is a weak solution of the above problem if and only ifa(u,v) = (f,v)v ∈ H_0^1(Ω)where (·, ·) denotes the L^2(Ω) inner product and a(u,v) = - ∑_i=1^n∑_j=1^n∫_Ωθ_ij∂ u∂ x_j∂ v∂ x_idx.We assume that there exist C_Θ,1 and C_Θ,2 depending on Θ for whichC_Θ,1 u ^2_H_1≤ a(u,u) ≤ C_Θ,2 u ^2_H_1,u ∈ V_k ⊂ H_0^1(Ω).This means that both (· , ·) and a(·, ·) are SPD bilinear forms on V_k,as needed in the previous convergence theory. Moreover, since the trace of V_k is zero on the boundary of Ω, we have that a(· , ·) is also equivalent to |·|_H^1(Ω) on V_k due to the Poincaré inequality.Let 𝒫_1 be the space of linear polynomials,then the multigrid spaces V_k in (<ref>)will be considered to be the finite element spacesof continuous piecewise-linear functions built on triangulations 𝒯_k of Ω,V_k = { v ∈ H_0^1(Ω) : v|_τ∈𝒫_1, ∀τ∈𝒯_k } k=0, …, J.Such triangulations will be definedby using either uniform or local midpoint refinement. In the case where such refinement procedure is performed only on a subdomain of Ω (local refinement), hanging nodes will be introduced in the mesh and the triangulation will be referred to as irregular (or non-conforming). Hanging nodes (also called slave nodes by some authors) are vertices of some element τ_1 ∈𝒯_kthat lie on the interior of an edge of some other element τ_2 ∈𝒯_k without being a node for τ_2.A more formal description of hanging nodes can be found in <cit.>.Continuity constraints can be added in the definition of the finite element spacesto make sure that no additional degrees of freedom are introduced for the hanging nodes.Therefore for all k=0, …, J, V_k has a nodal basis that consists of functions associated toall vertices of 𝒯_k excluding the hanging nodes. In practice, a possible way to obtain a continuous nodal basis is given in <cit.>, where shape functions of elements with hanging nodes in the element corners are modified. In <cit.>, the support of a basis function associated to a regular node n is the union of elements that share this node or the potential hanging nodes on the edges that have node n. We point out that the local refinement applications covered by our theory allow the presence of edges with an arbitrary number of hanging nodes. Usually, only 1-irregular meshes are considered <cit.>, namely meshes where at most one hanging node is allowed on any edge of the triangulation. §.§ Uniform refinement: overlapping non-nested subdomains,subproblems on regular gridsand exact subsolversWe first describe a case of uniform refinement by defining the triangulations and the corresponding subdomains on each of them. Let 𝒯_0 be a quasi-uniform coarse triangulation of Ω of size h_0 ∈ (0,1]. Assume 𝒯_k-1 has been obtained, then 𝒯_k is derived from 𝒯_k-1 by means of midpoint refinement. It follows that the size h_k of 𝒯_k will be h_k = 2^-k h_0 and that𝒯_0 ⊂𝒯_1 ⊂⋯𝒯_k,in the sense that any τ∈𝒯_k can be written as the union of elements in 𝒯_k+1 <cit.>. Let {Ω^i_k}_i=1^p_k be a collection of non-overlapping open subdomains of Ω whose boundaries align with the mesh triangulation 𝒯_k, such that Ω = ⋃_i=1^p_kΩ^i_k. For i=1, …, p_k, let Ω^i_kbe overlapping subsets of Ωwhose boundaries still align with the triangulation and are defined by Ω^i_k = { x ∈Ω|(x, Ω^i_k) ≤h_0 } .Notice that the number of subdomains p_k varies with the level. An example of a subdomain described in the above definition is shown in Figure <ref>.For this application,the multigrid spaces V_k in (<ref>),the subspaces V_k^i andthe subsolvers R_k^i are defined as follows. Given the triangulations 𝒯_k in Definition <ref> and the overlapping subdomains Ω^i_k in Definition <ref>,we set for k = 0, …, J and for i=0, …, p_kV_kin (<ref>), built on 𝒯_k as in Definition <ref> ,V^i_k:= V_0fori = 0 {v ∈ V_k |supp(v) ⊆Ω^i_k}fori = 1, …, p_k,R_k^i:= (A_k^i)^-1 .We point out that the V_k defined in (<ref>) satisfy the nestedness condition (<ref>). The following lemma describes a decomposition of V_k.Given V_k and V^i_k in Definition <ref>, we haveV_k = ∑_i=0^p_kV^i_k .Moreover, if we denote with v_i ∈ V^i_k the components of any v ∈ V_k(i.e., such that v = ∑_i=0^p_k v_i),there is a constant C_0 independent of h_0, h_k and p_k such that∑_i=0^p_k a(v_i , v_i) ≤ C_0 a(v,v)∀ v ∈ V_k .A proof of this result can be found in <cit.> and <cit.>.The choice of R_k^i implies that R^i_k A^i_k = I for all i=0, …, p_k and k=0, …, J and so we have w_1,k = w_1 = 1. Assumption <ref> is then satisfied. Now we can look at Assumptions <ref> and <ref>by showing the existence of the parameters K_0 and K_1 for this application. Let V_k be as in Definition <ref>.Then, there exists a constant K_0 satisfying Assumption <ref>. Let v ∈ V_k and consider the decomposition of V_k provided by Lemma <ref>. Thenwe have by (<ref>)∑_i=0^p_k ((R^i_k)^-1v_i , v_i)= ∑_i=0^p_k (A^i_k v_i , v_i) = ∑_i=0^p_k a(v_i , v_i) ≤ C_0 a(v,v).This shows that K_0 exists and K_0 = C_0.Let V^i_k and R^i_k as in Definition <ref>.Then, there exists a constant K_1 satisfying Assumption <ref>.Here we follow a variation of a procedure in <cit.>, Section 2.5. Given the subdomains Ω_k^1, …, Ω_k^p_k as in Definition <ref>, we define the symmetric p_k × p_k matrix G byG_i j = {[1 Ω_k^i ∩Ω_k^j ≠∅,;0 Ω_k^i ∩Ω_k^j = ∅;] g_0 = max_i=1,…,p_k(∑_j=1^p_k G_i j) = || G ||_∞Note that g_0 represents the maximum number of neighbors intersecting a subdomain (counting self intersections)and it does not depend on p_k but only on the geometry of the triangulation. Unlike <cit.>,the summation in the definition of the constant g_0 does not exclude the i term. Also, by the choice of the subdomains, g_0 will be uniformly bounded. Let S ⊂{0,1, … , p_k}×{0,1, … , p_k}, and consider the decomposition as in <cit.>, namelyS= S_00 ∪S_10 ∪S_01 ∪S_11 ,S_00 = {(i,j) ∈S | i = 0 , j = 0},S_10 = {(i,j) ∈S | 1 ≤i≤p_k , j = 0} ,S_01 = {(i,j) ∈S | i = 0 , 1 ≤j ≤p_k} ,S_11 = {(i,j) ∈S | 1 ≤i , j ≤p_k} . Let u_i , v_i ∈ V_k for i = 0,1, … , k, then the sum over S can be split as ∑_(i,j) ∈ S |a(T^i u_i,T^j v_j)|= ∑_(i,j) ∈ S_00|a(T^i u_i , T^j v_j)| + ∑_i:(i,0) ∈ S_10 |a(T^i u_i , T^0 v_0)|+ ∑_j:(0,j) ∈ S_01 |a(T^0 u_0 , T^j v_j)| +∑_(i,j) ∈ S_11 |a(T^i u_i,T^j v_j)| Let us consider one summand at a time. By the Cauchy-Schwarz inequality and Lemma <ref> we have that( ∑_(i,j) ∈ S_00 |a(T^i u_i , T^j v_j)| )^2≤(∑_i=0^p_ka(T^i u_i,u_i))(∑_j=0^p_ka(T^j v_j,v_j)). If for given i and j, Ω_k^i ∩Ω_k^j = ∅, then a(T^i u_i , T^j v_j) = 0.Hence for the last summand we have( ∑_(i,j) ∈ S_11 |a(T^i u_i,T^j v_j)| )^2 = ( ∑_(i,j) ∈ S_11 G_i j|a(T^i u_i,T^j v_j)| )^2≤( ∑_(i,j) ∈ S_11 G_ij√(a(T^i u_i,T^i u_i))√(a(T^j v_j,T^j v_j)))^2= ( ∑_(i,j) ∈ S_11 G_ij√(a(T^i u_i,u_i))√(a(T^j v_j,v_j)))^2(by def of P^i_k) ≤( ∑_i=1^p_k∑_j=1^p_k G_ij√(a(T^i u_i,u_i))√(a(T^j v_j,v_j)))^2 ≤ρ(G)^2(∑_i=1^p_ka(T^i u_i,u_i))(∑_j=1^p_ka(T^j v_j,v_j))(by (4.12) in <cit.>) ≤ g_0^2(∑_i=0^p_ka(T^i u_i,u_i))(∑_j=0^p_ka(T^j v_j,v_j)), where ρ(G) denotes the spectral radius of G which satisfies ρ(G) ≤ ||G||_∞. Considering that a(T^iu_i , T^0v_0)=0 anytime Ω_k^i ∩Ω_k^0 = ∅, for the second term of the sum in (<ref>) we have( ∑_i:(i,0) ∈ S_10 |a(T^iu_i , T^0v_0)| )^2 ≤( ∑_i:(i,0) ∈ S_10 G_0i√(a(T^iu_i , T^iu_i ))√(a(T^0v_0 , T^0v_0)))^2 = ( ∑_i:(i,0) ∈ S_10 G_0i√(a(T^iu_i , T^iu_i )))^2a(T^0v_0 , T^0v_0)= ( ∑_i:(i,0) ∈ S_10 G_0i√(a(T^iu_i , T^iu_i )))^2a(T^0v_0 , T^0v_0) ≤(∑_i=1^p_k G_0i) ( ∑_i:(i,0) ∈ S_10 a(T^iu_i , T^iu_i )) a(T^0v_0 , T^0v_0)≤g_0(∑_i=0^p_ka(T^i u_i,u_i)) a(T^0v_0 , v_0) (by def of P^i_k and g_0)≤g_0(∑_i=0^p_ka(T^i u_i,u_i))(∑_j=0^p_ka(T^jv_j , v_j)).Similarly, for the last term of the sum we have( ∑_j:(0,j) ∈ S_01 |a(T^0u_0 , T^jv_j)| )^2 ≤g_0(∑_i=0^p_ka(T^i u_i,u_i))(∑_j=0^p_ka(T^jv_j , v_j)).Combining these four inequalities, it follows that(∑_(i,j) ∈ S |a(T^i u_i,T^j v_j)| )^2 ≤ 4 (1 + 2 g_0 + g_0^2)(∑_i=0^p_ka(T^i u_i,u_i))(∑_j=0^p_ka(T^j v_j,v_j)).This shows that K_1 exists and K_1 = 2 (1+g_0).The next result follows immediately from Lemmas <ref> and <ref>. It shows how the assumption about the non-increasing behavior of ψ_k in (<ref>) is satisfied.Let V^i_k and R^i_k as in Definition <ref>.Then Assumption <ref> is satisfied withδ_k = 1 - 1C_0(3+ 2 g_0)^2 , ψ_k =m_kC_0(3+ 2 g_0)^2 ,if and only if m_k is non-increasing. Here, C_0 is the constant from Lemma <ref> and g_0 is defined in (<ref>). Notice that the constant δ_k is independent of k. Hence, we have the convergence result. If m_k is non-increasing, the multigrid algorithm <ref> converges with γ_k = C_0 (3 + 2 g_0)^2C_0 (3 + 2 g_0)^2 + 2 m_k,where γ_k are the constants defined in (<ref>).Moreover, if m_1 = m_2 = … = m_J, the error bound is optimalin the sense that it does not deteriorate as the number of multigrid spaces J increases.Notice that convergence can be achieved even by performing only one smoothing iteration,but a larger m_k can further lower the error bound.§.§ Local refinement:overlapping nested subdomains, subproblems on regular gridsand approximate subsolversNow we move to an application involving a locally refined grid. Let {Ω_k}_k=0^J be a collection of closed subdomains of Ω such thatΩ_J ⊂Ω_J-1⊂⋯Ω_0 ≡Ω.Let 𝒯_0 be a coarse quasi-uniform triangulation of Ω of size h_0∈ (0,1].Assume 𝒯_k-1 has been defined, then 𝒯_k is obtained performing midpoint refinement only on those elements of 𝒯_k-1 that belong to Ω_k.This process introduces hanging nodes, causing the grid 𝒯_k to become irregular, for all k=1,…,J. However, restricted to Ω_k, 𝒯_k is a regular grid without hanging nodes and size h_k =2^-k h_0. We observe that the sequence {𝒯_k}_k=0^J is nested in the sense that an element T ∈𝒯_k-1 can be written as the union of elements in 𝒯_k <cit.>. Moreover, by construction we have that h_0 = max_T ∈𝒯_k h_T, where h_T denotes the size of one element T ∈𝒯_k.Figure <ref> sketches an example of triangulation for this case. Concerning the spaces V_k, the subspaces V^i_k and the corresponding subsolvers R^i_k we choose the following. Given the overlapping subdomains Ω_iand the triangulations 𝒯_k in Definition <ref>,we set for k = 0, …, J and for i=0, …, kV_k= { v ∈ H_0^1(Ω) ∩ C^0(Ω) : v|_τ∈𝒫_1, ∀τ∈𝒯_k,}, where 𝒯_k is as in Definition <ref> ,V^i_k:=V_0,i = 0,{ v ∈ V_i |supp(v) ⊆Ω_i} , i = 1, …, k,R^i_k:= A_0^-1 , i = 0, 1λ^i kI , i = 1, …, kwhere λ^i denotes the spectral radius of A^i.We point out that the V_k satisfy by construction the nestedness condition (<ref>). Moreover, the continuity requirement in the definition of V_k implies that its nodal basis will have no function associated to hanging nodes of 𝒯_k. Also, since the support of the functions in each V^i_k is contained in Ω_i, the subproblems are all defined on uniformly refined grids without hanging nodes,although 𝒯_k is irregular. This considerably simplifies the implementation since no actual constraints have to be addedand no change in the nodal basis is required to obtain a continuous numerical solution. If v ∈ V^i_k, it will be a linear combination of the basis functionsassociated with the interior nodes of Ω_i. The following lemma is a consequence of the choice of the subspaces introduced in Definition <ref>.See also <cit.> for more on this decomposition. Given V_k and V^i_k in Definition <ref>, we haveV_k = ∑_i=0^k V^i_k.The result follows if for given v ∈ V_k we can find a decompositionv = ∑_i=0^k v_i such that v_i ∈ V^i_k.To do this, we will consider a result from <cit.>that relies on the construction of a sequence of operators Q_i : V_k → V_i. Let V_i be the space obtained by taking Ω_0 = Ω_1 = … = Ω_i,namely the space built over a uniformly refined triangulation of size h_i = h_0 2^-i and let Q_i be the L^2(Ω) projection operator onto V_i. Set Q_k = I and for i=0, …, k-1 define Q_i v = w as the unique function on V_i that satisfiesw = { [ Q_i v ,; v . ]It has been shown in <cit.> that (Q_i - Q_i-1) vis a function in V^i_k for all i = 1, …, k and that ((Q_i - Q_i-1)v , (Q_i - Q_i-1)v) ≤C_1h^2_0a(v,v)i =1, …, k ,((Q_i - Q_i-1)v , (Q_i - Q_i-1)v) ≤ C_1 λ_i^-1a(v,v)i =1, …, k, a(Q_i v , Q_i v) ≤ C_2 a(v,v)i = 0, …, k-1,where λ_i denotes the spectral radius of the operator A_i and C_1, C_1 and C_2 do not depend on i. It then follows that for all v∈V_kv = Q_0 v + ∑_i=1^k (Q_i - Q_i-1) v = ∑_i=0^k v_i ,wherev_i :=Q_0 v,i = 0,(Q_i - Q_i-1) v, i = 1, …, k, with v_i∈ V^i_k for all i.In this case p_k = k. This means that at each level k, the number of subdomains is fixed and equal to k as well. At the given level k, notice that V^i_k ⊈ V^j_k, ∀ i>j, since the trace of V^i_k on ∂Ω_i is zerowhile the trace of V^j_k is not. Moreover, it follows from Definition <ref> that the V^i_k are independent of k. Consequently so will be A^i, in the sense that A^i_i = A^i_i+1 = … = A^i_k. Note that with the choice of R^i_k in (<ref>) we have ρ(R_k^0 A_0) = 1 and ρ(R^i_k A^i) = 1/k for all i = 1, …, k. This implies that w_1,k = w_1 = 1,so that Assumption <ref> is satisfied. Now we can show the existence of the parameters K_0 and K_1.Let V^i_k and R^i_k as in Definition <ref>.Then, there exists a constant K_0 satisfying Assumption <ref>.Using the definition of R_k^i together with (<ref>) and (<ref>) we have∑_i=0^k ((R_k^i)^-1 v_i, v_i) = (A_0 v_0 , v_0) +k∑_i=1^k λ^i ((Q_i - Q_i-1)v , (Q_i - Q_i-1)v) ≤ a(v_0 , v_0) + k∑_i=1^k C_1 λ^iλ_ia(v,v) ≤ C_2 a(v,v) + k^2 C_1 a(v,v) ≤max{C_1 , C_2}(1 +k^2) a(v,v) = C_3 (1 + k^2) a(v,v).This shows that K_0 exists and K_0 = C_3 (1 + k^2).Let us now show the existence of K_1 for this application.Let V^i_k and R^i_k as in Definition <ref>. Then, there exists a constant K_1 satisfying Assumption <ref>. For i = 1, …, k and u ∈ V_k we have0 ≤ a(T^i u,T^i u)= a(R_k^iQ^iA_k u,R_k^iQ^iA_k u)= ( 1λ^i k)^2 a(Q^iA_k u,Q^iA_k u)= ( 1λ^i k)^2 (Q^iA_k u,A^iQ^iA_k u)≤( 1λ^i k)^2 λ^i (Q^iA_k u,Q^iA_k u)= ( 1λ^i k^2) (Q^iA_k u,A_k u)=1k(R_k^iQ^iA_k u,A_k u)=1k(T^i u,A_k u) =1ka(T^i u, u) .For i=0 we have R_k^0 = A_0^-1 so that T^0 = P^0 and a(T^0 u , T^0 u)= a(P^0 u , P^0 u) = a(P^0 u ,u) = a(T^0 u ,u). In summarya(T^i u ,T^i u)= a(T^i u ,u) , i = 0,≤1ka(T^i u, u) , i = 1, …, k. Let S ⊂{0,1, … , k}×{0,1, … , k},and consider the decomposition of such set as before with p_k = k, namelyS= S_00∪ S_10∪ S_01∪ S_11 ,S_00= {(i,j) ∈ S | i = 0 , j = 0} ,S_10= {(i,j) ∈ S | 1 ≤i≤ k , j = 0} , S_01= {(i,j) ∈ S | i = 0 , 1 ≤ j ≤ k} ,S_11= {(i,j) ∈ S | 1 ≤i , j ≤ k} .Let u_i , v_i ∈ V_k for i = 0,1, … , k, then∑_(i,j) ∈ S |a(T^i u_i,T^j v_j)|= ∑_(i,j) ∈ S_00 |a(T^i u_i , T^j v_j)| + ∑_i:(i,0) ∈ S_10 |a(T^i u_i , T^0 v_0)| +∑_j:(0,j) ∈ S_01 |a(T^0 u_0 , T^j v_j)| +∑_(i,j) ∈ S_11 |a(T^i u_i,T^j v_j)|.Let us consider one summand at a time. By the Cauchy-Schwarz inequality with the a(·,·) inner product, (<ref>) and Lemma <ref> we have that( ∑_(i,j) ∈ S_00 |a(T^i u_i , T^j v_j)| )^2≤ a(T^0 u_0 ,u_0) a(T^0 v_0 ,v_0) ≤(∑_i=0^ka(T^i u_i,u_i))(∑_j=0^ka(T^j v_j,v_j)). For the last summand we have,using again the same properties,( ∑_(i,j) ∈ S_11 |a(T^i u_i,T^j v_j)| )^2≤( ∑_(i,j) ∈ S_11√(a(T^i u_i,T^i u_i)) √(a(T^j v_j,T^j v_j)))^2 ≤1k^2( ∑_i:(i,j) ∈ S_11√(a(T^i u_i,u_i)))^2( ∑_j:(i,j) ∈ S_11√(a(T^j v_j,v_j)))^2 ≤1k^2 ( (∑_i=1^ka(T^i u_i,u_i)) k ) ( (∑_j=1^k a(T^j v_j,v_j))k ) ≤(∑_i=0^ka(T^i u_i,u_i))(∑_j=0^ka(T^j v_j,v_j)).For the second term of the sum in (<ref>) usethe Cauchy-Schwarz inequality with the a(·,·) inner product,and (<ref>), ( ∑_i:(i,0) ∈ S_10 |a(T^iu_i , T^0v_0)| )^2 ≤( ∑_i:(i,0) ∈ S_10√(a(T^iu_i , T^iu_i ))√(a(T^0v_0 , T^0v_0)))^2 = ( ∑_i:(i,0) ∈ S_10√(a(T^iu_i , T^iu_i )))^2a(T^0v_0 , T^0v_0) ≤(∑_i:(i,0) ∈ S_10 1 ) (∑_i:(i,0) ∈ S_10 a(T^iu_i , T^iu_i ) )a(T^0v_0 , v_0) ≤ k1k(∑_i:(i,0) ∈ S_10 a(T^iu_i , u_i ) )a(T^0v_0 , v_0) ≤(∑_i=0^ka(T^i u_i,u_i))(∑_j=0^ka(T^j v_j,v_j)).Similarly, for the last term of the sum we have( ∑_j:(0,j) ∈ S_01 |a(T^0u_0 , T^jv_j)| )^2≤(∑_i=0^ka(T^i u_i,u_i))(∑_j=0^ka(T^j v_j,v_j)).Combining these four inequalities, it follows that(∑_(i,j) ∈ S |a(T^i u_i,T^j v_j)| )^2 ≤ 4(∑_i=0^ka(T^i u_i,u_i))(∑_j=0^ka(T^j v_j,v_j)).This shows that K_1 exists and K_1 = 2. Let V^i_k and R^i_k as in Definition <ref>.Then, Assumption <ref> is satisfied withδ_k= 1 - 1C_4 (1 + k^2), ψ_k =m_kC_4 (1 + k^2) ,if and only if m_k is chosen so that ψ_k is non-increasing. Here, C_4 = 9 C_3 and C_3 is the constant in (<ref>). Note that δ_k is now increasing.Various choices of m_k guarantee Assumption <ref>:constant m_k = 1,decreasing m_k = J + 1 - k,increasing m_k = 1 + k.We now state the convergence result.If m_k is chosen so that ψ_k is non-increasing, the multigrid algorithm <ref> converges with γ_k = C_4 (1 + k^2)C_4 (1 + k^2) + 2 m_k,where γ_k is defined in Theorem <ref>, k = 0, 1, …, J.Moreover, the error bound is optimal (in the sense that it does not depend on the number of multigrid spaces J) if and only if m_k = q (1+k^2) for some q ∈ℕ, and is given byγ_1 = γ_2 = … = γ_J = C_42 q + C_4 . We observe that the number of smoothing iterations appears in the error boundand this was not shown in <cit.>.Although the choice m_k = q (1+k^2) is not optimal in terms of computational cost,since more smoothing steps are needed on finer grids,nevertheless it guarantees that the error bound is independent of the number of levels.§.§ Local refinement: overlapping non-nested subdomains, subproblems on irregular gridsand exact subsolvers We now describe another local refinement application. We keep the same triangulations as in Definition <ref> and the same definition of V_k as in the previous local refinement application. However, the subdomains are chosen as in Definition <ref>. This will lead to a different characterization of the space V_k. A sketch of the subdomains involved in this application is visible in Figure <ref>.Moreover, unlike Section<ref>, the overlapping subdomains Ω^k_i at each level k are not nested. Let us now define the spaces V_k and choose the subspaces for its decomposition, and the subsolvers R^i_k. Given the triangulations 𝒯_k in Definition <ref> and the subdomains Ω_i in Definition <ref>,we set for k = 0, …, J and for i=0, …,p_kV_k= { v ∈ H_0^1(Ω) ∩ C^0(Ω) : v|_τ∈𝒫_1, ∀τ∈𝒯_k,} where 𝒯_k is as in Definition <ref> ,V^i_k:=V_0, i = 0,{v ∈ V_k |supp(v) ⊆Ω^k_i} , i = 1, …, p_k, ,R_k^i:= (A_k^i)^-1 . Since the definition of V_k in Definition <ref> coincides with Definition <ref>,the V_k again satisfy the nestedness condition (<ref>) and the nodal basis does not have any functionassociated to the hanging nodes, because of the continuity requirement. Here, we give another characterization ofV_k based on the non-nested subdomains.Given V_k and V^i_k in Definition <ref>, we haveV_k = ∑_i=0^p_kV^i_k.Moreover, if we denote with v_i ∈ V^i_k the components of any v ∈ V_k (such that v = ∑_i=0^p_k v_i), then there is a constant C_k dependent only on k such that∑_i=0^p_k a(v_i , v_i) ≤ C_k a(v,v)∀ v ∈ V_k .We are going to construct a set of functions {v_i}_i=0^p_k∈ V^i_k ⊆V_k such that every v ∈ V_k can be expressed as their sum. To this end, let {θ^k_i}_i=1^p_k be a smooth partition of unity subordinate to the cover {Ω^k_i}_i=1^p_k. This means that ∑_i=1^p_kθ^k_i = 1, 0 ≤θ^k_i(x) ≤ 1 for all x ∈Ω^k_i and supp(θ^k_i) ⊂Ω^k_i, for all i=1 …, p_k.Let V^j be the subspace of V_k defined in Definition <ref> in the previous local refinement application. Then we know that V_k = ∑_j=0^kV^j, so that any v in V_k can be written as v = ∑_j=0^kv_j,where v_j ∈V^j are given by (<ref>). Define ℐ^j_h to be the standard nodal interpolant of the finite element spaceV^j for all j=1,…,k. Note that this is well defined since each V^j is built on a quasi-uniform grid. Then, for v ∈ V_k, setv_0 =v_0, v_i = ∑_j=1^k ℐ^j_h( θ^k_iv_j) ,i = 1, …,p_k.Notice that all the terms in the sum that defines v_i are functions in V^j ⊂ V_k andhave support in Ω^k_i, therefore they all belong to V_k^i.Moreover, using the fact that the ℐ^j_h are linear and projections we havev= ∑_j=0^k v_j = v_0 + ∑_j=1^k v_j=v_0 + ∑_j=1^k ℐ^j_h ( v_j ) =v_0 + ∑_j=1^k ℐ^j_h (∑_i=1^p_kθ^k_i v_j )= v_0+ ∑_j=1^k∑_i=1^p_kℐ^j_h( θ^k_iv_j) = ∑_i=0^p_k v_i.To prove the second part of the lemma, let us proceed one summand at a time.If T ∈𝒯_k ∩Ω^k_i, then using an inverse estimate (see <cit.>) we get| ℐ^j_h( θ^k_iv_j) |^2_H^1(T) ≤ h_0^-2|| ℐ^j_h( θ^k_iv_j) ||^2_L^2(T)≤ h_0^-2 C ||θ^k_iv_j ||^2_L^2(T)≤ h_0^-2 C || v_j ||^2_L^2(T),where the constant C is the bound for the operator norm of ℐ^j_h and it only depends on the reference element <cit.>. Summing over all T ∈𝒯_k ∩Ω^k_i (remember that we assumed the subdomains align with the triangulation) we obtain|v_i|^2_H^1(Ω) = |v_i|^2_H^1(Ω^k_i)≤∑_T ∈𝒯_k ∩Ω^k_i(∑_j=1^k |ℐ^j_h( θ^k_iv_j)|_H^1(T))^2 ≤∑_T ∈𝒯_k ∩Ω^k_i k ∑_j=1^k |ℐ^j_h( θ^k_iv_j)|^2_H^1(T)≤k ∑_j=1^kh_0^-2 C || v_j ||^2_L^2(Ω^k_i) . Summing over the subdomains Ω^k_i, and considering that each point in Ω is covered only a finite number of times <cit.> we obtain ∑_i=1^p_k |v_i|^2_H^1(Ω)≤k ∑_j=1^kh_0^-2C || v_j ||^2_L^2(Ω) .Thanks to (<ref>) we can say that|| v_j ||^2_L^2(Ω)≤C_1h_0^2 |v|^2_H^1(Ω).Therefore, using the previous results and the Poincaré inequality we have∑_i=1^p_k a(v_i,v_i) ≤C∑_i=1^p_k |v_i|^2_H^1(Ω)≤ k^2 C |v|^2_H^1(Ω)≤ k^2 C a(v,v).Again by (<ref>) we know that a(v_0 , v_0) ≤ C_2 a(v , v), hence if we let C_0 = max{C_2 , C} we can conclude with∑_i=0^p_k a(v_i,v_i) ≤C_0(1+k^2) a(v,v).The proof of this lemma for a uniform refinement case relies on the uniform boundedness of the standard nodal interpolator on V_k.In the case where an irregular grid is employed a nodal interpolator in the classical sense cannotbe defined on V_k.An alternative to the solution we adopted in our proof could be to use interpolation operatorsspecifically designed for irregular grids as in <cit.>.For the subsolvers we clearly have that R^i_k A^i = I for all i=0, …, p_k and so again we have w_1,k = w_1 = 1. Assumption <ref> is then true. However, from a practical point of view, defining problems on irregular grids actually requires theimplementation of the constraints that make the nodal basis of V_k continuous, as in <cit.>.Now we can show the existence of K_0 and K_1.Let V^i_k and R^i_k as in Definition <ref>.Then, there exists a constant K_0 satisfying Assumption <ref>. Considering the decomposition of v given by Lemma <ref>, we have ∑_i=0^p_k ((R^i_k)^-1v_i , v_i) = ∑_i=0^p_k (A^i_k v_i , v_i) = ∑_i=0^p_k a(v_i , v_i)≤C_0 (1+k^2) a(v,v).This shows that K_0 exists and K_0 = C_0 (1+k^2).Let V^i_k and R^i_k as in Definition <ref>.Then, there exists a constant K_1 satisfying Assumption <ref>. The existence of K_1 can be carried out exactly as for Lemma <ref>concerning the case of uniform refinement. Therefore K_1 exists and K_1 = 2 (1+g_0). The next lemma immediately follows.Let V^i_k and R^i_k as in Definition <ref>.Then, Assumption <ref> is satisfied withδ_k = 1 - 1C_0(1+k^2)(3+ 2 g_0)^2,ψ_k =m_kC_0(1+k^2)(3+ 2 g_0)^2 ,if and only if m_k is chosen so that ψ_k is non-increasing. Here, C_0 is the constant in (<ref>) and g_0 is defined in (<ref>).The constant δ_k is again increasing. Consequently the convergence bound for the multigrid algorithm is obtained.If m_k is chosen so that ψ_k is non-increasing,the multigrid algorithm <ref> converges with γ_k = C_5 (1 + k^2)C_5 (1 + k^2) + 2 m_k ,where γ_k is defined in Theorem <ref> and C_5 = C_0(3+ 2 g_0)^2.Moreover, the error bound is optimal (in the sense that it does not depend on the number of multigrid spaces J) if and only if m_k = q (1+k^2) for some q ∈ℕ, and is given byγ_1 = γ_2 = … = γ_J = C_52 q + C_5 . § CONCLUSIONS In this paper we performed a convergence analysisof a multigrid algorithmfor symmetric elliptic PDEs under no regularity assumptions with smoothers of SSC type. In particular, we focused on the dependence of the multigrid error boundon the number of smoothing steps. This represents a novel result for the case of no-regularity assumptions.We provided an analysis that can be used for any smoothing procedure of symmetric SSC type. We then utilized this framework to addressuniform and local refinement applications and study convergence bounds for the multigrid error. Our theory allows an arbitrary number of hanging nodes on a given edge of the triangulation. A judicious choice of the subdomain solvers and of the number of smoothing steps at each level can avoid the dependence of the multigrid error bound on the total number of multigrid levels. To this end, proper decompositions of the finite element spaces had to be derived in the analysis. For the uniform refinement case, a uniform bound for the multigrid error can be obtained, even regardless of the choice of the smoothing steps. For the local refinement applications, we describedtwo different subspace decompositionsof the multigrid space using overlapping nested or non-nested subdomains that correspond to different ways of enforcing the continuityof the finite element space when hanging nodes are present. In both cases, we show that convergence can be obtained and optimality can be guaranteedby appropriately choosing the number of smoothing steps for each refinement level.A computational analysis of the methods proposed in this paper will be subject to future investigation. § ACKNOWLEDGMENTS This work was supported by the National Science Foundation grant DMS-1412796. § REFERENCES | http://arxiv.org/abs/1709.09628v1 | {
"authors": [
"Eugenio Aulisa",
"Giorgio Bornia",
"Sara Calandrini",
"Giacomo Capodaglio"
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"published": "20170927170631",
"title": "Convergence estimates for multigrid algorithms with SSC smoothers and applications to overlapping domain decomposition"
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0cm 0cm 0cm 16cm defiDefinitionthm[defi]Theorem ex[defi]Example rem[defi]Remarkprop[defi]Proposition lemme[defi]Lemma cor[defi]Corollary | http://arxiv.org/abs/1709.09482v1 | {
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"Bruno Colbois",
"Ahmad El Soufi",
"Said Ilias",
"Alessandro Savo"
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"published": "20170927130611",
"title": "Eigenvalue upper bounds for the magnetic Schroedinger operator"
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theoremTheorem lemmaLemma propProposition corCorollary definition[1][Definition]#1equationsection plain thmTheorem[section] University of Connecticut and Carnegie Mellon UniversityProjective, Sparse, and Learnable Latent Position Network Models Neil A. Spencer and Cosma Rohilla Shalizi================================================================When modeling network data using a latent position model, it is typical to assume that the nodes' positions are independently and identically distributed. However, this assumption implies the average node degree grows linearly with the number of nodes, which is inappropriate when the graph is thought to be sparse. We propose an alternative assumption—that the latent positions are generated according to a Poisson point process—and show that it is compatible with various levels of sparsity. Unlike other notions of sparse latent position models in the literature, our framework also defines a projective sequence of probability models, thus ensuring consistency of statistical inference across networks of different sizes. We establish conditions for consistent estimation of the latent positions, and compare our results to existing frameworks for modeling sparse networks.§ INTRODUCTION Network data consist of relational information between entities, such as friendships between people or interactions between cell proteins. Often, these data take the form of binary measurements on dyads, indicating the presence or absence of a relationship between entities.Such network data can be modeled as a stochastic graph, with each individual dyad being a random edge. Stochastic graph models have been an active area of research for over fifty years across physics, sociology, mathematics, statistics, computer science, and other disciplines <cit.>.Many leading stochastic graph models assume that the inhomogeneity in connection patterns across nodes is explained by node-level latent variables. The most tractable version of this assumption is that the dyads are conditionally independent given the latent variables. In this article, we focus on a subclass of these conditionally independent dyad models—the distance-based latent position network model (LPM) of <cit.>. In LPMs, each node is assumed to have a latent position in a continuous space. The edges follow independent Bernoulli distributions with probabilities given by a decreasing function of the distance between the nodes' latent positions. By the triangle inequality, LPMs exhibit edge transitivity; friends of friends are more likely to be friends. When the latent space is assumed to be ℝ^2 or ℝ^3, the inferred latent positions can provide an embedding with which to visualize and interpret the network.Recently, there has been an effort to classify stochastic graph models into general unified frameworks. One notable success story has been that of the graphon for exchangeable networks <cit.>. The graphon characterizesall stochastic graphs invariant under isomorphism as latent variable models. LPMs can be placed within the graphonframework by assuming the latent positions are random effects drawn independently from the same (possibly unknown) probability distribution. However, graphons can be inappropriate for some modeling tasks, due to their asymptotic properties. The typical asymptotic regime for statistical theory of network models considers the number of nodes growing to infinity in a single graph. Implicitly, this approach requires the network model to define a distribution over a sequence of increasingly sized graphs. There are several natural questions to ask about this sequence. Prominent questions include: * At what rate does the number of edges in these graphs grow? * Is the model's behavior consistent across networks of different sizes? * Can one eventually learn the model's parameters as the graph grows?For all non-trivial[The only exception is an empty graph, for which all edges are absent with probability one. ]models falling within the graphon framework, the answer to question 1 is the same; the expected number of edges grows quadratically with the number of nodes <cit.>. Such sequences of graphs—in which the average degree grows linearly—are called dense. In contrast, many real-world networks are thought to have sub-linear average degree growth. This property is known as sparsity <cit.>). For sparse graphs, graphon models are unsuitable. Accordingly, recent years have seen an effort to develop sparse graph models that preserve the advantages of graphons. In particular, the sparse graphon framework <cit.> and the graphex framework <cit.> both provide straightforward ways to modify network models from the dense regime to accommodate sparsity. In this article, we add to the sparse graph literature by formulating a new sparse LPM. We target three criteria: sparsity (<ref>), projectivity (<ref>) and learnablity (<ref>). Projectivity of a model ensures consistency of the distributions it assigns to graphs of different sizes, and learnability ensures consistent estimation of the latent positions as the number of nodes grows. As we outline in Section <ref>, the existing methods for sparsifying graphons of <cit.> and <cit.> do not satisfy these criteria; they either violate projectivity or make it difficult to establish learnability. We thus take a more specialized approach to develop our sparse LPMs, turning to non-exchangeable network models for inspiration. Specifically, our new LPM framework extends the Poisson random connection model <cit.>—a specialized LPM framework in which the nodes' latent positions are generated according to a Poisson process. We modify the observation window approach proposed by <cit.> to allow our LPMs to exhibit arbitrary levels of sparsity without sacrificing projectivity. To obtain learnability results for our LPM framework, we develop and modify a combination of results related to low rank matrix estimation <cit.>, the Davis-Kahan Theorem <cit.>, and eigenvalues of random Euclidean distance matrices. Our proof strategy culminates in a concentration inequality for a restricted maximum likelihood estimator of the latent positions that applies to wide a variety of LPMs, providing a straightforward sufficient conditions for LPM learnability.The remainder of this article is organized as follows. Section <ref> defines sparsity (<ref>) and projectivity (<ref>) for graph sequences. It also defines the LPM, establishing sparsity and projectivity results for its exchangeable (<ref>) and random connection model (<ref>) formulations. Section <ref> describes our new framework for modeling projective sparse LPMs, and includes results that demonstrate that the resultant graph sequences are projective and sparse. Section <ref> defines learnability of latent position models, and provides conditions under which sparse latent position models are learnable. Finally, Section <ref> elaborates on connections between our approach, sparse graphon-based LPMs, and the graphex framework. It also includes a discussion of the limitations of our work. All proofs are deferred to Appendix <ref>. § BACKGROUND §.§ SparsityLet (Y^n)_n=1,…, ∞ be a sequence of increasingly sized (n × n) random adjacency matrices associated with a sequence of increasingly sized simple undirected random graphs (on n nodes). Here, each entry Y^n_ij indicates the presence of an edge between nodes i and j for a graph on n nodes. We say the sequence of stochastic graph models defined by (Y^n)_n=1,…, ∞ is sparse in expectation iflim_n →∞𝔼( ∑_i=1^n ∑_j=1^n Y^n_ij/n^2) = 0.In other words, a sequence of graphs is sparse in expectation if the expected number of edges scales sub-quadratically in the number of nodes. Recall that a node's degree is defined as the number of nodes to which it is adjacent. Sparsity in expectation is equivalent to the expected average node degree growing sub-linearly. If instead the average degree grows linearly, we say the graph is dense in expectation. In this article, we are also interested in distinguishing between degrees of sparsity. We say that a graph is e(n)-sparse in expectation iflim_n →∞𝔼(∑_i=1^n ∑_j=1^n Y_ij/e(n)) = Cfor some constant C ∈ℝ_+. That is, the number of edges scales Θ(e(n)). A dense graph could also be called n^2-sparse in expectation.Note that sparsity and e(n)-sparsity are asymptotic properties of graphs, defined for increasing sequences of graphs but not for finite realizations. These definitions differ from the informal use of “sparse graph” to refer to a single graph with few edges. It also differs from the definition of sparsity for weighted graphs used in <cit.>. In practice, we typically observe a single finite realization of a graph, but the notion of sparsity remains useful because many network models naturally define a sequence of networks. §.§ ProjectivityLet (ℙ^n)_n=1…∞ denote the probability distributions corresponding to a growing sequence of random adjacency matrices (Y^n)_n=1,…, ∞ for a sequence of graphs. We say that the sequence (ℙ^n)_n=1…∞ is projective if, for any n_1 < n_2, the distribution over adjacency matrices induced by ℙ^n_1 is equivalent to the distribution over n_1 × n_1 sub-matrices induced by the leading n_1 rows and columns of an adjacency matrix following ℙ^n_2. That is, (ℙ^n)_n=1,…, ∞ is projective if for any y ∈{0,1}^n_1 × n_1,ℙ^n_1(Y^n_1 = y)= ℙ^n_2(Y^n_2∈ X),where X = {x ∈{0,1}^n_2 × n_2: x_ij = y_ij if1 ≤ i,j ≤ n_1 }.Projectivity ensures a notion of consistency between networks of different sizes, provided that they are generated from the same model class. This property is particularly useful for problems of superpopulation inference <cit.>, such as testing whether separate networks were drawn from the same population, predicting the values of dyads associated with a new node, or pooling together estimates from separate networks in a hierarchical model. Such problems require that parameter inferences be comparable across differently sized graphs. Without projectivity, it is unclear how to make comparisons without additional assumptions.Projectivity has thus received considerable attention recently in the networks literature <cit.>. Our definition of projectivity departs from others in the literature in that it depends on a specific ordering of the nodes. Other definitions require consistency under subsampling of any n_1 nodes, not just the first n_1 nodes. The two definitions coincide when exchangeability is assumed, but differ otherwise.§.§ Latent Position Network Models The notion that entities in networks possess latent positions has a long history in the social science literature. The idea of a “social space” that influences the social interactions of individuals traces back to at least the seventeenth century <cit.>. A thorough history of the notions of social space and social distance as they pertain to social networks is provided in <cit.>.In the statistical network modeling literature, assigning continuous latent positions to nodes dates back to the 1970s, in which multi-dimensional scaling was used to summarize similarities between nodes in the data <cit.>. However, it was not until <cit.> that the modern notion of latent continuous positions were used to define a probabilistic model for stochastic graphs in the statistics literature. In this article, we focus on this probabilistic formulation, with our definition of latent position models (LPMs) following that of the distance model of <cit.>. Consider a binary graph on n nodes. The LPM is characterized by each node i of the network possessing a latent position Z_i in a metric space (S,ρ). Conditional on these latent positions, the edges are drawn as independent Bernoulli random variables followingℙ(Y_ij = 1|Z_i, Z_j)= K(ρ(Z_i, Z_j)).Here, K:ℝ_+ → [0,1] is known as the link probability function; it captures the dependencyof edge probabilities on the latent inter-node distances. For the majority of this article, we assume K is independent of n (<ref> is an exception). Furthermore, we focus on link probability functions that smoothly decrease with distance and are integrable on the real line, such as expit(-ρ^2), exp(-ρ^2) and (1 + ρ^2)^-1. Though the general formulation of the LPM in <cit.> allows for dyad-specific covariates to influence connectivity, our exposition assumes that no such covariates are available. We have done this for purposes of clarity; our framework does not specifically exclude them. §.§ Exchangeable Latent Position Network Models Originally, <cit.> proposed modeling the nodes' latent positions as independent and identically distributed random effectsdrawn from a distribution f of known parametric form. This approach remains popular in practice today, with S assumed to be a low-dimensional Euclidean space ℝ^d and f typically assumed to be multivariate Gaussian or a mixture of multivariate Gaussians <cit.>.We refer to this class of models as exchangeable LPMs because they assume the nodes are infinitely exchangeable. Exchangeable latent position network models are projective, but must be dense in expectation. Exchangeable latent position network models define a projective sequence of models. Provided in <ref>. Exchangeable latent position network models define dense in expectation graph sequences. Provided in <ref>. Consequently, LPMs with exchangeable latent positions cannot be sparse. To develop sparse LPMs, we must consider alternative assumptions.§.§ Poisson Random Connection Model Instead of the latent positions being generated independently from a distribution over S, we can treat them as drawn according to a point process over S. This approach—known as the random connection model—has been well-studied in the context of percolation theory <cit.>. Most of this focus has been on random geometric graphs <cit.>, a version of a LPMs for which K is an indicator function of the distance (i.e. K(ρ(Z_i, Z_j)) ∝ I(ρ(Z_i, Z_j) < ϵ)). Here, we instead study the random connection model as a statistical model, focusing the case where K is a smoothly decaying and integrable function. In particular, we consider the Poisson random connection model <cit.>, for which the point process is assumed to be a homogeneous Poisson process <cit.> over S ⊆ℝ^d. Because Poisson random connection models on finite-measure S are equivalent to exchangeable LPMs, the interesting cases occur when S has infinite measure, such as ℝ^d. In these cases, the expected number of points is almost-surely infinite, resulting in an infinite number of nodes. These infinite graphs can be converted into a growing sequence of finite graphs via the following procedure. Let G denote an infinite graph generated according to a Poisson random connection model on S. LetS_1 ⊂ S_2 ⊂⋯⊂ S_n ⊂⋯⊂ S denote a nested sequence of finitely-sized observation windows in S. For each S_i, define G_i to be the subgraph of G induced by keeping only those nodes with latent positions in S_i. Because these positions form a Poisson process, each G_i consists of a Poisson distributed number of nodes with mean given by the size of S_i. Each G_i is thus almost-surely finite, and the sequence of graphs (G_i)_i =1, …∞ contains a stochastically increasing number of nodes. For many choices of S, such as ℝ^d, this approach straightforwardly extends to a continuum of graphs by considering a continuum of nested observation windows of (S_t)_t ∈ℝ_+. In such cases, the number of nodes follows a continuous-time stochastic process, stochastically increasing in t.As far as we are aware, the above approach was first proposed by <cit.> in the context of defining a growing sequence of geometric random graphs. Their exposition concentrated on a one-dimensional example with S = ℝ_+ and observation windows given by S_t = [0,t]. For this example, one would expect to observe n nodes if t = n, with the total number of nodes for a given t being random. As noted by <cit.>, the formulation can be altered to ensure that n nodes are observed by treating n as fixed and treating the window size t_n as the random quantity. Here, t_n it equal to the smallest window width such that [0, t_n] contains exactly n points.These two viewpoints (random window size and random number of nodes) are complementary for analyzing the same underlying process. Under the appropriate conditions, the one-dimensional Poisson random connection model results in networks which are n-sparse in expectation. We formalize this notion as Proposition <ref>. The finite window approach approach also defines a projective sequence of models, as stated in Proposition <ref>.For a Poisson random connection model on ℝ_+ with an integrable link probability function, the graph sequence resulting from the finite window approach is n-sparse in expectation.Provided in <ref>Consider a Poisson random connection model on ℝ_+ with link probability function K. Then, the graph sequence resulting from the finite window approach is projective. Provided in <ref>. These results indicate that the Poisson random connection model restricted to observation windows is capable of defining a sparse graph sequences, but only for a specific sparsity level if the link probability function is integrable.For our new framework, we extend this observation window approach to higher dimensional S. By including an auxiliary dimension, we achieve all rates between n-sparsity and n^2-sparsity (density) in expectation. § NEW FRAMEWORKWhen working in a one-dimensional Euclidean latent space S = ℝ_+, the observation window approach for the Poisson random connection model is straightforward—the width of the window grows linearly with t, with nodes arriving as the window grows. As shown in Proposition <ref>, this process results in graph sequences which are n-sparse in expectation whenever K is integrable. However, extending to d dimensions (ℝ^d) provides freedom in defining how the window grows; different dimensions of the window can be grown at different rates.We exploit this extra flexibility to develop our new sparse LPM model. Specifically, through the inclusion of an auxiliary dimension—an additional latent space coordinate which influences when a node becomes visible without influencing its connection probabilities—we can control the level of sparsity of the graph by trading off how quickly we grow the window in the auxiliary dimension versus the others.In this section, we formalize this auxiliary dimension approach, showing that it allows us to develop a new LPM framework for which the level of sparsity can be controlled while maintaining projectivity. Our exposition consists of two parts: first, we present the framework in the context of a general S. Then, we concentrate on a special subclass with S = ℝ^d for which it is possible to prove projectivity, sparsity, and establish learnability results. We refer to this special class as rectangular LPMs. §.§ Sparse Latent Position Model Our new LPM's definition follows closely with that of the Poisson random connection model restricted to finite windows: the positions in the latent space are given by a homogeneous Poisson point process, and the link probability function K is independent of the number of nodes. The main departure from the random connection model is formulating K such that it depends on the inter-node distance in just a subset of the dimensions—specifically all but the auxiliary dimension. The following is a set of ingredients to formulate a sparse LPM. * Position Space: A measurable metric space (S, 𝒮, ρ) equipped with a Lebesgue measure ℓ_1. * Auxiliary Dimension: The measure space (ℝ_+, ℬ, ℓ_2) where ℬ is Borel and ℓ_2 is Lebesgue.* Product Space: The product measure space (S^*, 𝒮^*, λ) on (S ×ℝ_+, 𝒮×ℬ), equipped with λ = ℓ_1 ×ℓ_2, the coupling of ℓ_1 and ℓ_2.* Continuum of observation windows: A function H: ℝ_+ →𝒮^* such that t_1 < t_2 ⇒ H(t_1) ⊂ H(t_2) and |H(t)| = t.* Link probability function: A function K : ℝ_+ → [0,1]. Jointly, we say the triple ((S,𝒮, ρ), H, K) defines a stochastic graph sequence called a sparse LPM. The position space plays the role of the latent space as in traditional LPMs, with the link probability function K controlling the probability of an edge given the corresponding latent distance. The auxiliary dimension plays no role in connection probabilities. Instead, a node's auxiliary coordinate—in conjunction with its latent position and the continuum of observation windows—determines when it appears. Specifically, a node with position (Z,r) is observable at time t ∈ℝ_+ if and only if (Z,r) ∈ H(t). Here, time need not correspond to physical time; it is merely an index for a continuum of graphs as in the case for the Poisson random connection model. We refer to t_i—defined as the smallest t ∈ℝ_+ for which (Z_i,r_i) ∈ H(t)—as the arrival time of the ith node where (Z_i,r_i) are the corresponding latent position and auxiliary value for node i.Considered jointly, the coordinates defined by the latent positions and auxiliary positions assigned to nodes can be viewed as a point process over S ×ℝ_+. As in the Poisson random connection model, we assume this point process is a unit-rate Poisson. The continuum of observation windows H(t) controls the portion of the point process which is observed at time t. Since the size of H is increasing in t, this model defines a growing sequence of graphs with the number of nodes growing stochastically in t as follows.* Generate a unit-rate Poisson process Ψ on (S^*, 𝒮^*).* Each point (Z, r) ∈ S ×ℝ_+ in the process corresponds to a node with latent position Z and auxiliary coordinate r.* For a dyad on nodes with latent positions Z_i and Z_j, include an edge with probability K(ρ(Z_i, Z_j)).* At time t the subgraph induced by by restricting Ψ to H(t) is visible.A graph of size n can be obtained from the above framework by choosing any t_n such that |Ψ∩ H(t_n)| = n. Each t_n < t_n + 1 with probability one (by Lemma <ref>). Thus, the above generative process is well-defined for any n, and the nodes are well-ordered by their arrival times.Due to its flexibility, the above framework defines a broad class of LPMs. For instance, the exchangeable LPM can be viewed as a special case of the above framework in which the observation window grows only in the auxiliary dimension. However, the full generality of this framework makes it difficult to establish general sparsity and learnability results. For this reason, we have chosen to focus on a subclass of sparse LPMs to derive our sparsity, projectivity, and learnability proofs. We refer to this class as rectangular LPMs. We have chosen this class because it allows us to emphasize the key insights in the proofs without having to do too much extra bookkeeping. §.§ Rectangular Latent Position Model For rectangular LPMs, we impose further criteria on the basic sparse LPM. The latent space is assumed to be Euclidean (S = ℝ^d). The continuum of observation windows H(t) are defined by the nested regionsH(t)= [- g(t), g(t)]^d ×[0, t/(2g(t))^d]where g(t) = t^p/d for 0 ≤ p≤ 1 controls the rate at which the observation window grows for the latent position coordinates. The growth rate in the auxiliary dimension is chosen to be 2^-d t^1- p to ensure that the volume of H(t) is t. We further assume that0< ∫_0^∞ u^d-1 K(u) du < ∞to ensure that the average distance between a node and its neighbors remains bounded as n grows. We now demonstrate the projectivity and sparsity of rectangular LPMs as Theorems <ref> and <ref>.Rectangular sparse latent position network models define a projective sequence of models. Provided in <ref>A d-dimensional rectangular latent position network model is n^2-p-sparse in expectation, where g(n) = n^p/d. Provided in <ref> By specifying the appropriate value of p for a rectangular LPM, it is thus possible to obtain any polynomial level of sparsity within n-sparse and n^2-sparse (dense) in expectation. Other intermediate rates of sparsity such as nlog(n) can also be obtained considering non-polynomial g(n). We now investigate for which levels of sparsity it is possible to do reliable statistical inference of the latent positions.§ LEARNABILITY§.§ Preliminaries Recall that the edge probabilities in a LPM are controlled by two things: the link probability function K and the latent positions Z ∈ S^n. In this section, we consider the problem of consistently estimating the latent positions for a LPM using the observed adjacency matrix. We focus on the case where both K and S = ℝ^d are known, relying on assumptions that are compatible with rectangular LPMs.In the process of establishing our consistent estimation results for Z, we also establish consistency results for two other quantities: the squared latent distance matrix D^Z∈ℝ^n × n defined by D^Z_ij = Z_i - Z_j^2 and the link probability matrix P^Z ∈ [0,1]^n × n defined by P^Z_ij = K((D^Z_ij)^1/2). These results are also of independent interest because—like Z—the distance matrix and link probability matrix also characterize a LPM when K is known.We use the following notation and terminology to communicate our results. Let ·_F denote the Frobenius norm of a matrix,denote convergence in probability, 𝒪_d denote the space of orthogonal matrices on ℝ^d × d, and 𝒬_nd⊂ℝ^n × d denote the set of all n × d matrices with identical rows. We say that a LPM has learnable latent positions if there exists an estimator Ẑ(Y^n) such thatlim_n →∞inf_O ∈𝒪_d, Q ∈𝒬_ndẐ(Y^n)O- Q -Z_F^2/n 0.That is, a LPM has learnable positions if there exists an estimator Ẑ(Y^n) of the latent positions such that the average distance between Ẑ(Y^n) and the true latent positions converges to 0. The infimum over the transformations induced by O ∈𝒪_d and Q ∈𝒬_nd is included to account for the fact that the likelihood of a LPM is invariant to isometric translations (captured by Q) and rotations/reflections (captured by O) of the latent positions <cit.>. We say that a LPM has learnable squared distances if there exists an estimator Ẑ(Y^n) such thatlim_n →∞D^Ẑ(Y^n) - D^Z_F^2/n^2 0.That is, a LPM has learnable squared distances if the average squared difference between the estimator for the matrix of squared distances induced by Ẑ(Y^n) and the true matrix of squared distances D^Z converges to 0. Unlike the latent positions, D^Z is uniquely identified by the likelihood; there is no need to account for rotations, reflections, or translations. Finally, we say a LPM that is e(n)-sparse in expectation has learnable link probabilities if there exists an estimator Ẑ(Y^n) such thatlim_n →∞P^Ẑ(Y^n) - P^Z_F^2/e(n) 0.Note that a scaling factor of e(n) is used instead of n^2 to account for the sparsity. Otherwise the link probability matrix for a sparse graph could be trivially estimated because n^-2P^Z_F^2 0. §.§ Related Work on Learnability Before presenting our results, we summarize some of the existing work on learnability of LPMs in the literature.<cit.> considered the problem of estimating LPMs from a classical statistical learning theory perspective. They established bounds on the growth function and shattering number for LPMs with link function given by K(δ) = (1 + expδ)^-1. However, we have found that their inequalities were not sharp enough to be helpful for proving learnability for sparse LPMs. <cit.> provide regularity conditions under which LPMs have learnable positions on general spaces S, assuming that the link probability function K is known and possesses certain regularity properties. Specifically, they require that the absolute value of the logit of the link probability function is slowly growing, which does not necessarily hold in our setting. Our learnability results more closely resemble those of <cit.>, who consider a latent variable network model of the form logit(ℙ(A_ij = 1)) = α_i + α_j + β X_ij + Z_i^T Z_j, originally due to <cit.>. Here, α_i denote node-specific effects, X_ij denote observed dyadic covariates and β denotes a corresponding linear coefficient. If there are no covariates and α_i = Z_i^2/2, their approach defines a LPM with K(δ) = expit(-δ^2). <cit.> provide algorithms and regularity conditions for consistent estimation of both the logit-transformed probability matrix and Z^TZ under this model, using results from <cit.>. Here, we will use similar concentration arguments to establish Lemmas <ref> and <ref>, but our results differ in that we consider a more general class of link functions, and also establish learnability of latent positions via an application of the Davis-Kahan theorem.Our learnability of latent positions result (Lemma <ref>) resembles that of <cit.>, who establish that the latent positions for dot-product network models can be consistently estimated. The dot product model—a latent variable model which is closely related to the LPM—has a link probability function defined by K(Z_i, Z_j) = Z_i · Z_j with Z_j, Z_j ∈ S. The latent space S ⊂ℝ^d is defined such that all link probabilities must fall with [0,1]. Our proof technique follows a similar argument as the one used to prove their Proposition 4.3.It should be noted that learnability of the link probability matrix for the sparse LPM could be established by applying results from Universal Singular Value Thresholding <cit.>. However, it is unclear how to extend such estimators to establish learnability of the latent positions; estimated probability matrices from universal singular value thresholding do not necessarily translate to a valid set of latent positions for a given link function.Other related work includes <cit.>, which considers the problem of estimating latent distances between nodes when the functional form of the link probability function is unknown. They show that, if the link probability function is non-increasing and zero outside of a bounded interval, the lengths of the shortest paths between nodes can be used to consistently rank the distances between the nodes. <cit.> and <cit.> also propose estimators in similar settings with more specialized link functions. None of these approaches are appropriate for our case—we are interested in recovering the latent positions under the assumption K is known with positive support on the entire real line.§.§ Learnability Results Our learnability results assume the following criteria for a LPM: * The link probability function K is known, monotonically decreasing, differentiable, and upper bounded by 1-ϵ for some ϵ > 0.* The latent space S⊆ℝ^d.* There exists a known differentiable function G(n) such that I(Z_n≤ G(n)) 1. We refer to the above conditions as regularity criteria and refer to any LPM that meets them as regular. Criterion 3 implies that the sequence of latent positions is tight <cit.>. The class of regular LPMs contains several popular LPMs. Notably, both rectangular and exchangeable LPMs due to <cit.> are regular, as shown in Lemmas <ref> and Lemma <ref>. For a rectangular LPM, the G(n) in criterion 3 is closely related to g(t)—the width of the observation window. Specifically, it is established in Lemmas 10 and 11 in A.1 that a rectangular LPM with g(t) = t^p/d satisfies criterion 3 with G(n) = 2√(d) n^p/d. Here, t refers to the size of observation window (i.e. the expected number of observed nodes), and n refers directly to the number of observed nodes.Our approach for establishing learnability of Z involves proposing a particular estimator for Z which meets the learnability requirement as n grows. Our proposed estimator is a restricted maximum likelihood estimator for Z, provided by the following equation:Ẑ(Y^n) = argmax_z: z_i≤ G(n) ∀ i ∈ 1:n L(z: Y^n)where L(z: Y^n) denotes the log likelihood of latent positions z = (z_1, … z_n) ∈ℝ^n × d for a n × n adjacency matrix Y^n. We use D^Ẑ(Y^n) and P^Ẑ(Y^n) to denote the corresponding estimates of the squared distance matrix and link probability matrix. Note that the log likelihood L(z:Y^n) is given byL(z: Y^n)= ∑_i=1^n ∑_j =1^n Y^n_ijlog(K(z_i - z_j)) + (1-Y^n_ij)log(1- K(z_i - z_j)). To establish consistency, we first provide a concentration inequality for the maximum likelihood estimate of Z in Lemma <ref>. En route to deriving Lemma <ref>, we also derive inequalities for the associated squared distance matrix D^Z∈ℝ^n × n defined by D^Z_ij = Z_i - Z_j_F^2 (Lemma <ref>) and the link probability matrix P^Z ∈ [0,1]^n × n defined by P^Z_ij = K((D^Z_ij)^1/2) (Lemma <ref>). We combine these results in Theorem <ref> to provide conditions under which it is possible to consistently estimate Z, D^Z, and P^Z. Our results are sensitive to the particular choices of link probability function K and upper bounding function G. For this reason, we introduce the following notation to communicate our results.α^K_n = sup_0 ≤ x ≤ 2G(n)|K'(x)|/|x| K(x)ϵ, β^K_n = sup_0 ≤ x ≤ 2G(n)x^2 K(x)/K'(x)^2,where K'(x) denotes the derivative of K(x) and ϵ is given by the criteria on K imposed by regularity criterion 1. Consider a sequence adjacency matrices Y^n generated by a regular LPM with Z_n≤ G(n) for all n. Let P^Ẑ(Y^n) denote the estimated link probability matrix obtained via Ẑ(Y^n) from (<ref>). Then,ℙ(P^Ẑ(Y^n) - P^Z_F^2 ≥ 16 e α^K_n G(n)^2 n^1.5 (d + 2) ) ≤C/n^2for some constant C > 0. Provided in <ref>.Consider a sequence adjacency matrices Y^n generated by a regular LPM with Z_n≤ G(n) for all n. Let D^Ẑ(Y^n) denote the matrix of estimated squared distances obtained via Ẑ(Y^n) from (<ref>). Then,ℙ(D^Ẑ(Y^n) - D^Z_F^2 ≥ 2^9 e α^K_n β^K_n G(n)^2 n^1.5 (d + 2) ) ≤C/n^2for some constant C > 0. Provided in <ref>. Establishing concentration of the estimated latent positions is complicated by the need to account for the minimization over all possible rotations, translations, and reflections. The following matrix, known as the double-centering matrix, is a useful tool to account for translations: 𝒞_n = I_n - 1/n 1_n1_n^THere, I_n denotes the n-dimensional identity matrix and 1_n denotes n × 1 matrix consisting of ones.To establish our concentration of the estimated latent positions, we place conditions on the eigenvalues of the matrix 𝒞_n Z Z^T 𝒞_n.For a regular LPM, let λ_1 ≥⋯≥λ_d denote the d nonzero eigenvalues of 𝒞_n Z Z^T 𝒞_n and define λ_d+ 1 := 0. For functions a:ℕ→ℝ andb:ℕ→ℝ, we say that a LPM possesses a(n)-b(n) distinctly bunched eigenvalues if there exists a k ∈{1, …, d} and integers i_1, …, i_k+1 satisfying 1= i_1 < i_2 < ⋯ < i_k < i_k+1 = d+1 such that(λ_i_j+1 - 1 - λ_i_j)^2/λ_i_j+1 - 1≤ a(n)and (λ_i_j+1 - 1 - λ_i_j+1)^2/λ_i_j≥ b(n)for all j ∈{1, …, k}. In this definition, i_1,…, i_k+1 are boundary indices partitioning the eigenvalues λ_1, …, λ_d. Eigenvalues within the same subset of the partition can be thought of as remaining close to each other as n increases, whereas those from different subsets are distinguishable from each other as n grows. The levels of a(n) and b(n) dictate the level of proximity and distinguishability. Corollary <ref> in Appendix <ref> establishes that rectangular LPMs possess a(n)-b(n) distinctly bunched eigenvalues with a(n) and b(n) depending on the level of sparsity—sparser graphs require larger a(n) and smaller b(n)'s. Similarly, Corollary <ref> establishes that exchangeable LPMs due to <cit.> possess a(n)-b(n) distinctly bunched eigenvalues with a(n) = O(1) and b(n)^-1 = O(n^-1).Consider a sequence adjacency matrices Y^n generated by a regular LPM possessing a(n)-b(n) distinctly bunched eigenvalues with Z_n≤ G(n) for all n. Then,ℙ(inf_O ∈𝒪_dQ ∈𝒬_ndẐ(Y^n)O - Z - Q^2_F ≥3 a(n)/(d - 1)^-1 + 2^9 e α^K_n β^K_n G(n)^2 n^1.5/(50 (d + 2))^-1 b(n) ) ≤C/n^2for C > 0, where 𝒪_d denotes the space of orthogonal matrices on ℝ^d × d, 𝒬_nd⊆ℝ^n × d is the set of matrices with n identical d-dimensional rows,and Ẑ(Y^n) is obtained via (<ref>). Provided in <ref>.These three concentration results can be translated into sufficiency conditions for learnability. We summarize these in Theorem <ref>.A regular LPM that is e(n)-sparse in expectation has: * learnable link probabilities if α^K_n e(n)^-1 n^1.5 G(n)^2 → 0 as n grows.* learnable squared distances if β^K_n α^K_n n^-0.5 G(n)^2 → 0 as n grows.* learnable latent positions if it possesses a(n)-b(n) distinctly bunched eigenvalues witha(n)/n→ 0 andβ^K_n α^K_n n^0.5 G(n)^2/b(n)→ 0as n grows. Provided in <ref>. It may seem counter-intuitive that the conditions for learnability of Z, P^Z and D^Z differ, even though their estimators are all derived from the same quantity. For example, if β^K_n grows quickly enough, the LPM may have learnable link probabilities but not squared distances. This disparity can be understood by considering the metrics implied by each form learnability. Suppose that δ_ij = Z_i - Z_j is very large. Then mis-estimating δ_ij by a constant c>0(i.e. δ̂_ij = δ_ij + c) contributes (2δ_ijc + c^2)^2 to the error in D^Ẑ - D^Z_F^2. This contribution to the error is sizable, and can hinder convergence if made too often. However, the influence of the same mistake on P^Ẑ - P^Z_F^2 is minor; because the probability K(δ) is already small for large δ, (K(δ +c) - K(δ))^2 does not contribute much to the error. For small distances, the opposite may be true; a small mistake in estimated distance may lead to a large mistake in estimated probability. Thus, learnability of squared distances penalizes mistakes differently than learnability of link probabilities. However, there are typically far more large distances than small distances, meaning that the distance metric imposed by learnability of link probabilities is typically less stringent than for learnability of squared distances.Theorem <ref> can be used to establish Corollary <ref>, a learnability result for rectangular LPMs.Consider a d-dimensional rectangular LPM with g(n) = n^p/d and link probability function K(δ) = (C + δ^2)^-q for some C > 0, where q > max({d/2, 1}) and 0 ≤ p ≤ 1. Such a network has learnable* link probabilities if 2p < ( 1 + 2/d)^-1,* distances if 2p < d( 2q+6)^-1, * latent positions if 2p < d( 2q+4)^-1. Thus, for any s ∈ (1.5, 2], it is possible to construct a LPM that is projective, n^s-sparse in expectation, and has learnable latent positions, distances, and link probabilities.Provided in <ref>.Corollary <ref>, combined with the projectivity of rectangular LPMs, guarantees the existence of a LPM that is projective, learnable, and sparse for any sparsity level that is denser than n^3/2-sparse in expectation. Thus, we have shown that we have met our desiderata for LPMs laid out in the introduction.Perhaps surprisingly, our result in Corollary <ref> depends upon the dimension of the latent space. The higher the dimension, the richer the levels of learnable sparsity. Moreover, the learnability results in Theorem <ref> only apply to rectangular LPMs with link functions that decay polynomially. The β^K_n term is too large for the exponential-style decays that are commonly considered in practice <cit.>. We elaborate on these points in <ref>.In contrast, it is possible to prove learnability of exchangeable LPMs with exponentially decaying K. Corollary <ref> guarantees learnability of the exchangeable LPM for two exponential-style link functions. As far as we are aware, these are the first result learnability results for the latent positions for the original exchangeable LPM. Consider a LPM on S = ℝ^d with each latent position independently and identically distributed according to the multivariate Gaussian distribution with mean zero and diagonal variance matrix Σ. Let σ^2_1, …, σ^2_d denote the entries along the diagonal of Σ, with σ_1 ≥σ_2 ≥⋯≥σ_d > 0. Suppose that the link probability function is given by eitherK(δ) = (1 + exp(δ^2))^-1or K(δ) = τ e^-δ^2.for τ∈ (0,1). Such a network has learnable link probabilities, distances, and latent positions provided that σ_1^2 < 1/4. Provided in <ref>. Notably, the set of link functions in Corollary <ref> does not include the traditional expit link function that was suggested in the original paper LPM by <cit.>. The expit class of link functions implies a value α_n^k—defined as in (<ref>)—that is unbounded (see Table <ref> in Appendix <ref> for a summary of the α^k_n and β^K_n values for various link functions), meaning that Lemma <ref> cannot be applied to prove learnability for this class of LPMs. This does not necessarily mean that expit LPMs are not learnable, just that determining their learnability remains an open problem. Note however, that some classes of sparse LPMs (such as the example considered in Theorem <ref> (<ref>)) are provably unlearnable. We elaborate on this point in <ref>. The results in Theorem <ref> can also be used to obtain learnability results for more specialized LPMs such as sparse graphon-based LPMs. We provide such a result in <ref> when comparing sparse graphons with our approach. § COMPARISONS AND REMARKSExisting tools for constructing sparse graph models, such as the sparse graphon framework <cit.> or the graphex framework <cit.> can be used to develop suitably sparse latent position models. However, both approaches introduce sparsity in ways that have undesirable side effects for LPMs. We now describe both the sparse graphon framework (<ref>) and the graphex framework (<ref>), with discussion of how these frameworks fail to meet our desiderata of projectivity, learnability, and other useful properties for LPMs such as edge transitivity. Finally, we conclude by making some remarks on the results we have derived this article (<ref>). §.§ Sparse Graphon-based Latent Position Models <cit.> proposed a modification of graphon models to allow sparse graph sequences. Seeing as exchangeable LPMs are within the graphon family, it is straightforward to specialize this approach to define sparse graphon-based LPMs.As in exchangeable latent position models, the latent positions for a sparse graphon-based LPM are each drawn from a common distribution f, independently of each other the number of nodes n. However, the link probability function ℙ(Y_ij = 1|Z_i, Z_j) = K_n(ρ(Z_i, Z_j)) is allowed to depend on n. Specifically, K_n(x) = min({s_nK(x), 1}) where (s_n)_1…∞ is a non-increasing sequence and K: ℝ_+ →ℝ_+ satisfies 𝔼(K(ρ(Z_i, Z_j))) < ∞ for Z_i, Z_j ∼ f. These models express sparse graph sequences, with the sequence (s_n)_1…∞ controlling the sparsity of the resultant graph sequence. Sparse graphon-based latent position models define a n^2 s_n-sparse in expectation graph sequence.Proof provided in <ref>. Moreover, the learnability results in Theorem <ref> can be used to establish learnability results for sparse graphon-based versions of popular LPMs. Consider the following sparse graphon-based version of the exchangeable LPM. Let S = ℝ^d with the latent positions distributed according an isotropic Gaussian random vector with any variance σ^2 < 1/4. Suppose that the link probability function is given by eitherK_n(δ) = n^-p(1 + exp(δ^2))^-1or K_n(δ) = τ n^-p e^-δ^2for τ∈ (0,1), 0 ≤ p ≤ 1. Such a network has learnable link probabilities, squared distances, and latent positions if p < 1/2 - 2σ^2(1+c) for c>0. Thus, given an appropriate σ^2, this LPM can be both n^b-sparse and learnable for b∈(1.5,2]. Proof provided in <ref>As such, many sparse graphon-based LPMs achieve learnability under the same sparsity rate derived for rectangular LPMs in Corollary <ref>. Additionally, learnability can be established for link probability functions with lighter tails, as well as for latent spaces of arbitrary dimension d. These findings suggest a potential trade-off between projectivity and learnability under lighter-tailed link probability functions.Despite these advantages, there are practical ramifications of sparse graphon-based LPMs that limit their applicability as statistical models for a network. To start, the resultant sparse network sequences are not projective.Sparse-graphon latent position models do not define a projective sequence of models if (s_n)_n=1…∞ is not constant. Proof provided in <ref>.As noted in <ref>, inferences drawn using non-projective network models can be difficult to interpret, especially when the statistical application requires super-population inference <cit.>. As such, extra care must be taken to ensure sparse graphon-based inferences are reliable for a given application.In the specific context of LPMs, another ramification stems from the particular way the non-projective link function K_n(x) = min({s_nK(x), 1}) is defined. In particular, consider the probability of edge transitivity in sparse graphon-based LPMs as n increases. Edge transitivity—that is, the extra tendency for two nodes in a network to be connected given a shared neighbor—is one of the main selling points identified by <cit.> in their initial proposal of the LPM was as a useful statistical model. Notably, the triangle inequality for distances combines with the strictly decreasing LPM link probability function to promote transitivity in virtually all commonly-used exchangeable LPMs. Somewhat surprisingly, however, is the fact that this fact does not hold for sparse graphon-based versions of popular LPMs. Under fairly general conditions, the conditional probability of two nodes being connected given a shared neighbor declines to zero as n grows. We formally state this result as Theorem <ref>.Consider a sparse graphon-based latent position model on the latent space ℝ^d equipped with the Euclidean distance. Suppose that the sequence of link functions is given by K_n(δ) = min({s_nK(δ), 1}) where (s_n)_n ∈ℕ is a non-increasing sequence with a limit of 0 (i.e. the resultant LPM is sparse), and K(δ) is a non-negative, continuous, strictly decreasing function satisfying 𝔼(K(ρ(Z_i, Z_j)) K(ρ(Z_i, Z_k)) K(ρ(Z_j, Z_k)) ) < ∞ for Z_i, Z_j, Z_k ∼ f. Under these conditions, the resultant LPM will satisfy(Y^n_ij =1| Y^n_ik=1, Y^n_jk = 1) →_p 0as n →∞ for any arbitrarily chosen node indices (i,j,k). That is, the probability of edge transitivity will go to 0 as the number of nodes goes to infinity. Proof provided in <ref>. The conditions required for Theorem <ref> are quite general. Notably,𝔼(K(ρ(Z_i, Z_j)) K(ρ(Z_i, Z_k)) K(ρ(Z_j, Z_k)) ) < ∞is guaranteed to hold for any bounded function K, such as the standard choices expit(-ρ^2), exp(-ρ^2) and (1 + ρ^2)^-1, as well as many choices of unbounded K. For this reason, it may be undesirable to consider sparse-graphon based LPMs to model real-world networks in which edge transitivity is expected to be present, at least for standard link functions.In contrast, the sparse LPMs presented in Section <ref> exhibit nonzero probabilities of edge transitivity as n increases under standard assumptions on K. We establish this fact as Theorem <ref>. Let π be a permutation on (1, …, n) chosen uniformly at random from the set of permutations on (1, …, n) for n ∈ℕ. A d-dimensional regular rectangular latent position network model will satisfylim_n →∞(Y^n_π(i)π(j) =1| Y^n_π(i) π(k)=1, Y^n_π(j)π(k) = 1) ≠ 0.That is, the probability of edge transitivity does not goes to 0 as the number of nodes goes to infinity. Proof provided in <ref>.As such, our projective sparse LPMs are more suitable for modeling networks where edge transitivity is expected to be present. Seeing as edge transitivity is a primary selling point of LPMs, we argue that our LPMs are thus more suitable in most practical applications.Finally, It is also worth acknowledging that the sparse graph representation of <cit.> is more general than the sparse graphon representation described above. It allows for latent variables assigned defined through a point process rather than generated independently from the same distribution. For LPMs, this set-up equates to the traditional random connection model (<ref>). §.§ Comparison with the Graphex FrameworkBeyond the random connection model <cit.>, there has been a recent renewed interest in using point processes to define networks. This was primarily spurned by the developments in <cit.> and <cit.> in which they propose a new graph framework—based on point processes—for infinitely exchangeable and sparse networks. This approach was generalized as the graphex framework in <cit.>.Other variants and extensions of this work include <cit.>.In the graphex framework, a graph is defined by a homogeneous Poisson process on an augmented space ℝ_+ ×ℝ_+, with the points representing nodes. The two instances of ℝ_+ play the roles of the parameter space and the auxiliary space. The parameter space determines the connectivity of nodes through a function W:ℝ_+^2 → [0,1]. Connectivity is independent of the auxiliary dimension ℝ_+ that determines the order in which the nodes are observed. Clearly, our sparse LPM set-up shares many similarities with the graphex framework. Both assign latent variables to nodes according to a homogeneous Poisson process defined on a space composed of a parameter space to influence connectivity and an auxiliary space to influence order of node arrival. The graphex is defined in terms of a one-dimensional parameter space, but it can be equivalently expressed as a multi-dimensional parameter space as we do for the sparse LPM. The link probability function K for the sparse LPM depends solely on the distance between points, but it would be straightforward to extend to the more general set-up for W as in the graphex. However, it would take additional work to determine the sparsity levels and learnability propertiesof such graphs. The major difference between our framework and the graphex framework is how a finite subgraph is observed. To observe a finite graphex-based graph, one restricts the point process to a window ℝ_+ × [0, ν]. Here, the restriction is limited to the auxiliary space, with the parameter space remaining unrestricted. This alone is not enough to lead to a finite graph, as a unit rate Poisson process on ℝ_+ × [0,ν] still has an infinite number of points almost-surely. To compensate, an additional criterion for node visibility is included. A node is visible only if it has at least one neighbor. For some choices of W, this results in a finite number of visible nodes for a finite ν.<cit.> show that the expected number of nodes n_ν and edges e_ν are given by𝔼(n_ν)= ν∫_0^∞1 - exp( - ν∫_ℝ_+ W(x,y) dy )dx, 𝔼(e_ν)= 1/2ν^2 ∫_0^∞∫_0^∞ W(x,y) dx dyrespectively. Thus, the degree of sparsity in the graph is controlled through the definition of W. Clearly, for a finite-node restriction to be defined, the two dimensional integral over W in (<ref>) must be finite. Otherwise, the number of nodes is infinite for any ν.A sparse graphex-based LPM cannot be implemented in the naive manner because, if W is solely a function of distance between nodes, the two dimensional integral (<ref>) is infinite. One modification to prevent this to modify W to have bounded support, e.g. W(x,y) = K(|x-y|) I(0 ≤ x,y ≤ C). However, this framework is equivalent to the graphon framework and results in dense graphs <cit.>. It does not define a sparse LPM.Alternatively, we could relax the graphex such that latent positions are generated according to an inhomogeneous point process over the parameter sparse. This can be done though the definition of W. For instance, considerW(x,y)= K(|exp(x ) - exp(y )|).with K being the link probability function as defined in the traditional LPM. In this set-up, W can be viewed as the composition of two operations. First, an exponential transformation is applied to the latent positions resulting in an inhomogeneous rate function given by f(x)= 1/(1+x). Then, we proceed as if it were a traditional LPM in this new space, connecting the nodes according to K on their transformed latent positions. Finally, the isolated nodes are discarded. This approach defines a sparse and projective latent position network model, with the level of sparsity controlled by K. Though (<ref>) and (<ref>) provide a means with which to calculate the sparsity level, these expressions do not yield analytic solutions for most K. As a result, the graphex framework is far more difficult to work with when defining sparse LPMs; they lack the straightforward control over the level of sparsity provided by the growth function g(t) in rectangular LPMs.Furthermore, it is difficult to apply the tools derived in Theorem <ref> to establish learnability for graphex-based LPMs. The difficulty stems from the fact that regularity requires a probability bound on the maximum of the distances between the origin and the first n observed nodes. That is, we need a bound on max_i ≤ nX_iwhere X_1 …, X_n denote the latent positions of the first n observed nodes. Because of the irregular sampling scheme in which isolated nodes are discarded, it is difficult to establish such a bound for the graphex. Furthermore, any such bound is usually large due to the fact the latent positions at any n are generated according to an improper distribution. For this reason, whether or not such graphex-based LPMs are learnable is an open problem.§.§ Remarks We have established a new framework for sparse and projective latent position models that enables straightforward control the level of sparsity. The sparsity is a result of assuming the latent positions of nodes are a realization of a Poisson point process on an augmented space, and that the growing sequence of graphs is obtained by restricting observable nodes to those with positions in a growing sequence of nested observation windows.The notion of projectivity we consider here is slightly weaker than the one usually considered in the literature (e.g. <cit.>). Our definition requires consistency under marginalization of the most recently arrived node, rather than consistency under marginalization of any node. We do not consider this to be a major limitation—if the entire sequence of graphs were observed, the order of the nodes would be apparent. In practice, only a single network of finite size is available when conducting inference. However, in these cases the order of nodes is not required—we make no use of it when defining the maximum likelihood estimator. A finite observation from our new sparse LPM is equivalent to finite observation from an equivalent exchangeable LPM with f given by the shape of H(n_t). This follows from Lemma <ref> which indicates that the distribution of latent positions can be viewed as iid after conditioning on the number of nodes and randomly permuting the ordering. This means that the analysis and inference tools developed for exchangeable LPMs extend immediately to our approach when analyzing a single, finite network. From this viewpoint, we have merely proposed a different asymptotic regime for studying the same classes of models available under the exchangeability assumption. Theorem <ref> provides some consistency results under this asymptotic regime. However, the rates of learnability we achieved are upper bounds—the inequalities in Lemmas <ref>-<ref> are not necessarily tight. They are derived to hold even for the worse-case regular LPMs regardless of how the latent positions are generated. We demonstrate in Theorem <ref> (<ref>) that there are some classes regular LPMs for which it is impossible to learn the latent positions. This class of models includes any regular LPMs with G(n) = n^p/d and K exponentially decreasing. In these cases, it is possible for the LPM to result in graphs which are disconnected with probability trending to one by clustering the latent positions at two extreme points of the space.Though the regularity criteria technically allow for such instances by placing no assumptions on the distribution of Z besides bounded norms, these clusters arise with vanishing probability when the latent positions are assumed to follow a homogeneous Poisson process such as in rectangular LPMs. For this reason, a future research direction to explore is to establish better learnability rates for rectangular LPMs by tightening the bounds Lemmas <ref>-<ref> through assumptions on the distribution of the latent positions.§ PROOFS OF RESULTS AND SUPPORTING LEMMAS§.§ Intermediary Results The following are useful lemmas toward establishing the main results in this article. Restriction Theorem in <cit.> Let Λ be a Poisson process with mean measure μ on S, and let S_1 be a measurable subset of S. Then the random countable setΛ_1 = Λ∩ S_1can be regarded as a Poisson process on S with mean measureμ_1 (A) = μ(A ∩ S_1)or as a Poisson process on S_1 possessing a mean measure that is the restriction of μ to S_1. For a rectangular LPM, the number of nodes which are visible at time t is Poisson distributed with mean t. According to Lemma <ref>, the latent positions of nodes visible at time t follow a unit-rate Poisson process over H(t). Therefore, the number of nodes is Poisson distributed with expectation equal to the volume of H(t), which is t.Let t_n denote the arrival time of the nth node in a sparse LPM. Then, t_n ∼ Gamma(n, 1) if H(t) has volume t. Moreover, t_n/n1. Let n_t = |Ψ∩ H(t)| where Ψ denotes the unit rate Poisson process of latent positions. Then, it is straightforward to verify that n_t follows a one-dimensional homogeneous Poisson process on the positive real line. Note that t_n can be equivalently expressed as t_n = inf{t ≥ 0: |Ψ∩ H(t) | = n}.That is, t_n is the index of the smallest observation window containing n nodes for all positive integers n. Under this perspective, t_n can be viewed as a stopping time of n_t. It is well-known that t_1, the first arrival time of a unit-rate Poisson process, follows an exponential distribution with rate 1. Then, by the strong Markov property of Poisson processes t_n - t_n-1 is identical in distribution to t_1. Thus, t_n is equivalent to the sum of n independent exponential distributions, meaning it follows Gamma(n, 1). The fact that t_n/n a.s.→ 1 follows from the strong law of large numbers because t_n is the sum of n independent exponential random variables with mean one.Consider a sparse rectangular LPM. Let z denote the latent position of a node chosen uniformly at random of the nodes visible at time t. Then z follows a uniform distribution over [-g(t) , g(t)]^d. If a node is visible at time t, its latent position and auxiliary coordinate pair (z,r) are a point in a unit-rate Poisson process restricted H(t). By Lemma <ref>, this point process is a Poisson process with unit rate over the restricted space. Thus, if a node is visible at time (z, r), it is uniformly distributed over H(t) = [-g(t),g(t)]^d × [0,t/(2g(t))^d]. Marginalizing r provides the result.Let K be a decreasing non-negative function such that0<∫_0^∞ r^d-1K(r) dr < ∞,for some d ∈ℤ_+. Then,0< ∫_y ∈ [-B,B]^d K(x - y) dy < ∞for any B∈ℝ_+. Note that for all decreasing positive functions K, the functionR(x) = ∫_y ∈ [-B,B]^d K(y-x) dyis maximized when x is at the origin. Thus, for all x ∈ℝ^d,∫_y ∈ [-B,B]^d K(y-x) dy≤∫_y ∈ [-B,B]^d K(y) dy≤∫_y ∈ℝ^d:y < √(d) B K(y) dy∝∫_0^√(d) B r^d-1 K(r) dr< ∞.The positivity follows from K being non-negative and the positivity of the expression in (<ref>). Consider a rectangular LPM, with t_i denoting the arrival time of the ith node. Let π denote permutation chosen uniformly at random from all permutations on {1,…, n-1}. Then, conditional on t_n = T, each t_π(i)'s marginal distribution is uniform on [0, T] for i=1,…, n-1. Consequently, the latent position Z^π(i) of node π(i) is uniformly distributed on [-g(T), g(T)]^d.Let (w_i)_i=1,…,n denote the inter-arrival of times of the nodes. That is, w_1 = t_1 and w_i = t_i - t_i-1. As argued in the proof of Lemma <ref>, each w_i is exponentially distributed. Thus, the density of t_1, …, t_n-1 given t_n = T satisfies:f(t_1, …, t_n-1|t_n = T) ∝ I(0 ≤ t_1 ≤ t_2 ≤⋯≤ t_n-1≤ t_n)which is the same density as the order statistics of a uniform distribution on [0,T]. Thus, a randomly chosen waiting time t_π(i) is uniformly distributed on [0,T]. Let r^π(i) denote the auxiliary coordinate of node π(i). It follows that ℙ((Z^π(i), r^π(i)) ∈ [-g(a), g(a)]^d × [0,a/g(a)^d]) = a/T for all 0 ≤ a ≤ T. It follows that Z^π(i) is uniformly distributed on [-g(T), g(T)]^d. Consider a rectangular sparse LPM model restricted to H(t_n) such that n nodes are visible. Let {Z_1, …, Z_n } denote the latent positions of these nodes. Let δ^(n) = max_i=1,… nZ_i denote the largest Euclidean distance between a visible node's latent position and the origin. Then,ℙ(δ^(n) >√(d)g(n +√(nlog(n))))≤log(n)^-1indicating thatlim_n →∞ℙ(δ^(n) > √(d) g(n + √(nlog(n)))) → 0. Consequently,lim_n →∞ℙ(δ^(n) > 2√(d) g(n)) → 0. Let Z_ij denote the jth latent coordinate of node i. By construction, Z_ij≤ g(t_n) for any i≤ n, j ≤ d. Thus, δ^(n)≤√(d) g(t_n). By Lemma <ref>, know that t_n ∼ Gamma(n, 1). By Chebyshev inequality,ℙ(|t_n - n| > √(nlog(n)))≤log(n)^-1 ⇒ℙ(t_n > n +√(nlog(n)))≤log(n)^-1 ⇒ℙ(g(t_n) > g(n +√(nlog(n))))≤log(n)^-1 ⇒ℙ( d^-1/2δ^(n) > g(n +√(nlog(n))))≤log(n)^-1 ⇒ℙ( δ^(n) >√(d)g(n +√(nlog(n))))≤log(n)^-1The result in (<ref>) follows from taking the limit, and the result in (<ref>) follows from g(n +√(nlog(n))) ≤ 2 g(n) for all non-decreasing g(n) = n^p/d and n ≥ 1.Rectangular LPMs are regular with G(n) = 2√(d) n^p/d. Criteria 1 and 2 of a regular LPM hold by definition of a rectangular LPM. Lemma <ref> guarantees that satisfaction of criterion 3. Consider a LPM on S = ℝ^d with each latent position independently and identically distributed according to a multivariate Gaussian distribution with mean zero and diagonal variance matrix Σ. Let σ^2_1, …, σ^2_d denote the entries along the diagonal of Σ, with σ_1 ≥σ_2 ≥⋯≥σ_d > 0.If the link probability function is upper bounded by 1 -ϵ, then the LPM is regular with G(n) =√(2 σ_1^2 (1+c) log(n)) for any c > 0. Criteria 1 and 2 for regularity hold trivially. Thus, it is sufficient to prove criteria 3 for the prescribed G(n). Let Z_1, …, Z_n denote the latent positions. Define X_1, …, X_n such that for i ∈{1, …, n} and j ∈{1, …, d}, X_ij := σ_1/σ_j Z_ij.It follows from σ_1 ≥σ_j that X_i≥ Z_i. Moreover, each X_i is a mean zero isotropic Gaussian random vector with variance σ_1^2 in each dimension. Therefore, X_i^2/σ_1^2 follows a χ^2 distribution with parameter d. We can apply the concentration inequality on χ^2 random variables implied by <cit.>, to conclude, for any t > 0ℙ(X_i > σ_1 √(d + 2t + 2 √(d t)))≤exp(-t) ⇒ℙ(X_i > √(2)σ_1 (u + √(d)) )≤exp(-u^2)for any u > 0. Applying the union bound results inℙ(max_1 ≤ i ≤ nX_i >√(2)σ_1 (u + √(d)))≤ nexp(-u^2).So long as u^2 ≥ (1+c)log(n), for c > 0, the above probability goes to 0. Note that √(2 σ_1^2 (1+c) log(n)) dominates √(2d σ_1^2) as n grows.Because X_i≥ Z_i, a choice of G(n) =√(2 σ_1^2 (1+c) log(n)) yields the desired result for c > 0.Symmetrization LemmaLetΩ = {X ∈ℝ^n × d: X_i≤ G(n) ∀ i ∈ [n] }for G(n) ∈ℝ_+. Let L(x: Y^n) denote the log likelihood of the latent positions x ∈Ω as defined in (<ref>) for a link function K. Let (x) = L(x: Y^n) - L(0:Y^n) and ((x)) denote its expectation. Then, for h ≥ 1,(sup_x ∈Ω |(x) - ((x))|^h )≤ 2^h ( sup_x ∈Ω|∑_j=1^n ∑_i=1^n R_ij(Y^n_ijlog(K(δ^x_ij)/K(0)) + (1 - Y^n_ij)log(1- K(δ^x_ij)/1- K(0)) ) |^h)where R denotes an array of independent Rademacher random variables and δ^x_ij = x_i - x_j. This proof follows the same argument of that of <cit.>. Let _ij(x) denote the contribution of Y^n_ij to the standardized log likelihood. Thus, (x)= ∑_i=1^n ∑_j=1^n _ij(x)and (x) - ((x))= ∑_i=1^n ∑_j=1^n ℓ_ij(x)where each ℓ_ij(x) = _ij(x) - (_ij(x)) is a zero mean random variable. For each i,j, let ℓ'_ij(x) denote a random variable that is independently drawn from the distribution of ℓ_ij(x). Then, ℓ_ij(x) - ℓ'_ij(x) is a symmetric zero mean random variable with the same distribution as R_ij(ℓ_ij(x) - ℓ'_ij(x)). Moreover, we can view sup_x ∈Ω |f(x)| as defining a norm on the Banach space of functions f: Ω→ℝ. These facts, along with the convexity of exponentiating by h, imply the following. (sup_x ∈Ω|(x) - ((x))|^h )= (sup_x ∈Ω|∑_i=1^n ∑_j=1^n ℓ_ij(x)|^h )≤(sup_x ∈Ω|∑_i=1^n ∑_j=1^n ℓ_ij(x) - ℓ'_ij(x) |^h )(cf. <cit.>)= (sup_x ∈Ω|∑_i=1^n ∑_j=1^nR_ij(ℓ_ij(x) - ℓ'_ij(x)) |^h )= ( |sup_x ∈Ω∑_i=1^n ∑_j=1^nR_ij(_ij(x) - '_ij(x)) |^h )≤( (sup_x ∈Ω|∑_i=1^n ∑_j=1^n R_ij_ij(x)| + sup_x ∈Ω|∑_i=1^n ∑_j=1^n R_ij'_ij(x) |)^h )≤( 1/2sup_x ∈Ω|2∑_i=1^n ∑_j=1^n R_ij_ij(x)|^h + 1/2sup_x ∈Ω|2∑_i=1^n ∑_j=1^n R_ij'_ij(x) |^h )by convexity of exponentiating by h=1/2( sup_x ∈Ω|2∑_i=1^n ∑_j=1^n R_ij_ij(x)|^h) + 1/2𝔼(sup_x ∈Ω|2∑_i=1^n ∑_j=1^n R_ij'_ij(x) |^h ) = 2^h (sup_x ∈Ω|∑_i=1^n ∑_j=1^nR_ij_ij(x) |^h ).The result follows from the definitions of the _ij. Contraction Theorem <cit.>. Let F: ℝ_+ →ℝ_+ be convex and increasing. Let ϕ_i:ℝ→ℝ for i ≤ N satisfy ϕ_i(0) = 0 and |ϕ_i(s) - ϕ_i(t)| ≤ |s-t| for all s,t ∈ℝ. Then, for any bounded subset Ω⊂ℝ,𝔼(F (1/2sup_t ∈Ω^N| ∑_i=1^N R_i ϕ_i(t_i) | ) )≤𝔼(F (sup_t ∈Ω^N| ∑_i=1^N R_i t_i | ) )where R_1, …, R_N denote independent Rademacher random variables.Let R denote an n× n array of independent Rademacher random variables, K: ℝ_+ → [0, 1-ϵ] denote a link function that satisfies the regularity criteria in <ref> (i.e. monotonically decreasing, differentiable function that is upper bounded by 1- ϵ for some ϵ), and Ω = {X ∈ℝ^n × d: X_i≤ G(n)for alli ∈ [n] }with G(n) ∈ℝ_+, and Y^n ∈{0,1}^n × n. Define α_n^K as in (<ref>). That is,α^K_n = sup_0 ≤ x ≤ 2G(n)|K'(x)|/|x| K(x)ϵ.Then,( sup_x ∈Ω|∑_j=1^n ∑_i=1^n R_ij(Y^n_ijlog(K(δ^x_ij)/K(0)) + (1 - Y^n_ij)log(1- K(δ^x_ij)/1- K(0)) ) |^h) ≤ (2α^K_n)^h (sup_x ∈Ω(|∑_j=1^n ∑_i=1^n R_ijx_i - x_j^2 |^h) )for h ≥ 1, where δ^x_ij = x_i - x_j. We can apply Lemma <ref> to obtain this result as follows.For all x ∈Ω, i,j ∈ [n], we know by the triangle inequality that x_i - x_j^2 ≤ 4G(n)^2. Moreover, K(2G(n)) ≤ K(x_i - x_j) ≤ 1 - ϵ because K is regular. A Taylor expansion of log(K(√(·))) around 0 reveals that log(K(√(u))) - log(K(√(0)))= u K'(√(w))/2√(w) K(√(w)) for some w∈[0,4G(n)^2]= u K'(v)/2 v K(v) for some v∈[0,2G(n)].Taking the supremum over possible values of v, it follows that| log(K(√(u))) - log(K(√(0)))/α^K_n|≤ u.Similarly, a Taylor expansion of log(1 -K(√(·))) around 0 yieldslog(1-K(√(u))) - log(1-K(√(0)))= -u K'(√(w))/2v(1-K(√(w))) for some w∈[0,4G(n)^2] = -u K'(v)/2v(1-K(v)) for some v∈[0,2G(n)].Similarly, taking the supremum over possible values of v yields| log(1-K(√(u))) - log(1-K(√(0)))/α^K_n|≤ u.Together, we have| Y^n_ijlog(K(u)/K(0)) + (1 - Y^n_ij)log(1- K(u)/1- K(0)) /α^K_n|≤ u.Moreover, for any i,j, the function on the lefthand side is 0 at u=0. Thus, the function meets the criteria required of the ϕ functions in Lemma <ref> and the result follows from convexity of exponentiating by h. Let Σ, Σ̂∈ℝ^n × n be symmetric, with eigenvalues λ_1≥λ_2 ≥⋯≥λ_n and λ̂_1 ≥⋯≥λ̂_n, respectively. Fix 1 ≤ r ≤ s ≤ n and assume that min(λ_r-1 - λ_r, λ_s - λ_s + 1) > 0 where λ_0 := ∞ and λ_n + 1 := - ∞. Let q := s - r + 1. Let V, V̂∈ℝ^n × q have orthonormal columns satisfying Σ V_j = λ_j V_j and Σ̂V̂_j = λ̂_jV̂_j for j ∈{1, …, q}. Then, there exists an orthogonal matrix O ∈ℝ^q × q such thatV̂ O - V^2_F≤2^3Σ̂ - Σ^2_F/min(λ_r-1 - λ_r, λ_s - λ_s + 1)^2.This follows from the Davis-Kahan Theorem <cit.>. §.§ Projectivity Proofs §.§.§ Proof of Proposition <ref> Let Y^n_1 and Y^n_2 denote random graphs with n_1 and n_2 nodes (n_1 < n_2) generated according to an exchangeable LPM, and let ℙ^n_1 and ℙ^n_2 be their corresponding distributions. Let Z^j_i denote the random latent position of node i in Y^j for j=n_1,n_2. By definition, Z^n_1_i and Z^n_2_i are iid draws from the same distribution f on S. Thus, the (Z^j_i)_i=1… n_1 have identical distributions for each j. As a result, K(ρ(Z^n_1_i_1, Z^n_1_i_1)) has the same distribution as K(ρ(Z^n_1_i_2, Z^n_1_i_2)) for any 1 ≤ i_1, i_2 ≤ n_1. Because the distributions for each dyad coincide, the distributions over adjacency matrices coincide. §.§.§ Proof of Proposition <ref> Let Y^n_1 and Y^n_2 denote random graphs distributed with n_1 and n_2 nodes (n_1 < n_2) obtained by the finite window approach on the Poisson random connection model on ℝ_+, and let ℙ^n_1 and ℙ^n_2 be their corresponding distributions. Let Z^j_i denote the random latent position of node i in Y^j for j=n_1,n_2. For both cases, the random variables Z^j_1 - 0, Z^j_2 - Z^j_1, …, Z^j_n_1 - Z^j_n_1 - 1 are iid exponential random variables, by the interval theorem for point processes <cit.>. Thus, the (Z^j_i)_i=1… n_1 have identical distributions for each j. The rest follows identically as for Proposition <ref>. §.§.§ Proof of Theorem <ref> Let Y^n_1 and Y^n_2 denote random graphs distributed with n_1 and n_2 nodes (n_1 < n_2) obtained from a rectangular LPM on ℝ_+^d. Let ℙ^n_1 and ℙ^n_2 be their corresponding distributions. Let t_i^j, denote the arrival time for the ith node in Y^j for j=n_1,n_2. Following, Lemma <ref>, both t_i^n_1 and t_i^n_2 are equally distributed. Therefore, Z_i^n_1 and Z_i^n_2 must also be equally distributed. The rest follows as in the proofs for Proposition <ref>. §.§.§ Proof of Proposition <ref> Suppose (s_n)_n=1…∞ is not constant. Then there is an n_2 > n_1 ≥ 2 such that s_n ≠ s_n_2. Let Y^n_1 and Y^n_2 denote random graphs with n_1 and n_2 nodes. Notice that the marginal distribution of Y^n_12 in a graph with n nodes is given byℙ^n(Y_12=1)= 𝔼(ℙ(Y^n_12=1|Z_1, Z_2)) =𝔼(s_nK(ρ(Z_1, Z_2) ) )= s_n𝔼(K(ρ(Z_1, Z_2) ) ).Clearly,ℙ^n_1(Y_12=1)≠ℙ^n_2(Y_12=1) because s_n_1≠ s_n_2 and Z_1,Z_2 ∼ f independently of k. Thus the model cannot be projective.§.§ Sparsity Proofs§.§.§ Proof of Proposition <ref>Let n be the number of nodes in the latent position network model. Then the expected number of edges ∑_i=1^n ∑_j=1^n Y_ij is given by𝔼(∑_i=1^n ∑_j=1^n Y_ij/n^2)= 1/n^2∑_i=1^n ∑_j=1^n 𝔼( 𝔼(Y_ij| Z_i, Z_j ))= 1/n^2∑_i=1^n ∑_j=1^n𝔼 K(ρ(Z_i, Z_j))= 𝔼 K(ρ(Z_i, Z_j))where 𝔼 K(ρ(Z_i, Z_j)) is constant due to Z_i being independent and identically distributed. Thus, as long as the network is not empty, it is dense. §.§.§ Proof of Proposition <ref>A special case of Theorem <ref> with d=1 and g(t) = t.§.§.§ Proof of Theorem <ref> Let π be a permutation on (1, …, n) chosen uniformly at random from the set of permutations on (1, …, n). Then,∑_i=1^n∑_j=1^n Y_ij = ∑_i=1^n∑_j=1^n Y_π(i)π(j).Let Ω = [-g(t_n+1), g(t_n+1)]^d. By Lemma <ref>,𝔼(Y_π(i)π(j)|t_n+1)= ∫_z,z' ∈Ω K(z - z') 1/2^dg(t_n+1)^ddz'1/2^dg(t_n+1)^ddz ≤1/4^d g(t_n+1)^2d∫_z ∈Ω C dz∝1/2^dg(t_n+1)^dfor some C ∈ℝ_+ by Lemma <ref>. Similarly,𝔼(Y_π(i)π(j)|t_n+1)= ∫_z,z' ∈Ω K(z - z') 1/2^dg(t_n+1)^ddz'1/2^dg(t_n+1)^ddz ≥C'/2^dg(t_n+1)^dfor some C' ∈ℝ_+ by Lemma <ref>. Thus,𝔼(g(n)^d/n^2∑_i=1^n ∑_j=1^n Y_ij| t_n+1)= 𝔼(g(n)^d/n^2∑_i=1^n ∑_j=1^n Y_π(i)π(j)| t_n+1)= g(n)^d 𝔼(Y_π(i)π(j)| t_n+1)∝g(n)^d/g(t_n+1)^d.We can analytically integrate over possible t_n+1 because t_n+1 follows Gamma(n+1, 1), as given by Lemma <ref>.𝔼(g(n)^d/n^2∑_i=1^n ∑_j=1^n Y_ij)= 𝔼(𝔼(g(n)^d/n^2∑_i=1^n ∑_j=1^n Y_ij| t_n+1) )∝𝔼(g(n)^d/g(t_n+1)^d)= n^p∫_0^∞ t^-p1/Γ(n+1) t^nexp(-t)dt= n^pΓ(n-p+1)/Γ(n+1)which converges to one as n goes to infinity. §.§.§ Proof of Proposition <ref>Let n be the number of nodes in the LPM. Then the expected number of edges ∑_i=1^n ∑_j≠ i Y_ij is given by𝔼(∑_i=1^n ∑_j≠ iY_ij)= ∑_i=1^n ∑_j≠ i𝔼( 𝔼(Y_ij| Z_i, Z_j ))= ∑_i=1^n ∑_j≠ i𝔼(K_n(ρ(Z_i, Z_j)))= n(n-1) 𝔼(K_n(ρ(Z_i, Z_j)))where 𝔼(K_n(ρ(Z_i, Z_j))) takes the same value for all i≠ j because the Z_i are independent and identically distributed. By definition of K_n, we have thats_n K(ρ(Z_i, Z_j)) ≥ K_n(ρ(Z_i, Z_j)) ≥s_n s_1^-1 K_1(ρ(Z_i, Z_j)).Therefore,𝔼(K_n(ρ(Z_i, Z_j)) )≤ s_n 𝔼 (K(ρ(Z_i, Z_j))) 𝔼(K_n(ρ(Z_i, Z_j))) ≥ s_n s_1^-1𝔼 (K_1(ρ(Z_i, Z_j))).Since both 𝔼 (K(ρ(Z_i, Z_j)) and s_1^-1𝔼 (K_1(ρ(Z_i, Z_j))) are constants that are independent of n, the expected number of edges must be of order s_n n^2.§.§ Learnability Proofs §.§.§ Proof of Lemma <ref> Much of the argument provided here can be viewed specialization of the results established in <cit.>. For clarity, we include the entirety of the argument, illustrating our non-standard choices for many of the components, as well as some small differences such as using a restricted maximum likelihood estimator. The notation for our proofs is simplified by working with the following standardized version of the likelihood(z: Y^n)= L(z: Y^n) - L(z=0: Y^n)= ∑_i=1^n ∑_j =1^n Y^n_ijlog(K(δ^z_ij)/K(0)) + (1- Y^n_ij) log(1- K(δ^z_ij)/1 - K(0))where δ^z_ij = z_i - z_j. Note that the standardized likelihood and non-standardized version of the likelihood are maximized by the same value of z for a given Y^n. Going forward, we use the shorthand (z); Y^n is implied.In order to establish concentration of P^ẑ(Y^n) - P^z_F^2, we first establish concentration of related quantities. Specifically, Lemma <ref> establishes concentration of KL(P^ẑ(Y^n), P^z). Here, KL(P, Q) denotes the Kullback-Leibler divergence <cit.> between two link probability matrices P and Q. It is a non-negative and given byKL(P, Q)= ∑_i=1^n ∑_j=1^n log(P_ij/Q_ij) + (1-P_ij) log(1 - P_ij/ 1- Q_ij).Consider a sequence of adjacency matrices Y^n generated by a LPM meeting the regularity criteria provided in <ref>. Further assume that the true latent positions are within Ω = {X ∈ℝ^n × d: X_i≤ G(n) }where X^i denotes the ith row of X. Let P^Ẑ(Y^n) denote the estimated link probability matrix obtained via Ẑ(Y^n) from (<ref>). Then,ℙ(KL(P^ẑ(Y^n), P^z) ≥ 16 e α^K_n G(n)^2 n^1.5 (d + 2) ) ≤C/n^2for some C > 0.Note that for any z_0 ∈Ω, we have(z_0) - (z)= ((z_0) - (z)) + (z_0) - ((z_0)) - ((z) - ((z)))≤((z_0) - (z))+ | (z_0) - ((z_0))| + |((z) - ((z)))|≤((z_0) - (z)) + 2 sup_x ∈Ω |((x) - ((x)))|≤ -KL(P^z_0, P^z) + 2 sup_x ∈Ω |((x) - ((x)))|.Let z_0 = Ẑ(Y^n) denote the maximum likelihood estimator given in (<ref>). Then, because (z_0) - (z) ≥ 0,KL(P^ẑ(Y^n), P^z)≤ 2 sup_x ∈Ω |((x) - ((x)))|.So we can upper bound KL(P^ẑ(Y_n), P^z) by boundingsup_x ∈Ω |((x) - ((x)))|. Let h be an arbitrary positive integer (we will later let it be 2log(n)). Applying the Markov inequality for sup_x ∈Ω |((x) - ((x)))|^h yields:(sup_x ∈Ω |((x) - ((x)))|^h > c(n)^h)≤ (sup_x ∈Ω |((x) - ((x)))|^h) /c(n)^hfor a positive function c: ℕ→ℝ_+. To bound the expectation, we use a symmetrization argument (provided as Lemma <ref> in <ref>) followed by a contraction argument (stated as Corollary <ref> in <ref>). (sup_x ∈Ω |((x) - ((x)))|^h ) ≤ 2^h ( sup_x ∈Ω|∑_j=1^n ∑_i=1^n R_ij(Y^n_ijlog(K(δ^z_ij)/K(0)) + (1 - Y^n_ij)log(1- K(δ^z_ij)/1- K(0)) ) |^h) by Lemma <ref>≤ (4α^K_n)^h (sup_x ∈Ω(|∑_j=1^n ∑_i=1^n R_ijx^i - x^j^2 |^h) )by Corollary <ref>.= (4α^K_n)^h ( sup_x ∈Ω| ⟨ R, D^x ⟩|^h )where R = (R_ij) is a matrix of independent Rademacher random variables, D^x denotes the matrix of squared distances implied by x, and α^K_n is defined as in (<ref>). Let ·_o denote the operator norm and ·_* denote the nuclear norm. To bound ( sup_x ∈Ω⟨ R,D^x ⟩^h ), we make use of the fact that |⟨ A, B ⟩| ≤A_o B_*. Then,( sup_x ∈Ω| ⟨ R, D^x ⟩|^h )≤(sup_ΩR_o^h D^x^h_*)= (R_o^h)sup_x ∈ΩD^x_*^h≤C n^h/2sup_x ∈ΩD^x_*^h where (R_o^h) was bounded using <cit.> and C > 0 is a constant provided that h ≤ 2 log(n).Recall that the rank of a squared distance matrix D^x is at most d+2 where d is the dimension of the positions x. Moreover, each of the eigenvalues of D^x must be upper bounded by the product of maximum distance in D^x and n (where n is the number of points). For x ∈Ω, the maximum entry in D^x is at most 4G(n)^2. Thus, sup_x ∈ΩD^x_*≤ (d+2) 4nG(n)^2. Therefore, ( sup_x ∈Ω| ⟨ E, D^x ⟩|^h ) ≤ C n^3h/2(2G(n))^2h (d + 2)^h. Combining the above results yields(KL(P^ẑ(Y_n), P^z) ≥ c(n))≤2^4h C n^3h/2(α_n^K)^hG(n)^2h(d + 2)^h/c(n)^h.Let c(n) = 2^4 C_0 α^K_n G(n)^2 (d+2) n^3/2 for some constant C_0. Then, by letting h = 2log(n), we get(KL(P^ẑ(Y_n), P^z) ≥ c(n))≤ C C_0^-h= C C_0^-2log(n)= C /n^2 log(C_0)The result follows from letting C_0 = e. We can now leverage Lemma <ref> into Corollary <ref>, a concentration bound on the squared Hellinger distance d^2_H(P^ẑ(Y^n), P^z). Here, d^2_H(P, Q) denotes the squared Hellinger distance between two link probability matrices P and Q given byd^2_H(P, Q)= ∑_i=1^n ∑_j=1^n (√(P_ij) - √(Q_ij))^2 + (√(1 - P_ij) - √(1 - Q_ij))^2.Consider a sequence adjacency matrices Y^n generated by a LPM meeting the criteria provided in Section <ref>. Further assume that the true latent positions are within Ω = {X ∈ℝ^n × d: X_i≤ G(n) }.Let P^Ẑ(Y^n) denote the estimated link probability matrix obtained via Ẑ(Y^n) from (<ref>). Then,ℙ(d^2_H(P^ẑ(Y^n), P^z) ≥ 16 e α^K_n G(n)^2 n^1.5 (d + 2) ) ≤C/n^2(<ref>) Follows from Lemma <ref> and the fact that Kullback-Leibler divergence upper bounds the squared Hellinger distance <cit.>.Finally, the Frobenius norm P - Q_F^2 between P and Q is upper bounded by the squared Hellinger distance. We can thus proceed with our proof of Lemma <ref>.The result follows from Corollary <ref> because the squared Hellinger distance between two link probability matrices upper bounds the squared Frobenius norm between them. This follows from the fact that u ≤√(u) for all u ∈ [0,1]. Note that, rather than immediately upper bounding P - Q_F^2 by KL(P, Q), we introduce d^2_H(P, Q) in Corollary <ref> due to its utility in proving Lemma <ref>.§.§.§ Proof of Lemma <ref>Let d̂_ij and d_ij denote the (i,j)th entry of D^Ẑ(Y^n) and D^Z respectively. Then, d^2_H(P^Ẑ(Y^n), P^Z) = ∑_i=1^n ∑_j=1^n (K(√(d_ij))^1/2 - K(√(d̂_ij))^1/2)^2 + ((1-K(√(d_ij)))^1/2 - (1-K(√(d̂_ij)))^1/2)^2can be lower-bounded as follows. Notice that for any a,b ∈ℝ, a^2 + b^2 ≥1/2(a-b)^2. Therefore, d^2_H(P^Ẑ(Y^n), P^Z) ≥ ∑_i=1^n ∑_j=1^n 1/2((K(√(d_ij))^1/2 - K(√(d̂_ij))^1/2) - ((1-K(√(d_ij)))^1/2 - (1-K(√(d̂_ij)))^1/2))^2= ∑_i=1^n ∑_j=1^n 1/2((K(√(d_ij))^1/2 -(1-K(√(d_ij)))^1/2) - ( K(√(d̂_ij))^1/2 - (1-K(√(d̂_ij)))^1/2))^2. Let γ(t) = √(K(√(t))) - √(1- K(√(t))). Taylor expanding γ(t) around t = d_ij reveals that γ(d̂_ij)=γ(d_ij) + γ'(v) (d̂_ij - d_ij)for some v ∈ [0,4G(n)^2]withγ'(v)= K'(√(v))/4√(v)(1/√(K(√(v))) + 1/√(1 - K(√(v))))and K'(v) denoting the derivatives of γ and K with respect to v, respectively. Noting that |γ'(v)|≤|K'(√(v))|/4√(v)(1/√(K(√(v)) (1 - K(√(v))))),we can combine these results with (<ref>) to obtain a bound:d^2_H(P^Ẑ(Y^n), P^Z) ≥∑_i=1^n ∑_j=1^n 1/2inf_v ∈ [0,4G(n)^2]γ'(v)^2 (d̂_ij - d_ij)^2 = 1/2(inf_v ∈ [0,4G(n)^2]γ'(v)^2 )D^Ẑ(Y^n) - D^Z _F^2≥1/32(inf_θ∈ [0,2G(n)]K'(θ)^2/θ^2 K(θ)(1-K(θ))) D^Ẑ(Y^n) - D^Z _F^2= 1/32(sup_θ∈ [0,2G(n)]( θ^2 K(θ)(1-K(θ))/K'(θ)^2))^-1D^Ẑ(Y^n) - D^Z _F^2≥1/32 β^K_nD^Ẑ(Y^n) - D^Z _F^2where β^K_n is defined as in (<ref>). Combining this inequality with Corollary <ref> yieldsℙ(d^2_H(P_ẑ, P_z) ≥ 16 e α^K_n G(n)^2 n^1.5 (d + 2) )≤C/n^2 ⇒ℙ(1/32β^K_nD^Ẑ(Y^n) - D^Z _F^2 ≥ 16 e α^K_n G(n)^2 n^1.5 (d + 2) )≤C/n^2 ⇒ℙ(D^Ẑ(Y^n) - D^Z _F^2 ≥ 512eβ^K_nα^K_n G(n)^2 n^1.5 (d + 2) )≤C/n^2.§.§.§ Proof of Lemma <ref> Before proving Lemma <ref>, it is useful to first summarize our general strategy and introduce some notation. Our proof involves translating our concentration inequality for the latent squared distances (provided as Lemma <ref>) to an analogous one for the latent positions. To do this, we combine results from classical multidimensional scaling <cit.>, Weyl's inequality <cit.>, and the Davis-Kahan theorem (Lemma <ref>). Classical multi-dimensional scaling recovers a set of positions Z ∈ℝ^n × d corresponding to a squared distance matrix D ∈ℝ^n × n. It does so through an eigendecomposition of the double centered distance matrix -0.5 𝒞_n D 𝒞_n with 𝒞_n defined in (<ref>). Here,Z= V Λ^1/2is used to denote the recovered positions with Λ∈ℝ^d× d denoting a diagonal matrix consisting of the d nonzero eigenvalues of -0.5 𝒞_n D 𝒞_n and V ∈ℝ^n × d denoting a matrix with columns comprised of the corresponding eigenvectors. This technique is guaranteed to recover Z exactly (up to translations, rotations and reflections).Multidimensional scaling of both D^Z and D^Ẑ(Y^n) recovers the versions of the true latent positions and maximum likelihood estimatesZ= V Λ^1/2 Ẑ(Y^n)= V̂Λ̂^1/2.We resolve the identifiability issues (stemming from translations, rotations, and reflections) of these objects by minimizing over O ∈𝒪_p (rotations and reflections) and Q ∈𝒬_nd (translations). We avoid explicitly minimizing over translations in our proof by considering the centered estimates obtain from multi-dimensional scaling. That is, the versions of Z and Ẑ(Y^n) obtained from multi-dimensional scaling end up being sufficient andinf_O ∈𝒪_dQ ∈𝒬_ndẐ(Y^n)O - Z - Q^2_F≤inf_O ∈𝒪_dV̂Λ̂^1/2 O - V Λ^1/2^2_F. A few properties of orthonormal matrices are useful in our proof. By definition, the columns of V̂ and V have norm 1. Multiplying a matrix by O, V, or V̂ does not modify its Frobenius norm because their columns are orthonormal. Similarly, centering a matrix cannot increase its Frobenius norm, so pre-multiplying or post-multiplying by 𝒞_n does not increase the Frobenius norm. Finally, the following bits of notation are useful for keeping the proof succinct. For any m ∈ℕ, we let [m] = { 1, 2, …, m}. Also, the direct sum of matrices M_1 and M_2 is defined asM_1 ⊕ M_2:= [[ M_1 0; 0 M_2 ]]. We can now proceed with the details of the proof. Let Z = V Λ^1/2 and Ẑ = V̂Λ̂^1/2 be obtained from multidimensional scaling on D^Z and D^Ẑ(Y^n). Recall that the diagonal entries in Λ and Λ̂ are arranged in decreasing order such that λ_1 ≥λ_2 ≥⋯≥λ_d > 0 with the same decreasing structure for λ̂ (but allowing for λ̂_d = 0). Let O ∈𝒪_p denote a generic orthogonal matrix. First, we note that for any O ∈𝒪_p,ẐO - Z^2_F= V̂Λ̂^1/2O - V Λ^1/2^2_F = V̂Λ̂^1/2O - V̂Λ^1/2O + V̂Λ^1/2O- VΛ^1/2_F^2≤ 2V̂Λ̂^1/2O - V̂Λ^1/2O_F^2 + 2V̂Λ^1/2O- VΛ^1/2_F^2= 2 V̂ (Λ̂^1/2 - Λ^1/2) O^2_F + 2V̂Λ^1/2O- VΛ^1/2_F^2= 2 Λ̂^1/2 - Λ^1/2^2_F + 2V̂Λ^1/2O- VΛ^1/2_F^2 Furthermore, for an arbitrary matrix Φ∈ℝ^d × d, we haveV̂Λ^1/2O- VΛ^1/2_F^2= V̂(Λ^1/2 - Φ)O- V(Λ^1/2 - Φ) + V̂Φ O- V Φ_F^2 ≤ 3 Λ^1/2 - Φ^2_F + 3 Λ^1/2 - Φ^2_F + 3 V̂Φ O- V Φ_F^2= 6 Λ^1/2 - Φ^2_F + 3 V̂Φ O- V Φ_F^2.Substituting this back into (<ref>), we haveẐO - Z^2_F≤2 Λ̂^1/2 - Λ^1/2^2_F+ 12 Λ^1/2 - Φ^2_F+ 6 V̂Φ O- V Φ_F^2. We begin by establishing a bound on the first summand (<ref>). Weyl's inequality <cit.> tells us that|λ̂_i - λ_i |≤V̂Λ̂V̂^T - V Λ V^T_F for all i ∈ [d].Because centering a matrix cannot increase its Frobenius norm, this implies thatV̂Λ̂V̂^T - V Λ V^T_F= 𝒞_n(D^Ẑ(Y^n) - D^Z)𝒞_n_F≤D^Ẑ(Y^n) - D^Z _F.Noting that |λ̂^1/2_i - λ_i^1/2| = | λ̂_i - λ_i/λ_i^1/2 + λ̂_i^1/2|≤| λ̂_i - λ_i/λ_d^1/2|,it thus follows from Weyl's inequality thatΛ̂^1/2 - Λ^1/2^2_F= ∑_i=1^d (λ̂^1/2_i - λ_i^1/2)^2≤∑_i=1^d (λ̂_̂î - λ_i)^2/λ_d≤d/λ_dD^Ẑ(Y^n) - D^Z ^2_F.The result in (<ref>) is enough to for us to establish concentration of the first summand in (<ref>). Next, we consider the second and third summands from (<ref>) and (<ref>).It will be convenient to build a diagonal structure on the matrix Φ as follows. Recall our assumption that the LPM possesses a(n)-b(n) distinctly bunched eigenvalues for functions a(n) and b(n). Define k ∈ℕ and i_1, …, i_k, i_k+1 so as to satisfy the requirement of a(n)-b(n) distinctly bunched eigenvalues. For notational convenience, we also define i_0 := 0, λ_0 := ∞.Given k and the i_j's, we define the diagonal matrix Φ∈ℝ^d × d asΦ := λ^1/2_i_1 I_i_2 - i_1⊕λ^1/2_i_2 I_i_3 - i_2⊕⋯⊕λ^1/2_i_k-1 I_i_k - i_k-1⊕λ^1/2_i_k I_i_k +1 - i_kwhere I_d^* denotes the d^*× d^* identity matrix for any d^* ∈ℕ. It follows that Λ^1/2 - Φ^2_F= ∑_j=1^k ∑_i = i_j^i_j+1 - 1(λ_i^1/2 - λ_i_j^1/2)^2 ≤∑_j=1^k (i_j+1 - i_j - 1) (λ_i_j+1 - 1^1/2 - λ_i_j^1/2)^2 = ∑_j=1^k (i_j+1 - i_j - 1) (λ_i_j+1 - 1 - λ_i_j)^2/(λ_i_j+1 - 1^1/2 + λ_i_j^1/2)^2 ≤∑_j=1^k (i_j+1 - i_j - 1) (λ_i_j+1 - 1 - λ_i_j)^2/4 λ_i_j+1 - 1≤ (d - k) sup_j ∈ [k](λ_i_j+1 - 1 - λ_i_j)^2/4 λ_i_j+1 - 1≤(d - 1)/4 a(n), where the last line follows from the fact that the LPM possesses a(n)-b(n) distinctly bunched eigenvalues, as well as the fact that k ≥ 1.This provides a bound for (<ref>). Finally, let's consider(<ref>). Let's impose some additional structure on O by assuming O = O^Φ, where O^Φ is a block diagonal orthogonal matrix such that O^Φ = O^1⊕ O^2⊕⋯⊕ O^k,where for each j ∈ [k], O^j∈𝒪_i_j+1 - i_j. Let 𝒪^Φ denote the class of matrices possessing this block diagonal orthogonal structure, and note that 𝒪^Φ⊆𝒪_p. Because Φ and O^Φ possess the same block diagonal structure with each block of Φ being a scalar matrix, we have commutativity of the multiplication O^ΦΦ = Φ O^Φ. Therefore,V̂Φ O^Φ - V Φ_F^2 = V̂ O^ΦΦ - V Φ_F^2=∑_j=1^k λ_i_jV̂_i_j:(i_j+1 - 1) O^j - V_i_j:(i_j+1 - 1)_F^2by Lemma <ref>, where V_i_j:(i_j+1 - 1) and V̂_i_j:(i_j+1 - 1) are the submatrices of V and V̂ consisting of columns i_j through i_j+1 - 1.This format of (<ref>) is amenable to the application of the Davis-Kahan Theorem as stated in <cit.> (Lemma <ref>). Recall that this theorem tells us that, for any 1 ≤ r ≤ s ≤ d and q = s - r + 1,inf_O ∈𝒪_qV̂_r:s O - V_r:s_F^2≤2^3ẐẐ^T- Z Z^T ^2_F/min(λ_r-1 - λ_r, λ_s - λ_s+1)^2 Recall that ẐẐ^T- Z Z^T ^2_F ≤D^Ẑ(Y^n) - D^Z ^2_F. For a fixed choice of Φ, we thus haveinf_O ∈𝒪_pV̂Φ O- V Φ_F^2≤inf_O ∈𝒪^ΦV̂Φ O- V Φ_F^2=∑_j=1^k λ_i_jinf_O ∈𝒪_i_j+1 - i_jV̂_i_j:(i_j+1 - 1) O - V_i_j:(i_j+1 - 1)_F^2≤ ∑_j=1^k λ_i_j2^3D^Ẑ(Y^n) - D^Z ^2_F/min(λ_i_j - 1 - λ_i_j, λ_i_j+1 - 1 - λ_i_j+1)^2 ≤ 2^3 k D^Ẑ(Y^n) - D^Z ^2_F sup_j ∈ [k]λ_i_j/(λ_i_j+1 - 1 - λ_i_j+1)^2 ≤ 2^3 d D^Ẑ(Y^n) - D^Z ^2_F b(n)^-1,due to the assumption that the LPM possesses a(n)-b(n) distinctly bunched eigenvalues and that k ≤ d.Bringing together our bounds on the equations (<ref>), (<ref>), (<ref>), we obtain the boundinf_O ∈𝒪_dẐ O - Z^2_F≤3 (d - 1) a(n) + d (2/λ_d +48/b(n))D^Ẑ(Y^n) - D^Z ^2_F≤3 (d - 1) a(n) + 50 d b(n)^-1D^Ẑ(Y^n) - D^Z ^2_Fbecause λ_d ≥ b(n) by the definition a(n)-b(n) distinctly bunched eigenvalues.The final result then follows from applying Lemma <ref> to bound D^Ẑ(Y^n) - D^Z ^2_F.This concludes the proof of Lemma <ref>.Consider d, k ∈ℕ such that k ≤ d and ℓ_1, …, ℓ_k ∈ℕ such that ∑_j=1^k ℓ_j = d. Define O= O^1⊕ O^2⊕⋯⊕ O^k, Φ = ϕ_1 I_ℓ_1⊕ϕ_2 I_ℓ_2⊕⋯⊕ϕ_k I_ℓ_k,where O^1, …, O^k are orthogonal matrices satisfying O^1 ∈𝒪_ℓ_1, …. O^k ∈𝒪_ℓ_k and ϕ_1, …, ϕ_k ∈ℝ. Define i_1 := 1 and i_j =: i_j-1 + ℓ_j-1for j ∈{2, …, k}. Then,U O Φ - V Φ_F^2 = ∑_j=1^k ϕ_j^2 U_i_j:(i_j + ℓ_j - 1) O^j - V_i_j:(i_j + ℓ_j - 1)_F^2for any two matrices U, V ∈ℝ^n × d satisfying U^T U = I_d and V^T V = I_d. Here, U_a:b and V_a:b denote the submatrices of U and V consisting of columns a through b.Define A^1, …, A^k∈{0,1}^d × d such that for each j ∈ [k], A^j := c^j_1 I_ℓ_1⊕ c^j_2 I_ℓ_2⊕⋯⊕ c^j_k I_ℓ_kand for i,j ∈ [k], c^j_i = 1 if i = j, and is 0 otherwise. Note that each A^j is symmetric, diagonal, and idempotent with∑_j=1^k A^j = I_d.Furthermore, each A^j can be decomposed according toA^j = a^j.a^j.^Twhere, for j ∈ [k], a^j∈{0,1}^d ×ℓ_j such thata^j_ℓ m =1 if ℓ∈{i_j, i_j + 1 …, i_j + ℓ_j - 1} andm = ℓ - i_j + 10 otherwise .For any matrix M with d columns, it follows thatM_i_j:(i_j + ℓ_j - 1) = M a^j.Each of A^1, …, A^k, Φ, and O share the same block diagonal structure. Moreover, each block of A^1, …, A^k, and Φ is a scalar matrix. Therefore, we have that A^jΦ = Φ A^j, A^j O =O A^j, and Φ O = O Φ for any j ∈ [k].Applying these properties, as well as the trace definition of the Frobenius norm, we have thatU Φ O - V Φ^2_F= Tr(O^T Φ^T U^T U Φ O + Φ^T V^T V Φ - 2 Φ^T V^T U Φ O) = Tr(Φ^2) + Tr(Φ^2) - Tr(2 Φ^T V^T U Φ O ) = 2Tr( Φ^2 ) - 2Tr(Φ^2 V^T U O ). Focusing on the Tr(Φ^2 V^T U O ) term, we have thatTr(Φ^2 V^T U O )= Tr(I_d Φ^2 V^T U O ) = Tr(∑_j=1^k A^jΦ^2V^T U O ) = ∑_j=1^k Tr(A^jΦ^2 V^TU O ) = ∑_j=1^k Tr(A^j A^j A^jΦ^2 V^T U O ) = ∑_j=1^kTr(A^jΦ^2 A^jV^T U A^j O ). Next, we apply the decomposition A^j = a^j.a^j.^T described above. ∑_j=1^kTr(A^jΦ^2 A^jV^T U A^j O )= ∑_j=1^kTr(a^j.a^j.^TΦ^2 a^j.a^j.^T V^T U a^j.a^j.^T O ) = ∑_j=1^k Tr( .a^j.^TΦ^2 a^j.a^j.^T V^TU a^j.a^j.^TO a^j) = ∑_j=1^k Tr(Φ^2_i_j:(i_j + ℓ_j - 1)(V_i_j:(i_j + ℓ_j - 1))^T U_i_j:(i_j + ℓ_j - 1) O^j) = ∑_j=1^k ϕ_j^2 Tr( (V_i_j:(i_j + ℓ_j - 1))^T U_i_j:(i_j + ℓ_j - 1) O^j) Recognizing that Tr( Φ^2 ) = ∑_j=1^k ϕ_j^2 Tr(I_ℓ_j), we thus have: U Φ O - V Φ^2_F=∑_j=1^k ϕ_j^2 2Tr(I_ℓ_j - (V_i_j:(i_j + ℓ_j - 1))^T U_i_j:(i_j + ℓ_j - 1) O^j)= ∑_j=1^k ϕ_j^2 U_i_j:(i_j + ℓ_j - 1) O^j - V_i_j:(i_j + ℓ_j - 1)_F^2by the definition of the Frobenius norm. §.§.§ Proof of Theorem <ref> We first consider the learnability of the link probabilities. Let δ_n = α^K_n G(n) n^1.5 e(n)^-1 and suppose that δ_n → 0. Then,ℙ(P^ẑ(Y^n) - P^z^2_F/e(n) > δ_n)≤ℙ(P^ẑ(Y^n) - P^z^2_F/e(n) > δ_n e(n) |sup_1 ≤ i ≤ nz_i≤ G(n)) + ℙ(sup_1 ≤ i ≤ nz_i≥ G(n))≤C/n^2 + ℙ(sup_1 ≤ i ≤ nz_i≥ G(n))by Lemma <ref>. By the third regularity assumption, this expression converges to 0 in probability. Thus,P^ẑ(Y^n) - P^z^2_F/e(n) 0because δ_n = o(1). The proof follows the same reasoning for squared distances and latent positions, simply swapping out Lemma <ref> for Lemma <ref> and Lemma <ref>, respectively. §.§.§ Proof of Corollary <ref> The results follow from Theorem <ref>, and the following observations. Because g(n) = n^p/d and K is integrable by Lemma <ref>, Theorem <ref> imples that e(n) = n^2-p. Corollary <ref> implies that the LPM almost surely possesses a(n)-b(n) distinctly bunched eigenvalues with a(n) = O(n^2p/d - 1 + ϵ) for any ϵ > 0 (i.e. a(n) = o(n) for p < d) and b(n)^-1 = O(n^-1 - 2p/d). Table <ref> shows that for K(x) = (c + x^2)^-q, α_n^K ∼Θ(1) and β_n^K ∼Θ(G(n)^2q + 4).Recall that G(n) = Θ(n^p/d) by Lemma <ref>. Thus,β^K_n = Θ( n^p/d(2q+4)). Inserting these values into Theorem <ref> provides the results in points (1) through (3).Letting d grow large while simultaneously setting q to be the smallest integer larger than d/2 allows for learnability of all three targets for values of p that are arbitrarily close to 0.5. Thus, we can have learnability arbitrarily close to e(n) = n^1.5.The proof above relied on Corollary <ref>. Before presenting this result, we will first establish lemma <ref> to control the behavior of the eigenvalues λ_1, …, λ_d. This lemma and its proof were inspired by the work of <cit.>.Let λ_i denote the ith largest eigenvalue of 𝒞_nZZ^T𝒞_n, where 𝒞_n Z ∈ℝ^n × d is the centered matrix of n latent positions associated with a rectangular LPM generated with g(n) = n^p/d. Then, if p < d and i ≤ d,λ_1 - λ_d= O(g(n)^2n^ϵ)almost surelyfor any ϵ > 0 andλ_i/ng(n)^21/12.Recall that both Z^T𝒞_n𝒞_n Z and 𝒞_nZZ^T𝒞_n have the same d non-zero eigenvalues λ_1, …, λ_d. Furthermore,Z^T𝒞_n 𝒞_n Z - 𝔼(Z^T𝒞_n𝒞_nZ)_F ≤Z^TZ - 𝔼(Z^TZ)_F + Z^T1_nZ - 𝔼(Z^T 1_nZ)_F/nwhere 1_n ∈ℝ^n × n denotes a matrix filled with ones. Suppose t_n+1 is known, and π is a random permutation on 1, …, n. Then, each Z^π(i) is uniformly distributed on [-g(t_n+1), g(t_n+1)]^d by Lemma <ref>. Thus, after randomly permuting the row indices in Z, they can be treated as independent samples from this distribution. Going forward, in a slight abuse of notation, we assume that the rows of Z have been randomly permuted, meaning each row can be treated as an iid uniform sample on [-g(t_n+1), g(t_n+1)]^d.Therefore, (Z^TZ)_ij = ∑_k=1^n Z_ki Z_kj is the sum of n iid random variables, and each summand's absolute value is upper bounded by ng(t_n+1)^2. The Hoeffding bound <cit.> provides us with the following concentration result.ℙ(|(Z^TZ)_ij - 𝔼((Z^TZ)_ij)| > δ) ≤ 2 exp(- 2δ^2/ g(t_n+1)^4).Combining this with a union bound providesℙ((Z^TZ) - 𝔼((Z^TZ))_F > d^2δ) ≤ 2d^2 exp(- 2δ^2/ g(t_n+1)^4). Furthermore, (Z^T1_n Z)_ij = (∑_k=1^n Z_ki) (∑_k=1^n Z_kj) with both factors in this product being identically distributed. Moreover, the entries in either summand are bounded in absolute value according to|Z_kj| ≤ g(t_n+1). Therefore, applying the Hoeffding bound to each entry in Z^TZ yieldsℙ(|(Z^T1_n Z)_ij - 𝔼((Z^T1_n Z)_ij)| > δ)≤ 2ℙ(|(∑_k=1^n Z_ki) - 𝔼(∑_k=1^n Z_ki)) | > √(δ))≤ 4 exp(- 2 δ/ g(t_n+1)),achieved through a union bound on the two summands differing from their means by √(δ). Another union bound over all matrix entries results inℙ(Z^T1_nZ - 𝔼(Z^T 1_nZ)_F > d^2 δ) ≤ 4d^2 exp(- 2 δ/ g(t_n+1)). Combining Equations (<ref>), (<ref>), and (<ref>) yieldsℙ(Z^T𝒞_n 𝒞_n Z - 𝔼(Z^T𝒞_n𝒞_nZ)_F >2d^2 δ) ≤ 2d^2 exp(- 2δ^2/ g(t_n+1)^4) + 4d^2 exp(- 2 δ/ g(t_n+1)).To translate this into a result for the eigenvalues, we can apply Weyl's inequality <cit.> to obtainℙ(|λ_i(Z^T𝒞_n 𝒞_n Z) - λ_i(𝔼(Z^T𝒞_n 𝒞_n Z))| > 2d^2 δ) ≤ 2d^2 exp(- 2δ^2/ g(t_n+1)^4) + 4d^2 exp(- 2 δ/ g(t_n+1))for 1≤ i ≤ d.We can analytically determine the values of λ_i by noting that𝔼(Z^T𝒞_n 𝒞_n Z)_ii = (n-1)g(t_n+1)^2/12 𝔼(Z^T𝒞_n 𝒞_n Z)_i ≠ j = 0,which indicates that λ_i(𝔼(Z^T𝒞_n 𝒞_n Z))= (n-1)g(t_n+1)^2/12for i ≤ d, 0 otherwise. Substituting these values into (<ref>) we obtainℙ(|λ_i(Z^T𝒞_n 𝒞_n Z) - (n-1)g(t_n+1)^2/12| > 2d^2 δ) ≤ 2d^2 exp(- 2δ^2/ g(t_n+1)^4) + 4d^2 exp(- 2 δ/ g(t_n+1))for 1≤ i ≤ d. Choosing δ = ϵ (n-1)g(t_n + 1)^2 for ϵ > 0yields:ℙ(|λ_i(Z^T𝒞_n 𝒞_n Z)/(n-1)g(t_n+1)^2 - 1/12| > 2d^2 ϵ) ≤ 2d^2 exp(- 2ϵ^2 (n-1)^2 ) + 4d^2 exp(- 2(n-1) ϵ g(t_n+1)).Due to its reliance of the random quantity g(t_n+1), this bound is difficult to directly use. Instead, let us upper bound it with a deterministic quantity. Lemma <ref> established that t_n+1 follows Gamma(n+1, 1). Therefore,∑_n=1^∞ℙ(t_n≤ 1 )= ∑_n=1^∞∫_0^1 1/Γ(n) t^n-1 e^-tdt≤∑_n=1^∞1/Γ(n)=e.The Borel-Cantelli lemma thus implies that ℙ(t_n ∈ℕ≤ 1infinitely often ) = 0. Therefore, there almost-surely exists an n^* < ∞ such that g(t_n) ≥ 1 for all n ≥ n^*. Moreover, note that g(t_n) ∈ (0, 1] for n < n^*. In turn, we can incorporate the upper bound (<ref>) to determine∑_n=1^∞ℙ(|λ_i(Z^T𝒞_n 𝒞_n Z)/(n-1)g(t_n+1)^2 - 1/12| > 2d^2 ϵ) ≤ ∑_n=1^∞(2d^2 exp(- 2ϵ^2 (n-1)^2 ) + 4d^2 exp(- 2(n-1) ϵ g(t_n+1))) ≤ ∑_n=1^∞2d^2 exp(- 2ϵ^2 (n-1)^2 ) + 4d^2(∑_n=1^n^* 1 + ∑_n=n^*^∞exp(- 2(n-1) ϵ g(t_n+1)))≤ ∑_n=1^∞ 2d^2 exp(- 2(n-1)^2 ϵ^2 ) + 4d^2 (n^* + ∑_n=n^*^∞exp(- 2(n-1) ϵ))<∞ almost surely. We can thus apply the Borel-Cantelli Lemma to establish that λ_i(Z^T𝒞_n 𝒞_n Z)/(n-1)g(t_n+1)^2 1/12.This result implies (<ref>) by the following reasoning. Lemma <ref> established that t_n+1 (n+1)^-1 1. Therefore,g(t_n+1/n+1) = t_n+1^p/d/g(n) g(1) = 1by the continuous mapping theorem. Recognizing that n/(n-1) → 1 and g(n)/g(n-1) → 1 as n goes to infinity, we thus have thatλ_i(Z^T𝒞_n 𝒞_n Z)/ng(n)^2 1/12,giving us the result in (<ref>).Next, we will provide the result in (<ref>). We recognize that ℙ(|λ_1 - λ_d| > 4d^2 δ)≤ℙ(|λ_1 - (n-1)g(t_n+1)^2/12| + |λ_ d - (n-1)g(t_n+1)^2/12|> 4d^2 δ) ≤ℙ(|λ_1 - (n-1)g(t_n+1)^2/12|> 2d^2 δ) + ℙ(|λ_d - (n-1)g(t_n+1)^2/12|> 2d^2 δ)≤ 4d^2 exp(- 2δ^2/ g(t_n+1)^4) + 8d^2 exp(- 2 δ/ g(t_n+1))by (<ref>). Choosing δ = g(t_n+1)^2n^ϵ for ϵ > 0 implies thatℙ(|λ_1 - λ_d|/g(t_n+1)^2 + ϵ > 4d^2 )≤ 4d^2 exp(- 2n^2 ϵ) + 8d^2 exp(- 2g(t_n+1) n^ϵ)Recall that there almost-surely exists an n^* < ∞ such that g(t_n) ≥ 1 for all n ≥ n_*. In turn, we can once again apply Borell-Cantelli in the same manner∑_n=1^∞ℙ(|λ_1 - λ_d|/g(t_n+1)^2 n^ϵ > 4d^2 )≤4d^2 ∑_n=1^∞(exp(- 2n^2 ϵ) + 2 exp(- 2g(t_n+1) n^ϵ)) ≤4d^2 (∑_n=1^∞exp(- 2n^2 ϵ) + 2n^* + ∑_n=n^*^∞exp(- 2n^ϵ))<∞ almost surely.The Borell-Cantelli lemma thus implies the result in (<ref>). We can now use these results to determine that rectangular latent position network models possess a(n)-b(n) distinctly bunched eigenvalues. We state this result as Corollary <ref>. A d-dimensional rectangular latent position network model generated with g(n) = n^p/d almost surely possesses a(n)-b(n) distinctly bunched eigenvalues for a(n) = O(n^2p/d - 1 + ϵ) and b(n)^-1 = O(n^-1 - 2p/d) for any ϵ > 0. Lemma <ref> established thatλ_i/n^1 + 2 p/d1/12 for all i = 1, …, d. The continuous mapping theorem thus implies that 1/λ_d^2 = O(n^-2 - 4 p/d), λ_1/λ_d^2 = O(n^-1 - 2p/d)almost surely.Moreover, (<ref>) from Lemma <ref> implies that, almost surely,(λ_1 - λ_d)^2 = O( n^4p/d + ϵ),for any ϵ > 0. Therefore,(λ_1 - λ_d)^2/λ_d = O(n^2p/d - 1 + ϵ).The LPM is thus almost surely meets the criteria for possessing a(n)-b(n) distinctly bunched eigenvalues with k=1, i_1 = 1, a(n) = O(n^2p/d - 1 + ϵ) and b(n)^-1 = O(n^-1 - 2p/d) for ϵ > 0. §.§.§ Proof of Corollary <ref> Note that this LPM is regular with G(n) = √(2 σ_1^2 (1+c) log(n)) for any c > 0 by Lemma <ref>, and e(n) = n^2. Furthermore, Corollary <ref> establishes that this LPM is almost surely possesses a(n)-b(n) distinctly bunched eigenvalues for a(n) = O(1) and b(n)^-1 = O(n^-1). Consulting Table <ref>, we see that for both link functions α^K_n = Θ(1) and β^K_n = Θ(e^G(n)^2) = Θ(n^2σ^2 (1+c)). Thus, applying Theorem <ref> indicates that we have learnable latent positions and distances provided that 2σ_1^2(1+c) < 1/2.The above proof relied Corollary <ref> to establish that the LPM with Gaussian latent positions possesses a(n)-b(n) distinctly bunched eigenvalues. Before presenting this result, we first present Lemma <ref>, as it facilitates its proof. Let the rows of Z ∈ℝ^n × d independently follow a multivariate Gaussian distribution with mean zero and diagonal variance matrix Σ. Let σ^2_1, …, σ^2_d denote the entries along the diagonal of Σ, with σ_1 ≥σ_2 ≥⋯≥σ_d > 0. Let λ_i denote the ith largest eigenvalue of 𝒞_nZZ^T𝒞_n, where 𝒞_n is defined as in (<ref>). Then,λ_i/nσ_i^2.Moreover, |λ_i - λ_j| = O(1) almost surely whenever σ_i = σ_j. The proof proceeds very similarly as for Lemma <ref>. Recall that both Z^T𝒞_n𝒞_n Z and 𝒞_nZZ^T𝒞_n have the same d non-zero eigenvalues λ_1, …, λ_d. Furthermore,Z^T𝒞_n 𝒞_n Z - 𝔼(Z^T𝒞_n𝒞_nZ)_F ≤Z^TZ - 𝔼(Z^TZ)_F + Z^T1_nZ - 𝔼(Z^T 1_nZ)_F/nwhere 1_n ∈ℝ^n × n denotes a matrix filled with ones. Note thatZ^TZ - 𝔼(Z^TZ)_F ≤∑_i=1^d ∑_j=1^d |Z^TZ_ij - 𝔼(Z^TZ)_ij|Applying <cit.> and the union bound yieldsℙ(Z^TZ - 𝔼(Z^TZ)_F > d^2δ) ≤ 4d^2 exp(-n δ^2/3200 σ_1^2). Furthermore, note that (Z^T1_n Z)_ij/n = (∑_k=1^n Z_ki/√(n)) (∑_k=1^n Z_kj/√(n)) can be viewed as the product of two Gaussian distributed random variables with variances σ_i^2 and σ_j^2 respectively. This means that (Z^T1_n Z)/n can be viewed as u u^T where u is a d-dimensional Gaussian vector with mean 0 and variance matrix Σ. Again applying <cit.> and the union bound yieldsℙ((Z^T1_n Z)_ij - 𝔼((Z^T1_n Z)_ij)_F > d^2δ)≤ 4d^2 exp(- nδ^2/3200 σ_1^2). Combing Equations (<ref>), (<ref>), and (<ref>) yieldsℙ(Z^T𝒞_n 𝒞_n Z - 𝔼(Z^T𝒞_n𝒞_nZ)_F >2d^2 δ) ≤ 8d^2 exp(-n δ^2/3200 σ_1^2).To translate this into a result for the eigenvalues, we can apply Weyl's inequality <cit.>. This results inℙ(|λ_i(Z^T𝒞_n 𝒞_n Z) - λ_i(𝔼(Z^T𝒞_n 𝒞_n Z))| > 2d^2 δ) ≤ 8d^2 exp(-n δ^2/3200 σ_1^2)for 1≤ i ≤ d.We can analytically determine the values of λ_i by noting that 𝔼(Z^T𝒞_n 𝒞_n Z)_ii = (n-1)σ_i^2 𝔼(Z^T𝒞_n 𝒞_n Z)_i ≠ j = 0,which indicates that λ_i(𝔼(Z^T𝒞_n 𝒞_n Z)= (n-1)σ^2_ifor i ≤ d, 0 otherwise. Substituting this into (<ref>) and setting δ = (2d^2)^-1 (n-1) ϵ for ϵ > 0, we see thatℙ(|λ_i(Z^T𝒞_n 𝒞_n Z)/n-1 - σ_i^2 | > ϵ) ≤ 8d^2 exp(-n (n-1)^2 ϵ^2/12800 d^4 σ_1^2)It follows that∑_n=1^∞ℙ(|λ_i(Z^T𝒞_n 𝒞_n Z)/n-1 - σ_i^2 | > ϵ)≤∑_n=1^∞ 8d^2 exp(-n (n-1)^2 ϵ^2/12800 d^4 σ_1^2) < ∞.The Borel Cantelli lemma thus implies thatλ_i(Z^T𝒞_n 𝒞_n Z)/n σ_i^2because n^-1(n-1) → 1 as n goes to infinity.Moreover, suppose that σ_i = σ_j for some pair i,j ∈{1, …, d}. Then, using the triangle inequality, a union bound, and (<ref>), we see thatℙ(|λ_i(Z^T𝒞_n 𝒞_n Z) - λ_j(Z^T𝒞_n 𝒞_n Z) | > 2d^2 δ)≤ℙ(|λ_i(Z^T𝒞_n 𝒞_n Z) - (n-1)σ_i^2 | > 2d^2 δ) + ℙ(|λ_j(Z^T𝒞_n 𝒞_n Z) - (n-1)σ_i^2 | > 2d^2 δ)≤ 16d^2 exp(-n δ^2/3200 σ_1^2).Setting δ = (2d^2)^-1ϵ for ϵ > 0, we have that∑_n=1^∞ℙ(|λ_i(Z^T𝒞_n 𝒞_n Z) - λ_j(Z^T𝒞_n 𝒞_n Z) | > ϵδ)≤∑_n=1^∞ 16d^2 exp(-n ϵ^2/12800d^4 σ_1^2 ) < ∞.Again applying the Borel Cantelli lemma, we thus have that |λ_i(Z^T𝒞_n 𝒞_n Z) - λ_j(Z^T𝒞_n 𝒞_n Z) | = O(1)almost surely. We can now proceed with the proof of Corollary <ref>. Consider a LPM on S = ℝ^d with each latent position independently and identically distributed according to the multivariate Gaussian distribution with mean zero and diagonal variance matrix Σ. Let σ^2_1, …, σ^2_d denote the entries along the diagonal of Σ, with σ_1 ≥σ_2 ≥⋯≥σ_d > 0.This LPM almost surely possesses a(n)-b(n) distinctly bunched eigenvalues for a(n) = O(1) and b(n)^-1 = O(n^-1).Lemma <ref> established thatλ_i/nσ_i^2 for all i = 1, …, d. The continuous mapping theorem thus implies that n λ_i/(λ_i - λ_j)^2 σ_i^2/(σ_i^2 - σ_j^2)for any i, j ∈{1, …, d} for which σ_i ≠σ_j. Moreover, Lemma <ref> also implies that(λ_i - λ_j)^2 = O(1),almost surely whenever σ_i = σ_j. Therefore,(λ_i - λ_j)^2/λ_j = O(1/n)almost surely when σ_i = σ_j. Let k ≤ d be given by number of distinct values in that the sequence σ_1, …, σ_d. Let σ_(1) > σ_(2) > ⋯ > σ_(k) denote these distinct values. For j = 1, …, k define i_j to be the smallest element of {1, …, d} such that σ_i_j = σ_(j). Using these values of k and i_1, …, i_k, the LPM is thus almost surely meets the criteria for possessing a(n)-b(n) distinctly bunched eigenvalues with a(n) = O(1) and b(n)^-1 = O(n^-1).§.§.§ Proof of Corollary <ref>The proof set-up is almost identical to that of Corollary <ref>, with the sole departure being that β^K_n is also scaled by the inverse of sparsity term s(n)^-1 = n^p resulting in β^K_n = Θ(n^2σ^2(1+c) + p) requiring that 2p < 1- 4σ^2(1+c). By choosing a small value of σ^2, we can have p get arbitrarily close to 1/2. §.§ More Details regarding Sparse graphon-based LPMs§.§.§ Proof of Theorem <ref>For any δ > 0, the definition of K_n implies thats_n s_1^-1 K_1(δ) ≤ K_n(δ) ≤ s_n K(δ).Therefore, (Y^n_ij =1, Y^n_ik=1, Y^n_jk = 1)= 𝔼(K_n(δ(Z_i, Z_j)) K_n(δ(Z_i, Z_k)) K_n(δ(Z_j, Z_k))) ≤ s_n^3 𝔼(K(δ(Z_i, Z_j)) K(δ(Z_i, Z_k)) K(δ(Z_j, Z_k))) = C_1 s_n^3for some C_1 > 0 that is independent of n where Z_i, Z_j, Z_k ∼ f independently. Moreover,(Y^n_ij =1, Y^n_ik=1)= 𝔼(K_n(δ(Z_i, Z_j)) K_n(δ(Z_i, Z_k)))) ≥ s_n^2 s_1^-1𝔼(K_1(δ(Z_i, Z_j)) K_1(δ(Z_i, Z_k)))) = C_2 s_n^2for some C_2 > 0 independent of n because K_1(δ) ≤ 1 and s_1 ∈ℝ+.Turning to the probability of triadic closure, we thus have that(Y^n_ij =1, Y^n_ik=1, Y^n_jk = 1) = (Y^n_ij =1,Y^n_ik=1, Y^n_jk = 1)/(Y^n_ij =1, Y^n_ik=1)≤C_1 s_n^3/C_2 s_n^2∝ s_n → 0 as n →∞. §.§.§ Proof of Theorem <ref> Note that because we have assumed the link function is regular, it must be continuous, non-negative, non-increasing, and not identically zero. Therefore, there must exist a c ∈ (0, 1] and ϵ > 0 such that r ≤ 2 ϵ implies K(r)≥ c. Let Ω(n) = [-g(t_n+1), g(t_n+1)]^d and Ω_+(n) = [0, g(t_n+1)]^d. Following Lemma <ref>, we have that(Y^n_π(i)π(j) =1, Y^n_π(i) π(k)=1, Y^n_π(j)π(k) = 1)= ∫_x,y,z ∈Ω(n)1/2^3d g(t_n+1)^3d K(x - y) K(x - z) K(y-z) dz dy d x ≥ ∫_x,y,z ∈Ω(n)1/2^3d g(t_n+1)^3d c^3 I(x - y≤ϵ) I(x - z≤ϵ)dz dy d x ≥ ∫_x,y,z ∈Ω_+(n)1/2^3d g(t_n+1)^3d c^3 I(x - y≤ϵ) I(x - z≤ϵ)dz dy d x ≥ ∫_x,y,z ∈Ω_+(n)1/2^3d g(t_n+1)^3d c^3 I(y≤ϵ) I(z≤ϵ)dz dy d x ∝ 1/g(t_n+1)^2d∫_y,z ∈Ω_+(n)I(y≤ϵ) I(z≤ϵ)dz dy= π^d ϵ^2d/g(t_n+1)^2d 2^2dΓ(d/2 + 1)^2 ∝g(t_n+1)^-2d.Moreover, note that the function∫_y,z ∈Ω(n)1/2^3d g(t_n+1)^3d K(x - y) K(x - z) dz dyis maximized at x=0. Therefore, (Y^n_π(i)π(j) =1, Y^n_π(i) π(k)=1)= ∫_x,y,z ∈Ω(n)1/2^3d g(t_n+1)^3d K(x - y) K(x - z) dz dy d x ≤ ∫_x,y,z ∈Ω(n)1/2^3d g(t_n+1)^3d K(y) K(z) dz dy d x= 1/2^2d g(t_n+1)^2d∫_y,z ∈Ω(n) K(y) K(z) dz dy ∝g(t_n+1)^-2dby Lemma <ref>. Turning to the probability of triadic closure, we thus have that(Y^n_π(i)π(j) =1 | Y^n_π(i) π(k)=1, Y^n_π(j)π(k) = 1)= (Y^n_π(i)π(j) =1, Y^n_π(i) π(k)=1, Y^n_π(j)π(k) = 1) /(Y^n_π(i)π(j) =1, Y^n_π(i) π(k)=1) ≥ ∫_x,y,z ∈Ω_+(n)1/2^3d g(t_n+1)^3d c^3 I(y < ϵ) I(z < ϵ)dz dy d x /1/2^2d g(t_n+1)^2d∫_y,z ∈Ω(n) K(y) K(z) dz dy∝ g(t_n+1)/g(t_n+1) = 1,thus establishing the result.§.§ Towards a Negative Learnability Result In this section, we establish conditions under which regular LPMs are not learnable. Consider a regular LPM. Let n denote the number of nodes. Supposelim_n →∞ n^2 K(G(n)/1+c) → 0.for some c > 0, then this class of LPMs do not have learnable latent positions.Recall LeCam's theorem <cit.>, in the form it is used to determine minimax estimation rates: Let 𝒫 be a set of distributions parameterized by θ∈Θ. Let Θ̂ denote the class of possible estimators for θ∈Θ. For any pair P_θ_1, P_θ_2∈𝒫, inf_θ̂∈Θ̂sup_θ∈Θ𝔼_P_θ(d(θ̂, θ)) ≥Δ/8exp(-KL(P_θ_1, P_θ_2)),where Δ = d(θ_1, θ_2) for some distance d(·, ·), and KL denotes the Kullback-Leibler divergence <cit.>.Let Θ be the set of possible latent positions and 𝒫 be the distributions over graphs implied by a regular LPM with link probability function K. Without loss of generality, consider the latent space to be S = ℝ^1. We require that the latent positions Z_1, …, Z_n ∈ℝ be such |Z_i| ≤ G(n) for some differentiable and non-decreasing function G(n). Suppose G(n) = (1+c) g(n) for some non-decreasing differentiable function g, and let c > 0 be a small constant. To get a decent lower bound for this setting via LeCam, we choose two candidate sets of positions θ_1, θ_2 ∈Θ corresponding to probability models that do not differ much in KL-divergence, but have embeddings differing by a non-shrinking amount in n. To accomplish this, we exploit the fact that the larger the distance between two nodes, the smaller the change in the connection probability (and thus the KL divergence) due to a small perturbation in the distance. Consider θ^n_1 ∈ℝ^n according to θ^n_1 = (0, g(n), 0, g(n), …). That is, every odd-indexed latent position is 0, and every even-indexed latent position is g(n). Similarly, define θ^n_1 ∈ℝ^n according to θ^n_1 = (0, G(n), 0, G(n), …). Let the distance metric on Θ^2 follow from the definition of learnable latent positions. That is,d(θ^n_1, θ^n_2)= inf_O ∈𝒪_1, Q ∈𝒬_n1θ^n_1T- Q -θ^n_2_F^2/n= c/2.Here, 𝒪_1 and 𝒬_n1 capture all possible isometric transformations. Notice that this distance is constant in n. However,KL(P_θ_2^n, θ_1^n)≤n^2/2 K(g(n)) log(K(g(n))/K((1 + c)g(n))) → 0as n goes to infinity due to the assumption in (<ref>). Thus, by Lemma <ref>, this class of LPMs is not learnable.§ ACKNOWLEDGEMENTSWe are grateful to the members of the CMU Networkshop for feedback on our results and their presentation, and to conversations with Creagh Briercliffe, David Choi, Emily Fox, Alden Green, Peter Hoff, Jeannette Janssen, Dmitri Krioukov, and Cristopher Moore. plainnat | http://arxiv.org/abs/1709.09702v4 | {
"authors": [
"Neil A. Spencer",
"Cosma Rohilla Shalizi"
],
"categories": [
"math.ST",
"stat.TH"
],
"primary_category": "math.ST",
"published": "20170927190254",
"title": "Projective, Sparse, and Learnable Latent Position Network Models"
} |
lemmaLemma corollaryCorollaryGlobal modes and nonlinear analysis of inverted-flag flappingA. Goza, T. Colonius, and J. Sader 1 Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA 91125, USA 2 ARC Centre of Excellence in Exciton Science, School of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia 3 Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA Global modes and nonlinear analysis of inverted-flag flappingAndres Goza1 [email protected], Tim Colonius1, John E. Sader2,3 ============================================================================An inverted flag has its trailing edge clamped and exhibits dynamics distinct from that of a conventional flag, whose leading edge is restrained. We perform nonlinear simulations and a global stability analysis of the inverted-flag system for a range of Reynolds numbers, flag masses and stiffnesses. Our global stability analysis is based on a linearisation of the fully-coupled fluid-structure system of equations. The calculated equilibria are steady-state solutions of the fully-coupled nonlinear equations. By implementing this approach, we (i) explore the mechanisms that initiate flapping, (ii) study the role of vortex shedding and vortex-induced vibration (VIV) in large-amplitude flapping, and (iii) characterise the chaotic flapping regime. For point (i), we identify a deformed-equilibrium state and show through a global stability analysis that the onset of flapping is due to a supercritical Hopf bifurcation. For large-amplitude flapping, point (ii), we confirm the arguments of <cit.> that for a range of parameters this regime is a VIV. We also show that there are other flow regimes for which large-amplitude flapping persists and is not a VIV. Specifically, flapping can occur at low Reynolds numbers (<50), albeit via a previously unexplored mechanism. Finally, with respect to point (iii), chaotic flapping has been observed experimentally for Reynolds numbers of O(10^4), and here we show that chaos also persists at a moderate Reynolds number of 200. We characterise this chaotic regime and calculate its strange attractor, whose structure is controlled by the above-mentioned deformed equilibria and is similar to a Lorenz attractor. These results are contextualised with bifurcation diagrams that depict the different equilibria and various flapping regimes. § INTRODUCTION Uniform flow past a conventional flag—where the flag is pinned or clamped at its leading edge with respect to the oncoming flow—has been studied widely beginning with the early work of <cit.> (see <cit.> for a recent review). By contrast, studies of flow past an inverted flag, in which the flag is clamped at its trailing edge, have only been reported recently. The inverted-flag system displays a wide range of dynamical regimes <cit.>, many of which are depicted in figure <ref>. This figure is produced from the numerical simulations described in section <ref>.One of the dynamical regimes depicted is large-amplitude flapping (figure <ref>), which is associated with a larger strain energy than that of conventional flag flapping. These large bending strains make the inverted-flag system a promising candidate for energy harvesting technologies that convert strain energy to electricity, e.g., by using piezoelectric materials. <cit.> studied this energy harvesting potential in detail by performing numerical simulations of a fully-coupled fluid-structure-piezoelectric model.Transitions between the various regimes in figure <ref> depend on the Reynolds number (Re), dimensionless mass ratio (M_ρ), and dimensionless bending stiffness (K_B), defined asRe = ρ_f U L/μ, M_ρ = ρ_s h/ρ_f L, K_B = EI/ρ_f U^2 L^3where ρ_f (ρ_s) is the fluid (structure) density, U is the freestream velocity, L is the flag length, μ is the shear viscosity of the fluid, h is the flag thickness, and EI is the flexural rigidity of the flag. In experiments, regime transitions aretriggered by increasing the flow rate <cit.>. This coincides with a decrease in K_B and an increase in Re for fixed M_ρ, by virtue of (<ref>). In contrast, numerical simulations often decrease the flag's stiffness at fixed Re and M_ρ, which isolates the effect of various parameters and facilitates comparison to previous numerical studies of flow-induced vibration.Simulations show that for moderate Reynolds numbers (≲ 1000), a systematic decrease in K_B causes a change from a stable undeformed equilibrium state (figure <ref>) to a small-deflection stable state (figure <ref>). This is followed by a transition to small-deflection deformed flapping (figure <ref>), then to large-amplitude flapping (figure <ref>), and finally to a deflected-mode regime (figure <ref>) <cit.>. These simulations have been performed primarily for M_ρ≤ O(1) (heavy fluid loading), though <cit.> considered large values of M_ρ. The same regime transitions persist at higher Reynolds numbers, Re ∼ O(10^4), except that the small-deflection stable and small-deflection deformed flapping regimes discussed above are no longer present. That is, the undeformed equilibrium directly gives way to large-amplitude flapping <cit.>. Moreover,<cit.> experimentally identified a chaotic flapping regime (not shown in figure <ref>) at these higher Reynolds numbers that has yet to be reported using numerical simulations with Re ≤ O(1000).At low Reynolds numbers (Re < 50), numerical simulations have shown that the inverted flag's dynamics can change significantly: no flapping occurs, with the only observed regimes being the undeformed equilibrium and stable deflected states <cit.>. These simulations were performed over a wide range of K_B for only one value of M_ρ, and the system's dependence on these two parameters remains an open question at these lower Reynolds numbers.Several driving mechanisms of the various regimes illustrated in figures <ref>(a)–(e) have been identified. The bifurcation from the undeformed equilibrium is caused by a divergence instability (i.e., the instability is independent of M_ρ). This mechanism was originally suggested by <cit.>, and subsequently found computationally <cit.> and mathematically via a linear stability analysis <cit.>. For large-amplitude flapping, <cit.> used experiments and a scaling analysis to argue that this regime is a vortex-induced vibration (VIV) for a distinct range of parameters. The primary role of vortex shedding in large-amplitude flapping is further evidenced by the above-mentioned observation of <cit.> that flapping does not occur below Re ≈ 50 (for certain values of M_ρ). Based on a scaling analysis, <cit.> also predicted that VIV should cease as the mass ratio, M_ρ, increases—a prediction that is yet to be verified.With respect to the deflected-mode regime, small-amplitude flapping about a large mean-deflected position occurs, and <cit.> showed that the flapping frequency is identical to that of the vortex shedding caused by the flag's bluffness.In this article, we use high-fidelity nonlinear simulations and a global linear stability analysis to further characterise the regimes in figure <ref> and explore their driving physical mechanisms. We emphasise that our global stability analysis is based on a linearisation of the fully-coupled fluid-structure system of equations. Moreover, the computed equilibria are steady-state solutions of the fully-coupled nonlinear equations described in section <ref>. Our results are presented for Reynolds numbers of 20 and 200, various values of K_B, and values of M_ρ spanning four orders of magnitude.Using this approach, we (i) study the mechanisms responsible for the onset of small-deflection deformed flapping, (ii) probe the role of vortex shedding and VIV in large-amplitude flapping, and (iii) investigate whether chaotic flapping occurs at low-to-moderate Reynolds numbers (Re = 20 and 200). To explore (i), we first demonstrate that the small-deflection stable state is an equilibrium of the fully-coupled fluid-structure system. Through a global stability analysis, we show that the subsequent transition to small-deflection deformed flapping (figure <ref>) as the bending stiffness decreases is a supercritical Hopf bifurcation of this deformed equilibrium. For point (ii), we confirm the arguments of <cit.> that large-amplitude flapping is a VIV for the higher Reynolds number of Re = 200 and lower values of the mass ratio, M_ρ < O(1). VIV is also shown to cease for sufficiently large M_ρ, consistent with the scaling analysis of <cit.>. Moreover, we show that large-amplitude flapping persists at these large values of M_ρdespite the absence of VIV, albeit by a previously unidentified mechanism. This non-VIV large-amplitude flapping regime also occurs for large M_ρ at the lower Reynolds number of Re = 20. Consistent with the simulation results of <cit.>, we find no flapping at this low Reynolds number for M_ρ < O(1). Finally, with respect to (iii), we confirm that chaotic flapping persists at moderate Reynolds numbers (Re = 200) for light flags with M_ρ < 1, and demonstrate that the structure of the associated strange attractor is controlled by a combination of the large-amplitude and deflected-mode regimes. Chaos does not occur for heavy flags at Re = 200 or for any mass ratio considered at Re=20. Thus, chaos is associated with parameters for which VIV flapping occurs. We contextualise the simulation results over this wide range of parameters using bifurcation diagrams. These provide an overview of theequilibria, their stability, and the flapping dynamics. Figure <ref> shows an illustrativebifurcation diagram that summarises what will be shown in later sections. In these bifurcation diagrams, the leading edge transverse displacement (tip displacement) is plotted versus the flag flexibility (1/K_B) for a particular choice of Re and M_ρ (see the caption for details). Note that even though the undeformed and deformed equilibria become unstable with a decrease in K_B, they nonetheless remain as equilibria of the system. We demonstrate below through a global stability analysis that these unstable deformed equilibria are key to understanding a variety of flapping behaviour of the inverted-flag system.Two-dimensional (2D) simulations are presented throughout. As mentioned above, many similarities exist between the 3D experiments of <cit.> and 2D simulations of <cit.>. This suggeststhat features of the 2D dynamics persist in 3D (though <cit.> demonstrated that substantial differences occur for low-aspect ratio flags). Exploring these similarities and differences between 2D and 3D geometries is a subject of future work and is not considered here. Quantities presented below are dimensionless, with length scales, velocity scales, and time scales nondimensionalised by L, U, and L/U, respectively.§ NUMERICAL METHODS: NONLINEAR SOLVER AND GLOBAL STABILITY ANALYSISOur nonlinear simulations use the immersed boundary method of <cit.>. The method treats the fluid with the 2D Navier-Stokes equations, and the flag with the geometrically nonlinear Euler-Bernoulli beam equation. The method is strongly-coupled (i.e., it accounts for the nonlinear coupling between the flag and the fluid), and therefore allows for arbitrarily large flag displacements and rotations. We have validated our method against a variety of test problems involving conventional and inverted flags <cit.>. The global stability analysis is based on a linearisation of the nonlinear, fully-coupled flow-structure interaction system, and therefore reveals instability-driving mechanisms in both the flag and the fluid.In what follows, we review the nonlinear solver (see <cit.> for more details) and derive the linearised equations. We then describe the global mode solution approach, the procedure used to compute equilibria of the flow-flag system, and the grid spacing and domain size used for our simulations. §.§ Nonlinear solver We define the fluid domain as Ω and the flag surface as Γ. We let x denote the Eulerian coordinate representing a position in space, and χ(θ,t) be the Lagrangian coordinate attached to the body Γ (θ is a variable that parametrizes the surface). The dimensionless governing equations are written as∂u/∂ t = - u·∇u - ∇ p + 1/Re∇^2 u + ∫_Γf(χ(θ,t)) δ(χ(θ,t) - x) dθ ∇·u = 0ρ_s/ρ_f∂^2 χ/∂ t^2 = 1/ρ_f U^2∇·σ + g(χ) - f(χ)∫_Ωu(x)δ(x - χ(θ,t)) dx = ∂χ(θ,t)/∂ tIn the above, (<ref>) expresses the Navier-Stokes equations in an immersed boundary formulation, (<ref>) is the continuity equation for the fluid, (<ref>) represents the structural equations governing the motion of the flag (g is a body force term), and (<ref>) is the no-slip boundary condition enforcing that the fluid velocity matches the flag velocity on the flag surface. Note that f represents the effect from the flag surface stresses on the fluid, and is present in both (<ref>) and (<ref>) since by Newton's third law its negative imparts the fluid stresses on the flag surface <cit.>. In (<ref>), the time derivative is a Lagrangian derivative and the stress tensor is the Cauchy tensor in terms of the deformed flag configuration. The fluid equations are spatially discretised with the immersed boundary discrete-streamfunction formulation of <cit.>, which removes the pressure and eliminates the continuity equation. The flag equations are treated with a finite element corotational formulation <cit.>. The spatially discrete, temporally continuous equations written as a first order system of differential-algebraic equations areC^TCṡ = - C^T𝒩(s) + 1/ReC^TLCs - C^TE^T(χ)f M ζ̇ = -R(χ) + Q(g + W(χ)f)χ̇ = ζ0 = E(χ)Cs - ζwhere χ and f are discrete analogues to their continuous counterparts, s is the discrete streamfunction, ζ is the flag velocity, and all other variables are defined below.Equation (<ref>) represents the Navier-Stokes equations written in a discrete-streamfunction formulation, (<ref>) is the geometrically nonlinear Euler-Bernoulli beam equation, (<ref>) matches the time derivative of the flag position to the flag velocity, and (<ref>) is the interface constraint that the fluid and flag must satisfy the no-slip boundary condition on the flag surface.In (<ref>)–(<ref>), C and C^T are discrete curl operators that mimic ∇× (·); 𝒩(s) is a discretization of the advection operator u·∇u written in terms of the discrete streamfunction <cit.>; L is a discrete Laplacian associated with the viscous diffusion term; E^Tf is a “smearing” operator (arising from the immersed boundary treatment) that applies the surface stresses from the flag onto the fluid; M is a mass matrix associated with the flag's inertia; R(χ) is the internal stress within the flag; Qg is a body force term (e.g., gravity); and QWf is the stress imposed on the flag from the fluid. Equation (<ref>) is discretised in time using an Adams Bashforth AB2 scheme for the convective term and a second order Crank-Nicholson scheme for the diffusive term. The flag equations (<ref>)–(<ref>) are discretized using an implicit Newmark scheme. The method is strongly coupled, so the constraint equation (<ref>) is enforced at each time step including the present one.A novel feature of our method is the efficient iterative procedure used to treat the nonlinear coupling between the flag and fluid. Many methods use a block-Gauss Seidel iterative procedure, which converges slowly (or not at all) for light structures <cit.>. Other methods use a Newton-Raphson scheme, which exhibits fast convergence behaviour but requires the solution of linear systems involving large Jacobian matrices <cit.>. Our method employs the latter approach, but we use a block-LU factorization of the Jacobian matrix to restrict all iterations to subsystems whose dimensions scale with the number of discretisation points on the flag, rather than on the entire flow domain. Thus, our algorithm inherits the fast convergence behaviour of Newton-Raphson methods while substantially reducing the cost of performing an iteration.§.§ Linearised equations and global modes For ease of notation, we define the state vector y = [s, ζ, χ, f]^T and let r(y) be the right hand side of (<ref>)–(<ref>). We write the state as y = y_b + y_p, where y_b=[s_b, ζ_b, χ_b, f_b]^T is a base state and y_p=[s_p, ζ_p, χ_p, f_p]^T is a perturbation. Plugging this expression for y into (<ref>)–(<ref>), Taylor expanding about y_b, and retaining only first order terms in the perturbation variables gives the linearised equations:Bẏ_p = A(y_b)y_pwhere B = [ C^TC ; M;I ;0 ], A(y_b) = [J^ss 0-J^χ s -C^TE^T; 0 0 -K + J^χχQW; 0 I 0 0;EC-I J^χ c 0 ]_y = y_band the remaining sub-blocks of the Jacobian matrix A are given in index notation as(J^ss)_ik = -(C^TC)_ik^2 - C^T_ij∂𝒩_j/∂ s_k(J^χ s)_ik = C^T_ij∂ E^T_jl/∂χ_k (f_b)_l(J^χχ)_ik = Q_ij∂ W_jl/∂χ_k(f_b)_l (J^χ c)_ik = ∂ E_ij/∂χ_kC_jl(s_b)_lNote that we used Bẏ_b = r(y_b) in arriving at the linearised equations (<ref>).Global modes are eigenvectors v of the generalised eigenvalue problem Av = λ Bv, where λ is the corresponding eigenvalue. We build and store A and B sparsely and solve the generalised eigenvalue problem using an implicitly restarted Arnoldi algorithm (see <cit.> for more details). In the results below, 1×10^-10 was used as the tolerance for convergence of the computed eigenvalues and eigenvectors. Global eigenfunctions are unique to a scalar multiple, and were scaled to unit norm, || y ||_2 = 1. §.§ Equilibrium computations Undeformed and deformed equilibria are steady state solutions to the fully-coupled equations (<ref>)-(<ref>) with all time derivate terms set to zero; i.e., these equilibria satisfy 0 = r(y), where y = [s, ζ, χ, f]^T is the state vector and r(y) is the right hand side of (<ref>)-(<ref>). This is a nonlinear algebraic system of equations that we solve using a Newton-Raphson method. With this method, the k^th guess for the base state, y^(k), is updated as y^(k+1) = y^(k) + Δ y, where Δ y = -( A(y^(k)) )^-1 r( y^(k))Note that the Jacobian matrix A in (<ref>) is the same matrix as in (<ref>) evaluated at y=y^(k).The guess for the state y is updated until the residual at the current guess is less than a desired threshold (i.e., until ||r(y^(k))||_2 / ||y^(k)||_2< ϵ). In the results shown below we used ϵ = 1×10^-6.§.§ Domain size and grid resolution The flow equations are treated using a multidomain approach: the finest grid surrounds the body and grids of increasing coarseness are used at progressively larger distances <cit.>. In all computations below, the domain size of the finest sub-domain is [-0.2, 1.8] × [-1.1, 1.1] and the total domain size is [-15.04, 16.64] × [-17.44, 17.44]. The grid spacing on the finest domain is h = 0.01 and the grid spacing for the flag is Δ s = 0.02. For computations involving time marching, the time step is Δ t = 0.001, which gives a maximum Courant-Friedrichs-Levy number of ≈ 0.15. To determine the suitability of these parameters, we performed a grid convergence study of the nonlinear solver using Re = 200, M_ρ = 0.5, K_B = 0.35. For these parameters the flag enters limit cycle flapping of fixed amplitude and frequency. Using the grid described above, the amplitude and frequency of these oscillations were a = ± 0.81, f = 0.180, respectively.Refining the grid spacing to h = 0.0075 on the finest domain and increasing the domain such that the finest sub-domain size was [-0.2, 2.8]×[-1.5,1.5] and the total domain size was [-22.58, 25.18]×[-23.88,23.88] changed these values to a = ± 0.80, f = 0.183, respectively.§ DYNAMICS FOR RE = 200We now consider the inverted-flag system for Re = 200. We demonstrate the existence of a deformed equilibrium that is stable over a small range of stiffnesses and becomes unstable as K_B is decreased. The transition to small-deflection deformed flapping associated with this decrease in K_B is shown through a global stability analysis to be a supercritical Hopf bifurcation of the deformed equilibrium. We then consider the large-amplitude flapping regime, and confirm the arguments of <cit.> that this regime is a VIV for small values of M_ρ. VIV is shown to cease for larger mass ratios, consistent with the scaling analysis of <cit.>, and we demonstrate that large-amplitude flapping persists despite the absence of VIV. The potential mechanisms associated with this non-VIV large-amplitude flapping regime are discussed. We then use a global stability analysis to confirm the argument of <cit.> that small-amplitude flapping in the deflected-mode regime is driven by the bluff-body vortex-shedding instability. Finally, we show that for a range of K_B, light flags with M_ρ≤ O(1) exhibit chaotic flapping characterised by switching between large-amplitude flapping and the deflected-mode state. No chaotic flapping is observed for heavy flags, i.e., M_ρ > O(1).§.§ Bifurcation diagrams and general observationsFigure <ref> shows bifurcation diagrams at four different masses for Re = 200. Each plot gives the transverse leading edge displacement (tip deflection, δ_tip, nondimensionalised by the flag length L) as a function of the reciprocal stiffness (1/K_B). Solid lines represent stable equilibria, and dashed lines correspond to unstable equilibria. Information for unsteady regimes is conveyed through the markers. A set of markers at a given stiffness corresponds to tip deflection values from a single nonlinear simulation at moments when the tip velocity is zero (i.e., when the flag changes direction at the tip). From a dynamical systems perspective, the markers correspond to zero tip velocity Poincaré sections of a tip velocity-tip displacement phase portrait. All nonlinear simulations were started with the flag in its undeflected position and the flow impulsively started to its freestream value. A small body force was introduced at an early time to trigger any instabilities in the system. All simulations contain a minimum of 15 flapping cycles except for the chaotic flapping regime, where a minimum of 55 cycles were used. To avoid representing transient behaviour in the figures, we omit the first several flapping cycles in the diagrams. The bifurcation diagrams were insensitive to starting conditions—the results were unchanged by running a corresponding set of simulations with the flag initialised in its deformed equilibrium state. To illustrate the meaning of the markers in figure <ref> further, consider 1/K_B ≈4 for M_ρ = 0.5. The system enters into large-amplitude limit cycle flapping with a fixed amplitude of ≈± 0.8, and the bifurcation diagram reflects this with a marker at these peak tip displacements, which are the only tip displacement values where the tip velocity is zero. Note that there are actually several markers superposed onto one another at this stiffness since multiple flapping periods were used to plot these diagrams, though only one marker is visible because of the limit cycle behaviour exhibited. As another example, the bifurcation diagram at 1/K_B ≈ 6 for M_ρ = 0.05 depicts chaotic flapping. Many markers are visible at this stiffness because the flag changes direction at several different values of δ_tip. The value of using zero tip-velocity Poincaré sections for the bifurcation diagrams is seen through chaotic flapping: these Poincaré sections demonstrate the variety of transverse locations where the flag changes direction— a fact not captured through, for example, plotting the peak-to-peak flapping amplitudes at a given stiffness.The bifurcation diagrams in figure <ref> depict the undeformed equilibrium (I), deformed equilibrium (II), small-deflection deformed flapping (III), large-amplitude flapping (IV), deflected mode (VI), and chaotic flapping (V) regimes. In small-deflection deformed flapping, flapping is seen about the upward deflected equilibrium. There is a corresponding deformed equilibrium with a negative flag deflection, and different initial conditions would result in flapping about this equilibrium. We refrain from plotting this behaviour to avoid confusion with large-amplitude flapping.The undeformed equilibrium becomes unstable with decreasing stiffness due to a divergence instability (the critical stiffness for instability is independent of the mass ratio) <cit.>. We see from figure <ref> that this instability causes a transition to a regime where the flag is in a steady deflected position. As stiffness is decreased, this steady deflected state is characterised by increasingly large tip deflections (see figure <ref>). This regime was first observed by <cit.>, and we note that it represents a deformed equilibrium state (i.e., in the notation of section <ref> it satisfies the steady state equations r(y) = 0). Moreover, even for masses where flapping occurs, the deformed equilibrium still exists as an unstable steady-state solution to the fully-coupled equations (<ref>)–(<ref>). Note also that for a given stiffness the tip deflection of the deformed equilibrium is constant for all masses, since the equilibrium is a steady state solution of (<ref>)–(<ref>) and therefore does not depend on flag inertia. Figure <ref> provides illustrations of deformed equilibria for various stiffnesses (some of which are unstable).Figure <ref> gives the peak flapping frequency for the various regimes where flapping occurs. For all masses considered, small-deflection flapping is associated with a low frequency that is not indicative of VIV behaviour: using the maximal tip displacement as the length scale, the largest Strouhal number over all masses is 0.02. We show in the next section that this regime is caused by the transition to instability of the leading global mode of the deformed equilibrium. Note that the frequency is dependent on flag mass in this regime, illustrating the fully-coupled nature of the problem. Large-amplitude flapping is qualitatively different for light flags (M_ρ = 0.05, 0.5) and heavy flags (M_ρ = 5, 50). Light flags: the flapping frequency is roughly constant across an order-of-magnitude change in M_ρ, which demonstrates the flow-driven nature of flapping in this regime. <cit.> found that for a range of parameters large-amplitude flapping exhibits several properties of a VIV, and we confirm below that for light flags the fluid forces on the flag synchronise with the flag's motion to form a VIV. Heavy flags: the flapping frequency is decreased relative to light flags, and we show in a later section that there is a corresponding de-synchronisation between flapping and vortex shedding. Thus, for heavy flags large-amplitude flapping is not a VIV. This confirms the scaling analysis of <cit.> that VIV behaviour should cease for sufficiently heavy flags. We note from region IV of the bifurcation diagrams in figure <ref> that for M_ρ = 5, 50 large-amplitude flapping persist despite the absence of a VIV. We discuss the mechanism for flapping in this non-VIV regime in section <ref>.For all masses, the deflected-mode regime (occuring at low stiffness/ high flow rate) has a peak frequency that matches the bluff-body shedding frequency (depicted by the dashed lines). This regime is therefore flow-driven and caused by the canonical bluff-body wake instability irrespective of flag mass <cit.>. We show in section <ref> that the global stability analysis reflects this behaviour.Finally we note that for light flags (M_ρ = 0.05, 0.5), large-amplitude flapping bifurcates (with decreasing stiffness) to chaotic flapping before entering into the deflected-mode regime. The frequency plot for M_ρ = 0.05, 0.5 in figure <ref> illustrates this transition further: in the large-amplitude flapping regime (region IV), decreasing stiffness leads to a corresponding decrease in flapping frequency. Eventually the decrease in frequency becomes significant enough that de-synchronisation between flag flapping and vortex shedding occurs. At this point, chaotic flapping (region V) characterised by multiple frequencies ensues. This regime is discussed in more detail in section <ref>.§.§ Small-deflection deformed flapping Table <ref> demonstrates that the bifurcation to small-deflection deformed flapping is a supercritical Hopf bifurcation of the deformed equilibrium. For all four mass ratios (M_ρ), the onset of flapping is associated with the transition to instability of the leading mode of the deformed equilibrium. Table <ref> also shows that the leading mode accurately captures the flapping frequency observed in the nonlinear simulations near this stability boundary where flapping amplitudes remain small.Figures <ref> and <ref> show the leading mode of the deformed equilibrium near the critical stiffness values where bifurcation occurs for M = 0.5 and M =5, respectively. Flapping is associated with vortical structures isolated near the flag surface. Note that the vortical structures are longer for M_ρ = 5 (figure <ref>) than for M_ρ = 0.5 (figure <ref>), which is commensurate with the lower flapping frequency seen for the more massive case.To demonstrate how the leading mode manifests itself in the nonlinear simulations, we show in figure <ref> snapshots during a flapping period of a flag with M_ρ = 0.5, K_B = 0.41 (i.e., in the small-deflection flapping regime). Note in particular the absence of vortex shedding—the entire flapping period is associated with long vortical structures that extend from the flag into the wake. This is distinct from the large-amplitude flapping behaviour discussed in the next section, and emphasises that even at the moderate Reynolds number of Re = 200, inverted flags have an intrinsic flapping mechanism devoid of vortex shedding.§.§ Large-amplitude flapping Decreasing stiffness in the small-deflection deformed flapping regime is associated with an increasingly unstable leading mode (see table <ref>) and a corresponding increase in flapping amplitude. Eventually, the amplitude is sufficiently large for the flag to reach past the centreline (δ_tip = 0) position, and large-amplitude flapping ensues. At this Reynolds number, the large-amplitude behaviour is associated with sufficient bluffness to the flow that vortex shedding occurs. As discussed in section <ref>, the resulting dynamics are dependent on flag mass: light flags undergo the VIV behaviour identified by <cit.> and heavy flags do not, and we therefore consider them separately in what follows. §.§.§ Large-amplitude flapping of light flags For sufficiently light flags (M_ρ≤ 0.5 in our studies), flapping was shown to synchronise with specific vortex-shedding patterns <cit.>, and <cit.> used experiments and a scaling analysis to argue that this regime is a VIV. To illustrate the synchronisation of vortex shedding and flapping, we show snapshots from a half-period of large-amplitude flapping for M_ρ = 0.05, 0.5 in figure <ref>. Note that despite an order of magnitude change in mass, the vortex-shedding pattern in the top two rows of the figure is similar: when the flag reaches its peak amplitude the leading edge vortex formed during the upstroke grows (left plot); as the flag begins its downstroke the leading edge vortex is released and a trailing edge vortex forms (second from left plot); the vortices grow in size as the flag reaches its centreline position (second from right plot); while the leading and trailing edge vortices advect downstream to form a P vortex pair (see <cit.> for a description of this vortex characterisation), a leading edge vortex forms as the flag continues its downstroke (rightmost plot). When the flag reaches its peak position, an analogous process to the one just described occurs during the upstroke (with oppositely signed vorticity).To confirm that this regime is a VIV, we show in figure <ref> time traces of the coefficient of lift and tip displacement for M_ρ = 0.05, 0.5. The lift and tip displacement are synchronised, and therefore satisfy the definition of VIV <cit.>. §.§.§ Large-amplitude flapping of heavy flags The interaction between vortex shedding and flapping is qualitatively different as the flag mass is increased further. Figure <ref> shows that for M_ρ = 5, 50 additional vortices are shed per half-period. The flapping cycle for M_ρ = 5 is similar to the lighter flag cases, except that during the downstroke an additional leading edge vortex forms (first row, second from rightmost plot) and advects downstream along with the original leading and trailing edge vortices (rightmost plot). The additional vortex leads to a P + S wake structure. For M_ρ = 50, even more vortices are shed during the downstroke: the first leading and trailing edge vortices are shed when the flag is still near its peak amplitude (second row, leftmost plot); the flag begins its downstroke and another leading-trailing edge vortex pair are formed (second from leftmost plot); as the flag nears its centreline position, a third leading edge vortex forms (second from rightmost plot); this leading edge vortex combines with a newly formed trailing edge vortex during the downstroke phase to form a third pair that is advected downstream; at the end of the downstroke phase, new vortices form at the leading and trailing edge as the flag reaches its peak amplitude (rightmost plot).This desynchronisation is illustrated further through time traces of the coefficient of lift and tip displacement (see figure <ref>). Unlike the in-phase behaviour between lift and tip displacement seen for light flags, heavy flags contain multiple lift peaks for a given peak in tip displacement. Moreover, for M_ρ = 50, there is a slight departure from periodicity that can be observed in figure <ref> (and is reflected in the bottom right bifurcation diagram in figure <ref>). The break in synchronisation of vortex shedding and flapping demonstrates that sufficiently heavy flags do not undergo a VIV—a result argued by <cit.> through a scaling analysis. We emphasise that large-amplitude flapping still occurs for heavy flags despite the absence of a VIV. The mechanism responsible for flapping can be explained through the deformed equilibrium of the system. In this regime, the deformed equilibrium is unstable and has a sufficiently large saturation amplitude for the flag to flap past the centreline position and into the region of attraction of the deformed equilibrium on the other side of the centreline. This newly sampled deformed equilibrium is also associated with a saturation amplitude that leads the flag to flap past the centreline, so an indefinite process ensues with flapping occurring around these two equilibria. We show in section <ref> that large-amplitude flapping also occurs for massive flags at Re = 20, which attests to the presence of non-VIV flapping mechanisms and to the potential for large-amplitude flapping in the absence of any vortex shedding. §.§ Deflected-mode regime For low stiffnesses the system transitions to a large-deflection state about which small-amplitude flapping occurs. As seen in figure <ref>, this flapping is not centered about the deformed equilibrium position (i.e., the mean and equilibrium states are different). The nature of flapping in this regime is qualitatively distinct from that of small-deflection deformed flapping and large-amplitude flapping. <cit.> observed that when the flapping frequency is scaled by the freestream velocity and mean tip amplitude (i.e., the mean projected length to the flow), it agrees well with the classical 0.2 value found for vortex shedding past bluff bodies <cit.>. They used this finding to argue that the bluff-body wake instability is responsible for the small-amplitude flapping in this regime. The global stability analysis of the deformed equilibrium confirms this previous conclusion. Figure <ref> shows that the least damped mode is characterised by a vortical structure in the wake of the flag similar to the leading mode of a rigid stationary cylinder <cit.>. This is in contrast to the leading mode found for small-deflection deformed flapping and large-amplitude flapping, which has isolated vortical structures near the flag surface[The analogous mode to the leading global mode for small-deflection deformed flapping and large-amplitude flapping is also unstable in the deflected-mode regime, but is associated with a smaller growth rate.]. Moreover, as seen in table <ref>, the flapping frequency of this leading mode is independent of mass ratio (M_ρ). This is also distinct from the mass-dependent flapping frequency of the least damped mode for the previously discussed regimes. The similar vortical structure of the mode to other bluff-body flows and the independence of the structural parameters on the modal frequency demonstrate that the leading instability is associated with vortex shedding and is flow-driven. The presence of vortex shedding in this deflected-mode regime is also the cause of the difference between the mean and equilibrium flapping positions. Vortex shedding is associated with an increase in the mean forces on the flag compared with the equilibrium state (which is devoid of vortex shedding; c.f., figure <ref>). To demonstrate this increase in fluid forces, we ran a simulation with the flag fixed in the deformed equilibrium position corresponding to K_B = 0.1. With the flag fixed in this position, the bluffness of the body causes the flow to enter limit-cycle vortex shedding with a mean lift and drag of 0.356 and 0.583, respectively. By contrast, when the fully-coupled system is in the deformed equilibrium and vortex shedding is absent, the lift and drag forces are 0.192 and 0.344, respectively. The increase in mean forces causes a corresponding increase in flag deflection, and thus in the nonlinear simulations flapping occurs about a mean position that is raised from the equilibrium state.We emphasise that this vortex shedding mode is stable for small-deflection deformed flapping and large-amplitude flapping found at higher stiffnesses (and studied in the previous sections). For example, the growth rate of the vortex shedding mode for the large-amplitude flapping parameters K_B = 0.2, M_ρ = 0.5 is -0.392. The vortex-shedding mode is therefore not the cause of instability of the deformed equilibrium in those regimes.§.§ Chaotic flappingFor sufficiently light flags (M_ρ≤ 0.5 in our studies), large-amplitude flapping (region IV in the bifurcation diagrams of figure <ref>) bifurcates to chaotic flapping (region V) before entering into deflected-mode flapping (region VI). Figure <ref> shows that the time trace of the tip displacement is aperiodic and associated with broadband frequency content.To demonstrate mathematically that this behaviour is chaotic, we compute the Lyapunov exponent of the system using the time-delay method of <cit.>(this approach was also used by <cit.> to identify chaotic flapping of conventional flags). The method computes an approximation of the distance in time, d(t), of two trajectories starting close to one another at an initial time t_0. The evolution of this distance is written asd(t) = d(t_0) e^γ (t - t_0)where γ is the Lyapunov exponent that represents the departure or convergence of the two trajectories. A zero value of γ corresponds to a stationary state where the system is in limit cycle behaviour; a positive value of γ corresponds to divergence of the two trajectories, and thus to chaotic flapping. Table <ref> shows the Lyapunov exponent computed for various values of M_ρ and K_B. For large-amplitude and deflected-mode flapping, the exponent is approximately zero, coincident with limit-cycle flapping. In the chaotic regime that occurs at stiffnesses between large-amplitude and deflected-mode flapping, the exponent is positive and larger by an order of magnitude, indicative of a transition to chaotic behaviour in this regime.The bifurcation diagrams in figure <ref> demonstrate that increasing mass reduces the chaotic behaviour. In moving from M_ρ = 0.05 to M_ρ = 0.5, there were certain stiffnesses within the chaotic flapping regime that exhibited periodic flapping instead of chaotic flapping (see figure <ref>). We believe that this is an artifact of only running the simulations for finite time, but the absence of chaotic flapping over a minimum of 55 flapping periods for certain stiffnesses at M_ρ = 0.5 speaks to the effect of increasing inertia on reducing the chaotic behaviour. For the heavier flag cases of M_ρ = 5, 50, chaotic flapping disappears altogether. Thus, chaotic flapping is only associated with mass ratios (M_ρ) for which VIV flapping occurs. To elucidate the nature of chaotic flapping, we show in figure <ref> phase portraits of tip velocity versus tip displacement for inverted flags in the large-amplitude flapping, chaotic flapping, and deflected-mode regimes. The figures demonstrate that the chaotic flapping phase portrait contains both the large periodic orbit of large-amplitude flapping and the small-amplitude large-deflection periodic orbit of deflected-mode flapping. Thus, chaotic flapping is a regime in which large-amplitude flapping and the deflected mode hybridise to form a new strange attractor involving both states. The chaotic nature of the regime is associated with the apparent randomness in switching between these two orbits.§ DYNAMICS FOR RE = 20Previous numerical simulations of <cit.> demonstrated the absence of flapping for flags with M_ρ≤ O(1) and Re < 50. We now consider Re =20 to investigate the stability and dynamics of the inverted-flag system below this previously identified critical Reynolds number. In agreement with <cit.>, we find that light flags with M_ρ = 0.05, 0.5 do not flap. Heavy flags with M_ρ = 5, 50 are shown to undergo both small-deflection deformed flapping and large-amplitude flapping. Such behaviour has yet to be reported, and we demonstrate that for this heavy flag case neither flapping regime is a VIV. As was observed for Re = 200, small-deflection flapping is caused by a supercritical Hopf bifurcation of the deformed equilibrium associated with the transition to instability of the least damped mode of the deformed equilibrium. Large-amplitude flapping is characterised by an increase in saturation amplitude of small-deflection flapping until eventually the flag swings past the centreline and begins a process where it samples both deflected equilibria. Finally, we show that at this low Reynolds number the deflected-mode state is not associated with flapping, and is instead a formal stable equilibrium of the fully-coupled system. §.§ Bifurcation diagrams and general observations Figure <ref> gives bifurcation diagrams of the inverted-flag system at four different masses. These figures were plotted as described in section <ref>. The bifurcation diagrams reveal four distinct regimes: a stable undeformed equilibrium (I), a stable deformed equilibrium (II), small-deflection deformed flapping (III), and large-amplitude flapping (IV). While many of the same bifurcations found at Re = 200 remain for Re = 20, there are also distinctions between them that are visible through the bifurcation diagrams. First, flapping does not occur for all masses considered at this lower Reynolds number, which demonstrates the stabilising effect of fluid diffusion for inverted-flag dynamics. Second, the deflected-mode state no longer corresponds to flapping at this lower Reynolds number, and is instead a formal stable equilibrium of the fully-coupled fluid-structure equations of motion. Finally, chaotic flapping does not occur for any of the considered values of M_ρ at this lower Reynolds number. As was seen for Re = 200, the divergence instability of the undeformed equilibrium (caused by decreasing K_B) leads to a stable deformed equilibrium that is independent of the mass ratio, M_ρ. This follows from the fact that the deformed equilibria are steady state solutions and therefore do not depend on flag inertia.As stiffness is decreased, light flags remain in this deformed equilibrium regime—no flapping occurs at this Reynolds number for M_ρ =0.05, 0.5. Moreover, since the equilibrium states do not depend on flag inertia, their bifurcation diagrams are identical. By contrast, with decreasing stiffness heavy flags transitioned from the deformed equilibrium to (respectively) small-deflection flapping and large-amplitude flapping before returning at even lower stiffnesses to a stable deformed equilibrium. To demonstrate the non-VIV nature of flapping at this low Reynolds number, we show in figure <ref> the peak flapping frequency for the parameters corresponding to the bifurcation diagrams in figure <ref>. For all cases, the flapping frequency from the nonlinear simulations (denoted by the markers) is substantially different from the bluff-body vortex-shedding frequency. In the remainder of this section we explore the physical mechanisms behind the various regimes and the transitions between them.§.§ Small-deflection deformed flapping We show in table <ref> that the transition from the deformed equilibrium to small-deflection deformed flapping is associated with the least damped global mode of the deformed equilibrium becoming unstable. Thus, as was seen for Re = 200, small-deflection deformed flapping is a supercritical Hopf bifurcation of the deformed equilibrium state.Table <ref> also shows that the corresponding eigenvalue accurately predicts the flapping frequency of the nonlinear simulations near the stability boundary. To illustrate the vortical structures and flag shapes associated with the instability mechanism at this lower Reynolds number, we plot the real and imaginary parts of the leading global mode near the critical stiffness for M_ρ = 5 in figure <ref> (the plot is similar for M_ρ = 50). Flag motion is associated with four vortical structures isolated near the flag surface. We emphasise that a linear stability analysis of the undeformed equilibrium state is associated with a zero-frequency (non-flapping) unstable mode, and therefore does not capture the flapping behaviour observed in the nonlinear simulations. This demonstrates that the divergence instability derived by <cit.> for inviscid fluids persists at lower Reynolds numbers. Figure <ref> shows the leading global mode of the undeformed equilibrium. The mode has a similar flag shape and set of vortical structures to the real part of the leading mode of the deformed equilibrium. A noticeable distinction between the two, however, is that the vortical structures of the undeformed equilibrium mode are symmetric about the equilibrium flag position while those of the deformed equilibrium mode are not. The presence of asymmetry associated with flapping is indicative of the interplay between fluid forces, flag inertia, and internal flag stresses necessary to sustain flapping. To explore this interplay, consider a perturbation of the deformed equilibrium that sets the flag into motion in the direction of increasing deflection. This causes an increase in internal flag stresses that act to restore the flag to its deformed equilibrium. These stresses are opposed by the flag inertia and by forces from the oncoming fluid, which tend to destabilise the system further away from its deformed equilibrium. By contrast, if the flag is set into motion the other direction (towards the undeformed state), the fluid forces act to restore the flag to its deformed equilibrium and the flag inertia and internal flag stresses act as destabilising forces. The exchange of internal flag stresses and fluid forces as destabilising quantities is a unique feature of flapping about the deflected state—an analogous perturbation to a flag in the undeformed equilibrium results in flag stresses that are always restoring and fluid forces that are always destabilising. §.§ Large-amplitude flapping We now consider the transition from small-deflection deformed flapping to large-amplitude flapping. Within the small-deflection deformed flapping regime, the bifurcation diagrams show that decreasing stiffness causes an increase in flapping amplitude. This is associated with an increase in growth rate of the leading mode (see table <ref>). The mechanism through which the increasingly unstable leading mode develops into large-amplitude flapping is similar to what was discussed for heavy flags at Re = 200. Eventually, the growth in saturation amplitude leads the flag to deform past the centreline position and into the region of attraction of the deformed equilibrium on the other side of the centreline. This newly sampled deformed equilibrium is also associated with a saturation amplitude that leads the flag to flap past the centreline, and indefinite flapping occurs around these two equilibria. We emphasise that at this low Reynolds number, vortex shedding does not occur, and the flapping frequency from figure <ref> demonstrates that flapping is not a VIV in this regime. The nonlinear behaviour characterised by flapping around the two unstable deformed equilibria provides the necessary non-VIV flapping mechanism.We note that this non-VIV flapping mechanism is distinct from what is observed in large-amplitude oscillations of elastically mounted cylinders at subcritical Reynolds numbers. In the elastically mounted cylinder case, <cit.> showed through nonlinear simulations and a global stability analysis that VIV persists at subcritical Reynolds numbers for certain parameters, and that large-amplitude vibrations are a result of this VIV. They moreover demonstrated that for these parameters the vibration frequency matched the bluff-body shedding frequency. By contrast, in the case of large-amplitude inverted-flag flapping, the flapping frequency is substantially smaller than the bluff-body vortex-shedding frequency. §.§ Large-deflection equilibrium (deflected-mode regime) A continued decrease in stiffness leads to a bifurcation from large-amplitude flapping back to a stable deformed equilibrium with large deflection. This transition corresponds to the re-stabilisation of the leading global mode (e.g., for M_ρ = 5, K_B = 0.17 the growth rate of the leading mode is -0.032). Note that this deflected-mode state is distinct from that found at higher Reynolds numbers, where the flag undergoes small-amplitude oscillations driven by vortex shedding <cit.>. Since vortex shedding is absent at Re = 20, the deflected-mode regime is a formal equilibrium of the fully-coupled equations of motion. § CONCLUSIONS We used 2D high-fidelity nonlinear simulations and a global linear stability analysis of inverted-flag flapping to (i) investigate the physical mechanisms responsible for the onset of flapping, (ii) study the role of vortex shedding in large-amplitude flapping, and (iii) further characterise various regime bifurcations that were previously identified and explored <cit.>. We performed studies at Re = 20 and 200 for a wide range of K_B and over a four-order-of-magnitude range of M_ρ. For Re = 20 and M_ρ≤ O(1), no flapping occurs and the flag transitions with decreasing stiffness from an undeformed equilibrium to a deformed equilibrium. For all other combinations of Re and M_ρ considered, with decreasing flag stiffness the system transitions from a stable undeformed equilibrium to a stable deflected equilibrium via a divergence instability, to an unstable deformed equilibrium through a supercritical Hopf bifurcation that exhibits small-amplitude flapping, to large-amplitude flapping, and finally to a deflected-mode state. Below we summarise the key features of each of these regimes. Stable deflected equilibrium: we demonstrated that for all parameters considered the stationary deflected state identified by <cit.> is a formal equilibrium of the fully-coupled equations, and that even when flapping occurs this equilibrium persists as an unstable steady-state. A similar deformed equilibrium was found at Re = O(30,000) by <cit.> through the addition of damping; establishing similarities between these findings is an area for future work.Small-deflection deformed flapping: the deformed equilibrium becomes unstable and transitions to small-deflection deformed flapping with decreasing stiffness (K_B). This occurred at Re =200 for all mass ratios considered and at Re=20 for heavy flags (M_ρ > O(1)). This transition was shown to be initiated by a supercritical Hopf bifurcation of the deformed equilibrium state (i.e. a complex-conjugate set of eigenvectors becomes unstable). For all parameters that exhibited this small-deflection flapping regime, the leading mode and ensuing nonlinear behaviour are both devoid of vortex shedding and have a flapping frequency that is not commensurate with a VIV.Large-amplitude flapping: light flags (M_ρ <O(1)) at Re = 200 exhibit VIV behaviour in which the fluid forces on the flag are synchronised to the flag motion. This coincides with the arguments of <cit.> based on experimental measurements and a scaling analysis. By contrast, heavy flags (M_ρ > O(1)) did not exhibit VIV behaviour—also consistent with the scaling analysis of <cit.>. Yet, we found that large-amplitude flapping persists at these large values of M_ρ despite the absence of VIV, and demonstrated that this flapping is due to the instability of the two deformed equilibria on either side of the centreline. To further highlight the potential for large-amplitude flapping without VIV mechanisms, we also showed that large-amplitude flapping occurs for heavy flags at Re = 20. No flapping was observed for flags with M_ρ < O(1) at Re = 20, which is in agreement with the simulations of <cit.>.Deflected-mode: for Re = 200 we used a global stability analysis to confirm the argument of <cit.> that this regime is driven by the canonical bluff-body wake instability. For all masses considered, the leading mode has vortical structures similar to the leading global mode found in canonical bluff-bodies <cit.> and a flapping frequency commensurate with the St ∼ 0.2 bluff-body scaling <cit.>. We then showed that the deflected mode does not exhibit any flapping at any mass ratio for Re =20, and the system is instead in a large-deflection equilibrium state. Chaotic flapping: we identified chaotic flapping for light flags at Re = 200 and characterised this regime by switching between large-amplitude flapping and the deflected-mode regime. No chaotic flapping was observed at Re = 200 for M_ρ > O(1) or at Re = 20 for any of the mass ratios considered. These findings demonstrate a wide range of physical mechanisms that drive the various dynamical regimes of the inverted flag system. Moreover, they highlight that the system dynamics depend on both the Reynolds number and mass ratio. At the same time, these results motivate future work that compares our low-to-moderate Reynolds number computational findings with results at higher Reynolds numbers and in three dimensions.§ ACKNOWLEDGMENTS AJG and TC gratefully acknowledge funding through the Bosch BERN program and through the AFOSR (grant number FA9550-14-1-0328). JES thanks the ARC Centre of Excellence in Exciton Science and the Australian Research Council Grants Scheme.jfm | http://arxiv.org/abs/1709.09745v1 | {
"authors": [
"Andres Goza",
"Tim Colonius",
"John Elie Sader"
],
"categories": [
"physics.flu-dyn"
],
"primary_category": "physics.flu-dyn",
"published": "20170927215229",
"title": "Global modes and nonlinear analysis of inverted-flag flapping"
} |
[ Neil Mañibo Accepted . Received; in original form============================================This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2.This is a general setting for the understanding of the degeneration of the sixth Painlevé equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form,through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system.The problem of analytic classification is also addressed.1Key words: #1 · § INTRODUCTIONWe consider a parametric family of non-autonomous Hamiltonian systems of the formx(x-ϵ)dy_1dx = ∂ H∂ y_2(y,x,ϵ)x(x-ϵ)dy_2dx =-∂ H∂ y_1(y,x,ϵ),(y,x,ϵ)∈(^2××,0),shortly written asx(x-ϵ)dydx=J(D_yH), J=([01; -10 ]),with a singular Hamiltonian function H(y,x,ϵ)/x(x-ϵ), where H(y,x,ϵ) is an analytic germ such that H(y,0,0) has a non-degenerate critical point (Morse point) at y=0:D_yH(0,0,0)=0, D_y^2 H(0,0,0)≠ 0.The last condition means that the y-linear terms of the right side of (<ref>) are of the form A(x,ϵ)y for A=J D_y^2H, whereA(0,0)∼([λ^(0)(0) 0; 0 -λ^(0)(0) ]) for some λ^(0)(0)≠ 0.For ϵ≠ 0 the system (<ref>) has two regular singular points at x=0 and x=ϵ.At each one of them, the local information about the system is carried by a formal invariant and a monodromy (holonomy) operator.On the other hand, for ϵ=0 the corresponding information about the irregular singularity at x=0 is carried by a formal invariant and by a pair of non-linear operators.Our main goal is to explain the relation between these two distinct phenomena, and to show how the Stokes operators are related to the monodromy operators. The principal thesis is, that while the monodromy operators diverge when ϵ→ 0, they each accumulate to a 1-parameter family of “wild monodromy operators” which encode the Stokes phenomenon(Theorem <ref>). It is expected that this “wild monodromy” should have Galoisian interpretation.Along the way, we provide a formal normal form and a sectoral normalization theorem for the family (Theorem <ref>), an analytic classification (Theorem <ref>), and a decomposition of the monodromy operators (Theorem <ref>).In Section <ref>, we illustrate all this on the example of traceless 2× 2 linear differential systemsx(x-ϵ)dydx=A(x,ϵ)y, A(x,ϵ)∈sl_2(), A(0,0)≠ 0,for which our description follows from the more general work of Lambert and Rousseau <cit.>.Here the relation between the monodromy and the Stokes phenomenon can be summarized as: When ϵ→ 0 the elements of the monodromy group of the system (<ref>) accumulate to generators of the wild monodromy group of the limit system (that is the group generated by the Stokes operators and the exponential torus). The linear case can be kept in mind as a leading example of which the general non-linear case is a close analogy. An important example of a confluent family of systems (<ref>), which in fact motivated this study, is the degeneration of the sixth Painlevé equation to the fifth one,presented in Section <ref>. A more detailed treatment of this confluence will be the subject of an upcoming article <cit.>.§.§ AcknowledgmentsThis paper was inspired by the works of C. Rousseau and L. Teyssier <cit.>, C. Lambert and C. Rousseau <cit.>, and A. Bittmann <cit.>. It was written during my stay at Centre de Recherches Mathématiques at Université de Montréal.I want to thank Christiane Rousseau for her support and the CRM for its hospitality.§ THE FOLIATION AND ITS FORMAL INVARIANTSThe family of systems (<ref>) defines a family of singular foliations in the (y,x)-space, leaves of which are the solutions. We associate to (<ref>) a family of vector fields tangent to the foliationsZ_H,ϵ(y,x)=x(x-ϵ)∂_x+X_H,x,ϵ(y),whereX_H,x,ϵ(y)=∂ H∂ y_2(y,x,ϵ)∂_y_1-∂ H∂ y_1(y,x,ϵ)∂_y_2.The vector field Z_H,0(y,x) has a saddle-node type singularity at (y,x)=0, i.e. its linearization matrix has one zero eigenvalue, corresponding to the x-direction. It follows from the Implicit Function Theorem that, for small ϵ≠ 0, Z_H,ϵ has two singular points (y_0(0,ϵ),0) and (y_0(ϵ,ϵ),ϵ) bifurcating from (y_0(0,0),0)=0 and depending analytically on ϵ. The aim of this paper is a study of their confluence when ϵ→ 0.The two singularities of Z_H,ϵ have each a strong invariant manifold _0={(y,x):x=0}, resp. _ϵ={(y,x):x=ϵ}.Away of these invariant manifolds the vector field Z_H,ϵ is transverse to thefibration with fibers _c={(y,x):x=c}. The (y,x)-space is endowed with a Poisson structure associated to the 2-form ω=dy_1∧ dy_2,the restriction of which on each fiber _c is symplectic. The vector field Z_H,ϵ is transversely Hamiltonian with respect to this fibration, the form ω, and the Hamiltonian function H(y,x,ϵ).§.§ Fibered changes of coordinates We consider the problem of analytic classification of families of systems (<ref>), or orbital analytic classification of vector fields (<ref>), with respect to fiber-preserving (shortly fibered) changes of coordinates (y,x,ϵ)=(Φ(u,x,ϵ),x,ϵ).Such a change of coordinate transforms a system (<ref>)to a systemx(x-ϵ)du/dx =(D_uΦ)^-1 J(D_yH)∘Φ-x(x-ϵ)(D_uΦ)^-1∂Φ∂ x=( D_uΦ)^-1 J(D_u(H∘Φ))-x(x-ϵ)(D_uΦ)^-1∂Φ∂ x,using the identity PJP= P· Jfor any 2×2 matrix P. We call a fibered transformation Φ transversely symplectic if (D_uΦ)≡ 1, i.e. if it preserves the restriction of ω to each fiber _x.Two systems (<ref>) with Hamiltonian functions H(y,x,ϵ)/x(x-ϵ) and H̃(u,x,ϵ)/x(x-ϵ) are called analyticallyequivalent if there exists an analytic germ of a transversely symplectic transformation y=Φ(u,x,ϵ)that is analytic in (u,x,ϵ) and transforms one system to another:Φ^*Z_H,ϵ=Z_H̃,ϵ.If a transformation y=Φ(u,x,ϵ) is transversely symplectic, then the transformed system (<ref>) is transversely Hamiltonian w.r.t.ω=du_1∧ du_2. It is enough to show that the system du/dx=(D_uΦ)^-1∂Φ∂ x is transversely Hamiltonian, that is, denoting([ f_1; f_2 ]):=(D_uΦ)^-1∂Φ∂ x, to show that ∂u_1f_1+∂_u_2f_2=0. Using the identity (<ref>), we can express f_1=1 D_uΦ(∂_u_2Φ)J∂_xΦ,f_2=-1 D_uΦ(∂_u_1Φ)J∂_xΦ, hence∂_u_1f_1+∂_u_2f_2 = 1 D_uΦ[ (∂_u_2Φ)J∂_x∂_u_1Φ -(∂_u_1Φ)J∂_x∂_u_2Φ]=1 D_uΦ∂_x[(∂_u_2Φ)J∂_u_1Φ]=1 D_uΦ∂_x D_uΦ=0.We will therefore consider a larger class of systems of the formx(x-ϵ)dy/dx=F(y,x,ϵ),whereF(y,x,ϵ)=J(D_yH) x(x-ϵ),(D_y F(0,0,0)=(D_y^2 H(0,0,0))≠ 0,for some germ H as before, which are transversely Hamiltonian modulo O(x(x-ϵ)). This class is closed to analytic transversely symplectic transformations. Correspondingly, we consider also families of vector fieldsZ_ϵ(y,x)=x(x-ϵ)∂_x+F_1(y,x,ϵ)∂_y_1+F_2(y,x,ϵ)∂_y_2,with Z_ϵ__0=X_H,0,ϵ, Z_ϵ__ϵ=X_H,ϵ,ϵand with L_∂_xZ_0__0=X_∂_x· H,0,0 for ϵ=0. §.§ The formal invariant χ(h,x,ϵ)LetH:(^2,0)→(,0) have a non-degenerate critical point at 0, and let ω be a symplectic volume form.There exists an analytic system of coordinates u=(u_1,u_2) in which ω=du_1∧ du_2,and H=G_H(u_1u_2).The function G_H is uniquely determined by the pair (H,ω) up to the involutionG_H(u_1u_2)↦ G_H(-u_1u_2),induced by the symplectic change of variable J:(u_1,u_2)↦(u_2,-u_1). The pair (G_H,ω) is called the Birkhoff-Siegel normal form of the pair (H,ω). Moreover, if (H,ω) depend analytically on a parameter, then so does G_H and the change of coordinates.While not explicitly stated, the existence part of the theorem is originally proved by Siegel in <cit.>. See also <cit.> and <cit.>. The uniqueness can be seen by expressing G_H(h) in terms of a period map over a vanishing cycle, see Section <ref> below.– The Theorem <ref>provides the existence of an analytic transformation of a Hamiltonian vector field ẏ=J(D_yH) in dimension 2 to its Birkhoff normal form u̇=J(D_uG_H),with G_H(h)=λ h+…,where ±λ≠ 0 are the eigenvalues of the linear part JD_y^2H(0). The involution (<ref>) corresponds to the freedom of choice of the eigenvalue λ.– The change of coordinates is far from unique. Indeed, the flow of any vector fieldξ=a(u_1u_2)(u_1∂_u_1-u_2∂_u_2) preserves the normal form. Let H=H(y,x,ϵ) be our germ. By the implicit function theorem, for each small (x,ϵ), the functionH(·,x,ϵ) has an isolated non-degenerate critical point y_0(x,ϵ), depending analytically on (x,ϵ). Let y=Φ(u,x,ϵ) be the transformation to the Birkhoff-Siegel normal form for the function y↦ H(y,x,ϵ) and the form ω=dy_1∧ dy_2, depending analytically on (x,ϵ), i.e.H(·,x,ϵ)∘Φ(u,x,ϵ)=G_H(u_1u_2,x,ϵ), D_uΦ(u,x,ϵ)≡ 1.By (<ref>), it brings the system (<ref>) to a prenormal formx(x-ϵ)du/dx =J(D_uG_H)+O(x(x-ϵ))=χ(u_1u_2,x,ϵ)([10;0 -1 ]) u+O(x(x-ϵ)),whereχ(h,x,ϵ)=χ^(0)(h,ϵ)+xχ^(1)(h,ϵ):= ∂ G_H∂ h(h,x,ϵ) x(x-ϵ),h=u_1u_2, or equivalently, χ(h,x,ϵ)= {[ 1ϵ[x ∂ G_H∂ h(h,ϵ,ϵ)-(x-ϵ)∂ G_H∂ h(h,0,ϵ)],ϵ≠ 0,;∂ G_H∂ h(h,0,0)+x∂^2G_H∂ h∂ x(h,0,0),ϵ=0. ].The function χ(h,x,ϵ) is called a formal invariant of the system (<ref>). For ϵ≠ 0 the formal invariant χ is completely determined by the functions G_H(·,0,ϵ) and G_H(·,ϵ,ϵ) ,which are analytic invariants of the autonomous Hamiltonian systems X_H,0,ϵ, X_H,ϵ,ϵ (<ref>) on the strong invariant manifolds _0,_ϵ. The formal invariant χ(h,x,ϵ) is well-defined up to the involutionJ^*: χ(h,x,ϵ)↦ -χ(-h,x,ϵ),induced by the symplectic transformation u↦ Ju. It is uniquely determined by the polar part of the Hamiltonian H(y,x,ϵ)/x(x-ϵ), and it is invariant with respect to fibered transversely symplectic changes of coordinates. Letλ(x,ϵ)=χ(0,x,ϵ).Then±λ(x,ϵ) are the eigenvalues of the matrix A(x,ϵ)=JD_y^2H(0,x,ϵ) modulo x(x-ϵ), see Example <ref>, and the involution (<ref>) corresponds to the freedom of choice of the eigenvalue λ. [Traceless linear systems] A traceless linear systemx(x-ϵ)dy/dx=A(x,ϵ)y,with A(x,ϵ)=0 and A(0,0)∼([λ^(0)(0) 0; 0 -λ^(0)(0) ]) for some λ^(0)(0)≠ 0, is of the form (<ref>) for the quadratic form H(y,x,ϵ)=12yJA(x,ϵ)y. Let ±λ̃(x,ϵ) be the eigenvalues of A(x,ϵ), and let C(x,ϵ) be a corresponding matrix of eigenvectors of A(x,ϵ), depending analytically on (x,ϵ) and normalized so that C(x,ϵ)=1. The change of variable y=C(x,ϵ)u, brings the system (<ref>) tox(x-ϵ)du/dx=λ̃(x,ϵ)([10;0 -1 ])u+ x(x-ϵ)C^-1dC/dx.Denoting λ(x,ϵ):=(λ^(0)(ϵ)+xλ^(1)(ϵ))=λ̃(x,ϵ) x(x-ϵ), then we have χ(h,x,ϵ)=±λ(x,ϵ). §.§.§ Geometric interpretation of the invariant χ. For each small (x,ϵ), the functionH(·,x,ϵ) has an isolated non-degenerate critical point y_0(x,ϵ), depending analytically on (x,ϵ), with a critical value h_0(x,ϵ). For (x,ϵ) fixed, h∈(,h_0), consider the germ of the level set S_h(x,ϵ)={y∈(^2,y_0(x,ϵ)): H(y,x,ϵ)=h}⊂_x.As a basic fact of the Picard–Lefschetz theory <cit.>, we know that if h is a non-critical value for H(·,x,ϵ), i.e. h≠ h_0, then S_h(x,ϵ) has the homotopy type of a circle.Let γ_h(x,ϵ) depending continuously on (x,ϵ) be a loop generating the first homology group of S_h(x,ϵ), the so called vanishing cycle. And let μ be a 1-form such thatω=dH∧μ;its restriction to a non-critical level S_h(x,ϵ) is called the Gelfand-Leray form of ω and is denoted μ=ω/dH.Its period function over the vanishing cyclep(h,x,ϵ):=12π i∫_γ_h(x,ϵ)ω/dH,is well-defined up to a sign change (orientation of γ_h), and depends analytically on (x,ϵ) <cit.>. Let G_H(·,x,ϵ) be the inverse of the function h↦∫_h_0(x,ϵ)^h p(s,x,ϵ) ds.Then (G_H,ω) is the Birkhoff-Siegel normal form of (H,ω). Indeed, the above formula for G_H is invariant with respect to analytic transversely symplectic changes of coordinates: Supposing that H=g(y_1y_2,x,ϵ) is in its Birkhoff-Siegel normal form, then the level sets are written asS_h={y_1≠ 0, y_2=g^∘(-1)(h,x,ϵ)y_1}, andω/dH=dy_1/y_1·∂ g/∂(y_1y_2)∘ g^∘(-1)(h,x,ϵ), and therefore p(h)=(∂ g/∂(y_1y_2))^-1∘ g^∘(-1)(h,x,ϵ), i.e. G_H=g. The above formula for the the Birkhoff-Siegel normal form and hence for the formal invariant χ involves a double inversion which makes it difficult to calculate.The following proposition, which will be proved in Section <ref>, allows to determine it in some special cases.This will be useful in the case of the fifth Painlevé equation (Section <ref>). Let H(y) be of the formH(y)=G(h)+y_iΔ(y_i,h), h=y_1y_2,for some i∈{1,2}, with G,Δ analytic germs, andG(h)=λ h+O(h^2),λ≠ 0.Then (G,ω) is the Birkhoff-Siegel normal form for the pair (H,ω), ω=dy_1∧ dy_2.For ϵ=0, suppose thatH(y,x,0)=G^(0)(h,0)+xH^(1)(y,0)+O(x^2), h=y_1y_2,where G^(0)(h,0)=λ^(0)(0) h+O(h^2), λ^(0)(0)≠ 0. Write H^(1)(y,0)=G^(1)(h,0)+y_1Δ_1(y_1,h,0)+y_2Δ_2(y_2,h,0),with Δ_i(y_i,h,0)=O(y_i). Thenχ(h,x,0)=∂∂ h(G^(0)(h,0)+x G^(1)(h,0))is the formal invariant of the vector field Z_H,0=x^2∂_x+X_H associated to H.Consider a deformation H(y,x,ϵ)=G^(0)(h,0)+xG^(1)(h,0)+xy_1Δ_1(y_1,h,0)+(x-ϵ)y_2Δ_2(y_2,h,0)+O(x(x-ϵ)),and calculate the Birkhoff-Siegel invariants for H(y,0,ϵ), H(y,ϵ,ϵ), using Proposition <ref>. §.§ Model system (formal normal form)Let χ(h,x,ϵ) be the formal invariant of the system (<ref>). The model family (formal normal form) for the the system (<ref>) is the family of systemsx(x-ϵ)du_1dx = χ(u_1u_2,x,ϵ)· u_1x(x-ϵ)du_2dx =-χ(u_1u_2,x,ϵ)· u_2,which is Hamiltonian with respect to the Hamiltonian function G(u_1u_2,x,ϵ)/x(x-ϵ),G(h,x,ϵ)=∫_0^h χ(s,x,ϵ) ds.The formal normal form of the family Z_H,ϵ is the associated family of vector fieldsZ_G,ϵ=x(x-ϵ)∂_x+χ(u_1u_2,x,ϵ)(u_1∂_u_1-u_2∂_u_2). The system (<ref>) is integrable with the function h(u)=u_1u_2 being its first integral, Z_G,ϵ· h=0. The general solutions of (<ref>) are of the formu_1(x,ϵ;c) =c_1 E_χ(c_1c_2,x,ϵ),u_2(x,ϵ;c) =c_2 E_χ(c_1c_2,x,ϵ)^-1, c=(c_1,c_2)∈^2,whereE_χ(h,x,ϵ)={[ x^-χ^(0)(h,ϵ)/ϵ(x-ϵ)^χ^(0)(h,ϵ)/ϵ+χ^(1)(h,ϵ), for ϵ≠ 0,;e^-χ^(0)(h,0)/xx^χ^(1)(h,0), for ϵ= 0. ].§ FORMAL AND SECTORAL NORMALIZATION THEOREMThroughout the text we will denote={|y|<δ_y},={|u|<δ_u},={|x|<δ_x},={|ϵ|<δ_ϵ},for some δ_y, δ_u, δ_x,δ_ϵ>0, and implicitly suppose that δ_ϵ<<δ_x so that the singular points x=0,ϵ are both well inside . Let η>0 bean arbitrarily small constant, and let δ_x>>δ_ϵ>0 be radii of small discs at 0 in the x-and ϵ-space. Let λ(x,ϵ)=χ(0,x,ϵ), and let_±:={ |ϵ|<δ_ϵ, |(±ϵλ(0,0))|<π-2η},be two sectors in the ϵ-space. For ϵ∈_± define a domain _±(ϵ)in the x-space as a simply connected ramified domain spanned by the complete real trajectories of the vector fields e^iω_±·x(x-ϵ)λ(x,ϵ)∂_xthat never leave the disc of radius δ_x, where the phase ω_± varies continuously in the interval{[ max{0,(±ϵ/λ(0,0))}-π/2+η<ω_±<min{0,(±ϵ/λ(0,0))}+π/2-η,for ϵ≠ 0,;|ω_±|<π/2-η,for ϵ=0. ].The constraints (<ref>) on the variation of ω_± are such that the real dynamics of the vector field (<ref>) and the asymptotic behavior of the solutions (<ref>) would not change drastically depending on ω_±. Namely, for ϵ≠ 0:* The point x=ϵ is repulsive when |ϵ/λ(x,ϵ) -ω_±|<π/2 and attractive when|-ϵ/λ(x,ϵ) -ω_±|<π/2, and vice-versa for the point x=0. * The u_1-component of the solution (<ref>) tendsto 0 along a negative real trajectory of (<ref>)and to ∞ along a positive real trajectory for |ω_±|<π2, and vice-versa for the u_2-component. For ϵ=0, the domain _+(0)=_-(0) consists of a pair of overlapping sectoral domains ^(0), ^(0) of opening 2π-2η with a common point at x=0. See Figure <ref>. Denoting x_1,±(ϵ) the attractive equilibrium point of (<ref>) and x_2,±(ϵ) the repulsive one,x_1,+(ϵ)=x_2,-(ϵ)=0,x_1,-(ϵ)=x_2,+(ϵ)=ϵ,then u_i(x,ϵ)→∞, u_j(x,ϵ)→ 0, (j=3-i),whenx→ x_i,±(ϵ) along a real trajectory of (<ref>). Before giving a general theorem on sectoral normalization for the parametric family (<ref>), let us first state it for the limit system with ϵ=0 which has an irregular singularity of Poincaré rank 1 at x=0.The system (<ref>) with ϵ=0 can be brought to its formal normal form (<ref>)through a formal transversely symplectic change of coordinates (y,x)=(Ψ̂(u,x,0),x),Ψ̂(u,x,0)=∑_k≥ 0ψ^(k)(u)x^k,where ψ^(k)(u) are analytic in u on a fixed neighborhood of 0. This formal series is generally divergent, but it is Borel 1-summable, with a pair of Borel sums Ψ^(u,x,0) and Ψ^(u,x,0) defined respectively above the sectors x∈^(0), ^(0) of Definition <ref> (for some 0<η<π2 arbitrarily small and some δ_x>0 depending on η), and u∈. The fibered sectoral transformations(y,x)=(Ψ^∙(u,x,0), ∙=,, are transversely symplectic and bring the system (<ref>) with ϵ=0 to its formal normal form. The Theorem <ref> is originally due to Takano <cit.> for systems (<ref>) whose formal invariant is of the form χ(h,x)=λ^(0)+xχ^(1)(h). In the case of the irregular singularity of the fifth Painlevé equation it was proved earlier by Takano <cit.>.Some similar and closely related theorems are due to Shimomura <cit.>, Yoshida <cit.>, and recently by Bittmann <cit.>, which apply to doubly resonant systemsx^2dy/dx=F(y,x), with D_yF(0,0)=λ^(0)(0)([1; -1 ]) under a condition on positivity of ∂/∂ x D_yF(0,0). This condition is not satisfied for Hamiltonian systems (<ref>)but nevertheless allows to treat Painlevé equations. Let Z_H,ϵ(y,x) be a family of vector fields (<ref>) and let χ(h,x,ϵ) be their formal invariant.(i) There exists a formal transversely sympectic change of coordinates(y,x,ϵ)=(Ψ̂(u,x,ϵ),x,ϵ)written as a formal power seriesΨ̂(u,x,ϵ)=ψ^(0)(u,ϵ)+xψ^(1)(u,ϵ)+x(x-ϵ)∑_k,l≥ 0ψ̃^(kl)(u)x^kϵ^l,with ψ^(0)(u,ϵ), ψ^(1)(u,ϵ) analytic in (u,ϵ), and ψ̃^(kl)(u) analytic in u on a fixed neighborhood={|u_1|,|u_2|<δ_u} of 0, which brings Z_ϵ to its formal normal form (<ref>). The coefficients ψ̃^(kl) grow at most factorially in k+l:max_u∈ψ̃^(kl)(u)≤ L^k+l(k+l)!for some L>0.(ii) There exists a transversely symplectic fibered change of coordinates (y,x,ϵ)=(Ψ_±(u,x,ϵ),x,ϵ), with Ψ_±(u,0,ϵ)=ψ^(0)(u,ϵ) (<ref>), defined for x in the spiraling domain _±(ϵ), ϵ∈_±, of Definition <ref> (for some 0<η<π2 arbitrarily small and some δ_x,δ_ϵ>0 depending on η), and for u∈, which brings Z_ϵ to its formal normal form (<ref>). It is uniformly continuous on _±={(x,ϵ)| x∈_±(ϵ)}and analytic on its interior. When ϵ tends radially to 0 with ϵ=β, thenΨ_±(u,x,ϵ) converges to Ψ_±(u,x,0) uniformly on compact sets of the sub-domainslim_ϵ→ 0 ϵ=β_±(ϵ)⊆(0). Note that in our notation Ψ_±(u,x,0) consists of a pair of sectoral transformations Ψ^(u,x,0) and Ψ^(u,x,0); it is a functional cochain using the terminology of <cit.>. (iii) Let Ψ(u,x,ϵ) be an analytic extension of the function given by the convergent seriesΨ̃(u,x,ϵ)=∑_k,l≥ 0ψ̃^(kl)(u)/(k+l)!x^kϵ^l, ψ̃^(kl) as in (<ref>). For each point (x,ϵ), for which there isθ∈ ]-π/2,π/2[ such that ⊆_±, with 𝐒_θ⊂ denoting the circlethrough the points 0 and 1 withcenter on e^iθ^+, we can express Ψ_±(u,x,ϵ) through thefollowing Laplace transform of Ψ̃:Ψ_±(u,x,ϵ)=ψ^(0)(u,ϵ)+xψ^(1)(u,ϵ)+x(x-ϵ)∫_0^+∞ e^iθΨ(u,sx,sϵ) e^-s ds.In particular, Ψ_±(u,x,0) is the pair of sectoral Borel sums Ψ^(u,x,0), Ψ^(u,x,0) of the formal series Ψ̂(u,x,0).As a consequence, Ψ_± and Ψ̂ satisfy the same (∂_u,∂_x,∂_ϵ)-differential relations with meromorphic coefficients. The proof will be given in Section <ref>. The transformations Ψ_± and Ψ̂ are unique up to left composition with an analytic symmetry of the model system, see Corollary <ref>.The system (<ref>) possesses:(i) a formal first integral given by h∘Ψ̂^∘(-1)(y,x,ϵ), where Ψ̂ as above and h(u)=u_1u_2 is a first integral of the model system,(ii) an actual first integral given by h∘Ψ_±^∘(-1)(y,x,ϵ) that is bounded and analytic on the domain _±. The solution y=Ψ_±(0,x,ϵ) is called ramified center manifold. It is the unique solution that is bounded on _±(ϵ) (cf. <cit.>).In the variable x=ϵ z, the system (<ref>) takes the form of a singularly perturbed systemϵ z(z-1)dydz=J(D_yH)(y,ϵ z,ϵ).The domains z∈1/ϵ_±(ϵ), ϵ∈_±, then correspond to the Stokes domains in the sense of exact WKB analysis <cit.>, where the Stokes curves would be the real separatrices of the point z=∞ of the vector field (<ref>) e^i(ω_±+ϵ)z(z-1)/λ(0,0)∂_z with a fixed phase ω_± (<ref>). § STOKES OPERATORS AND ACCUMULATION OF MONODROMYWe will define several operators acting as transversely symplectic fibered isotropies on the three following foliations given by three different vector fields:* Foliation in the (u,x)-space given by the model vector field Z_G,ϵ (<ref>).* Foliation in the (c,x)-space, c being the constant of initial condition in (<ref>), given by the rectified vector field Z_0,ϵ=x(x-ϵ)∂_x.Note that a fibered isotropy of Z_0,ϵ is necessarily independent of x; it acts on the c-space of initial conditions only.* Foliation in the (y,x)-space given by the original vector field Z_H,ϵ (<ref>). §.§ Symmetries of the model system: exponential torus A vertical infinitesimal symplectic symmetry (shortly infinitesimal symmetry) of the normal form vector field Z_G,ϵ (<ref>) is a germ of vector field ξ in the (u,x)-space that preserves: (i) the x-coordinate:L_ξx=ξ· x=0,(ii) the symplectic form ω=du_1∧ du_2: L_ξω=0,(iii) the vector field Z_G,ϵ: L_ξZ_G,ϵ=[ξ,Z_G,ϵ]=0. A vector field ξ is an infinitesimal symmetry of Z_G,ϵ if and only if ξ=X_f,x,ϵ is a Hamiltonian vector field with respect to ω for a first integral f(u,x,ϵ) of Z_G,ϵ: Z_G,ϵ· f=0.The conditions (i) and (ii) say that ξ=a_1(u,x,ϵ)∂_u_1+a_2(u,x,ϵ)∂_u_2 with∂ a_1/∂ u_1+∂ a_2/∂ u_2=0, i.e. a_1=∂ f/∂ u_2, a_2=-∂ f/∂ u_1 for some f, and ξ=∂ f/∂ u_2∂_u_1-∂ f/∂ u_1∂_u_2=X_f.The condition (iii) says that0=Z_G,ϵ· D_uf-X_f,x,ϵ· D_uG=D_u(Z_G,ϵ· f).Up to a translation f(u,x,ϵ)↦ f(u,x,ϵ)-f(0,x,ϵ) which does not affect X_f,x,ϵ, this condition is equivalent to Z_G,ϵ· f=0. The vector field Z_G,ϵ has the following obvious first integrals (cf. (<ref>)):c_1(u,x,ϵ)=u_1· E_χ(h,x,ϵ)^-1, c_2(u,x,ϵ)=u_2· E_χ(h,x,ϵ),andh(u)=u_1u_2=c_1c_2,where E_χ(h,x,ϵ) is as in (<ref>). Clearly, any function of c=(c_1,c_2)is again a first integral, and since c defines local coordinates on the space of leaves (space of initial conditions), the converse is also true.Note that the map c:u↦ c(u,x,ϵ) conjugates the vector field Z_G,ϵ to the “rectified” vector field Z_0,ϵ=x(x-ϵ)∂_x in the (c,x)-space:Z_G,ϵ=c^*(x(x-ϵ)∂_x).It turns out that analytic first integrals are functions of h=c_1c_2 only.If f(u,x,ϵ) is an analytic (resp. meromorphic) first integral of Z_G,ϵ on some neighborhood ×× of 0, then f=F(u_1u_2,ϵ) with F analytic (resp. meromorphic). On one hand, c_1(u,x,ϵ), c_2(u,x,ϵ) are local coordinate on the space of leaves, hence any first integral is a function of them (depending on ϵ). On the other hand, any analytic germ f(u,x,ϵ) is uniquely decomposed asf=f_0(h,x,ϵ)+u_1f_1(u_1,h,x,ϵ)+u_2f_2(u_2,h,x,ϵ), with f_l analytic. Writing u_i=c_i(u,x,ϵ)· E_χ^-(-1)^i(h,x,ϵ) (<ref>), we see that for f to be bounded when x→ x_i,±, we must have f_i=0, i=1,2. Therefore f=f_0 which must then be independent of x.A meromorphic function is a quotient of analytic ones. The statement remains true also if restricted to ϵ=0, or a generic fixed ϵ (such that λ^(0)(ϵ)ϵ∉). The Lie algebra of analytic infinitesimal symmetries of Z_G,ϵconsists of Hamiltonian vector fields ξ=a(u_1u_2,ϵ)(u_1∂_u_1-u_2∂_u_2)=a(h,ϵ)X_h,a(h,ϵ) analytic,and is commutative. It is also called the infinitesimal torus. The time-1 flow map of a vector field (<ref>) is given byu↦ T_a(u):=Φ^1_aX_h(u)= ([e^a(h,ϵ) 0; 0 e^-a(h,ϵ) ]) u.A (transversely symplectic fibered)isotropy of the model vector field Z_G,ϵ is a germ of symplectic transformation (u,x,ϵ)↦(ϕ(u,x,ϵ),x,ϵ) analytic in u∈, such that ϕ^*Z_G,ϵ=Z_G,ϵ. An isotropy that is analytic in x on a full neighborhoodof both singularities will be called a symmetry. For ϵ∈_±{0} define the left and right intersection sectors_i,±^∩(ϵ)={x∈_±(ϵ): x_i,±+e^2π i(x-x_i,±)∈_±(ϵ)}.and for ϵ=0 let _i±^∩(0) be their limits. They are the domains of self-intersection of _±(ϵ) attached to the points x_i,±(ϵ) (<ref>). Let ϕ_i,±(u,x,ϵ) be a sectoral isotropy of the normal form vector field Z_G,ϵ, analytic and bounded for x∈_i,±^∩(ϵ), u∈. Thenc_i∘ϕ_i,±=c_i· e^f_i(h,c_i,ϵ),and c_j∘ϕ_i,±=c_j· e^f_j(h,ϵ)+g_j(h,c_i,ϵ),for some analytic germs f_i,f_j,g_j.The isotropy ϕ_i,±(u,x,ϵ) is analytic in u on some neighborhood of u=0 and bounded when x→ x_i,±.In particular, the restriction of c∘ϕ_i,± to any fiber {x=cst≠ 0,ϵ} is analytic in u, and therefore, since c∘ϕ_i,±(c,ϵ) isindependent of x, it is an analytic function of c on some neighborhood of c=0.We have c_k=u_k E_χ(h,x,ϵ)^(-1)^k, k=1,2, with E_χ given by (<ref>), and lim_x→ x_i,±x∈_±(ϵ) E_χ(h,x,ϵ)^(-1)^i=0. Writing ϕ_i,±=(ϕ_1,i,±,ϕ_2,i,±), its k-th component is given by ϕ_k,i,±=(c_k∘ϕ_i,±)·(E^-(-1)^k∘ h∘ϕ S_i,±),and we see that the expansion of c_i∘ϕ_i,± in powers of c can contain only terms c_i^n_ic_j^n_j=u_i^n_iu_j^n_jE_χ(h,x,ϵ)^(-1)^i(n_i-n_j) with n_i≥ n_j+1,while the expansion of c_j∘ϕ_i,± in powers of c can contain only terms c_i^n_ic_j^n_j=u_i^n_iu_j^n_jE_χ(h,x,ϵ)^(-1)^i(n_i-n_j) with n_i≥ n_j-1. Since ϕ_i,±must be invertible D_c(c∘ϕ_i,±)_c=0≠ 0 from which it follows that c∘ϕ_i,± is of the form (<ref>). Note that the hypersurface {u_i=0}={c_i=0} consists of all leaves of Z_G,ϵ that are bounded when x→ x_i,± inside X_±(ϵ), and ϕ_i,± must preserve it.An isotropy of the normal form vector field Z_G,ϵ that is bounded and analytic on ×_±(ϵ) is a symmetry. It is given by a time-1 flow of some vector field (<ref>).An isotropy ϕ(u,x,ϵ) of the model systembounded and analytic on ×_± is in particular bounded and analytic on ×_i,±^∩(ϵ) for each ϵ∈_±, and therefore by Lemma <ref>, it is such thatc∘ϕ=(c_1e^f_1(h,ϵ),c_2e^f_2(h,ϵ)),i.e.ϕ(u,x,ϵ)=(u_1e^f_1(h,ϵ),u_2e^f_2(h,ϵ))for some analytic germs f_1,f_2. The transverse symplecticity condition is then rewritten asddh(h∘ϕ)=ddh(he^f_1(h,ϵ)+f_2(h,ϵ))=1,which implies that e^f_1(h,ϵ)+f_2(h,ϵ)=1, i.e. ϕ(u)=([e^f_1(h,ϵ) 0; 0 e^-f_1(h,ϵ) ]) u. The Lie group of (transversely symplectic fibered) symmetries (<ref>) of Z_G,ϵ is commutative and connected. It is called the exponential torus. A characterization of the Lie group of symmetries of a general system (<ref>) will be given in Proposition <ref>.The normalizing transformations Ψ̂ and Ψ_± of Theorem <ref> are unique modulo composition with elements of the exponential torus (i.e. flow maps of infinitesimal symmetries analytic in ϵ). They are uniquely determined by the analytic germ ψ^(0)(u,ϵ)=Ψ_+(u,0,ϵ)=Ψ_-(u,0,ϵ), cf. (<ref>), (<ref>).§.§ Canonical general solutionsThe model system has a canonical general solution u(x,ϵ;c) (<ref>), depending on an “initial condition” parameter c∈^2, uniquely determined by a choice of a branch of the function E_χ(h,x,ϵ) (<ref>). Correspondingly, y(x,ϵ;c)=Ψ_±(u(x,ϵ;c),x,ϵ) is a germ of general solution of the original system on ×_±(ϵ). In order for this solution to have a continuous limit when ϵ→ 0, one has to split the domain _±(ϵ) in two parts, corresponding to the two parts of _±(0), by making a cut in between the singular points x_1,±, x_2,± along a trajectory of (<ref>) through the mid-point ϵ2 (see Figure <ref>). Let us denote _±^(ϵ) the upper and_±^(ϵ) the lower part (with respect to the oriented line λ^(0)(0)) of the cut domain _±(ϵ)= _±^(ϵ)∪_±^(ϵ).The two parts of _±(ϵ) intersect in the left and right intersection sectors _i,±^∩(ϵ) (Definition <ref>) attached to {x_1,±, i=1,2, and for ϵ≠ 0 also in a central part along the cut. Now take two branches E_χ^(h,x,ϵ) and E_χ^(h,x,ϵ) of E_χ(h,x,ϵ) on the two parts of the domain, that agree on the rightintersection sector _2,±^∩, and have a limit when ϵ→ 0.Correspondingly they determine a pair of general solutions of the model systemu^∙(x,ϵ;c),∙=,, and a pair of canonical general solutions of the original systemy_±^∙(x,ϵ;c):=Ψ_±(u^∙(x,ϵ;c),x,ϵ),∙=,. Since the transformation Ψ_± is unique only modulo right composition with an exponential torus actionT_a(u,ϵ) (<ref>), which acts onu^∙(x,ϵ;c) asT_a(·,ϵ)∘ u^∙(x,ϵ;c)=u^∙(x,ϵ;·)∘ T_a(c,ϵ),the solutions y_±^∙ are determined only up to the same right action of T_a(c,ϵ). §.§ Formal monodromyThe formal monodromy operators are induced by monodromy acting on the solutions u^∙(x,ϵ;c), ∙=,, of the model system.For ϵ≠ 0 the induced action of formal monodromies along simple counterclockwise loops around each singular point x_i,±=0,ϵ on the 3 foliations is given by: * Monodromy operators of the model system N_x_i,±(·,x,ϵ)∘ u(x,ϵ;c)= u(e^2π i(x-x_i,±)+x_i,±,ϵ;c),acting on the foliation of the normal form vector field Z_G,ϵ commutatively by N_0: u ↦exp(-2π iχ^0(h,ϵ)ϵ([10;0 -1 ]))· u= T_-2π i/ϵχ_x=0(u),N_ϵ: u ↦exp(2π i[χ^0(h,ϵ)ϵ+χ^(1)(h,ϵ)] ([10;0 -1 ]))· u= T_2π i/ϵχ_x=ϵ(u).The total monodromy of the model systemis given byN= N_0∘ N_ϵ= N_ϵ∘ N_0:u↦exp(2π iχ^(1)(u_1u_2,ϵ) ([10;0 -1 ]))· u.* Formal monodromy operators N_x_i,±(·,ϵ)∘ c(u,x,ϵ)=c(·,x,ϵ)∘ N_i,±(u,x,ϵ),acting on the space of initial conditions c commutatively byN_0: c ↦exp(-2π iχ^0(h,ϵ)ϵ([10;0 -1 ]))· c= T_-2π i/ϵχ_x=0(c),N_ϵ: c ↦exp(2π i[χ^0(h,ϵ)ϵ+χ^(1)(h,ϵ)] ([10;0 -1 ]))· c= T_2π i/ϵχ_x=ϵ(c),and a formal total monodromyN=N_0∘ N_ϵ= N_ϵ∘ N_0:c↦exp(2π iχ^(1)(h,ϵ) ([10;0 -1 ]))· c.* Formal monodromy operators N_i,±(y,x,ϵ) acting on the foliation of the original vector field Z_H,ϵ:N_i,±(·,x,ϵ)∘Ψ_±(u,x,ϵ)=Ψ_±(·,x,ϵ)∘ N_i,±(u,x,ϵ),andN_±(·,x,ϵ)= N_1,±(·,x,ϵ)∘ N_2,±(·,x,ϵ)= N_2,±(·,x,ϵ)∘ N_1,±(·,x,ϵ).The canonical solutions u_±^, u_±^ of the model system on the domains _±^, _±^, are defined such that they agreeon the right intersection sector _2,±^∩. Therefore on the left intersection sector they are connected by the total formal monodromy operator u_±^(x,ϵ;c)= N(·,x,ϵ)∘ u_±^(x,ϵ;c)=u_±^(x,ϵ;·) ∘ N(c,ϵ), x∈_1,±^∩,and by the formal monodromy N_x_i,± on the central cut between the two domains for ϵ≠ 0 (cf. Figure <ref>). §.§ Stokes operators and sectoral isotropiesLet y=Ψ_±(u,x,ϵ) be the normalizing transformation on _±(ϵ). We call Stokes operators the operators that change the determination of Ψ_± over the left or right intersection sectors. If x∈_i,±^∩(ϵ), then for ϵ≠ 0 we denotex̅= e^2π i(x-x_i,±)+x_i,±the corresponding point in _±(ϵ) on the other sheet, and extend this notation by limit to ϵ=0.Namely,if x∈_1,±^∩(ϵ)⊂_±^(ϵ), then x̅∈_±^(ϵ),if x∈_2,±^∩(ϵ)⊂_±^(ϵ), then x̅∈_±^(ϵ).Then the Stokes operators are the operatorsΨ_±(u,x,ϵ)↦Ψ_±(u,x̅,ϵ), x∈_i,±^∩(ϵ),which for ϵ=0 are the Stokes operators in the usual sense that send the Borel sum of the formal x-series Ψ̂(u,x,0) in one non-singular direction to the Borel sum in a following non-singular direction.To each of these Stokes operators we associate sectoral isotropies of the 3 foliations.* Sectoral isotropies S_i,±(u,x,ϵ) of the normal form vector field Z_G,ϵ:Ψ_±(·,x,ϵ)∘ S_i,±(u,x,ϵ) =Ψ_±(u,x̅,ϵ), x∈_i,±^∩(ϵ).The pair ( S_1,±, S_2,±) is an analog of the Martinet-Ramis invariant of saddle-node singularity <cit.>.* Sectoral isotropies S_1,±(c,ϵ) and S_2,±(c,ϵ) of the rectified vector field Z_0,ϵ=x(x-ϵ)∂_x in the c-space:u^(x,ϵ;·)∘ S_1,±(c,ϵ) = S_1,±(·,x,ϵ)∘ u^(x,ϵ;c),x∈_1,±^∩(ϵ),u^(x,ϵ;·)∘ S_2,±(c,ϵ) = S_2,±(·,x,ϵ)∘ u^(x,ϵ;c),x∈_2,±^∩(ϵ).* Sectoral isotropies S_i,±(y,x,ϵ) of the original vector field Z_H,ϵ:S_i,±(·,x,ϵ)∘Ψ_±(u,x,ϵ)=Ψ_±(u,x̅,ϵ), x∈_i,±^∩(ϵ). Let S_i,±(c,ϵ)=(S_1,i,±(c,ϵ),S_2,i,±(c,ϵ)) be a Stokes sectoral isotropy (<ref>). Then* S_i,i,±(c,ϵ)=c_i+c_i^2·σ_i,i,±(h,c_i,ϵ) for an analytic germ σ_i,i,±,* S_j,i,±(c,ϵ)=c_j+σ_j,i,±(h,c_i,ϵ) for an analytic germ σ_j,i,±, j=3-i,subject to a condition D_c(S_i,±)=1.The term σ_j,i,±(0,0) is responsible for the ramification of the ramified center manifold y=Ψ_±(0,x,ϵ) of the original vector field Z_ϵ at the sector _i,±^∩(ϵ). The isotropy S_i,±(u,x,ϵ) is analytic in u on some neighborhood of u=0 and bounded in x with lim_x→ x_i,± S_i,±(u,x,ϵ)=u.By Lemma <ref>, c_i∘ S_i,±=c_i· e^f_i(h,c_i,ϵ),andc_j∘ S_i,±=c_j· e^f_j(h,ϵ)+g_j(h,c_i,ϵ)where f_i,f_j,g_j are some analytic functions of (h,c_i,ϵ).Knowing that lim_x→ x_i,± h∘ S_i,±(u,x,ϵ)=h,h∘ S_i,±=h+c_i·(…), where (…) is an analytic function of (h,c_i,ϵ), which implies that lim_x→ x_i,±E_χ^-1· (E_χ∘ h∘ S_i,±)=1.Writing S_i,±=( S_1,i,±, S_2,i,±), its k-th component is S_k,i,±=(c_k∘ S_i,±)·(E^-(-1)^k∘ h∘ S_i,±).We conclude that f_i=c_i·(…) and f_j=0. §.§ Symmetry group of the system The group of (analytic transversely symplectic fibered) symmetries of a system (<ref>) is either* isomorphic to the exponential torus: this happens if and only if the system is analyticallyequivalent to the model (<ref>), or* isomorphic to a finite cyclic group.If the symmetry group is non-trivial, then the system has an analytic center manifold (bounded analytic solution on a neighborhood of both singular points).If Φ(y,x,ϵ) is a symmetry of the system (<ref>), thenΦ(·,x,ϵ)∘Ψ_±(u,x,ϵ)=Ψ_±(·,x,ϵ)∘ϕ(u,ϵ) for some germ ϕ:u↦([e^a(h,ϵ); e^-a(h,ϵ) ]),from the exponential torus, and the analyticity of Φ means that this ϕ must commute with the Stokes operators S_i,± (<ref>) (note that ϕ acts the same way on c as on u). Using their characterization in Proposition <ref>, this means thatσ_i1,±(h,c_1)=e^aσ_i1,±(h,e^a c_1),σ_i2,±(h,c_2)=e^-aσ_i2,±(h,e^-a c_2), i=1,2.This can be satisfied only if* either σ_ij,±(h,c_j)=0 for all i,j, i.e. if S_1,±=𝕀, S_2,±=𝕀 and the system is analytically equivalent to its formal normal form,* or there is k∈ such that c_jσ_ij,±(h,c_j)=0 contains only powers of c_j^k for all i,j, and e^ka=1, i.e. a∈2π i/k.§.§ Analytic classification The collection (χ,{ S_1,+, S_2,+, S_1,-, S_2,-}) is called an analytic invariant of a system (<ref>). Two analytic invariants (χ,{ S_i,±}), (χ̃,{S̃_i,±}) are equivalent if* either χ=χ̃and there is an elementϕ(u,ϵ) of the exponential torus, analytic in ϵ, such that:S_i,±=ϕ∘S̃_i,±∘ϕ^∘(-1), i=1,2,* or χ(h,x,ϵ)=-χ̃(-h,x,ϵ)and there is an elementϕ(u,ϵ) of the exponential torus, analytic in ϵ, such that: S_i,±=Jϕ∘S̃_j,∓∘ (Jϕ)^∘(-1), i=1,2, j=3-i, where J:(u_1,u_2)↦ (u_2,-u_1). Note that the definition of _±, _± and x_i,± depends on λ(x,ϵ)=χ(0,x,ϵ), therefore the relation λ̃=-λ entails the renaming_±=_∓,_±^∙=_∓^∙,x̃_i,±=x_j,∓. By the construction, an analytic invariant of a system (<ref>) is uniquely defined up to the equivalence.Two systems (<ref>) are analytically equivalent (in the sense of Definition <ref>) if and only if their analytic invariants are equivalent.If y=Φ(ỹ,x,ϵ) is an analytic transformation from one system to another, then the sectoral normalizations y=Ψ_±(u,x,ϵ) andỹ=Ψ̃_±(u,x,ϵ)=Φ∘Ψ_± provide the same analytic invariant.Conversely, if the analytic invariants are equivalent, then up to modifying one of the normalizing transformation, one can suppose that they are in fact equal, in which case Φ_±=Ψ̃_±∘Ψ_±^∘(-1) are analytic transformations between the systems on _+ and _-. In fact Φ_+=Φ_- is an analytic on the whole ϵ-neighborhood .Indeed, the composition Φ_+∘Φ_-^∘-1 is a symmetry of the second system on the intersection _+∩_-, and as such it is determined by its value at x=0; but since Ψ̃_+_x=0=Ψ̃_-_x=0=ψ̃^(0)(u,ϵ) and Ψ_+_x=0=Ψ_-_x=0=ψ^(0)(u,ϵ) are analytic in ϵ (<ref>), this means that Φ_+∘Φ_-^∘-1_x=0=𝕀 and therefore Φ_+∘Φ_-^∘-1=𝕀.§.§ Decomposition of monodromy operators For ϵ≠ 0, let x_0∈_±(ϵ){0,ϵ} be a base-point, and let two counterclockwise simple loops around the singular points x_i,±, i=1,2, be as in Figure <ref>. Correspondingly, we have two monodromy operators M_x_i,± acting on the foliation by the solutions of the original system (<ref>) by analytic continuationalong the loops. Since the monodromy operators M_x_i,± act on the foliation, theyare independent of the choice of the two-parameter general solution on which they act on the left(a different general solution is related to it by a change of the parameter, independent of x and acting on the right). In particularM_x_1,+= M_x_2,-, M_x_2,+= M_x_1,-.For ϵ≠ 0, the monodromy operators M_x_i,± of the original foliation are well defined on some open neighborhood of the ramified center manifold y=Ψ_±(0,x,ϵ) in ×_±(ϵ). Their (left) action is given byM_x_i,±= S_i,±∘ N_i,±,where S_i,± are the Stokes operators (<ref>) andN_i,± are the formal monodromy operators (<ref>).HenceM_0 = S_1,+∘ N_1,+= S_2,-∘ N_2,-,M_ϵ = S_2,+∘ N_2,+= S_1,-∘ N_1,-.Their right action on analytic extension of the canonical general solutions y_±^∙ (<ref>) to the whole _±(ϵ) is given byM_x_i,±(y_±^∙(x,ϵ;c),x,ϵ)=y_±^∙(x,ϵ;·)∘ M_i,±^∙(c,ϵ),∙=,,whereM_1,±^ =N_x_2,±^∘(-1)∘ S_1,±∘ N, M_1,±^ =S_1,±∘ N_x_1,±,M_2,±^ =S_2,±∘ N_x_2,±,M_2,±^ =N_x_2,±∘ S_2,±,cf. Figure <ref>. The demonstration of the given formulas is straightforward, but we include it here for the sake of completeness.Let x̅:=x_i,±+e^2π i(x-x_i,±), then M_x_i,±(·,x,ϵ)∘ y_±(x,ϵ;c) =y_±(x̅,ϵ;c)=Ψ_±(·,x̅,ϵ)∘ u(x̅,ϵ;c)=( S_i,±∘Ψ_±)(·,x,ϵ)∘ u(x̅,ϵ;c)=( S_i,±∘Ψ_±∘ N_i,±)(·,x,ϵ)∘ u(x,ϵ;c)=( S_i,±∘ N_i,±∘Ψ_±)(·,x,ϵ)∘ u(x,ϵ;c)=( S_i,±∘ N_i,±∘ y_±)(x,ϵ;c),using (<ref>), (<ref>), (<ref>). Similarly, to calculate M_1,±^ for example,y_±^(x̅,ϵ;c) =Ψ_±(·,x̅,ϵ)∘ u^(x̅,ϵ;c)=Ψ_±(·,x̅,ϵ)∘ u^(x̅,ϵ;·)∘ N_x_2,±(c,ϵ)=Ψ_±(·,x̅,ϵ)∘ u^(x,ϵ;·)∘ N_x_1,±∘ N_x_2,±(c,ϵ)=Ψ_±(·,x,ϵ)∘ u^(x,ϵ;·)∘ S_1,±∘ N_x_1,±∘ N_x_2,±(c,ϵ)=Ψ_±(·,x,ϵ)∘ u^(x,ϵ;·) ∘ N_x_2,±^∘(-1)∘ S_1,±∘ N_x_1,±∘ N_x_2,±(c,ϵ)=y_±^(x,ϵ;·)∘ N_x_2,±^∘(-1)∘ S_1,±∘ N(c,ϵ),see Figure <ref>. The general solutions y_±^, y_±^ are related byy_±^(x,ϵ;c)=y_±^(x,ϵ;·)∘ N_x_2,±(c,ϵ).Note that in general, a composition of the two monodromies may not be defined if the image of the first does not intersect the domain of definition the second.§.§ Accumulation of monodromy * For ϵ≠ 0, the pseudogroup generated by the monodromy operators ⟨ M_0(·,x,ϵ),M_ϵ(·,x,ϵ)⟩ is called the (local) monodromy pseudogroup. The pseudogroup generated by the corresponding action on the initial condition c⟨ M_1,±^∙(·,ϵ), M_2,±^∙(·,ϵ) ⟩ is its representation with respect to the general solution y_±^∙(x,ϵ;c).* For ϵ= 0, the pseudogroup generated by the Stokes operators and by theelements of the exponential torus (pushed-forward by the sectoral transformations Ψ^∙):⟨ S_1,±(·,x,0),S_2,±(·,x,0), { T_a^∙(·,x,0)}_a ⟩, where T_a^∙(·,x,0)=Ψ_±^∙(·,x,0)_*T_a=Φ^1_(Ψ_±^∙)_*(a(h)X_h), is called the (local) wild monodromy pseudogroup. The pseudogroup generated by the corresponding action on the initial condition c⟨ S_1,±(·,0), S_2,±(·,0), { T_a(·)}_a⟩,is its representation with respect to the formal transseries solution ŷ(x,0;c)=Ψ̂(u(x,0;c),x,0). Note that the pseudogroup (<ref>) is independent of the freedom of choice of the sectoral normalizations Ψ_±^∙ of Theorem <ref>.One of the main goals of this paper is to understand the relation between the monodromy pseudogroup for ϵ≠ 0 and the wild monodromy pseudogroup for ϵ=0. Suppose that the formal invariant χ(h,x,ϵ) is such thatχ(h,x,0)=λ^(0)(0)+xχ^(1)(h,0),and thereforeχ(h,x,ϵ)=λ^(0)(0)+ϵ∂χ^(0)∂ϵ(h,0)+xχ^(1)(h,0)+O(xϵ)+O(ϵ^2).Let {ϵ_n}_n∈± be sequence in _±{0} defined byλ^(0)(0)ϵ_n=λ^(0)(0)ϵ_0+n,ϵ_0∈_±{0}along which the exponential factor e^2π iλ^(0)/ϵ in the formal monodromy (<ref>) stays constant, and denoteκ:=e^2π iλ^(0)/ϵ_0.Then the formal monodromy operators N_0(u,x,ϵ), N_ϵ(u,x,ϵ), resp. N_0(u,x,ϵ), N_ϵ(u,x,ϵ), converge along each such sequence to a symmetry of the model system (element of the exponential torus) -6ptN_0(κ;u,x):= lim_n→±∞ N_0(u,x,ϵ_n) : c↦([ κ^-1e^-2π i∂χ^(0)/∂ϵ(h,0) 0; 0κ e^2π i∂χ^(0)/∂ϵ(h,0) ]) c,-6ptN_ϵ(κ;u,x):= lim_n→±∞ N_ϵ(u,x,ϵ_n) : c↦([ κ e^2π i[∂χ^(0)/∂ϵ(h,0)+χ^(1)(h,0)]-15pt0;0 -15ptκ^-1 e^-2π i[∂χ^(0)/∂ϵ(h,0)+χ^(1)(h,0)] ]) c,-6ptκ∈^*.This implies that also the monodromy operators M_i,±^·(c,ϵ), resp. M_x_i,±(y,x,ϵ), converge along such sequences{ϵ_n}_n∈±⊂_±{0}. Denote N_i,±^∙(κ;y,x):= lim_n→±∞ N_i,±(y,x,ϵ_n), x∈_±^∙,∙=,. Suppose that the formal invariant of the form (<ref>). Then the monodromy operators of the system (<ref>) for ϵ≠ 0 accumulate along the sequences {ϵ_n}_n∈±(<ref>) to a 1-parameter family of wild monodromy operatorsM_1,±(κ;y,x):=lim_n→±∞ M_x_1,±(y,x,ϵ_n) = S_1,±(·,x,0)∘Ñ_x_1,±^(κ;y,x),x∈^∩_1,±(0),M_2,±(κ;y,x):=lim_n→±∞ M_x_2,±(y,x,ϵ_n) = S_2,±(·,x,0)∘Ñ_x_2,±^(κ;y,x),x∈^∩_2,±(0).In particular, if we replace κ by e^-2π i[∂χ^(0)/∂ϵ(h,0)+δ_i,±(0)χ^(1)(h,0)], δ_i,±(ϵ)=x_i,±(ϵ)/ϵ, so that Ñ_x_i,±^∙(κ;y,x) becomes an identity, we obtain the Stokes operatorsS_i,±(y,x,0)= M_i,±(e^-2π i[∂χ^(0)/∂ϵ(h,0)+δ_i,±(0)χ^(1)(h,0)];y,x).The vector fieldẏ=±(-1)^i(κ∂∂κM̃_i,±(κ;y,x)^∘(-1))∘M̃_i,±(κ;y,x)equals to the push-forward Ψ_±^∙(·,x,0)_* (X_h) of the vector field X_h=u_1∂_u_1-u_2∂_u_2, where ∙= if i=1 and ∙= if i=2,which “generates” the commutative Lie algebra of bounded infinitesimal symmetries on the sector ^∙(0). The knowledge of the limits M_i,±(κ;y,x), κ∈^*, allows to recover the infinitesimal symmetry Ψ_±^∙(·,x,0)_*(X_h) (<ref>), and hence its Hamiltonian, the bounded first integral Ψ_±^∙(·,x,0)_*(h) which vanishes at the singular points (y_0(x_i,±,ϵ),x_i,±), and therefore,knowing the formal invariant χ, also the formal monodromy operators Ñ_x_1,±^∙(κ;y,x), and finally the Stokes isotropies S_i,±(y,x,0).In the case when the assumption (<ref>) is not met, one can nevertheless get a similar “accumulation” result by replacing in (<ref>)λ^(0)(0) by χ(h,0,0) and defining κ(h)=e^2π iχ(h,0,0)/ϵ_0(h) (<ref>). § CONFLUENCE IN 2×2 TRACELESS LINEAR SYSTEMS AND THEIR DIFFERENTIAL GALOIS GROUP To illustrate the matter of the previous section, let us consider a confluence of two regular singular points to a non-resonant irregular singular point in a family of linear systemsx(x-ϵ)dy/dx=A(x,ϵ)y, y∈^2,where A is a 2× 2 traceless complex matrix depending analytically on (x,ϵ)∈(×,0), such thatA(0,0)≠ 0 has two distinct eigenvalues ±λ^(0)(0).The Theorem <ref> in this case can be found in the thesis of Parise <cit.> and in the work of Lambert and Rousseau <cit.> (see also <cit.>). It provides us with a canonical fundamental solution matricesY_±^∙(x,ϵ)=Ψ_±(x,ϵ)· U_±^∙(x,ϵ),∙=,,where the transformation matrix Ψ_±(x,ϵ) is bounded on _±, andU_±^∙(x,ϵ)=([E_λ^∙(x,ϵ) 0; 0 E_λ^∙(x,ϵ)^-1 ]),is a solution to the diagonal model systemx(x-ϵ)dy/dx=([λ(x,ϵ); -λ(x,ϵ) ])y.The solution basis Y_±^∙(x,ϵ) is also called a mixed basis: the first (resp. second) column spans the subspace of solutions that asymptotically vanish when x→ x_1,±(ϵ) (resp. when x→ x_2,±(ϵ)), and it is an eigensolution with respect to the corresponding monodromy operator M_x_1,± (resp. M_x_2,±) associated to its eigenvalue e^±2π iλ(x_1,±,ϵ)/ϵ (resp. e^±2π iλ(x_2,±,ϵ)/ϵ).A general solution is a linear combinationy_±^∙(x,ϵ;c)=Y_±^∙(x,ϵ)· c,∙=,. Letbe the field of meromorphic functions of the variable x on a fixed small neighborhood of 0, equipped with the differentiation d/dx. For a fixed small ϵ, the local differential Galois group (also called the Picard-Vessiot group) of the system (<ref>) is the group of -automorphisms of the differential field ⟨ Y(·,ϵ)⟩, generated by the components of any fundamental matrix solution Y(x,ϵ). The differential Galois group acts on the foliation associated to the system by left multiplication. Fixing a fundamental solution matrix Y=Y_±^∙, then each automorphism is represented by a right multiplication of Y_±^∙ by a constant invertible matrix, hence the differential Galois group is represented by an (algebraic) subgroup of _2() acting on the right.It is well known <cit.> that the differential Galois group is the Zariski closure of 12pt 0pt ϵ≠ 0: the monodromy group generated by the two monodromy operators around the singular points 0 and ϵ, ϵ= 0: the wild monodromy group[The name “wild monodromy”is borrowed from <cit.>.]generated by the Stokes operators and the linear exponential torus [For general linear systems one would need to add also the total formal monodromy N(0)=T_2π iλ^(1)(0), which in our case already belongs to the exponential torus.]which acts on the fundamental solutions Y_±^∙ asT_a^∙:Y_±^∙(x,0)↦ Y_±^∙(x,0)· T_a,whereT_a=([e^a ;e^-a ]), a∈.The question is how are these two different descriptions related?The monodromy matrices of Y_±^, Y_±^, around the points x_1,±(ϵ), x_2,±(ϵ), ϵ∈_±{0}, are given respectively byM_1,±^ =N_x_2,±^(-1) S_1,± N,M_2,±^ =S_2,± N_x_2,±M_1,±^ =S_1,± N_x_1,±, M_2,±^ =N_x_2,± S_2,±,where S_i,± are of the formS_1,±=([ 1 0; s_1,± 1 ]), S_2,±=([ 1 s_2,±; 0 1 ]).In particular M_1,±^∙ is lower-triangular and M_2,±^∙ is upper-triangular. When ϵ→ 0 along a sequence[The idea of taking limits of monodromy along such sequences can be found in the works of J.-P. Ramis <cit.> or A. Duval <cit.>.]1/ϵ_n=1/ϵ_0+n/λ^(0), n∈±, ϵ_0∈_±{0}, these monodromy converge respectively to M̃_i,±^∙(κ)=lim_n→±∞ M_i,±^∙(ϵ_n) given byM̃_1,+^(κ) =Ñ_ϵ(κ)^-1S_1(0)N(0), M̃_1,+^(κ) =S_1(0)Ñ_0(κ), M̃_2,+^(κ) =S_2(0)Ñ_ϵ(κ), M̃_2,+^(κ) =Ñ_ϵ(κ)S_2(0), M̃_1,-^(κ) =Ñ_0(κ)^-1S_1(0)N(0), M̃_1,-^(κ) =S_1(0)Ñ_ϵ(κ), M̃_2,-^(κ) =S_2(0)Ñ_0(κ), M̃_2,-^(κ) =Ñ_0(κ)S_2(0),with Ñ_0(κ)=([ κ^-1 ; κ ])T_-2π i dλ^(0)/dϵ(0),Ñ_ϵ(κ)=([κ ;κ^-1 ])T_2π i [dλ^(0)/dϵ(0)+λ^(1)(0)], N(0)=T_2π iλ^(1)(0).We call them wild monodromy matrices.The family of them {M̃_1,±^∙(κ), M̃_2,±^∙(κ) | κ∈^*} generates the same group, the representation of the wild monodromy group with respect to the formal solution Ŷ(x,ϵ), as does the collection ofthe Stokes matrices and the linear exponential torus{S_1(0),S_2(0)}∪{T_a | a∈}.Hence we have the following theorem, whose general idea was suggested by J.-P. Ramis <cit.>: When ϵ→ 0 the elements of the monodromy group of the system (<ref>) accumulate to generators of the wild monodromy group of the limit system. § CONFLUENT DEGENERATION OF THE SIXTH PAINLEVÉ EQUATION TO THE FIFTHThe sixth Painlevé equation isP_VI: q” =1/2(1/q+1/q-1+1/q-t)(q')^2- (1/t+1/t-1+1/q-t)q' +q(q-1)(q-t)/2 t^2(t-1)^2[(ϑ_∞-1)^2-ϑ_0^2t/q^2+ϑ_1^2(t-1)/(q-1)^2+(1-ϑ_t^2)t(t-1)/(q-t)^2],where ϑ=(ϑ_0,ϑ_t,ϑ_1,ϑ_∞)∈^4 are complex constants. It isa reduction to the q-variable of a time dependent Hamiltonian system <cit.>dq/dt= ∂ H_VI(q,p,t)/∂ p,dp/dt=-∂ H_VI(q,p,t)/∂ q,with a polynomial Hamiltonian function H_VI=1t(t-1)[q(q-1)(q-t)p^2-(ϑ_0(q-1)(q-t)+ϑ_1 q(q-t)+(ϑ_t-1)q(q-1) )p+ (ϑ_0+ϑ_1+ϑ_t-1)^2-ϑ_∞^24(q-t)].It has three simple (regular) singular points on the Riemann sphere ^1 at t=0,1,∞.The fifth Painlevé equation P_V [The equation (<ref>) is the fifth equation of Painlevé with a parameter η_1=-1. A general form of this equation would be obtained by a further change of variable t̃↦-η_1 t̃. The degenerate case P_V^deg with η_1=0 which has only a regular singular point at ∞ is not considered here.] P_V: q”=(1/2q+1/q-1)(q')^2- 1/t̃q'+ (q-1)^2/2t̃^2((ϑ_∞-1)^2 q-ϑ_0^2/q)+(1+ϑ̃_1)q/t̃-q(q+1)/2(q-1),is obtained from P_VI as a limit ϵ→ 0 after the change of the independent variablet= 1+ϵt̃,ϑ_t= 1/ϵ,ϑ_1= -1/ϵ+ϑ̃_1+1,which sends the three singularities to t̃=-1/ϵ,0,∞. At the limit, the two simple singular points -1/ϵ and ∞ merge into a double (irregular) singularity at the infinity.The change of variables (<ref>), changes the function ϵ· H_VI toH_VI,ϵ=1t̃(1+ϵt̃)[q(q-1)(q-1-ϵt̃)p^2-(ϑ_0(q-1)(q-1-ϵt̃)+ϑ̃_1 q(q-1-ϵt̃)+t̃q-ϵt̃q )p + (ϑ_0+ϑ̃_1)^2-ϑ_∞^24(q-1-ϵt̃)],and the Hamiltonian system todq/dt̃= ∂ H_VI,ϵ(q,p,t̃)/∂ p,dp/dt̃=-∂ H_VI,ϵ(q,p,t̃)/∂ q,whose limit ϵ→ 0 is a Hamiltonian system of P_V. In the coordinate x=1/t̃+ϵ, the above system is written as x(x-ϵ)dq/dx= ∂ H(q,p,x,ϵ)/∂ p, x(x-ϵ)dp/dx=-∂ H(q,p,x,ϵ)/∂ q,withH(q,p,x,ϵ) = -(1+ϵt̃) H_VI,ϵ(q,p,t̃)=-(ϑ_0+ϑ̃_1)^2-ϑ_∞^24((x-ϵ)q-x)+xϑ_0p+(1-ϵ-(x-ϵ)ϑ_0-x(ϑ_0+ϑ̃_1))qp +(2x-ϵ)(qp)^2 +(x-ϵ)(θ_0+ϑ̃_1)q^2p-xqp^2-(x-ϵ)q^3p^2,and Theorem <ref> can be applied. The formal invariant χ of the system (<ref>) isχ(h,x,ϵ)=1-ϵ-(x-ϵ)ϑ_0-x(ϑ_0+ϑ̃_1)+2(2x-ϵ)h.Let q̃ =q-xA, A =ϑ_01+ϵ(1+ϑ_0+ϑ̃_1),p̃ =p+(x-ϵ)B, B =(ϑ_0+ϑ̃_1)^2-ϑ_∞^24(1+ϵ(1-ϑ_0)),and let H̃(q̃,p̃,x,ϵ)=H(q,p,x,ϵ)-(ϑ_0+ϑ̃_1)^2+ϑ_∞^24tx.Then for ϵ≠ 0,H̃(q̃,p̃,ϵ,ϵ)= (1-ϵ(1+ϑ_0+ϑ̃_1))q̃p̃+ϵ(q̃+ϵ A)^2p̃^2-ϵ(q̃+ϵ A)p̃^2,hence by Proposition <ref> the Birkhoff-Siegel invariant of H(q,p,ϵ,ϵ) isG(h,ϵ,ϵ)=(1-ϵ(1+ϑ_0+ϑ̃_1))h+ϵ h^2,and H̃(q̃,p̃,0,ϵ)= (1-ϵ(1-ϑ_0))q̃p̃-ϵq̃^2(p̃+ϵ B)^2 -ϵ(ϑ_0+ϑ̃_1)q̃^2(p̃+ϵ B) +ϵq̃^3(p̃+ϵ B)^2,hence by Proposition <ref> the Birkhoff-Siegel invariant of H(q,p,0,ϵ) isG(h,0,ϵ)=(1-ϵ(1-ϑ_0))h-ϵ h^2,i.e.G(h,x,ϵ)=(1-ϵ-(x-ϵ)ϑ_0-x(ϑ_0+ϑ̃_1))h+ (2x-ϵ)h^2.The Theorem <ref> for the limit system ϵ=0 is in this case due to Takano <cit.>, see also <cit.>. A separate paper <cit.> will be devoted to a more detailed study of the confluence P_VI→ P_V and of the non-linear Stokes phenomenon in P_V through the Riemann-Hilbert correspondence.§ PROOF OF THEOREM <REF> AND OF PROPOSITION <REF>The proof of Theorem <ref> is loosely based on the ideas of Siegel's proof of Theorem <ref> <cit.>. We constructthe normalizing transformation y=Φ_±(u,x,ϵ) in a couple of steps as a formal power series in the u-variable with coefficients depending analytically on (x,ϵ)∈_±, and then show that the series is convergent. The main tool to prove the convergence is the Lemma <ref> below.Let ϕ_±(u,x,ϵ)=∑_| m|≥ 2ϕ_±, m(x,ϵ)u^ m, m=(m_1,m_2), u^ m=u_1^m_1u_2^m_2, | m|=m_1+m_2,be a power series in the u-variable with coefficients bounded and analytic on (x,ϵ)∈_±. We will write{ϕ_±}_ m:=ϕ_±, m.Denoting ϕ_±, m:=sup_(x,ϵ)∈_± |ϕ_±, m(x,ϵ)|the supremum norm over _±, letϕ_±(u)=∑_| m|≥ 2ϕ_±, mu^ m, be a majorant power series to ϕ. We will write ϕ_±(u)≺ψ_±(u)if{ϕ_±}_ m≤{ψ}_ mfor allm. The following lemma is the essential technique in Siegel's proof. Let ϕ=(ϕ_1,ϕ_2)=O(u^2) be a formal power series in u,and let r=(r_1,r_2)=O(u^2) be a convergent power series in u. If ϕ_j(u)≺ r_j(u+ϕ(u)), j=1,2, where ϕ=(ϕ_1,ϕ_2), then ϕ is convergent.See <cit.>. It can be also found implicitely in <cit.> and <cit.>.§.§ Step 1: Ramified straightening of center manifold and diagonalization of the linear partSuppose that the system is in a pre-normal form,x(x-ϵ)dy/dx=JD_yF(y,x,ϵ), JD_yF(y,x,ϵ)=χ(y_1y_2,x,ϵ) ([10;0 -1 ])y+O(x(x-ϵ)).We will show that there exists a ramified transversely symplectic change of variabley=T_±(x,ϵ)w+ϕ_±, 0(x,ϵ), T_±(0,ϵ)=I, ϕ_±, 0(x,ϵ)=O(x(x-ϵ))bounded and analytic on the domain _± of Definition <ref>, that brings the system to a formx(x-ϵ)dw/dx=χ(w_1w_2,x,ϵ) ([10;0 -1 ])w+x(x-ϵ) f_±(w,x,ϵ),with f_±(w,x,ϵ)=O(|w|^2), ∂ f_1,±/∂ w_1+∂ f_2,±/∂ w_2=0. The solution w=0 of the transformed system (<ref>),corresponds to a bounded ramified solution y=ϕ_±, 0(x,ϵ)of the system (<ref>). The paper <cit.>, see Theorem <ref> below, shows that there is a unique such solution on the domain _±; this it is the “ramified center manifold” of the corresponding foliation.The variable ỹ=y-ϕ_±, 0(x,ϵ) then satisfiesx(x-ϵ)dỹ/dx=JD_yF(ỹ+ϕ_±, 0,x,ϵ)-JD_yF(ϕ_±, 0,x,ϵ),whose linear part is A_±(x,ϵ):=J(D^2_yF)(ϕ_±, 0,x,ϵ)=λ(x,ϵ) ([10;0 -1 ])+x(x-ϵ)R(ϕ_±, 0,x,ϵ). The transformation matrix T_± (<ref>) must then satisfyx(x-ϵ)d T_±/dx=A_± T_±-λ T_±([10;0 -1 ]).The existence of such a transformation T_± bounded on _± is known <cit.> when A_± is analytic. In our case the matrix A_± is ramified, but their proof works anyway. We will obtain T_± directly using Theorem <ref>.Writing R=(r_ij)_i,j andT_±=([ 1 t_1,±; t_2,± 1 ])([ e^b_1,± 0; 0 e^b_2,± ]),then the terms t_i,±, i=1,2, are solutions to Riccati equationsx(x-ϵ)d t_i,±dx=(-1)^i-12λ t_i,± +x(x-ϵ)[r_ij+(r_ii-r_jj)· t_i,±-r_ji·(t_i,±)^2],and the terms b_i,± are solution to d b_i,±dx=(r_ii+r_ij t_j,±), i.e.b_i,±=∫_0^x r_ii+r_ijt_j,± dx. Combining the equations (<ref>) for ϕ_±, 0 and (<ref>) for t_±, in which r_ij=r_i,j(ϕ_±, 0,x,ϵ),we get an analytic system for which the existence of a unique bounded solution on _± is assured by the following theorem. Consider a system of the form x(x-ϵ)dϕ/dx=Mϕ + f(ϕ,x,ϵ),(ϕ,x,ϵ)∈^m××.with M an invertible m× m-matrix whose eigenvalues are all [This assumption is inessential, it is added here just to simplify the statement. See <cit.> for a general version of the statement.] on the line λ^(0), and f(ϕ, x,ϵ) analytic germ such that D_ϕf(0,0,0)=0, and f(0,x,ϵ)=O(x(x-ϵ)).(i) The system (<ref>) possesses a unique solution in terms of a formal power series in (x,ϵ):ϕ̂(x,ϵ)=∑_k,j=0^+∞ϕ_kjx^kϵ^j,ϕ_kj∈^m.This series is divisible by x(x-ϵ), and its coefficients satisfyϕ_kj≤ L^k+j(k+j)! for some L>0.(ii) The system (<ref>) possesses a unique bounded analytic solution ϕ_±(x,ϵ) on the domain _±(ϵ), ϵ∈_± of Definition <ref> (for some δ_x,δ_ϵ>0). It is uniformly continuous on _±={(x,ϵ)| x∈_±(ϵ)}and analytic on the interior of _±, and it vanishes (is uniformly O(x(x-ϵ))) at the singular points.When ϵ tends radially to 0 with ϵ=β, thenϕ(x,ϵ) converges to ϕ(x,0) uniformly on compact sets of the sub-domainslim_ϵ→ 0 ϵ=β_±(ϵ)⊆_±(0). (iii) Let Φ(x,ϵ) be analytic extension of the function given by the convergent seriesΦ(x,ϵ)=∑_j,kϕ_kj/(k+j)!x^kϵ^j. For each point (x,ϵ), for which there is θ∈ ]-π/2,π/2[ such that ⊆_±, with 𝐒_θ⊂ denoting the circlethrough the points 0 and 1 withcenter on e^iθ^+, we can express ϕ_±(x,ϵ) as thefollowing Laplace transform of Φ:ϕ_±(x,ϵ)=∫_0^+∞ e^iθΦ(sx,sϵ) e^-s ds.In particular, ϕ_+(x,0)=ϕ_-(x,0) is the functional cochain consisting of the pair of Borel sums of the formal series ϕ̂(x,0) in directions on either side of λ^(0).Since the trace of the linear part of both systems (<ref>) and (<ref>) is null, then by the Liouville–Ostrogradskii formula T(x,ϵ) is constant in x and equal to T(0,ϵ)=1. Therefore the transformation (<ref>) is transversely symplectic, and by Lemma <ref> the transformed system (<ref>) is transversely Hamiltonian.§.§ Step 2: NormalizationSuppose that the system is in the form (<ref>). We will show that there exists a ramified change of variable w=Φ_±(v,x,ϵ), Φ_±(·,0,ϵ)=𝕀, that will bring it to an integrable formx(x-ϵ)dv/dx=α_±(h,x,ϵ) ([10;0 -1 ])v, h=v_1v_2,for some germ α_±(h,x,ϵ), with α_±(0,x,ϵ)=λ(x,ϵ). The transformation Φ_± must satisfyx(x-ϵ)∂_xΦ_±+α_±·(v_1∂_v_1-v_2∂_v_2)Φ_±= χ∘Φ_±·([10;0 -1 ])Φ_±+x(x-ϵ) f_±∘Φ_±.We are looking for Φ_± written asΦ_±(v,x,ϵ)=v+Ψ_±,Δ+x(x-ϵ)Ψ_±,⋆(v,x,ϵ),Ψ_±,Δ+Ψ_±,⋆=:Ψ_±where the power expansion of the j-th coordinate of Ψ_±,Δ is equal toΨ_j,±,Δ=∑_n≥ 1{Ψ_j,±}_(n,n)+ e_jh^nv_j,e_j being the j-the elementary vector,while the power expansion of the j-th coordinate of {Ψ_±,⋆} does not contains any power h^nv_j, n≥ 0. In particular, (v_1∂_v_1-v_2∂_v_2)Ψ_±,Δ=([10;0 -1 ])Ψ_±,Δ. Therefore (<ref>) becomes∂_x Ψ_±,Δ +∂_x(x(x-ϵ)Ψ_±,⋆)+ λ(v_1∂_v_1-v_2∂_v_2-([10;0 -1 ]))Ψ_±,⋆=-α_±^* (v_1∂_v_1-v_2∂_v_2)Ψ_±,⋆+ χ^*∘Φ_±-α_±^*/x(x-ϵ)([10;0 -1 ])(v+Ψ_±,Δ)+ G_±,where G_±=(χ^*∘Φ_±)([10;0 -1 ])Ψ_±,⋆+f∘Φ_±, andχ^*=χ-λ, α_±^*=α_±-λ.Setα_±(h,x,ϵ)=∑_n≥ 0{χ∘Φ_±}_(n,n)h^n,and denoteK_±:=χ^*∘Φ_±-χ^*∘(v+Ψ_±,Δ)/x(x-ϵ),which is an analytic function of v+Ψ_±,Δ and Ψ_±,⋆ with coefficients depending on x,ϵ. Then {χ^*∘Φ_±-α_±^*}_(n,n)=0 for all n≥ 0, and{χ^*∘Φ_±-α_±^*}_ n=x(x-ϵ){K_±}_ n for all multi-indices n with n_1≠ n_2, since {α_±^*}_ n=0={χ^*∘(v+Ψ_±,Δ)}_ n.Expanding the j-th coordinate, j=1,2, of the equation (<ref>) in powers of v we get: * for m=(n,n)+ e_j:∂_x{Ψ_j,±}_(n,n)+ e_j={G_j,±}_(n,n)+ e_j, * for a multi-index m with m_1-m_2+(-1)^j≠ 0:∂_x {x(x-ϵ)Ψ_j,±}_ m + (m_1-m_2+(-1)^j) λ{Ψ_j,±}_ m=-(m_1-m_2){α_±^*·Ψ_j,±}_ m - (-1)^j{K_±·(v_j+Ψ_j,±,Δ)}_ m+ {G_j,±}_ m.The right sides of (<ref>) and (<ref>) are functions of{Ψ_±}_ k=({Ψ_1,±}_ k,{Ψ_2,±}_ k),with k_1≤ m_1, k_2≤ m_2, | k|<| m| only, which means that the equations for {Ψ_j,±}_ m can be solved recursively.The equation (<ref>) is solved by{Ψ_j,±}_(n,n)+ e_j(x,ϵ)=∫_0^x {G_j,±}_(n,n)+ e_jdx.The equation (<ref>) has a unique bounded solution {Ψ_j,±}_ m given by the integral{Ψ_j,±}_ m(x,ϵ)=e^-(m_1-m_2+(-1)^j)t_λ/x(x-ϵ)∫_x_i,±^x e^(m_1-m_2+(-1)^j)t_λ{F_j,±}_ mdx,where {F_j,±}_ m is the right side of (<ref>),t_λ(x,ϵ)= {[ -λ^(0)ϵlog x+(λ^(0)ϵ+λ^(1))log(x-ϵ),for ϵ≠ 0,; -λ^(0)/x+λ^(1)log x, for ϵ=0, ].is a branch of the rectifying coordinate for the vector fieldx(x-ϵ)/λ(x,ϵ)∂_x=∂_t_λ,on _±(ϵ), and the integration follows a real trajectory of the vector field (<ref>) in _±(ϵ) from a point x_*= {[ x_1,±, if m_1-m_2+(-1)^j>0,; x_2,±, if m_1-m_2+(-1)^j<0, ].x_i,± is as in(<ref>),to x, along which the integral is well defined.Note that the convergence of the constructed formal transformation Φ_± is equivalent to the convergence of Ψ_± (<ref>). We prove the convergence of the latter series using Lemma <ref>. For this we need to estimate the norms of (<ref>) and (<ref>).Let {Ψ_j,±}_ m be given by (<ref>). Then {Ψ_j,±}_ m≤c_⋆m_1-m_2+(-1)^j{F_j,±}_ m,for some c_⋆>0 independent of m.Let τ_k(x,ϵ)=kt_λ(x,ϵ)+log(x(x-ϵ)). If k=m_1-m_2+(-1)^j≠ 0, then for ϵ small enough the integrating path can be deformed so thatit corresponds to a ray τ_k(ξ,ϵ)∈τ_k(x,ϵ)-e^iω_±[0,+∞[, with ω_± as in (<ref>). We have dτ_k/dx(x,ϵ)=kλ(x,ϵ)+2x-ϵ/x(x-ϵ).Hence {Ψ_j,±}_ m≤1/cosω_±{F_j,±}_ m(x,ϵ)/kλ(x,ϵ)+2x-ϵ,with cosω_±>sinη>0, η as in (<ref>). Therefore * for m=(n,n)+ e_j:{Ψ_j,±}_(n,n)+ e_j≤ c{G_j,±}_(n,n)+ e_j, * for a multi-index m with m_1-m_2+(-1)^j≠ 0:{Ψ_j,±}_ m≤ c ({α_±^*Ψ_j,±}_ m +{K_±·(v_j+Ψ_j,±,Δ)}_ m + {G_j,±}_ m) ,for some c>0.ThereforeΨ_j,± ≺ c α_±^*Ψ_j,± +cK_±·(v_j+Ψ_j,±) +cG_j,±≺ 2c (χ^*∘Φ_±)·Ψ_j,± +ck_±·(v_j+Ψ_j,±)+cf_j,±∘Φ_± =: R_j,±(v+Ψ_±,Δ,Ψ_±,⋆)≺ R_j,±(v+Ψ_±,v+Ψ_±),where K_±=K_±(v+Ψ_±,Δ,Ψ_±,⋆), k_±= K_±(v+Ψ_±,Δ,Ψ_±,⋆), and R_j,±(w_1,w_2)=O(|w|^2), and we can conclude with Lemma <ref>.§.§ Step 3: Final reduction and transverse symplecticity of the transformationSuppose that the system is in the form (<ref>), and write α_±(h,x,ϵ)=χ̃_±(h,x,ϵ)+x(x-ϵ)β_±(h,x,ϵ),χ̃_±(h,x,ϵ)=χ̃_±^(0)(h,ϵ)+xχ̃_±^(1)(h,ϵ).Then the transformation v=e^∫_0^xβ_± dx ([10;0 -1 ])u will bring it to the normal form with formal invariant χ̃_±.Let us show that the transformationy=Ψ_±(u,x,ϵ) obtained as a composition of the transformations of Steps 1-3 is transversely symplectic and therefore χ̃_±=χ.Let u_±(x,ϵ;c) be a germ of a general solution of the normal form system with the formal invariant equal to χ̃_±, depending on an initial condition parameter c=(c_1,c_2)∈(^2,0), (D_cu_±)≠0, and let y_±(x,ϵ;c)=Ψ_±(u_±(x,ϵ;c),x,ϵ) be the corresponding solution germ of the system (<ref>). Then D_cy_±=D_uΨ_±· D_c u_± satisfies the linearized systemx(x-ϵ)dD_cy_±/dx=JD_y^2H· D_cy_±,and by the Liouville-Ostrogradskii formulax(x-ϵ)d/dx(D_cy_±)=(JD_y^2H)·(D_cy_±),but (JD_y^2H)=0, i.e. (D_cy_±)=(D_uΨ_±)·(D_c u_±) is constant in x.Similarly, (D_cu_±) is constant in x.Therefore (D_uΨ_±(u,x,ϵ)) is also constant in x, and equal to (D_uΨ_±(u,0,ϵ))=1 since Ψ_±(u,0,ϵ)=u.This terminates the proof of Theorem <ref>. §.§ Proof of Proposition <ref> We will construct a formal symplectic change of coordinate Φ=Φ(h,u_i), written as a formal power series in h and u_i, such that G=H∘Φ. The transformation Φ is constructed recursively as a formal limitΦ=lim_k→+∞Φ_k,1,Φ_k+1,1=lim_l→+∞Φ_k,l,Φ_0,1=𝕀, H∘Φ_k,l=G+O(h^ku_i^l).At each step (k,l), k≥ 0, l≥ 1, we want to get rid of the power h^ku_i^l in H∘Φ_k,l. We constructΦ_k,l+1=Φ_k,l∘Φ^1_f_k,lX_h^ku_i^l as a composition of Φ_k,l with the time-1 flowof a Hamiltonian vector fieldf_k,lX_h^ku_i^l=-(-1)^if_k,lh^k-1u_i^l[ku_i∂_u_i-(k+l)u_j∂_u_j] for some f_k,l∈. The flow sends both h and u_i to functions of (h,u_i),Φ^1_f_k,lX_h^ku_i^l: u_i↦ u_i+O(h^k-1u_i^l+1), h↦ h+(-1)^ilf_k,lh^ku_i^l+O(h^2k-1u_i^2l),where the terms O(h^2k-1u_i^2l) are null if k=0.If H∘Φ_k,l=G+H_k,lh^ku_i^l+O(h^ku_i^l+1) for some H_k,l∈, then we want (G+H_k,lh^ku_i^l)∘Φ^1_f_k,lX_h^ku_i^l-G(h) =O(h^ku_i^l+1),H_k,lh^ku_i^l+λ· (-1)^ilf_k,lh^ku_i^l=O(h^ku_i^l+1),f_k,l =-(-1)^iH_k,lλ l. Now that we have constructed the formal symplectic transformation Φ, we can conclude by the following Proposition. LetH,H̃:(^2,0)→(,0) be two germs with a non-degenerate critical point at 0, and let ω,ω̃ be germs of symplectic forms. Then the two pairs (H,ω), (H̃,ω̃) are analytically equivalent if and only if they are formally equivalent. By Theorem <ref>, the Birkhoff-Siegel normal form is, up to the involution (<ref>), a complete analytic invariant for each pair. Therefore it is enough to show that it is also a formal invariant. This can be seen from the invariance of a formalization of the formula (<ref>) of Section <ref>. AAAAA[AVG12]AVG V.I. Arnold, A.N. Varchenko, S.M. Gusein-Zade, Singularities of Differentiable Maps, Volume II: Monodromy and Asymptotics of Integrals, Birkhäuser, Boston, 2012.[Bit16a]Bit16iA. Bittmann, Doubly-resonant saddle-nodes in (^3,0) and the fixed singularity at infinity in the Painlevé equations: Formal classification,Qual. Theory Dyn. Syst. (2016). [Bit16b]Bit16iiA. Bittmann, Sectorial analytic normalization for a class of doubly-resonant saddle-node vector fields in (^3,0), preprint arXiv:1605.05052. [Bit16c]Bit16iiiA. Bittmann, Doubly-resonant saddle-nodes in (^3,0) and the fixed singularity at infinity in the Painlevé equations. Part III: Local analytic classification, preprint arXiv:1605.09683. [Du98]Du A. Duval, Confluence procedures in the generalized hypergeometric family, J. Math. Sci. Univ. Tokyo 5 (1998), 597–625.[FS94]FS94 J.-P. Françoise, M. Smaïli, Lemme de Morse transverse pour des puissances de formes de volume, Annales Fac. Sci. Toulouse, 6e Série 3 (1994), 81–89.[Glu01]Glu01 A. Glutsyuk, Confluence of singular points and the nonlinear Stokes phenomena, Trans. Moscow Math. Soc 62 (2001), 49–95.[HLR13]HLR J. Hurtubise, C. Lambert, C. Rousseau, Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k, Moscow Math. J. 14 (2013), 309–338.[IY08]IlYa Y. Ilyashenko, S. Yakovenko, Lectures on Analytic Differential Equations, Grad. Studies Math. 86, Amer. Math. Soc., Providence, 2008. [KT05]KT T. Kawai, Y. Takei, Algebraic analysis of singular perturbation theory, Translations of Mathematical Monographs 227, Amer. Math. Soc., 2005. [Kli16]Kl2 M. Klimeš, Confluence of singularities of non-linear differential equations via Borel-Laplace transformations, J. Dynam. Contr. Syst. 22 (2016), 285–324. [Kli17]Kl4 M. Klimeš, Non-linear Stokes phenomenon in the fifth Painlevé equation and a wild monodromy action on the character variety, arXiv:1609.05185 (2017). [LR12]LRC. Lambert, C. Rousseau, Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1, Moscow Math. Journal 12 (2012), 77–138. [MR82]MR1 J. Martinet, J.-P. Ramis, Problèmes de modules pour des équations différentielles non linéaires du premier ordre, Publ. Math. IHES 55 (1982), 63–164.[MR91]MR2 J. Martinet, J.-P. Ramis, Elementary acceleration and multisummability. I, Ann. Inst. Henri Poincaré (A) Physique théorique 54 (1991), 331–401. [MM80]MM J.-F. Mattei, R. Moussu, Holonomie et intégrales premières, Ann. Sci. É.N.S. 4e série 13 (1980), 469–523. [Oka80]Oka1 K. Okamoto, Polynomial Hamiltonians associated with Painlevé Equations. I, Proc. Japan Acad. Ser. A, Math. Sci. 56 (1980), 264–268.[Par01]PaL. Parise, Confluence de singularités régulières d'équations différentielles en une singularité irrégulière. Modèle de Garnier, thèse de doctorat, IRMA Strasbourg (2001). [http://irma.math.unistra.fr/annexes/publications/pdf/01020.pdf][Ram89]Ra1 J.-P. Ramis, Confluence and resurgence, J. Fac. Sci. Univ. Tokyo, Sec. IA 36 (1989), 703–716.[RR11]RR11 J. Rebelo, H. Reis, Local Theory of Holomorphic Foliations and Vector Fields, arXiv:1101.4309 (2011).[RT08]RT C. Rousseau, L. Teyssier, Analytical moduli for unfoldings of saddle-node vector fields,Moscow Math. J. 8 (2008), 547–614.[Shi83]Shi S. Shimomura, Analytic integration of some nonlinear ordinary differential equations and the fifth Painlevé equation in the neighborhood of an irregular singular point, Funkcialaj Ekvacioj 26 (1983), 301–338. [SM71]SM C.L. Siegel, J.K. Moser, Lectures on Celestial Mechanics, Grundlehren der mathematische Wissenschaften 187, Springer, 1971. [SP03]SP M. Singer, M. van der Put, Galois Theory of Linear Differential Equations, Grundlehren der mathematische Wissenschaften 328, Springer, 2003. [Tak83]Ta K. Takano, A 2-parameter family of solutions of Painlevé equation (V) near the point at infinity, Funkcialaj Ekvacioj 26 (1983), 79–113. [Tak86]Ta86 K. Takano, Reduction for Painlevé equations at the fixed singular points of the first kind, Funkcialaj Ekvacioj 29 (1986), 99-119. [Tak90]Ta90 K. Takano, Reduction for Painlevé equations at the fixed singular points of the second kind, J. Math. Soc. Japan 42 (1990), 423–443. [Tey04]Tey04 L. Teyssier, Équation homologique et cycles asymptotiques d'une singularité col-nœud, Bull. Sci. Math. 128 (2004), 167–187. [Yos84]Yo1 S. Yoshida, A general solution of a nonlinear 2-system without Poincaré's condition at an irregular singular point, Funkcialaj Ekvacioj 27 (1984), 367–391. [Yos85]Yo2 S. Yoshida, 2-parameter family of solutions of Painlevé equations (I)-(V) at an irregular singular point, Funkcialaj Ekvacioj 28 (1985), 233–248. [Vey77]Vey77 J. Vey, Sur le lemme de Morse, Inventiones Math. 40 (1977), 1–9. | http://arxiv.org/abs/1709.09078v1 | {
"authors": [
"Martin Klimes"
],
"categories": [
"math.CA"
],
"primary_category": "math.CA",
"published": "20170926150746",
"title": "Stokes phenomenon and confluence in non-autonomous Hamiltonian systems"
} |
firstpage–lastpage Fully dissipative relativistic lattice Boltzmann method in two dimensions [ Received July 19, 2017; Accepted September 21, 2017 =========================================================================We presentresults from the analysis of 2997 fundamental mode RR Lyrae variables located in the Small Magellanic Cloud (SMC). For these objects near-infrared time-series photometry from the VISTA survey of the Magellanic Clouds system (VMC) and visual light curves from the OGLE IV survey are available. In this study the multi-epochK_ s-band VMC photometry was used for the first time to deriveintensity-averaged magnitudes of the SMC RR Lyrae stars.We determined individual distances to the RR Lyrae stars from the near-infraredperiod-absolute magnitude-metallicity (PM_K_ sZ) relation, which has a number of advantages in comparison with the visual absolute magnitude-metallicity (M_V- [Fe/H]) relation, such as a smaller dependence of the luminosity on interstellar extinction, evolutionary effects and metallicity. The distances we have obtained were used to study the three-dimensional structure of the SMC. The distribution of the SMC RR Lyrae stars is found to be ellipsoidal.The actual line-of-sight depth of the SMC is in the range from 1 to 10 kpc, with an average depth of 4.3 ± 1.0 kpc. We found that RR Lyrae stars in the eastern part of the SMC are affected by interactions of the Magellanic Clouds.However, we do not see a clear bimodalityin the distribution of RR Lyrae stars as observed for red clump (RC) stars.Surveys – stars: variables: RR Lyrae – galaxies: Magellanic Clouds – galaxies: structure§ INTRODUCTION The Small Magellanic Cloud (SMC) is a nearbydwarf irregular galaxy. It is part of the Magellanic System (MS) which also comprises the Large Magellanic Cloud (LMC), the Magellanic Bridge (MB) and the Magellanic Stream.The SMC gravitationally interacts with the LMC and the Milky Way (MW). As a result it has a complex internal structure characterised by a disturbed shape and a large extent along the line-of-sight <cit.>.RR Lyrae stars are old (age >10 Gyr),low-mass (∼0.6 - 0.8 M_⊙), radially-pulsating variables located on the horizontal branch of the colour-magnitude diagram (CMD).They pulsate in the fundamental mode (RRab), first-overtone mode (RRc) or both modessimultaneously (RRd).RR Lyrae stars are abundant in globular clusters and in the halos of galaxies. Theycould serve to study the structure, interaction history and the distance to the parent systems because they follow an absolute magnitude-metallicity (M_V- [Fe/H]) relationin the visual band and infrared period-luminosity (PL) and PL-metallicity (PLZ) relations.The three-dimensional structure of the SMC as traced by RR Lyrae stars has been the subject of a number of studies (e.g. ; ;; ; ; ; , ). <cit.> analysed 76 SMC RR Lyrae stars finding that these objects are distributed smoothly and do not show a strong concentration in the bar or in the centre of the SMC.<cit.> published the catalogue of RR Lyrae stars observed in the SMCby the third phase of the Optical Gravitational Lensing Experiment (OGLE III) and concluded that the distribution of RR Lyrae stars in the galaxy is roughly round on the sky with two maxima near the centre.<cit.> analysed the relative positions of different regions of the SMC inferred from the V and I photometry of the RR Lyrae variables observed byOGLE III. According to this study the SMC RR Lyrae stars have an ellipsoidal distribution and the north-eastern part of the SMC is locatedcloser to us. Similarly, <cit.> used the SMC RR Lyrae starsfrom the OGLE III survey and found that the RR Lyrae stars show a spheroidal or ellipsoidal distribution with an off-centered and nearly bimodalpeak.<cit.> found that the north-eastern arm of the SMC islocatedcloser than the plane of the SMC main body.These authorsstudied the depth along the line-of-sight and concluded that it is larger for the central part of the SMC. <cit.> studied a sample of RRab stars located in 14 deg^2 of the SMC and found that the north-eastern part of the SMC has a greater depth. Furthermore, they studiedthe metal abundance and the spatial distribution of the RR Lyrae variables and suggested that metal-richer and metal-poorer objects in the sample belong to different dynamical structures. <cit.> studied the RR Lyrae stars using OGLE IV data, which covers almost entirely the SMC. They confirm the ellipsoidal shape of the SMC and do not find any sub-structures. However, they detect some asymmetry in the equal density contours in the eastern part of the SMC.Namely, the center of the ellipsoid shifts to east and towards the observer.More recently, <cit.> used the sample of the OGLE IV RRab stats to determine the distance, reddening and structural parameters of the SMC.In all studies mentioned above visual photometry was used to analyse the SMC RR Lyrae stars. In the presentstudy, for the first time,K_ s-band multi-epoch photometry fromthe VISTA survey of the Magellanic Clouds system (VMC) for 2997 RRab stars distributed across a large area(∼42 deg^2) of the SMC, was used to probe the galaxy's structure.The near-infrared (K or K_ s bands) period-absolute magnitude-metallicity(PM_KZ) relation has a number of advantages in comparison with the optical M_V- [Fe/H] relation,specifically,a smaller dependence of the luminosity on interstellar extinction (A_K=0.114A_V), evolutionary effects and metallicity. It was discovered by <cit.> and later studied by different authors from both a theoretical and an observational point of view (e.g. ;;; ; ; ; ; ). Here we apply the relation from <cit.> to derive individual distances to RR Lyrae stars in the SMC sample and study the structure of this galaxy. In Section <ref> we provide information about our sample of RR Lyrae variables in the SMC and the data available for these objects from the VMC and OGLE IV surveys.In Section <ref> we analysethe PK_ s relations for the whole sample of SMC RR Lyrae stars as well as for individual VMC tiles.We present the three-dimensional structure of the SMC as traced by RR Lyrae stars in Section <ref>.Finally, Section <ref> provides a summary of our results and main conclusions. § DATA §.§ The VMC survey VMC <cit.> is an ongoing imaging survey of the MS in the Y, J, K_ s passbands, centred at λ = 1.02, 1.25 and 2.15 μm, respectively. It started in 2009 and observations of the MS were 89% complete as ofSeptember 2017, while the observations of the SMC were 100% complete.The survey covers the LMC area (∼105 deg^2) with 68 tiles, the SMC (∼42 deg^2) with 27 tiles, the MB area (∼21 deg^2) with 13 tiles and part of the Stream (∼3 deg^2) with 2 tiles.The VMC K_ s-bandobservations are taken over 13 separate epochs: 11 times with an exposure time of 750 s (deep epochs) and twice with an exposure time of 375 s (shallow epochs). Additional epochs may be obtained if observations have to be repeated because the requested sky conditions are not met <cit.>.Every single deep epoch reaches a limiting magnitude of K_ s∼ 19.2 mag with a signal-to-noise ratio S/N = 5, in the Vega system. VMC reaches a sensitivity limit on the stacked images of K_ s=21.5 mag withS/N = 5.The VMC images are processed by the Cambridge Astronomical Survey Unit (CASU; ). The data are then sent to the Wide Field Astronomy Unit (WFAU) in Edinburgh where the single epochs are stacked, catalogued and ingested into theVISTA Science Archive (VSA;). The strategy, main science goals and the first data fromthe VMC survey were described in <cit.>. Analysis of variable stars based on VMC data was presented in <cit.>, <cit.>, <cit.> and <cit.> for classical Cepheids, <cit.> for Anomalous Cepheids, <cit.> for Type II Cepheids, in <cit.> for classical Cepheids, RR Lyrae stars and eclipsing binaries (EBs), in <cit.> for RR Lyrae stars and in <cit.> for EBs.Using the VMC observations of the red clump (RC) stars<cit.> and <cit.> studied the structure of the SMC and the 30 Doradus region in the LMC, respectively.<cit.> analysed the star formation history of the SMC using the VMC photometry. To study the K_ s-band light curves of the SMC RR Lyrae stars we used VMC observations in all 27 SMC tiles. We analysed the K_ s-band light curves with the GRaphical Analyser of TImes Series (GRATIS), custom software developed at the Observatory of Bologna by P. Montegriffo (see e.g. ). This software requires at least 11epochs to model the light curves and properly compute intensity-averaged magnitudes. However, even though all sources located in the analysed tiles are expected to have 13 good quality epochs, starsaffected by some problems, such as blending, may not necessarily be detected in all epochs and, thus, have the 11 data points necessary for the analysis with GRATIS.Hence, initially we used all available VMC epochs including observations obtained in nights with sky conditions (seeing and ellipticity)that did not meet the VMC requirements <cit.> and discarded data points that deviated significantly from the model fit during the analysis with GRATIS. For some stars the model line fits well all data points, hence, we used all available data in the analysis. The VMC coverage of the SMC is shown in Figure <ref>, where red boxes represent the 27 SMC tiles (∼42 deg^2) used in this study. The X and Y axes in Fig. <ref> are the coordinates of a zenithal equidistant projection as defined by <cit.>.The time sampling of the VMC survey and the significantly reduced amplitudes of the light variations in the K_ s passband allow us to determine mean K_ s magnitudes for RR Lyrae and other pulsating stars with great precision (). On the other hand,the small amplitude of pulsation in the K_ s band complicates the search for new variablesbased solely on near-infrared data.A method to identify variable stars in the MS based only on the VMC photometry was developed by <cit.>, but it is more suited to search for classical Cepheids. Hence, our study is based only on known SMC variable stars identified by optical microlensing surveys such as OGLE IV <cit.>.§.§ OGLE IV The initial goal of the OGLE survey was the search for microlensing events, but as a byproduct the survey also discovered a large number of variable stars in the MS and in the MW bulge. The OGLE IV catalogue comprises observations of about 650 deg^2 in the MS obtained betweenMarch 2010 and July 2015 with the 1.3 m Warsaw telescope at the Las Campanas Observatory, Chile <cit.>.Observations were performed in the Cousins I passband with the number of data points ranging from 100 to 750 andin the Johnson V band with the number of observations ranging from several to 260.The OGLE IV catalogue is publicly available from the OGLE website[http://ogle.astrouw.edu.pl] and contains, among others, 45451 RR Lyrae variables, of which 6369 stars are located towards the SMC <cit.>. The catalogue provides right ascension (RA), declination (Dec), mode of pulsation, mean V and I magnitudes, period of pulsation, I-band amplitude, parameters of the Fourier decomposition andtime-series V,I photometry.Among the SMC RR Lyrae variables we selected only RRab stars for our analysis, since RRc and RRd stars have smaller amplitudes and noisier light curves, hence, it is particularly complicated to fit their near-infraredlight curves and obtain reliable mean K_ s magnitudes. OGLE IV provides information on 4961 RRab stars in the SMC, of which 3484 variables have counterparts in the VMC catalogue within 1. Among 1477 variables that do not have VMC counterpart within 1, 1336 are located outside the VMC footprint, while 141 stars are in the VMC footprint but we did not find them likely because of coordinates' uncertainty.Adopting a cross-matching radius of 2 arcsec instead of 1 arcsec we were able to recoverVMC counterparts for100 among 141 stars, and 125 stars are recoveredwithin a radius of 5 arcsec. However,it is risky to add these stars to the sample, since wrong crossmatches could happen.Starsthat do not have a VMC counterpart even within 5 arcsec are aligned along the edges of the VMC footprint, located in the gap between VMC tiles 5_3 and 5_4 or are clustered in some kind of holes of the VMC coverage likely caused by saturatedMW stars in front of the SMC. However, the percentage of objects that we lose owing to these issues is rather small. We discarded from the sample of 3484 variables whichhave counterparts in the VMC catalogue within 1, stars that have fewer than 11 K_ s band epochs(see Section <ref>) and sources observed by VISTA detector 16. Detector 16 is affected by a time-varying quantum efficiency which makes accurate flat fielding impossible <cit.>. This effect is more significant in the shorter wavelength filters but also present in the K_ sband. After these cleaning procedures we are left with a sample of3121 RR Lyrae variables. We analysed the K_ s-band light curves of the3121 RR Lyrae stars with GRATIS,and derived intensity-averagedmagnitudes and amplitudes using the periods provided by OGLE IV. Examples of the K_ s-band light curves are shown in Fig. <ref>, whereblack and green dots represent the data points used and discarded from the analysis, respectively. Red lines represent the best fits obtained with GRATIS. The distribution of the3121 RR Lyrae stars in the K_ s magnitude versuslogarithmic period plane is shown in Fig. <ref>. The large majority of the starsfollow the PK_ s relation. However, there is a group of objects that do not follow the PK_ s relation and have significantly brighter magnitudes. They are likely MW members or sources blended with close companions. A division between SMC and MW RR Lyrae stars is not provided in the OGLE IV catalogue since<cit.> pointed outthat it was impossible to separate the MW and SMC old stellar populations owing to the interaction between the two galaxies. However, in the current analysis we want to focus on the SMC inner structure. Hence, we perform an approximate separation between the two populations and clean the sample from blended sources.In order to do this we performa linear non-weighted least squares fitto the whole sample of3121 RR Lyrae variables (red line in Fig. <ref>) by progressively discarding objects which deviate more than 5σ from the best fit line (122 sources in total, shown bygreen squares in Fig. <ref>).We chose a rather large scatter (5σ from the PK_ s relation) to distinguish objects that belong to the SMC because the actual dispersion of the PK_ s relation in Fig. <ref> includes not only the intrinsic dispersion of the relation, but also the scatter caused by the different distances spanned by the RR Lyrae stars in oursample (depth effect). Selecting stars located within a smaller interval, for example3σ, is risky, since it causes the removal of RR Lyrae variables that actually belong to the SMC but are significantly scattered from the PK_ s owing to the large extension of the SMC along the line-of-sight. Of the122 stars that scatter more than 5σ from the PK_ s relation72 are too bright also in the OGLE IV catalogue and significantly scattered from the PL relation in I band, hence, they are either MW members or stars blended in both, the I and K_ s passbands. We discarded them from the following analysis. Additional50 objects have V and I magnitudes consistent with the distance to the SMC, but too bright magnitudes in the K_ s passband. We checkedtheir OGLE and VMC images and found that all of them have close companionsand are blended in the K_ s band, hence should be discarded.We are thus left with a sample of2999 RR Lyrae stars. Classification of one of them(OGLE-SMC-RRLYR-1505) was marked as uncertain in the OGLE IV catalogue.Another object (OGLE-SMC-RRLYR-3630) represents two RRab stars with different periods located on the line of sight. We analysed the K_ s-band light curve of this star using both periods but we were not able to distinguish which of the two periods is the correct oneowing to the small number of data points in the K_ s band. After discarding starsOGLE-SMC-RRLYR-1505 and OGLE-SMC-RRLYR-3630 our final sample comprises 2997 RR Lyrae variables, which are shown as black dots in Fig. <ref>.Their main properties are presented in Table <ref>.§.§ ExtinctionThe reddening in the SMC has been studied by several authors(e.g. ; ; ; ; ,). In this study we have obtained our own estimation of the reddening for the2997 RR Lyrae variables in our sample from the difference between their intrinsic and observed colours:E(V-I)=(V-I)-(V-I)_0 <cit.> developed an empirical relation that connects the intrinsic colour (V-I)_0 of RRab stars to the V-band amplitude and the period: (V-I)_0=(0.65±0.02) - (0.07±0.01)Amp(V) + (0.36±0.06)log P The OGLE IV catalogue provides amplitudes in the I passband. We transformedthem to V-band amplitudes using the relation developed as a byproduct during the work for the Gaia first data release (DR1, ). The relation was derived using a large sample of RR Lyrae stars which also includes SMC's RR Lyrae variables (Ripepi et al. in preparation).Amp(V)=(1.487±0.013)Amp(I) + (0.042±0.007) The r.m.s. of the relation is 0.03 mag. We calculated the intrinsic colours of the2997 RR Lyrae variables in our sample using Eqs. <ref>-<ref>. The uncertaintiesin (V-I)_0 were calculated by error propagation adopting the r.m.s. of Eq. <ref> as an uncertainty in Amp(V). Then we calculated individual reddening values for all2997 variables using Eq. <ref> and V and I apparent magnitudes from OGLE IV. Uncertaintiesin apparent V and I magnitudes are not provided in the OGLE IV catalogue. Following <cit.> we assumed their valuesas 0.02 mag.Reddening values and related uncertaintiesobtained by this procedure are provided in Table <ref>. For27 RRab stars in the sample it was not possible to estimate the reddening since their V apparent magnitudes are not available in the OGLE IVcatalogue.The mean reddening of the remaining2970 RR Lyrae variables is ⟨ E(V-I) ⟩ =0.06±0.06 mag. We assigned this mean value to the27 stars for which a direct determination of reddening was not possible.Reddening estimates bear uncertainties which may lead to negative reddening values. For232 stars out of2997 (8% of the sample) we found negative reddening values. However,165 of them have reddening consistent with zero within the uncertainties, which means that only67 variables (2% of the sample) have really negative values.The lowest reddening in the sample is E(V-I)=-0.56 mag, whilethe median reddening of all stars with negative values isE(V-I)=-0.02 mag. If negative reddening values are ignored, this will skew the distribution towardspositive reddening values and lead to biases. By retaining negative reddening values, the uncertainties will be propagated properly. Hence, we keep the negative values as a reflection of the uncertainties in reddening in the following analysis. We have divided the SMC region in small sub-regions of equal area (0.6 × 0.5 deg^2). Only those sub-regions which have at least 10 RR Lyrae variables (3 × Poissonian error) are considered for the analysis. There are 110 sub-regions which satisfy this criterion. The number of stars in sub-regions ranges from 10 to 60. More specifically, the number of stars is 10-30 in the outer regions and 30-60 in the inner regions of the SMC. Fig. <ref> shows the distribution ofstars in each of the 110 sub-regions. In Fig. <ref> we show the mean extinction values of RR Lyrae variables located inthe sub-regions. The extinction is larger in the eastern/south-eastern parts of the SMC. Our extinction maps are very similar to the extinction maps produced by <cit.> and <cit.> using RC stars,and <cit.> using RR Lyrae stars. We discuss this in details in Section <ref>.In order to calculate dereddened K_ s,0 magnitudes we applied the relations E(V-I)=1.22E(B-V) and A_K=0.114A_V(, ) thus obtaining:K_ s,0=K_ s - 0.29E(V-I)The mean extinction A_K_ s of the2997 RR Lyrae stars in our sample is 0.02 mag, which is significantly smallerthan the typical uncertainty of the K_ s individual mean apparent magnitudes(0.07 mag). <cit.> estimated the typical reddening towards the SMC from the median dust emission in surrounding annuli and found the value E(B-V)=0.037 mag which corresponds to E(V-I)=0.045 mag. This value is smaller than the value of reddening estimated in this paper E(V-I)=0.06±0.06 mag, but is in agreement with it within the errors. <cit.> produced an extinction map across the SMC and studiedthe nature of extinction as a function of stellar population. In particular, these authors derived extinction values for cooler and older stars(5500 K≤ T_eff≤ 6500 K) and for hotter and younger stars (12000 K≤ T_eff≤ 45000 K). They found that the mean extinction is lower for the cooler population(A_V=0.18 mag). This corresponds to A_K=0.02 mag, which is equal to the mean extinction value that we found for our sample of 2997 RR Lyrae stars in the SMC. <cit.> estimated the mean internal extinction of the SMC: A_V=0.45 mag, which corresponds to A_K=0.05 mag, hence, higher than the mean extinction A_K=0.02 mag found in this paper for the RR Lyrae stars. <cit.> estimated thereddening towardsthe SMC using RC stars and found the mean valueE(V-I)=0.053±0.017 mag. <cit.>found the mean reddening of the SMC to beE(V-I)=0.04±0.06 mag using RC stars andE(V-I)=0.07±0.06 mag from the RR Lyrae stars. <cit.> estimated the mean reddening value of the SMC using OGLE IV RR Lyrae stars and found E(B-V)=0.056±0.019 mag, which corresponds toE(V-I)=0.068 mag. All the estimates of the reddening in the SMC based on RC and RR Lyrae stars appear to be consistent with the value found in the present study. Individual dereddened K_ s,0 magnitudesof 2997 RR Lyrae stars in the SMC are listed in column 7 of Table <ref>and used in the following analysis to derive the individual distance to each RR Lyrae star in the sample. § PERIOD-LUMINOSITY RELATIONWe performed anon-weighted linear least squares fit of the PK_ s,0 relation defined by the2997 SMC RR Lyrae variables in our sample:K_ s,0 = (-3.17 ± 0.08)logP + (17.68 ± 0.02) We used the non-weighted fit in order to avoid biasing by brighter objects which usually have smaller uncertainties.The fit is shown in Fig. <ref>. The r.m.s.of the relation is large (0.17 mag) and could be owing to: metallicity differences, intrinsic dispersion of the PK_ s,0 relation or depth effect.The dependence of the K-band magnitude on metallicity has been investigated in a number of studies with a tendency of the theoretical and semi-theoretical analyses (; ) to derive a steeper metallicity slopethan found bythe empiricalanalyses (; ; ; ). Literature empirical values for the metallicity slope vary from 0.03 ± 0.07 ( based on 70 field RR Lyrae variables in the bar of the LMC) to 0.12 ± 0.04 ( from the analysis of RR Lyrae stars in ω Cen).Thus, the dependence of the K_ s magnitudeon metallicity does not seem to be able to explain the large scatter seen in Eq. <ref>.Hence, we conclude that the large spread observed in the RR Lyrae PK_ s relation is mainly caused by the intrinsic dispersion of the relation and by depth effect.A detailed discussion of the SMC line-of-sight depth and the intrinsic dispersion of the PK_ s,0 relation is provided in Section <ref>. To reduce the scatter we calculated the PK_ s,0 relations as a non-weighted linear regression by progressively discarding objects that deviate more than 3σ,in each of the27SMC tiles selected for the present analysis, separately. The resulting relations are summarised incolumn 6 of Table <ref> and shown in Fig. <ref>. The r.m.s. of the relations remains significant (0.13-0.24 mag) even for single tiles and is systematically larger in the inner regions of the SMC. The slope of thePK_ s,0 relation in different tiles varies from -1.57 to -4.14. This provides hints of an elongated structure of the SMC, which we studied using individual distances to the2997 RR Lyrae stars in our sample. § STRUCTURE OF THE SMC§.§ Individual distances of RR Lyrae stars Using the Fourier parameters of OGLE IV light curves<cit.> estimated individual metallicities on the <cit.> and <cit.>metallicity scales for3560 RRab stars in the SMC.We found that2426 out of2997 RR Lyrae stars in our sample have individual metallicities estimated by <cit.>. We used the metallicity estimates on the Zinn & West metallicity scale from <cit.> for2426 RR Lyrae stars in the sample, and to the571 RR Lyrae stars without an individual metallicity estimate we assigned the mean metallicity of the SMC RRab stars determined by the same authors: [Fe/H]_ZW84=-1.85±0.33 dex.These 571 stars are distributed smoothly in the SMC with a slight concentration in the bar-like feature, where the crowding could have prevented an accurate estimation of metallicity. We use our dereddened intensity-averaged K_ s,0 magnitudes (Section <ref>), periods from the OGLE IV catalogue <cit.> and metallicities from <cit.> along with eq. 16 from <cit.> to determine individual distance modulifor each of the2997 SMC RR Lyrae stars. The PM_K_ sZ relation in <cit.> was obtained on the metallicity scale defined in<cit.>. <cit.> pointed out that there is no clear offset between the metallicity scales defined in their paper and <cit.> metallicity scale, and considered the 0.06 dex difference found for the three calibrating clusters used in their analysis as a possible offset. Hence, we added 0.06 dex to the metallicity values provided by <cit.>. However, the dependence on metallicity of the M_K_ s magnitudes is small (0.03, ), hence, a 0.06 dex offset will give only 0.002 mag error to the distance modulus, which is much smaller than typical errors of distance moduli(0.15 mag). Thus, the uncertainty in the offset between two metallicity scales will not affect our final results.Individual distance moduli obtained for the 2997 RR Lyrae stars in the sample are summarised in column 11 ofTable <ref>.The weighted mean distance modulus of the2997 RR Lyrae stars in our sample is (m-M)_0=18.88 mag, with a standard deviation of 0.20 mag.This is 0.08 mag shorter than derived by <cit.> based on a statistical analysis of the SMC distance estimates available in the literature,(m-M)_0=18.96±0.02 mag, but both determinations are entirely consistent with each other within the mutual uncertainties (standard deviations). It is also shorter than the estimates of the SMC's distance modulus based on classical Cepheids observed by VMC(m-M)_0=19.01±0.05 mag, (m-M)_0=19.04±0.06 mag <cit.> and (m-M)_0=19.01±0.08 mag <cit.>. Note that the mean distance modulus derived in the current study reflects only the statistical distribution of individual distance moduli in the sample, and is not a distance to the centre of the ellipsoid formed by RR Lyrae stars. The derived mean value is significantly affected by the spatial distribution of RR Lyrae stars andcannot be considered as a distance to the SMC. The mean distance modulus derived in the current study corresponds to the mean distance 60.0 kpc. The standard deviation of the mean value is ∼5 kpc and reflects the extension of the SMC RR Lyrae star distribution along the line-of sight (see Section <ref>).We also calculated mean distances to the RR Lyrae stars in each of the27 tiles, separately (Table <ref>). The derived distance moduli span the range from 18.78 to 18.94 mag.There is a clear trend of tiles located in the eastern and south-eastern parts of the SMC to be closer to us. However, distances to stars in a giventile calculated in this way are approximate, since the same distance is assigned to all stars in a tile. To perform a more accurate analysis of the SMC'sstructure we used the individual distances of the2997 RR Lyrae variables. In Fig. <ref> we present the distance modulus distributions defined bythe individual RR Lyrae stars in each tile. It shows that the eastern regions have asymmetric distributions of the RR Lyrae stars and are located closer to us. The two-dimensional distribution of distances in the SMC is shown in Fig <ref>. The upper-left, upper-right, lower-left and lower-right panelsshow respectively the closer RR Lyrae stars with (m-M)_0<18.68 mag, the more distant RR Lyrae stars with (m-M)_0 >19.08 mag, the sample within 1σ error of the mean distance modulus and the total sample. RR Lyrae stars at all distances are distributed smoothly. We cannot see any signature of the bar from the distribution of RR Lyrae stars.Similarly to Fig. <ref>, in Fig. <ref> we plot the two-dimensional distribution of extinction, A_K_ s, derived in Section <ref>.The upper-left and upper-right panels of Fig. <ref> show that more distant RR Lyrae stars in general have higher values of extinction than closer stars. The mean extinction of the closest RR Lyrae variables is ⟨ A_K_ s⟩ =0.017 mag which should be compared with ⟨ A_K_ s⟩ =0.023 mag derived for the most distant stars. The highest extinction is observed in the central part and in the active star-forming region in an eastern extension of the SMC called the Wing, located at α = 01^ h 15^ m and δ = -73^∘10^' <cit.>. The concentration of stars with high extinction in the central regions seen in all panels of Fig. <ref> extends from the north-east to south-west and outlinesthe bar-like feature of the SMC. Thus, although we could not detect the bar from the spatial distribution of RR Lyrae stars (Fig. <ref>), the bar-like feature is seen in the two-dimensionaldistribution of the extinction (Fig. <ref>). Note that a higher extinction in the Wing of the SMCis seen for all RR Lyrae stars except the closest ones, which could mean that the star-forming region is located at a greater distance than (m-M)_0=18.68 mag. These findings are in agreement with results obtained by <cit.> and <cit.> who measured the reddening in the SMCusing RC stars detectedby OGLE III andfound the main concentrations of higher reddening in the bar and Wing of the SMC.<cit.> determined individual reddening for OGLE IV RR Lyrae stars and found the south-eastern part of the SMC to contain theregions of the higher extinction, which is in agreement with our findings.We calculated the centroid of our sample as an average of RA and Dec values of the RR Lyrae stars, and found that it is located at α_0 = 00^ h 55^ m 50^ s.97 andδ_0 = -72^∘51^'29.^''27. We divided the whole sample of RR Lyrae stars according to RA in an eastern (RA > α_0) and awestern (RA < α_0) region based on their positions. There are1473 and 1524 RR Lyrae stars in the eastern and the western regions, respectively. The distance modulus distributions (for a bin size of 0.05 mag) of the RR Lyrae variables in the eastern (blue line) and western (red dotted line) regions of the SMC are shown in Fig. <ref>. The two distributions show a difference with an excess (15%) of closer RR Lyrae variables in the eastern region.We note that the error in individual distances has a range from 0.1 to 0.2 mag, with an average error of 0.15 mag. This is larger than the bin size used in Fig. <ref>. We changed the bin size from 0.05 to 0.2 mag and we do observe an excess of ∼ 15% closer RR Lyrae stars in the eastern regions for any choice of the bin size.The mean distances estimated to the eastern andwestern regions of the SMC are, respectively, 59.56 ± 0.14 kpc and 60.50 ± 0.14 kpc, where errors are calculated as the standard deviation divided by the square root of number of stars in the region. This suggests that the eastern region is∼ 0.94 ± 0.20 kpc closer to us than the western region.We considered whether this asymmetry could be caused by an overestimation of the extinction in the eastern region. In order to check this possibility we assigned the mean value of extinctionderived in Section <ref> ⟨ A_K_ s⟩=0.02 mag to all2997 stars in the sample and derived mean distances to the eastern and western regions of the SMC of 59.57±0.14 kpc and 60.40±0.14 kpc, respectively. These distances are in agreement with the values obtained by applying individual extinction corrections and proves that our distance determination from the K_ s magnitudes is not affected by a variation of extinction in the SMC. Furthermore, there is still a difference of 0.83±0.20 kpc between the mean distances of RR Lyrae stars in the eastern and western regions. Thus, the appearance of the eastern region closer to us is not owing to an overestimation of the extinction and rather reflects the actual spatial distribution of the SMC RR Lyrae stars. This is consistent with results obtained by <cit.>, <cit.> and<cit.> from the study of SMC RR Lyrae starsusing the OGLE III data. Recent results from <cit.> based on RR Lyrae stars using the OGLE IV cataloguedo not imply any sub-structuresbut show some asymmetry in the equal density contours in the eastern part of theSMC.They detect a shift in the centre of the ellipsoidal fit, to RR Lyrae stars in different density bins, towards the east and closer to the observer as a function of radius from the SMC centre. This again suggests that the eastern region of the SMC includes a largernumber of closer RR Lyrae stars in the entire OGLE IV data set, in agreement with our results.§.§ Mean distances and line-of-sight depth The variation of the line-of-sight depth across the SMC can be obtained by measuring the mean distance and standard deviation in smaller sub-regions. We divided the SMC region covered by the 27 VMC tiles into smaller sub-regions of equal area (0.6 × 0.5 deg^2), as anticipated in Section <ref> (110 sub-regions in total). The standard deviation with respect to the mean distance is a measure of the line-of-sight depth. Along with the actual line-of-sight depth in the distribution, the standard deviation has contributions from the intrinsic magnitude spread of RR Lyrae stars owing to errors in photometry and metallicity effect, as well as intrinsic luminosity variations owing to evolutionary effects. To perform an approximate estimation of the intrinsic magnitude spread we analysed the SMC globular cluster NGC 121. This is the oldest SMC cluster. It contains four RR Lyrae stars <cit.>,which were confirmed to be cluster's members based on their location close to the cluster's centre and their distribution on the Horizontal Branch of the cluster's CMD (Clementini, private communication).As part of our analysis we derived dereddened K_ s,0 magnitudes for three of them: V32, V35 and V37 (identification from ). The RR Lyrae starV36 was not included in our analysis since it turned out to be an RRc star according to the OGLE IV catalogue <cit.>. We usedK_ s,0 magnitudes from VMC, periods from the OGLE IV catalogue, the metallicity value of[Fe/H]=-1.51 dex <cit.> and eq. 16 from <cit.> to determine individual distance moduli of the RR Lyrae stars in NGC 121. Themean distance modulus of NGC 121 from the RR Lyrae stars is (m-M)_0=18.97±0.07 mag. Since the three variables belong to the same cluster we consider the depth effect negligible, hence, the standard deviation of the mean value,0.07 mag,represents the intrinsic magnitude spread caused by the effects described above.<cit.> estimated the intrinsicV-magnitude spread of 101 RR Lyrae stars in the LMC caused by the internal photometric errors, metallicity distribution and evolutionary effect as 0.10 mag. Since the dependence of the K_ s magnitudes on metallicity and evolutionary effect is smaller than in V band, our estimation of intrinsic dispersion in the SMC as 0.07 mag is reasonable, even though it is based only on three stars. The actual line-of-sight depth in the 110SMC sub-regions is estimated as σ^2_ los = σ^2_ measured - σ^2_ intrinsic. Figs. <ref> and <ref> show the two-dimensional distributions we obtain formean distance and actual line-of-sight depth, respectively. The mean distance map clearly shows that the eastern sub-regions are closer to us, as it was widely discussed above. The actual line-of-sight depth values range from1 to 10 kpc, with an average depth of4.3 ± 1.0 kpc. Fig. <ref>shows that the largest depth is found in the central regions of the SMC. The standard deviation associated with the mean distance modulus of our entire sample is 0.2 mag (Section <ref>). After correcting for the intrinsic width, this dispersion corresponds to a line-of-sight depth of 5.2 kpc. The tidal radius of the SMC is estimated as 4 – 9 kpc by <cit.>, 7 – 12 kpc by <cit.> and <cit.> estimated the SMC edge in the eastern direction to be at a radius of 6 kpc. <cit.> found the presence of old/intermediate-age populations at least up to a radius of 9 kpc.All these values are comparableto the 1σ depth of 10 kpc observed in the central regions of the SMC.§.§ Three dimensional structure of the SMC The Cartesian coordinates corresponding to each RR Lyrae star can be obtained using the star's RA, Dec coordinates and the distance modulus. We assume the x-axis antiparallel to the RA axis, the y-axis parallel to the Dec axis, and the z-axis along the line-of-sight with values increasing towards the observer. The distance modulus is used to obtain the distance to each star in kpc. The RA, Dec and the distance are converted into x, y, z Cartesian coordinates using the transformationequations given by <cit.> and assuming the origin of the system at α_0 = 00^ h 55^ m 50^ s.97 andδ_0 = -72^∘51^'29.^''27. The distance to the origin of the SMC is taken as 60 kpc with standard deviation of 5 kpc (Section <ref>).RA and Dec of each star correspond to the VMCcoordinates. The positional accuracy of the VMC survey is ∼ 0.02^''. The errors associated with x, y and z are calculated using error propagation.Based on both observations and theoretical studies ( and references therein), the old and intermediate-age stellar populations in the SMC are suggested to be distributed in a spheroidal/ellipsoidal system. Our sample of SMC RR Lyrae stars shows a smooth distribution on the sky. Thus we modelled the RR Lyrae stellar distribution as a triaxial ellipsoid. The parameters of the ellipsoid, such as the axes ratio, the position angle of the major axis of the ellipsoid projectionon the sky (ϕ) and the inclination of the longest axis with respect to the sky plane (i) are estimated using a method of inertia tensor analysis similar to that described in<cit.> (also refer to ; ). The basic principle is to create an inertia tensor of the x, y, z coordinates and estimate the Eigen vectors and Eigen values. The Eigen vectors correspond to the spatial directions and the square roots of the Eigen values correspond to the axes ratio.First, we applied the method to the (x,y) coordinates and found that the stars are located in an elongated distribution with an axes ratio of1:1.12(±0.01) and the major axis has a position angleϕ = 85^∘.5±0^∘.1. We repeated the procedure with (x,y,z) coordinates. The z values have a large scatter, owing to large uncertainty in distances, compared to good positional accuracy (0.02^''). As described in <cit.>, to correct for this internal scatter in the z value, we subtracted the internal scatter of 0.07 mag from the z-component (see Section <ref>).We obtained axes ratio, ϕ and i values of1:1.11(±0.01):3.30(±0.70), 84^∘.6±0^∘.2and 2^∘.1±0^∘.6,respectively. The small inclination angle of the longest axis with respect to the sky plane suggests that the longest axis of the ellipsoidal distribution of the RR Lyrae stars is oriented nearly along the line-of-sight. A compilation of structural parameters of the SMC ellipsoid from the RR Lyrae stars available in the literature is provided in Table <ref>. In the majority of these studies the RR Lyrae stars from the OGLE III catalogue were used, while<cit.> analysed RRab stars in the OGLE IV catalogue, which covers almost entirely the SMC. In the current study we used RR Lyrae variables observed by OGLE IV which also have counterparts in the VMC catalogue. Qualitatively all previous results indicate that the RR Lyrae stars in the SMC are distributed in an ellipsoid with the longest axis oriented almost along the line-of-sight. Our results also support these findings, but quantitatively there are some differences. This could be mainly owing to the difference in the spatial coverage of the data used in the various studies.Fig. <ref> shows the distribution of RRab stars used in the this paper and RRab stars from the OGLE III and OGLE IV catalogues. OGLE IV covers a larger area than VMC and OGLE III, which are basically focused on the central regions of the SMC, where the majority of RR Lyrae stars are concentrated.§.§ Effect of interaction of the Magellanic Clouds Mutual interactions between the Magellanic Clouds and the resultant tidal stripping of material from the SMC are believed to be the most probable scenario for the formation of the MB (; ). One of the challenges to this scenario is the lack of conclusive evidence for the presence of tidally stripped intermediate-age/old stars in the MB.<cit.> identified a closer (distance, D ∼ 55 kpc) stellar structure in front of the main body of the eastern SMC, which is located 4^∘ from the SMC centre. These authors suggested that it is the tidally stripped stellar counterpart of the H I in the MB. <cit.> also identified a foreground population (at ∼12±2 kpc in front of the main body of the SMC), whose most likely explanation is tidal stripping from the SMC. Moreover, they identified the inner region (at ∼ 2-2.5 kpc from the centre) from where the signatures of interactions start becoming evident, thus, supporting the hypothesis that the MB was formed from tidally stripped material from the SMC. Both these studies were based on RC stars. Recently, <cit.> found that the young (∼120 Myr) and old(∼220 Myr) Cepheids in the SMC have different geometric distributions. They also found closer Cepheids in the eastern regions which are off-centred in the direction of the LMC. They suggest that these results are owing to the tidal interaction between the Magellanic Clouds. The RR Lyrae stars which are older (age ≥ 10 Gyr) than RC stars (age ∼ 2 – 9 Gyr) and Cepheids (100 – 300 Myr) are also expected to be affected by tidal interactions. The RR Lyrae stars are well identified and we have accurate distances to each individual star. The presence of tidal signatures in the oldest populations would provide strong constraints to theoretical models which explain the formation of the MS. Hence, we explore our data set in more detail. In the present study we see a shift in the mean distance of the RR Lyrae stars in the easternregion compared to the western region. We also see that RR Lyrae starsin the eastern tiles haveasymmetric distance distributions (Fig. <ref>) andthat the mean distances of the eastern sub-regions are shorter (Fig. <ref>). Fig. <ref> presents the radial variation of the mean distance to the various sub-regions. Blue filled hexagons representsub-regions in the eastern and red filled triangles represent sub-regions in the western parts of the SMC. The plot shows that the eastern sub-regions are located at shorter distances compared to the western sub-regions. Beyond 2^∘ in radius from the centre, the majority of the eastern sub-regions have shorter distances. This angular radius corresponds to a linear radius of ∼ 2.2 kpc from the SMC centre and is similar to the radius from where the signatures of interactions, in the form of bimodality in the distribution of RC stars, start to become evident <cit.>. We point out that the typical error associated with the mean distances of the sub-regions is ∼ 1 kpc (as shown in the top-right of the plot). Thus, our results indicate that the oldest stellar population in the eastern part of the SMC, in the direction of the MB, is affected by the interactions of the Magellanic Clouds, 200-300 Myr ago,which is believed to be mainly responsible for the formation of the MB.However, in our present study using RR Lyrae stars, we do not see a clear bimodality found by <cit.>in the RC distribution, where two peaks separated by ∼ 12 ± 2 kpc are observedin the eastern part of the SMC. While comparing our Fig. <ref> with fig. 6 of <cit.>, the discrepancy is very evident for tiles 6_5 and 5_6. This implies that RC stars and RR Lyrae stars are distributed in different structural components in the eastern part of the SMC and/or a different origin of the foreground RC population. <cit.> results on the RC stars are based on only 13 VMC tiles covering the entire central part and some outer regions. We plan to address the discrepancy observed between RC and RR Lyrae stars distributions in the future based onthe whole sample of SMC tiles observed by VMC. Recently, <cit.> identified tidal tails around the LMC and the SMC using Gaia DR1 <cit.>. They found that the SMC's outer stellar density contours show a characteristic S-shape which is a typical signature of tidal stripping seen in satellite galaxies. They also identified a trailing arm (motion of the SMC with respect to the LMC) from the SMC which extends towards the LMC. This stellar tidal tail containscandidate RR Lyrae stars off-set by ∼ 5 deg from the gaseous MB. Both the old and young MB components originate in the south-eastern part of the SMC (∼ 6 deg east and 3 deg south from the SMC centre), where the authors claim see the trailing tail. Though our observed tiles do not cover the region where <cit.> identified the SMC's trailing arm, the eastern and south-eastern regions where we see closer RR Lyrae stars are in the direction of the newly identified SMC's trailing arm. The study of different stellar populations including RR Lyrae stars in the outer VMC tiles will be useful to confirm and eventually understand this newly identified tidal signature. According to the simulations, the line-of-sight velocities of the stellar debris from the SMC and the LMC are significantly different, hence, a detailed spectroscopic study of RR Lyrae stars and other distance indicators in the eastern regions of the SMC will also provide valuable information to understand the interaction history of the MS. § CONCLUSIONS In this study we analysed the structure of the SMC for the first time using multi-epochnear-infrared photometry of RR Lyrae stars observed bythe VMC survey. For this analysis2997 RRab stars distributed in27 VMC tiles with visual photometry and pulsation periods from the OGLE IV survey, were used.The well-sampled light curves in the K_ s band obtained by VMC allowed us to derive accurateintensity-averaged magnitudes for each RR Lyrae star in the sample. Individual reddening valueswere calculated using the V and I magnitudes from the OGLE IV survey. We fit the PK_ s relations of RR Lyrae stars in each tile, separately,and found a significant dispersion which is mostly owing to a depth effect. To study the three-dimensional structure of the SMC we derived individual distances to the 2997 RR Lyrae stars in the sample by applying the near-infrared PM_K_ sZ relation defined in <cit.> tointensity-averaged K_ smagnitudes from the VMC survey, periods from the OGLE IV catalogue and photometrically determined metallicities from <cit.> which we transformedto the metallicity scale defined by<cit.>. The RR Lyrae variables in the SMC are found tohave a roughly spheroidal or ellipsoidal distribution. We modelled the distribution of the SMC RR Lyrae stars as a triaxial ellipsoid and found the values of the axes ratio, position angle of the major axis of the ellipsoid projected on the skyandinclination of the longest axis with respect to the sky planeusinginertia tensor analysis. The parameters of the SMC structure are: axes ratio =1:1.11(±0.01):3.30(±0.70), 84^∘.6±0^∘.2and 2^∘.1±0^∘.6.The results obtained in this paperare generally consistent with previous studies. Existing discrepancies are mostly owingto differences in the areaof the SMC covered by the different studies. The actual line-of-sight depth of the SMC has values in the range from1 to 10 kpc, with an average depth of 4.3±1.0 kpc. The central parts of the SMC have larger depth. Taking into account the standard deviation associated with the mean distance modulus of our entire sample(m-M)_0=18.88±0.20 mag we estimated aline-of-sight depth of the SMC as 5.2 kpc. The spatial distribution of the RR Lyrae stars in our sample does not show features typical ofyoung stellar populations, such as a bar. However, from the two-dimensional distribution of the extinction at different distances we can see the bar-like feature and the star forming region in the eastern Wing of the SMC. From our analysis of the SMC structure we concluded that the eastern part of the SMC is located closer to us than the western part. In eastern regions, beyond 2 deg in radius from the centre, the majority of the sub-regions have shorter distances. The regions where we see alarge number of closer RR Lyrae stars are in the direction of SMC's trailing arm newly identified with Gaia <cit.>.All these resultsindicate that the oldest stellar population in the eastern part of the SMC is affected by interactions of the two Magellanic Clouds, occurred about 200-300 Myr ago which are believed to be mainly responsible for the formation of the MB. Further study of RR Lyrae stars in the MB and outer regions of the LMC and SMC using near-infrared photometrywill provide valuable information to understand the interaction history of the MS. § ACKNOWLEDGEMENTSSupport for this research has been provided by PRIN INAF 2014 (EXCALIBURS, PI G. Clementini). We thank the Cambridge Astronomy Survey Unit (CASU) and the Wide Field Astronomy Unit (WFAU) in Edinburgh for providing calibrated data products under the support of the Science and Technology Facility Council (STFC) in the UK. S.S acknowledges research funding support from Chinese Postdoctoral Science Foundation (grant number 2016M590013).M.-R.C. acknowledges support from STFC (grant number ST/M00108/1) and from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 682115). R. d. G. acknowledges financial support from the National Natural Science Foundation of China through grants 11373010, 11633005 and U1631102.The table is published in its entirety in the electronic version of the paper. A portion is shown here for guidance regarding its format and content. 99 [Belokurov et al.2017]Bel2017 Belokurov V., Erkal D., Deason A. J., Koposov S. E., De Angeli F., Evans D. W., Fraternali F., Mackey D.,2017, , 466, 4711[Besla et al.2012]Besla2012 Besla G., Kallivayalil N., Hernquist L., van der Marel R. P., Cox T. 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"authors": [
"T. Muraveva",
"S. Subramanian",
"G. Clementini",
"M. -R. L. Cioni",
"M. Palmer",
"J. Th. van Loon",
"M. I. Moretti",
"R. de Grijs",
"R. Molinaro",
"V. Ripepi",
"M. Marconi",
"J. Emerson",
"V. D. Ivanov"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170926144900",
"title": "The VMC survey - XXVI. Structure of the Small Magellanic Cloud from RR Lyrae stars"
} |
Modeling WiFi Traffic for White Space Prediction in Wireless Sensor Networks Indika S. A. Dhanapala^1,*, Ramona Marfievici^1, Sameera Palipana^1, Piyush Agrawal^2, Dirk Pesch^1 ^1Nimbus Centre for Embedded Systems Research, Cork Institute of Technology, Cork, Ireland ^2United Technologies Research Centre, Cork, Ireland ^*Contact Author: December 30, 2023 ==========================================================================================================================================================================================================================================================================================================Cross Technology Interference (CTI) is a prevalent phenomenon in the 2.4 GHz unlicensed spectrum causing packet losses and increased channel contention. In particular, WiFi interference is a severe problem for low-power wireless networks as its presence causes a significant degradation of the overall performance.In this paper, we propose a proactive approach based on WiFi interference modelling for accurately predicting transmission opportunities for low-power wireless networks. We leverage statistical analysis of real-world WiFi traces to learn aggregated traffic characteristics in terms of Inter-Arrival Time (IAT) that, once captured into a specific 2nd order Markov Modulated Poisson Process (MMPP(2)) model, enable accurate estimation of interference. We further use a hidden Markov model (HMM) for channel occupancy prediction. We evaluated the performance of [i)]* the MMPP(2) traffic model real-world traces and an existing Pareto model for accurately characterizing the WiFi traffic and,* compared the HMM based white space prediction to random channel access.We report encouraging results for using interference modeling for white space prediction.Wireless Sensor Networks, WiFi Traffic Modelling, Interference Prediction, Markov Modulated Poisson Process, Hidden Markov Model § INTRODUCTION Wireless technologies operating in unlicensed radio spectrum, such as the 2.4 GHz ISM band, suffer from Cross Technology Interference (CTI), which is the time and frequency overlapping of concurrent transmissions. The interference occurs due to the broadcast nature of wireless transmissions of co-located devices whose radios are based on different technologies such as WiFi (IEEE802.11), Bluetooth (IEEE802.15.1) or IEEE802.15.4 and who cannot coordinate their transmissions. As a device cannot decode a data packet or MAC frame sent by a device of a different technology, which uses different modulation and coding schemes, CTI reduces the devices' ability to decode its own receiving signals. Consequently, CTI creates packet/frame losses, increases channel contention which increases delay, and ultimately under–utilizes the scarce frequency spectrum. CTI caused by high power transmitters affects the lower power devices in particular, this is a problem for IEEE802.15.4 based WSNs in the 868 MHz and 2.4 GHz band as it reduces transmission opportunities and increases packet losses <cit.>.WiFi is a particular problem as it dominates the 2.4 GHz frequency band with much high transmission powers (20 dBm) than other technologies. In this study, we only consider coexistence of WiFi and IEEE802.15.4 based WSNs but the principles are fundamentally applicable to other technologies as well.As IEEE802.11 is a higher bandwidth technology than IEEE802.15.4, e.g. bandwidth ranges between 20 MHz and 40 MHz depending on the specific PHY layer specification, it creates interference for multiple 2 MHz IEEE802.15.4 channels simultaneously (see Figure <ref>). This effect is called aggregated WiFi interference and reduces white spaces (the transmission free time periods between consecutive MAC frame transmissions) for WSNs, reduces transmission opportunities for WSNs and degrades WSN performance in terms of reliability and lifetime. Furthermore, the higher data rate of WiFi compared to IEEE802.15.4, e.g. 1 Mbps to 300 Mbps (depending on PHY layer) versus 250 kbps creates rapid variations in time and frequency domain, which are hard to detect by the typical IEEE802.15.4 sensor node receiver hardware.Existing solutions can be divided into reactive and proactive approaches to address CTI between WiFi and WSNs.Reactive approaches include frequency domain solutions such as spreading and frequency hopping <cit.>, the use of multiple antennae and transceivers to allow switching between different frequency bands when the current operating frequency band is interfered <cit.>, forward error correction schemes <cit.> and other time domain solutions such as CSMA/CA based on listen before talk (LBT) <cit.>.Proactive approaches try to predict white spaces, e.g. transmission opportunities for the low power technology when the high power technology is not transmitting <cit.>. However, reactive approaches only deal with the problem when it occurs, thus negatively affecting performance and many approaches require changes in the respective standards, e.g. spreading codes, forward error correction or changes in the medium access protocol. On the other hand, proactive approaches can be used to quantify and predict when interference occurs if accurately modelled and can be used to predict when its best for the low power technology to transmit. In this paper, we propose a proactive time domain technique to address CTI for WSNs. Our approach is based on [i)]* an accurate WiFi traffic model to capture aggregated WiFi interference, and* a model-basedtechnique to predict white spaces for IEEE802.15.4 transmissions. The approach is motivated by our findings that WiFi traffic exhibits self–similarity at different time scales, which we found when examining real-world WiFi traffic traces and their packet Inter–Arrival Times (IAT) at different time resolutions (see Figure <ref>). We selected a 2nd order Markov Modulated Poisson Process (MMPP(2)) to model the WiFi packet IAT and a Hidden Markov Model (HMM) for predicting the white spaces. Our motivation for using an MMPP(2) model is that we need to model WiFi traffic for both saturated (i.e, busy) and unsaturated channel characteristics and we need to capture the self-similarity of WiFi traffic with only a few model parameters that are easily measurable. We learn WiFi aggregated traffic characteristics in terms of WiFi packet IAT from real-world WiFi traces and use those to calibrate the parameters of the MMPP(2) model, which provides us with an accurate model of aggregated WiFi interference. We then use the HMM for channel occupancy prediction, which is trained using the real-world WiFi traces and the MMPP(2) model. This provides us with a prediction when the channel is interference free and available to access. We have evaluated the performance of the MMPP(2) traffic model with respect to the traces we collected and an existing Pareto model <cit.> and compared our HMM based channel access with 0.5-persistent random access technique.Our analysis shows we can successfully predict the presence of white spaces. The rest of the paper is organised as follows. We provide a summary of existing techniques for interference detection and modelling for IEEE802.15.4 based WSNs in Section <ref>. Then, we discuss important properties of WiFi traffic and introduce our approach in Section <ref>. The performance evaluation of the approach is presented in Section <ref>. We discuss limitations of our work in Section <ref>, before the brief concluding remarks of Section <ref>. § RELATED WORKDetecting and classifying interferenceSeveral works aim to measure and understand the impact of interference on WSNs, and classify interfering sources <cit.>. Musaloiu and Terzis <cit.> used based features to quantify the interference of all 16 IEEE802.15.4 channels to select the least interfered channel. Noda et al. <cit.> compute the ratio of channel idle time and channel busy time for assessing channel quality in the presence of interference. SpeckSense <cit.> classifies bursts to characterize the channel as periodic, bursty or a combination of both. SoNIC <cit.> uses information from corrupted packets, therefore not incurring additional energy cost for sampling like the previous methods, to classify the sources of interference. However, these works succeed in detecting and identifying interference but it is not clear how these techniques are useful for autonomous interference mitigation due to the diversity of interferers. As a solution, TIIM <cit.> extracts features from corrupted packets to quantify the interference conditions instead of identifying the interferer, thereby, the interference condition can be mapped to a specific mitigation technique. Nonetheless, TIIM does not present an implementation of these mitigation techniques; only in their follow up work, CrossZig <cit.>, the authors have implemented an adaptive packet recovery and an adaptive FEC coding for this.All these solutions, however, are reactive, depending on the prevailing channel conditions, and do not aim to predict the white spaces through modeling,which is instead our goal in this paper.Modeling the interferenceCreating lightweight models of interference is not a trivial task. Several researchers have proposed models for channel occupancy <cit.> or models for emulating interference caused by WiFi and Bluetooth <cit.>.Stabellini and Zander <cit.> consider a general inter network coexistence scenario from a receiver centric perspective. They propose a two–state semi–Markov model, in which, at a given time instant, a channel is identified as Busy if a device transmits or Free otherwise. They exploit the model for each node to identify the less interfered channel and to switch accordingly.In comparison, we consider the WiFi traffic perspective, and not that of a sensor receiver centric approach, and identify only the available white spaces of a specific channel through modeling WiFi traffic in time domain. Boano et al. <cit.> define two–state semi–Markov model for channel occupancy and use noise measurements to measure the duration of the Free and Busy instants, and compute the CDF of Free and Busy periods. Based on the longest Busy period, the authors derive MAC protocols parameters such that the application requirements are met.JamLab <cit.> models and regenerates interference patterns of different sources such as WiFi, Bluetooth and microwave ovens using sensor nodes. For modeling WiFi traffic, the authors considered both saturated (always Busy) and unsaturated traffic scenarios. Saturated traffic is modeled with a Markov Chain model, whilefor the non–saturated WiFi a probability mass function of empirical data is used. In contrast, our goal is not to emulate WiFi traffic but to estimate it, and for this we use a single Markov Modulated Poisson Process (MMPP) to capture the existing traffic conditions, both saturated and non–saturated.The work presented in <cit.>, instead, is closely related to our work since it focuses on a model–based white space prediction mechanism for WSNs in the presence of WiFi interference. Nevertheless, the model technique they used, Pareto, is a heavy–tailed distribution which usually captures long–range dependent (LRD) traffic behavior. However, our collected WiFi traces indicate that WiFi traffic is not exhibiting long–range dependency in the long run. Therefore, we use an MMPP model as itis not based on the LRD assumption. § APPROACH We provide the required background in traffic characterization in Section <ref> and then describe in details our approach. We begin by describing in Section <ref> the elements that are critical when using our approach for predicting white spaces. An integral element is the understanding of the WiFi traffic characteristics described in Section <ref> and the actual modeling of the WiFi aggregated traffic, described in Section <ref>. Finally, in Section <ref> we introduce the HMM model supporting the prediction of white spaces.§.§ Preliminaries Self–similarity A number of empirical studies of traffic measurements <cit.> have convincingly demonstrated that thenetwork traffic is self–similar. Intuitively, self–similarity describes the phenomenon in which certain characteristics (i.e., structural patterns and statistical properties) of the traffic are preserved irrespective of scaling in space or time, which can actually be exploited to better infer traffic properties.To clarify our terminology, we briefly summarize the definition of a few basic concepts. Let X(t) = (X_t: t ≥ 0) be a stationary process, i.e., its joint distribution across a collection of times t_1, ⋯, t_N is invariant to time shifting. X is called self–similar if: X(at) = a^H X(t), a>0where the equality refers to equality in distributions, a is a scaling factor, and self–similarity parameter H is called the Hurst parameter. The proposed approach takes advantage of this property,depicted in Figure <ref> for a set of our traces, to infer traffic characteristics without incurring excessive data trace collection. Traffic statistical propertiesKey statistical metrics typically used to provide insights on network traffic are: mean (μ), standard deviation (σ) and, coefficient of variation (C) computed as σ/μ. Moreover, the predominant way to quantify self–similarity is through H, which is a scalar. H takes on values from 0 to 1, with a value of 0.5 indicating the absence of self–similarity, and the closer H is to 1, indicating the greater the self–similarity. Calculating this parameter is not that straightforward: firstly, it can only be estimated, secondly, although there are several methods to estimate it, they often produce conflicting results. In our approach we used Peng, Periodogram and Box–Periodogram methods to estimate H as they can produce a good estimate of H for a sample size as low as 7000 <cit.>. The median of the three estimated Hurst parameters is used, because the median is more stable than the mean even when one of the estimators fails. §.§ OverviewFigure <ref> illustrates the steps we follow for predicting white spaces for WSNs. To achieve our goal, we must start from assessingquantitatively the characteristics of the WiFi traffic in the operating environment. To this end, we collect a set of WiFi traces of length x seconds and, in case of overlapping WiFi channels, we merge them to replicate the WiFi interference as seen by WSNs. Then, we extract the packet IAT distribution of the aggregated trace and characterize it in terms of mean (M_1), coefficient of variation (C) and Hurst parameter (H). All the aforementioned statistical parameters are exploited to tune an MMPP(2) model for estimating the aggregated WiFi traffic for y seconds. Then, we develop a Hidden Markov Model (HMM) with two states, Free and Busy, which is key for predicting transmission opportunities at run–time. The initial state probabilities are determined using the steady state probabilities of the MMPP(2) model, while the training is done using a set of WiFi traces of length z seconds.§.§ Determining WiFi Traffic CharacteristicsThe approach we describe in this paper is based on an MMPP(2) model that: [i)] * assumes traffic exhibits self-similarity and,* uses empirical data for estimating its parameters. Therefore, we start by acquiring raw real–world WiFi traces from the operating environment and then process them in three steps: * aggregating: it aggregates WiFi traces of multiple overlapping WiFi channels as seen by WSNs, i.e., concatenating raw traces and ordering them as a function of their timestamps. The output of this step is a trace of aggregated WiFi traffic; * traffic characteristics extraction: the IAT distribution and its corresponding statistics in terms of mean (M_1), coefficient of variation (C) and Hurst parameter (H) are determined from the aggregated WiFi traces. * verifying self–similarity property: the self–similarity property of the aggregated WiFi trace is determined by checking if the Hurst parameter (H) satisfies the following condition: H ∈ (0.5,1). §.§ Estimating WiFi Traffic We now describe how we exploit the processing just described towards building estimates of aggregated WiFi interference.To this end we use an MMPP(2) model, as shown in Figure <ref>, whose defining parameters are (<cit.>):Q = [ -r_1 -r_1; -r_2 -r_2 ] ,Λ =[ λ_1 0; 0 λ_2 ],π = 1/r_1 + r_2[ r_2 r_1 ]where Q represents the infinitesimal generator, Λ is the matrix of the Poisson arrival rates, and π is the initial probability vector of the underlying Markov process.According to <cit.>, an MMPP(2) process can be approximated by a second-order hyperexponential distribution, with parameters μ_1, μ_2 and p, for fitting empirical packet IAT distribution to the model. As shown in Figure <ref>, μ_1, μ_2 are mean packet arrival rates and p represents the probability at which traffic is generated at a mean rate of μ_1.We exploit the output of the traffic characterization extraction, computed mean time between arrival (M_1) and the coefficient of variation (C), to automatically compute the parameters p, μ_1 and μ_2 of the hyperexponential distribution, through the balanced means method <cit.>: p= 1/2( 1 + √(C^2 - 1/C^2 + 1)), μ_1 = 2p/M_1, μ_2 = 2(1-p)/M_1Then, the parameters of the MMPP(2) model, λ_1, λ_2, r_1 and r_2 can be calculated from the estimated parameters p, μ_1, μ_2 and the Hurst parameter H, through: λ_1= 1/2[ p(1-β)(μ_1-μ_2) + βμ_1 + μ_2 + √(ξ)]λ_2= μ_1μ_2[λ_1 - p(μ_1 - μ_2) - μ_2]/λ_1μ_1 - λ_1p(μ_1 - μ_2) - μ_1μ_2 r_1= (μ_1 - λ_1)(μ_2 - λ_1)/λ_2 - λ_1 r_2= (λ_2 - μ_1)(λ_1 + r_1 - μ_1)/μ_1 - λ_1whereξ = [ p(1 - β)(μ_1 - μ_2) + βμ_1 + μ_2 ]^2 - 4βμ_1μ_2,β = 2 - 2HThe approximation expressed through Eq. <ref> can be applied if and only if WiFi packet IAT distribution satisfies both conditions: 0.5 < H < 1 and C > 1. In all the other cases, when 1/√(2)≤ C ≤ 1, a second–order Coxian distribution <cit.> must be used prior to the second order hyperexponential distribution. In this case, parameters p, μ_1 and μ_2 are computed, through:p = 1/2C^2, μ_1 = 2/M_1( p/1+p), μ_2 = 2/M_1 Note that C<1/√(2) and H<=0.5 case was not considered in this work.This stage enables us to estimate and generate aggregated WiFi interference considering the particular profile of the observing environment.§.§ Predicting White Spaces As discussed in Section <ref>, the contribution we put forth here is an approach for predicting transmission opportunities for WSNs in the presence of WiFi interference.Next, we describe how we exploit the output of the previous stage, the estimated aggregated interference, along with real–world WiFi traces, to train an HMM that enables us to predict the wireless channel state. We adopt the notation presented in <cit.> to indicate the complete parameter set of the HMM: * hidden (unobserved) statesS = {s_1, s_2}: correspond to two different regimes of the wireless channel, Free and Busy;* initial state probabilities π: in this context, they correspond to the steady state probabilities of the MMPP(2) model;* observationsO = {o_1, o_2}: correspond to two different values of the mean packet IAT of the MMPP(2) modeled trace, IAT_small and IAT_large. We define a threshold computed as the average over all samples' means for the training phase, and as a moving average of the mean WiFi packet IAT for the prediction phase. The choice of the moving average is motivated by the need to dynamically adapt the HMM to the operating environment. * state transition probability matrix A: models the evolution of the wireless channel as transitions among the set of unobserved states;* observation probability matrix B. For a more in–depth discussion on how the observations are obtained and training procedure of the model, the reader shall refer to Section <ref>.The model parameters A and B are properly initialized for the data set under consideration using uniformly distributed probabilities matrices and recomputed using the Baum–-Welch algorithm <cit.>. § PERFORMANCE EVALUATIONWe validate our approach for the white space prediction by: [i)]* conducting a statistical comparison between empirical data traces (training set) we collected in indoor deployments, traces from the MMPP(2) model, and traces from a state-of-the-art Pareto model; * comparing our predictions from HMM model with a 0.5–persistent random access method.First, we present the experimental setup and how we acquired in-field WiFi traces. Then, we report and discuss our experimental results and show how to improve the accuracy by system calibration.§.§ WiFi Traces and Their CollectionLocationIn our study, we considered two indoor environments, and . This choice allowed us to exploit environments with different WiFi traffic saturations and to validate our approach under different conditions. While the has bursty traffic, it exhibits less self-similarity than the one in ; confirmed by the values of H: 0.5 for and 0.7 for .Hardware/software platform The WiFi dataset we use has been acquired by USB WiFi dongles that are 802.11n compliant. In both environments, the dongles were connected to a line-powered USB hub connected to a PC.Because we are interested in exploiting IAT distributions, we created records of traces, i.e., pcap files, of timestamped WiFi traffic using the tcpdump tool. Then, a WiFi aggregated interference trace was obtained by merging traces of overlapping WiFi channels, i.e., traces from overlapping channels were appended and sorted in ascending order based on their timestamps. To this end, we used mergecap tool and then tshark to extract the IAT distribution. Data collection executionTo be useful, a model should accurately model traffic on any channel. In the environment we collected traces from four WiFi channels, i.e., 1–4, ensuring each WSN overlapped channel, i.e., 11–14, is affected by a different number of WiFi channels, as can be seen in Figure <ref>. In the environment we wanted to collect traces under the same conditions as in but the environment did not exhibit that much WiFi traffic. Therefore, we resorted to sniffing WiFi channels 7 to 13 which overlap with WSN channels 20 to 23, each being affected by four WiFi channels. Moreover, to induce diversity in traffic on measured channels in environment we generated a continuous trace of video-streaming, using a laptop, on WiFi channel 11 which only overlaps with WSN channels 21, 22, 23.The WiFi traces were collected during the day, for two hours, from 10:30AM to 12:30PM, in and . These data collection settings allow us to gather traces from channels in different environments that exhibit channel interference pattern diversity. §.§ System Calibration We start the evaluation by calibrating the training duration xand modelling duration y of MMPP(2) model, and the training duration z of the HMM model, defined as in Figure <ref>.MMPP(2) calibration The performance of the MMPP(2) model depends on parameters x and y.For a low x, the statistics from the training traces can not correctly capture the WiFi traffic behaviour, as the number of samples in such a small window is limited. Our logs indicate that a value ofx ≤ 1 second leads to this. A larger x is equivalent with an increase in the length of the training trace which is prohibited in resource-constrained settings like the ones we target. Because we are interested in predicting the short-term behaviour of WiFi traffic, y should be as small as possible. Moreover, in order to accurately model WiFi traffic,the MMPP(2) model must generate traffic from both its states at least once. This leads to define a lower bound for y denoted as y_lb=1/r_1+1/r_2, where 1/r_i represents the duration of a state in the MMPP(2) model. Therefore, during the calibration process, we used y as an integer multiple of its lower bound y_lb in order to find its optimum value (y=y_lb× k, k ∈ℤ^+).As our goal is to reduce the RMSE between the modeled traffic and the testing set, we first played with different values for x and y to get a better understanding of their impact on the RMSE. Next, we fixed y at k=1 in and k=2 in , and vary x as an exponential function starting from 60 seconds to 40 minutes.Overall, all of our evaluations, as depicted in Figure <ref>, show that a value of 1 for k and 300 seconds for x and 2 for k and 500 seconds for x provide a minimum RMSE in and , respectively. HMM calibration The performance of the HMM model depends on the size z of the training set. Our goal is to choose z that maximizes the hit rate and minimizes the FDR. For this we used the precision-recall curve analysis <cit.>, see Figure <ref>, looking at the impact of z on the variation of hit rate and the precision (1-FDR) metric. Overall, all our evaluations show that a value of 300 seconds for and 960 seconds for satisfy the aforementioned criteria. Since we want to avoid the dependency between parameter z and the WSN channel, we consider the average of hit rate and FDR over all channels.§.§ MMPP(2) Model ValidationTo evaluate the performance of our approach, we trained the MMPP(2) model with traces from the collected data traces from different environments and channels (i.e., length of training set: in for channels 21–23 is 6500 samples, for channel 20 is 3500 samples as it is not under the interference of the video-streaming traffic; in for channels 11–13 is 16000 samples, for channel 14 is 28000 samples as it is the channel affected by the highest number of WiFi interferers). We compare on the basis of RMSE statistics computed on the generated trace from the MMPP(2) model and a state-of-the-art Pareto model <cit.> versus an unseen trace (testing set) from the collected data traces.Both models were trained with the same data trace, moreover, Pareto's scaling parameter was set to the maximum packet duration of a WSN packet. i.e., 4.256 ms, representing the minimum white space duration. Figure <ref> shows the results from our comparison in terms of percentage RMSE. A few trends are clearly identifiable. First, the accuracy of the estimation decreases as one goes from to for both models–i.e., as the quantity of the interference decreases. However, this is not true for MMPP(2) on channel 14 in and 20 in .Second, the accuracy of the estimation of MMPP(2) model is always higher than the Pareto model. The difference is more marked on channel 11 in and channel 20 in , Figure <ref> showing a difference of 6.6 ms and 244.5 ms, respectively. However, channel 14 in shows a different behaviour, Pareto delivering best performance on this channel even if there is more traffic. From Table <ref>, one can see thaton channel 14 there is a higher traffic, i.e., higher number of samples and lower mean IAT 10.7 ms, than on the other channels, while the Hurst parameter has the lowest value 0.51, which translates to a less self-similar behaviour. The characteristics of the traffic on channel 14 are more favourable for Pareto which captures the mean IAT than for MMPP(2) which requires all the three parameters for traffic modelling. Moreover, on channel 20, which is not overlapped by the video-streaming traffic channel, the mean of IAT is almost double, i.e., 141.5 ms, than on other channels, i.e., 67–82 ms, as can be seen in Table <ref>. As Pareto is a heavy-tailed distribution, it fails to capture the unsaturated traffic on this channel, having an RMSE of 259 ms. On the other hand, MMPP(2) model performs the best, i.e, RMSE of 14.5 ms out of all channels.All these results confirm the versatility of the MMPP(2) for modelling both unsaturated and saturated traffic. §.§ HMM Model EvaluationWe now show that our HMM model approach provides accurate predictions of white spaces and compare it with a 0.5–persistent random access method (i.e., transmission attempt takes place with probability 0.5).We consider four metrics to assess the performance of our approach: [i)]* hit rate,* false discovery rate (FDR),* precision, computed as 1-FDR,and, * F1 score.All metrics are derived from the elements of the confusion matrix in Table <ref>, as follows:Hit rate = TP/TP+FN, FDR = FP/TP+FP, F1 score = 2(1-FDR) ×hit rate/2(1-FDR)+hit rate The first metric accounts for theability of the model to predict the channel is Free when it is reallyFree, while thesecond provides a direct assessment of the probability at which the model incorrectly identifies the channel as Free when it is Busy. In fact, FDR is a measure of the packet loss rate of the WSN when the prediction mechanism is being used. A good prediction mechanism should maximize the first metric, i.e., the hit rate, while minimizing the second, i.e., the FDR. The third metric, F1 score is defined as the harmonic mean between hit rate and precision metrics, balancing the two. Therefore, both metrics have to be high in order for F1 score to be high and the prediction to be good.For the HMM training, a set of training observations and initial state probabilities are required. The latter are obtained from the MMPP(2) model and are constituted by the steady state probabilities, while the former are obtained from the sniffed z seconds lengthWiFi traffic as depicted in Figure <ref>. This step, in addition, extracts samples of y seconds length every T seconds, where T is equal to the WSN data rate and y is the modelled traffic duration. Next, the mean IAT of each sample is computed and compared with the predefined threshold, as defined in Section <ref>, in order to obtain the training observations sequence. Then, both the state transitionand observation probability matrix are properly initialized for the data set under consideration using uniformly distributed probabilities matrices and recomputed using the Baum–Welch algorithm. Based on these values, the calibratedx, y and z values, and considering a WSN data rate T = 5 seconds, the HMM model is used for predicting white spaces every T seconds.We evaluated the performance of our approach using the four metrics and by comparing it with a 0.5–persistent random access method with no retransmissions as Pareto, previously used for comparison in Section <ref>, can not be used in conjunction with an HMM.The results are shown in Table <ref>. We begin our analysis with the FDR metric which, despite its simplicity, provides an indicator of how our approach performs in avoiding collisions, i.e., packet losses. The quantitative data in the leftmost part of Table <ref> shows that for all combinations of environments and channels our approach performs as good as the 0.5–persistent access method. Moreover, our approach consistently performs better in than , apart from channel 13. The reason for this behaviour is the fact that in the WiFi traffic exhibits heavy bursts which reduces the availability of white spaces compared to , while increasing the probability ofincorrectly discovering channel is Free. This behaviour is evident if one looks through the lens of the FPs values. On the other hand, a single FP is observed on channel 13, and our approach always predicts this channel is Free, succeeding every time except once.Moreover, looking at the hit rate metric, one grasps quickly that our approach performs much better in correctly predicting that the channel is Free than the random method. Differences are more marked in and can be as high as 50%, except channel 11.We conjecture that this is an effect of the HMM state transition matrix, although this aspect requires further, finer-grained investigation.Finally, the F1 score confirms that our approach is more efficient than the 0.5–persistent method in identifying available white spaces in both environments. § LIMITATIONS AND DISCUSSIONOur work is a first step towards CTI mitigation through WiFi aggregated traffic modeling and white spaces prediction.We validated our approach on real-world WiFi traffic traces from two environments each exhibiting different traffic characteristics, and show that it achieves good performance in both settings. Despite the encouraging results, future work is required to validate the approach in different scenarios, i.e., different times (day/night) and locations with distinct traffic characteristics.Despite the self-similar traffic behaviour, during the day, the statistics characterizing the traffic might change, triggering the need to calibrate the system parameters. These additional experiments are already on our research agenda, and will enable us to ascertain to what extent our approach can be generalized. In its current implementation, system calibration requires user control. Future work could optimize on this aspect and implement an approach that automatically recomputes the parameters when needed. Our approach requires both WiFi transceivers and a sensor node to be collocated for the latter to perceive the same interference as seen by WiFi. However, this is not a barrier with the emergence of new chips such as Qualcomm <cit.> where multiple radio transceivers are collocated, i.e., WiFi, Bluetooth, 802.15.4. Finally, a practical use of our technique would be its integration with both technologies.§ CONCLUSIONSCross technology interference has been a crucial problem in wireless communication notably for devices operating in the unlicensed bands. While most existing solutions focus on detecting and classifying interference, less work has been done on interference modeling specially for mitigation purpose. In this work, we proposed a solution for accurately modeling WiFi aggregated traffic and predicting the presence of interference on the channel. We validated our WiFi traffic model against real–traces and a state–of–the–art Pareto model, and evaluated the prediction mechanism against 0.5–persistent random access method, both for saturated and unsaturated traffic. Results show that we are better in both settings. Acknowledgements We would like to thank Renato Lo Cigno from University of Trento, Italy, for his insightful comments on modelling WiFi traffic. This work has been funded by the Irish Research Council in collaboration with United Technologies Research Centre, Cork, Ireland. IEEEtran | http://arxiv.org/abs/1709.08950v1 | {
"authors": [
"Indika S. A. Dhanapala",
"Ramona Marfievici",
"Sameera Palipana",
"Piyush Agrawal",
"Dirk Pesch"
],
"categories": [
"cs.NI"
],
"primary_category": "cs.NI",
"published": "20170926114056",
"title": "Modeling WiFi Traffic for White Space Prediction in Wireless Sensor Networks"
} |
APS/123-QEDDepartment of Physics & Astronomy, University of Denver, Denver, Colorado, USA [email protected] of Physics & Astronomy, University of Denver, Denver, Colorado, USA We theoretically analyze and experimentally measure the extrinsic angular momentum contribution of topologically structured darkness found within fractional vortex beams, and show that this structured darkness can be explained by evanescent waves at phase discontinuities in the generating optic. We also demonstrate the first direct measurement of the intrinsic orbital angular momentum of light with both intrinsic and extrinsic angular momentum, and explain why the total orbital angular momenta of fractional vortices do not match the winding number of their generating phases. Valid PACS appear here The Angular Momentum of Topologically Structured Darkness Mark E. Siemens December 30, 2023 ========================================================= Though it is well known that the orbital angular momentum (OAM) of light is a physical momentum that can be used to apply mechanical torque to small particles <cit.>, its classification as an intrinsic property of light remains a subject of discussion <cit.>. A clearly intrinsic property such as photon spin is independent of the transverse position of the calculation axis, but the photon OAM is more complicated. Zambrini et al. showed that while the average OAM of a beam is invariant under arbitrary linear translations, its OAM spectrum is not <cit.>. This led those authors to deem the OAM a quasi-intrinsic property. It has also been stated that the angular momentum of a beam is intrinsic if and only if there exists a choice of axis such that the mode has no net transverse momentum <cit.>. An extension of these conclusions is that to understand the natures of the intrinsic and extrinsic OAM, we must study a system in which both are present. In this letter we consider what is arguably the most natural of such systems: fractional vortex beams. It is well known that passing a Gaussian beam through a 2π m helical phase optic imparts the beam with an OAM of m ħ for any m∈ℤ. However it has only recently come to light that the OAM of fractional vortex beams is more subtle: the mode formed by Gaussian illumination of a phase ramp of topological charge μ∈ℝ does not in general have an average OAM of μħ per photon, as shown in Fig. <ref>.a. Though the expectation value of the OAM of these fractional vortex modes has been calculated <cit.> and experimentally verified <cit.>, there has been no physical explanation for the discrepancy between the topological charge of the generating phase structure and the OAM of the resulting mode. In this letter we show that the intrinsic OAM of a fractional vortex beam can be directly measured, and that it matches the topological charge of the generating phase, even for non-integer μ. This suggests that the fundamental action of the 2πμ spiral phase applied to a beam is not to impart the beam with μħ of total OAM, but to impart the beam with μħ of intrinsic OAM. This leads us to investigate the form and physical meaning of the other OAM component, that of extrinsic angular momentum. We do this by separating fractional vortex modes into coherent sums of intrinsic and extrinsic components, which in the case of fractional OAM modes means separating into light and dark parts.We start with a description of our direct optical measurement of the intrinsic OAM of light. In a prior work the authors demonstrated a technique for the quantitative measurement of total OAM by transforming the light with a cylindrical lens, recording an image with a camera at its focus, and calculating high-order moments <cit.>. This OAM measurement technique exploits the fact that at the focal plane of a cylindrical lens, a camera records the intensity of the 1D Fourier transform of a mode incident on the lens. From that image the OAM can be calculated asℓ_meas = 4 π/f λ∬_-∞^∞ I(x',y)_ℓ x'ydx'dy/∬_-∞^∞ I(x',y)_ℓ dx'dy.where I(x',y) is the spatially-resolved light intensity at the focal plane of the lens <cit.>. However, for modes without cylindrical symmetry in the transverse plane, the cylindrical lens orientation with respect to the mode under test affects which OAM components are measured. For fractional modes, there is mirror symmetry in the transverse plane, along the line of the lateral discontinuity, and to measure the total OAM, the cylindrical lens must be oriented at π/4 with respect to the line of symmetry <cit.>.Any extrinsic component to the OAM can be written as a net linear transverse momentum, and thus there must be exactly one orientation of the cylindrical lens for which all of the extrinsic component is not counted in the measurement. As the intrinsic OAM is all that is left to be measured and is by definition invariant on measurement orientation, there exists exactly one orientation of the cylindrical lens for which the intrinsic OAM can be measured directly. As it is known that upon propagation the line of symmetry moves in the ϕ̂ direction, we choose the orientation of the cylindrical lens to be oriented π/2 radians from the orientation of the discontinuity, so as to zero the measurement to extrinsic angular momentum by mapping the transverse extrinsic momentum contribution onto the optical axis in the Fourier domain.The result of our intrinsic OAM measurement is shown in blue in Fig. <ref>. This is the first direct measurement of intrinsic OAM in the presence of extrinsic OAM, and it confirms the nontrivial claim that the intrinsic OAM of fractional vortex modes goes as μħ. The extrinsic OAM can be calculated as the difference between the total and the intrinsic OAM, the result of which is shown in black in Fig. <ref>. Having shown that the intrinsic OAM of fractional vortex beams matches the topological charges of their generating phases, we now set out to understand the optical structure of these beams in connection with intrinsic and extrinsic OAM contributions.We start by constructing the form of the fractional vortex mode. We can write the general form of a fractional vortex in terms of LG modesψ_μ=∑_l=-∞^∞C_ℓ∑_p=0^∞C_ p(ℓ) LG_ℓ psuch that <cit.>C_ℓ=i (1-e^2 i π(μ -⌊μ⌋ )) e^i θ(μ -l)-i θ (μ -⌊μ⌋ )/2 π(μ -l) in which θ represents the orientation angle of ϕ=0, and thus specifies the direction of the lateral phase discontinuity in the final structure. For each ℓ in the sum, there is a hypergeometric response in the radial structure, as in Karimi et al <cit.>. Restricting the mathematical expression for the hypergeometric response to physical cases, it can be simplified so thatC_ p(ℓ)=ℓΓ(ℓ/2+p)/2 √(p! (ℓ+p)!). As can be seen in Fig. <ref>.c, the transverse momenta of the modes have a great deal of interesting structure. Gbur elegantly showed that as μ goes continuously from one integer to the next, a radial string of unit vortices of alternating charge forms until μ is a half integer and the string extends infinitely from the center of the mode <cit.>. As μ continues to increase, the chain collapses towards infinity, leaving a single additional vortex which moves into the center. The vortex chain can be seen in Fig. <ref>, and the movement of the `additional' vortex can be seen in Fig. <ref>.c: At μ=1.25 there is a clearly defined, local 2π phase wrap near the mode center, while at μ=1.50 the core opens up to accept the `new' vortex left by the infinite chain, and at μ=1.75 there is a well defined core phase wrap of 4π. Fig. <ref> compares theoretical and experimental fields for ψ_4.4, demonstrating that our model matches experimental observations of both the intensity and phase of fractional vortex modes. In our measurements, phase was experimentally measured using phase stepping holography <cit.>. To experimentally generate fractional vortex beams, we pass a laser with a Gaussian profile through a spatial light modulator encoded with a fractional spiral phase hologram, which takes the form of a forked grating with a lateral discontinuity, and take images with a CCD a distance of 1/20Z_R beyond the modulator. The experimental results accurately resolve the dominant phase structure and the lateral stripe from the linear phase discontinuity in the generating phase, which is composed of a line of unit vortices of alternating sign that extends outward from near the center of the mode <cit.>.Having built and experimentally confirmed the structure of fractional vortex beams ψ_μ, we now separate them into their intrinsic and extrinsic parts, in keeping with our measurements in Fig. <ref>. We define the intrinsic component of ψ_μ as ψ_iμ(r,ϕ)=|ψ_μ(r, θ̃+π)|e^iμϕ.This intrinsic component has a phase defined by the non-integer phase wrap of the generating optic, and its amplitude is a circularly symmetric wrap of the radial function of ψ_μ at the angle opposite the discontinuity, where there are no effects of the discontinuity. We are careful not to call this theoretical object a mode, as it is not a solution to the wave equation and cannot propagate independently. However we find it conceptually useful and it is essential for the development of our understanding of its extrinsic counterpart. The intrinsic component ψ_iμ possesses uniform average OAM of μħ per photon. In order to describe the complete optical mode ψ_μ, we introduce a second component, that of structured darkness, asψ_dμ=ψ_μ - ψ_iμ. We can evaluate the complex form of ψ_dμ numerically for any μ. We can also perform the same operation on experimentally measured vortex beams via phase reconstruction. Fig. <ref> demonstrates the match between the theory and experiment in the experimentally viable case of non-zero propagation from the generating phase optic. Both the numerical and experimental plots of the structured darkness associated with ψ_1.5 demonstrate that nearly all measurable transverse momentum takes the form of off axis, linear momentum which cancel and thus form a net zero extrinsic contribution. This is consistent with our result shown in Fig. <ref>: the extrinsic angular momentum is zero for half-integer μ. Similar analysis of modes with other values of μ were also consistent with Fig. <ref>. We now discuss a physical interpretation of structured darkness. It has been shown that the dark singularity at the center of an integer vortex beam, generated by the Gaussian excitation of a spiral phase, is the result of depletion of the Gaussian into non-propagating evanescent waves wherever the propagation angle of component plane waves is too high to be physical <cit.>. Similarly, we claim that the form of the structured darkness of a fractional vortex beam at z=0 represents that of evanescent waves excited along the lateral phase discontinuity of the generating phase optic. To prove that the structured darkness of fractional vortex modes represents the regions depleted by the excitation of evanescent waves at the generating phase optic, we demonstrate that we can numerically generate a fractional mode by calculating the propagating, non-evanescent part of a mode with Gaussian amplitude and fractional 2πμ spiral phase. This is done by taking the Fourier transform of the helically phased Gaussian, applying a circular aperture, and inverse Fourier transforming back. In the case that the aperture blocks only non-propagating evanescent waves, we can calculate the near field of the propagating field. By selecting a much smaller aperture in k-space, the far field of the mode can be calculated because this is mathematically equivalent to spatial filtering. Fig. <ref>.a-b shows the result of such far field calculations for μ=1 and μ=1.5. Both cases match the expectation in the far field in both phase and amplitude, without any a priori knowledge of the Laguerre-Gaussian compositions, or even of the form of any mode other than the input Gaussian. Fig. <ref>.c-d shows the amplitudes of the near field results for the same values of μ, zoomed in to the beam center. Even in the near field, there is a clearly defined lateral singularity along the discontinuity of the generating phase. Given that this behavior is apparent with only the assumption that evanescent waves do not propagate, we conclude that the region of structured darkness seen beyond the near field represents the effect of evanescent structure at the surface of the phase optic on the propagating mode. While this is somewhat counterintuitive as it is well known that evanescent waves do not propagate, it is clear that their structure can influence that of a surrounding, propagating field. The authors thank A.A. Voitiv for assisting in data acquisition, J.M. Knudsen, J.T. Gopinath and R.D. Niederriter for helpful discussions, and acknowledge financial support from the National Science Foundation (1509733, 1553905). ieeetr | http://arxiv.org/abs/1709.09299v1 | {
"authors": [
"Samuel N. Alperin",
"Mark E. Siemens"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170927012135",
"title": "The Angular Momentum of Topologically Structured Darkness"
} |
[ [ December 30, 2023 =====================empty empty In recent years, various shadow detection methods from a single image have been proposed and used in vision systems; however, most of them are not appropriate for the robotic applications due to the expensive time complexity. This paper introduces a fast shadow detection method using a deep learning framework, with a time cost that is appropriate for robotic applications. In our solution, we first obtain a shadow prior map with the help of multi-class support vector machine using statistical features. Then, we use a semantic-aware patch-level Convolutional Neural Network that efficiently trains on shadow examples by combining the original image and the shadow prior map. Experiments on benchmark datasets demonstrate the proposed method significantly decreases the time complexity of shadow detection, by one or two orders of magnitude compared with state-of-the-art methods, without losing accuracy.§ INTRODUCTION Dealing with shadows is one of the most fundamental issues in image processing, computer vision and robotics. Shadows are omnipresent in outdoor applications and must be taken into account in the solutions to standard computer vision and robotics problems such as image segmentation <cit.>, change detection <cit.>, place recognition <cit.>, background subtraction <cit.>, visual robot localization <cit.> and navigation <cit.>. Unfortunately in most of these cases images are strongly influenced by shadow at different times of a day, making it difficult to interpret or understand a scene. Although in some applications people have attempted to address this challenge by relying on robust features with impressive results, these features often do not provide sufficient invariance to shadow.The problem of detecting shadow is a well-studied research topic, and many methods have been proposed <cit.>. Existing methods can be categorized into two major groups. The first group of methods alleviate or remove the effect of shadow by providing an invariant representation of the image <cit.>. Most of these methods model the process of image formation to build shadow-free images. Although these methods are effective to some extent, all of them have the limitation in terms of dealing with non-Plankian source of light, narrow-band color camera and environment calibration. They also tend to lose information in the shadow-free representation that can be important for scene understanding. The other group of methods to deal with shadow rely on a learning framework <cit.>. These methods specifically focus on shadow detection while attempting to keep the original color and intensity of images. Unfortunately, these methods still have difficulty in robotics applications that we will elaborate in the next section. State-of-the-art shadow detection methods come from the second group above and are based on convolutional neural networks (CNN). In this paper we present a novel and fast method, also based on CNN. Our method detects shadows of an image without any assumptions about image quality or the camera and it is thereforeappropriate for the robotics applications. To develop an efficient shadow-detection algorithm, rather than labeling individual pixels, we work with super-pixels, obtained through segmentation. Subsequently, we extract color and texture features from each super-pixel and, with the help of a trained SVM, compute a shadow prior in terms of the probability of a super-pixel being shadow. Then, we use the combination of the original image and the obtained shadow prior as the input to a patch-level CNN to compute the improved shadow probability map of the image. The edge pixels between the super-pixels are further refined by running the same patch-level CNN a second time, to produce the final shadow detection result. We will show that the proposed method can provide comparable results with existing deep shadow detection methods due to the use of the combination of texture features and deep neural networks, but works much faster in both training and detection phases than existing CNN based methods, due to the use of super-pixels. This method enables us to detect shadow in an image in robotic applications, a task that was not possible before due to the high time complexity.The remainder of this paper is organized as follows. In Sections <ref>, we discuss related works in shadow detection in further detail. In Section <ref>, we introduce our novel method to this problem, focusing on improving the efficiency of an existing deep neural network based solution. Comparative experimental results on benchmark datasets are described in Section <ref>, and finally Section <ref> summarizes our method and concludes the paper.§ RELATED WORKSThe importantce of detecting shadow in a single image has been well investigated in computer vision and robotics community. One common approach in robotics community computes “intrinsic images" <cit.> by decomposing an image into its reflectance and illuminance constituents <cit.>. As discussed in the previous section, these methods have restrictive assumptions on illumination and sensor. To relax these assumptions, data-driven methods have been proposed, which work with images in grayscale or color space based on a learning framework to learn shadow in different situations <cit.>. Zhu et al. <cit.> proposed a method that classifies regions based on statistics of intensity, gradient, and texture, computed over local neighborhoods, and refines shadow labels by exploiting spatial continuity within a conditional random field (CRF) framework <cit.>. Lalonde et al. <cit.> find shadow boundaries by comparing the color and texture of neighboring regions and employing a CRF to encourage boundary continuity. To benefit from global information, Guo et al. <cit.> proposed a region based method, which can model long-range interaction between pairs of regions of the same material, with two types of pairwise classifiers, under similar/different illumination conditions. Then, they incorporated the pairwise classifier and a shadow region classifier into a conditional random field (CRF) via graph-cut optimization <cit.>. Vicente et al. <cit.> proposed a multi-kernel model to train a shadow support vector machine (SVM). Their multi-kernel model is a summation of base kernels, one for each type of local features. The main limitation of this model is the assumption of equal importance for all features. To avoid this assumption, they proposed leave-one-out kernel optimization <cit.> in which the parameters of the kernel and the classifier are jointly learned. They also used least square SVM (LSSVM), which has a closed form solution and therefore is faster than SVM. However, their approach is still computationally expensive.Although these methods reviewed above provide good accuracy in some cases, they run on the order of many minutes or seconds, and are not applicable in robotics due to this high time complexity.Recently, some end-to-end deep learning frameworks have been proposed to learn the most relevant features for shadow detection. They outperform the state-of-the-art methods that use hand crafted features. The first method in this category is proposed by Khan et al. <cit.>. They train two CNNs, one for detecting shadow regions and the other for detecting shadow boundaries. They also train a unary classifier by combining the two CNNs, and the per-pixel predictions are then fed to a CRF for enforcing local consistency. <cit.> proposes a structured deep edge detection for shadows and shows that using structured label information, local consistency over pixel labels can be improved. More recently, Vicente et al. <cit.> propose a method by combining an image level fully connected network (FCN) and a patch-based CNN. They train the FCN for semantically aware shadow prediction and use the output of the FCN as shadow prior with the corresponding input RGB image to train the patched-CNN from a random initialization. This method produces excellent result but is unsatisfactory in terms of its time complexity.In this paper, we propose a novel method based on deep learning with a shadow prior. Like <cit.>, our method can detect shadow from a single image, but at a much lower time complexity than all existing methods based on deep learning.The key insight of our method is that it performs shadow detection on a per super-pixel basis. Initial result from such a detection method would produce boundary effects between super-pixels.We overcome such artifacts with a post-processing step using edge refinement.§ PROPOSED METHOD In this section we describe our proposed method to detect shadows from a single image. Our method uses two steps to learn shadow from training images. First we obtain a prior map that we call shadow prior using a trained SVM on color and texture features, and then we train a patched-CNN using the original images and their shadow priors obtained from the first step. These two steps will be detailed in Section <ref> and <ref>, respectively. For the detection of shadows in a given image, we compute its shadow priorfirst with the SVM and use the prior and the image as input to the trained patched-CNN, considering only the center pixel of each super-pixel of the input as the representative in order to reduce the computational time. In doing so, however, the super-pixels near object and shadow boundaries tend to produce unreliable “edge effects". To overcome this problem, we refine prediction labels for the edge pixels along super-pixel boundaries, using the patch-CNN again. In other words, algorithm makes use of the trained patched-CNN twice. The output of the second patched-CNN provides the final detected shadow areas. The edge refinement step of our algorithm will be discussed in Section <ref>.§.§ Computing Shadow Prior Shadow prior computation involves steps shown in the first row of Fig. <ref>. We first segment the image using mean shift algorithm <cit.> to obtain superpixels. Second, using a trained classifier on texture and color features, we estimate the shadow probability of each region. Segmentation enables us to estimate the shadow probability for each region instead of each pixel and therefore, the computation time is significantly decreased.In general, a shadowed region is darker and less textured than a non-shadow region <cit.>. Therefore, the color and texture of a region can help to predict whether it is in shadow. We exploit this observation and represent color with a histogram in L*a*b space, with 21 bins per channel, as was successfully applied in <cit.>. We represent texture with the texton histogram provided by <cit.>. We train our SVM classifier with the color and texture features with a χ^2 kernel and slack parameter C = 1 <cit.>. We define the shadow prior of each super-pixel, as the log likelihood output of this trained classifier. In the subsequent step, we use this shadow prior as a critical input to our patched-CNN. §.§ Training Patched-CNN with the Shadow Prior In the next step of our shadow detection pipeline, we employ a patch-wise CNN to predict shadow. Current research in <cit.> showed that using patches of an image with a specific size has two benefits in the case of shadow recognition. First, these patches include enough local pattern of the image and more global information in a large-range of neighborhood pixels than pixel-based methods. Secondly, using patches we are able to provide more training samples with different patterns from a limited number of labeled images. As discussed, one of the challenging problems in the case of shadow detection is the number of training samples, which can significantly affect the accuracy of a deep neural network. Unfortunately, available shadow benchmark datasets are small due to the high cost of shadow annotation. This patch-wise structure enables us to provide a huge number of patches of shadow and non-shadow areas for training that can increase the overall accuracy of the network.We utilize the network architecture used in <cit.>. The deep network has six convolutional layers, two pooling layers, and one fully connected layer. The input of this network is a 32×32 RGBP patch selected from combining RGB image and shadow prior image P. The output is the shadow probability map of the patch. We select equal number of patches for training in three classes as follows.* Shadow patches: Since we are going to learn shadow areas, we first select patches from shadow regions.* Non-shadow patches: We select patches from non-shadow image locations randomly to include patches of various textures and colors. Also, these selected patches prevent overfitting.* Shadow-Edge patches: We also select patches on edges between shadow and non-shadow regions, to learn the shadow boundaries. Since the ground-truth is binary, locations of all shadow edges can be extracted accurately. Using this strategy, we are able to provide millions of patches from thousands of images. The loss function of the network is the average negative log-likelihood of the prediction of every pixel. §.§ Edge Refinement of Super-Pixel Labels For the detection of shadows in a given image, only the patches centered in the super-pixel center are used and the average value of each predicted patch is assigned to all pixels of the super-pixel. These predictions made by the patched-CNN are local, and the prediction results near shadow boundaries are poor. To improve the accuracy of our detection algorithm, higher level interaction between the regions is needed. Therefore, in this final step we process the edge pixels between the regions by patched-CNN once again, shown in the bottom row of Fig. <ref>. We only process those pixels that are on edges between the segments, labeled as R(S) and defined as: R(S) = {s_i ∈ S| s_i ≥αmax_1 ≤ i ≤ m(s_i) }R(S) contains those segments with the higher probability than a threshold of maximum shadow probability in the region-based prediction P'. α is a constant threshold (equal to 0.2 in our implementation) and m is the number of superpixels or regions in the image or the region-based prediction P'. Absolute non-shadow regions always provide a very low shadow probability in the shadow prior map, and (<ref>) only filters out those regions. This thresholding step will reduce the number of pixels to be refined and the total time of this step.For each boundary pixel (x, y) between segments that is included in R(S), a window patch with size 32 × 32 surrounding the pixel (x, y) from its shadow prior and corresponding original image are given to the patched-CNN to predict the shadow probability for that patch. Then we set the edge pixel (x, y) and its 8 neighbor pixels' probability values to be average probability value of these 9 pixels. This step can integrate the segmented probability maps obtained in previous step and the final shadow probability map becomes smooth.§ EXPERIMENTAL RESULTS In this section we perform a set of experiments to evaluate our proposed method and compare it with other state-of-the-art methods. We first use three challenging available datasets “UCF" <cit.>, “UIUC" <cit.>, and “SBU" <cit.> for shadow detection to evaluate quantitatively the proposed method. The number of images in each of dataset in our experiments is as follows. * UCF Dataset: This dataset contains 355 images with manually labeled pixel-based groundtruth. * UIUC Dataset: This dataset contains 108 images (32 train images and 76 test images) with pixel-based ground truth. * SBU Dataset: This new dataset contains 4,727 images (4,089 train images and 638 test images) with pixel-based ground truth. * Combined Dataset: Both UCF and UIUC datasets include an insufficient number of images, and to evaluate the propose method we need to select a portion of these datasets as training samples. Since our proposed method works on patches, we are able to create many patches from the datasets for the training phase. However, to be fair in comparison with other methods, we combine UCF, UIUC, and SBU datasets to train for all methods. The combined dataset includes 5,078 images. We randomly selected 25% of the images for testing, and the rest for training.In addition, we use two other datasets “UACampus" and “St. Lucia" <cit.> to obtain qualitative results of detecting shadows on roads, one of the common problems in robot applications. §.§ Evaluation Metrics To evaluate the proposed method in terms of detection accuracy, we use three evaluation metrics as follows.Shadow Accuracy =TP/all shadow pixels Non-shadow Accuracy = TN/all non-shadowpixels Total Accuracy = TP+TN/allpixels For comparison in terms of computational efficiency, we simply use the execution time of shadow detection in a single image as the performance metric. Therefore, a total of four performance metrics, three for accuracy and one for efficiency, are considered in our experimental evaluation. §.§ Results on Benchmark DatasetsIn this section we evaluate our proposed method and compare it with Stacked-CNN <cit.> and Unary-Pairwise <cit.>. We select Stacked-CNN as a recent shadow detection method based on deep learning framework that uses a shadow prior map. We choose the unary-pairwise method since it is one of the best statistical methods to detect shadows from a single image. Fig. <ref> shows example results of these methods and ours. In the third row, although the unary-pairwise method provides acceptable results in some cases, it completely fails in the first, third, fifth, and sixth columns. The fourth row of Fig. <ref> shows the results of Stacked-CNN, which in the most cases are comparable with our method (the fifth row). The last row shows the binary shadow mask of the proposed method by a constant threshold.Table <ref> shows the quantitative results on SBU dataset. The values shown are the average of the performance metrics on all test images. Although the total accuracy of the proposed method is not the best, with respect to shadow accuracy, our method outperforms the other methods. The goal of the proposed method is providing a fast shadow detection method without losing the accuracy. Thereore, in Table <ref> we show the execution time of training and testing phases of the proposed method. Results in both Tables <ref> and <ref> illustrate that the proposed method works an order of magnitude faster than the statistical methods and two orders of magnitude faster than the deep learning competing method with the comparable accuracy. This is almost an entirely a result of predicting shadow/non-shadow labels on super-pixels rather than pixels, even with the additional cost to pay for refining the boundary pixels. We also evaluate the proposed method and compare it with the other methods on the combined dataset to investigate the effect of increasing the number of images for training and testing. Table <ref> shows that our method is still comparable with other methods in terms of accuracy, and Table <ref> shows that the execution time of our method is significantly less than the other two methods, as was the case on individual datasets.§.§ Shadow Detection in Road Detection Application To illustrate the utility of our method in a real application, we consider the problem of detecting shadow on roads in this section. Cast shadows on roads can cause difficulty or mistakes in the scene interpretation or segmentation for this application. To determine the performance of our method in this application, we apply the proposed method on St. Lucia and UACampus datasets to show the potential of the method in detecting shadows on roads. Figs. <ref> and <ref> show the results of our method in terms of shadow probability maps of sample images.It is clear from these examples that our method is able to generate a probability map of high accuracy, and can serve as a useful building block for road detection in, for example, autonomous driving. We also apply the proposed method on aerial images to detect shadows, a common problem for many applications that rely on the aerial images. Fig. <ref> shows the qualitative results of the proposed method to detect shadows in the aerial images. Once again, the results are highly accurate, and our method can directly contribute to solutions in aerial imaging applications. § CONCLUSIONIn this paper, we have presented a method for accurately detecting shadow in a single image.Our method combinestraditional color and texture features and deep learning in a novel way, and achieves start-of-the-art performance in terms of detection accuracy and out-performance state-of-the-art in terms of computational efficiency. Our method uses color and texture features to compute a shadow prior map by training an SVM. The prior map and the original input image are then used as input to a patched-CNN to compute shadow probability map, one for each super-pixel, to achieve the desired computational efficiency. In the final step, we refine the prediction result of the patched-CNN by re-estimating the class labels of boundary pixels between super-pixels with the same patched-CNN. Extensive experimental results demonstrate that the proposed method works significantly faster than the existing deep or statistical methods, by one to two orders of magnitude, without losing the accuracy. 99segment J. Bruce, T. Balch, M. Veloso, “Fast and inexpensive color image segmentation for interactive robots", Intelligent Robots and Systems (IROS), IEEE/RSJ International Conference on. 200, pp. 2061:2066.shakeri_ICCV M. Shakeri, H. Zhang, “Moving object detection in time-lapse or motion trigger image sequences using low-rank and invariant sparse decomposition", IEEE International Conference on Computer Vision (ICCV), 2017, pp. 5123:5131.place_rec S. Lowry, N. Sunderhauf, P. Newman, J. Leonard, D. Cox, P. Corke, M. J. Milford, “Visual place recognition: A survey,” IEEE Trans. Robot., vol. 32, no. 1, pp. 1–19, Feb. 2015.shakeri_small M. Shakeri, H. 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Lin, “Libsvm: a library for support vector machines", ACM transactions on intelligent systems and technology (TIST), vol. 2, no. 3, p. 27, 2011.Vicente2 T. F. Y. Vicente, L. Hou, C. P. Yu, M. Hoai, D. Samaras, “Large-scale training of shadow detectors with noisily-annotated shadow examples", In European Conference on Computer Vision (ECCV), 2016, pp. 816:832.CRF J. Lafferty, A. McCallum, F. C. Pereira, “Conditional random fields: Probabilistic models for segmenting and labeling sequence data", In 18th International Conference on Machine Learning (ICML), 2011, pp. 282:289.Graph_cut Y. Boykov, O. Veksler, and R. Zabih. “Fast approximate energy minimization via graph cuts", IEEE transactions on pattern analysis and machine intelligence (PAMI), 23(11), 2001, pp. 1222:1239.Lucia A. Glover, W. Maddern, M. Milford, G. Wyeth, “FAB-MAP+RatSLAM: Appearance-based SLAM for Multiple Times of Day", International Conference on Robotics and Automation (ICRA), 2010. | http://arxiv.org/abs/1709.09283v2 | {
"authors": [
"Sepideh Hosseinzadeh",
"Moein Shakeri",
"Hong Zhang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170926232042",
"title": "Fast Shadow Detection from a Single Image Using a Patched Convolutional Neural Network"
} |
A Gradient Descent Method for Optimization of Model Microvascular Networks Shyr-Shea Chang Dept. of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, USA. Marcus Roper [1] Dept. of Biomathematics, University of California Los Angeles, Los Angeles, CA 90095, USA.Accepted ....... Received......; in original form ...... ======================================================================================================================================================================================================================== Within animals, oxygen exchange occurs within networks containing potentially billions of microvessels that are distributed throughout the animal's body. Innovative imaging methods now allow for mapping of the architecture and blood flows within real microvascular networks. However, these data streams have so far yielded little new understanding of the physical principles that underlie the organization of microvascular networks, which could allow healthy networks to be quantitatively compared with networks that have been damaged, e.g. due to diabetes. A natural mathematical starting point for understanding network organization is to construct networks that are optimized accordingly to specified functions. Here we present a method for deriving transport networks that optimize general functions involving the fluxes and conductances within the network. In our method Kirchoff's laws are imposed via Lagrange multipliers, creating a large, but sparse system of auxiliary equations. By treating network conductances as adiabatic variables, we derive a gradient descent method in which conductances are iteratively adjusted, and auxiliary variables are solved for by two inversions of O(N^2) sized sparse matrices. In particular our algorithm allows us to validate the hypothesis that microvascular networks are organized to uniformly partition the flow of red blood cells through vessels. The theoretical framework can also be used to consider more general sets of objective functions and constraints within transport networks, including incorporating the non-Newtonian rheology of blood (i.e. the Fahraeus-Lindqvist effect). More generally by forming linear combinations of objective functions, we can explore tradeoffs between different optimization functions, giving more insight into the diversity of biological transport networks seen in nature.§ INTRODUCTION The human cardiovascular network contains billions of vessels, ranging in diameters from centimeters to microns, and continuously carries trillions of blood cells. Cardiovascular networks are robust in some respects and fragile in others. They are robust in the sense that although each network is far more complex than even the largest traffic or hydraulic networks built by humans, in healthy organisms microvascular networks show remarkably little of the chronic patterns of traffic congestion that plague human-built networks. At the same time, the microvascular part of the network; made up of fine vessels less than 8 μm in diameter, is susceptible to accumulated damage from micro-occlusions<cit.> and micro-aneurysms<cit.>. This cardiovascular damage is a leading cause of aging related health problems. Systemic microvascular damage associated with diabetes mellitus, can lead to erectile dysfunction<cit.>, limb loss<cit.>, neuropathy<cit.> and dementia<cit.>. Although each of these forms of microvascular damage is diagnosed and treated in a completely different way, they may have a common physical basis. We therefore ask: What physical functions are microvascular networks organized to perform, and what forms of damage interfere with its ability to perform these functions?Techniques like plasticization have long enabled the largest vessels in the cardiovascular network to be mapped out. More recently Micro-optical Sectioning Tomography (MOST) has been used to map the blood vessels within rodent brains to micron resolution<cit.>, and mapping the blood vessels in the human brain is one of the central goals of the BRAIN initiative<cit.>. Meanwhile long working distance two photon microscopes can be used to directly measure blood flows within living rodent brains<cit.>. But using this data still requires understanding of the organizing principles for microvascular networks.A natural mathematical starting place for deriving organizing principles for transport networks is to frame the problem of network design as a problem in optimization. For example, in 1926 Murray first derived relationships between vessel radii and fluxes at different levels of the arterial network, assuming that the network minimizes a total cost made up of the viscous dissipation and a metabolic cost of maintaining the vessels that is proportional to their volume<cit.>. A particular consequence of this optimization, is that when a `parent' vessel within the network divides into two `daughters', the sum of the cubes of the daughter radii will equal the cube of the parent radius<cit.>, and this result has been validated in studies on real animals<cit.>. The notion of cardiovascular networks as optimizing transport has since found many applications, underlying theoretical models for how energy needs scale with organism size<cit.> as well as clinical computational fluid dynamics (CFD) studies in which different candidate surgical graft geometries are ranked by their transport efficiency<cit.>. Many (but not all, see Zamir<cit.>) studies of larger vessels (typically extending down to a few mm in diameter) show that they conform to Murray's law, suggesting that on a population level, these vessels are organized to minimize dissipation. However fine vessels account for a large share of the total network dissipation; for example in humans capillary beds and the arterioles that supply them, account for about a half of the total dissipation in the cardiovascular network[Since the total flux of blood is the same at each level of the vascular network, we can estimate the dissipation at each level from pressure measurements, such as those summarized in Guyton and Hall<cit.>]. Yet we are aware of no data that shows that principles of dissipation minimization extend to these vessels, which are typically arranged into topologically complex networks<cit.> (also see Figure <ref>). Indeed our own analysis of the zebrafish trunk microvasculature, which is a model system for studying vascularogenesis, showed that uniform partitioning of red blood cells between the many fine vessels perfusing the trunk, is a more likely candidate optimization principle for these networks than minimizing dissipation<cit.>. In fact we showed that the adaptations used within the zebrafish trunk network to ensure uniform perfusion directly lead to an 11-fold increase in dissipation within the network<cit.>.To understand the function of microvascular networks, and indeed to understand biological transport networks generally, which may be optimized for mixing<cit.>, resistance to damage<cit.>, or for the ability to accommodate high variations in flow<cit.>, it would be highly useful to have a framework for generating networks that optimize a particular target function, while respecting constraints. Before introducing our method for optimizing general functions we first describe previous methods for generating optimal transport networks (the relationship of this paper to these previous works is also presented in Table 1). Early methods for optimization followed Murray's original approach<cit.>, by optimizing transport within individual vessels, or at junctions in which single vessels bifurcate<cit.>. Although these methods allow local geometric optimization – i.e. of the position and angles of branching points within a network – they can only be used once the topology of the network, that is, the sequence in which vessels branch or fuse, has been defined. Banavar et al.<cit.> and Bohn and Magnasco<cit.> developed an iterative scheme that allowed optimization of entire networks linking a given set of sources to a given set of sinks given constraints on the total amount of material available to build the network. This approach made use of the fact that the laws governing flow in a network (Kirchoff's first and second laws, which will be described in more detail below), are automatically satisfied when dissipation is minimized within a network<cit.>. Katifori et al.<cit.> and Corson<cit.> later developed this theory to study networks that are designed to minimize dissipation given fluctuating set of source and sink strengths, or under variable damage (in which a random set of links within the network is eliminated). All of these works adopt an iterative approach, in which the conductances of network edges are iteratively updated until the dissipation is minimized: Corson<cit.> uses a relaxation method, while Katifori et al.<cit.> use gradient descent. In both cases, implicit use is made of the fact that the optimal network (i.e. the one that minimizes dissipation) will also obey Kirchoff's laws. Recent advances have focused on how structural adaptation (the process by which vessels within the transport network adjust their radii in response to the amount of flow that they carry) can be used to produce results equivalent to searching for a dissipation minimizing configuration by gradient descent<cit.>. These works also highlight that incorporating both growth and structural adaptation in a network can reliably find global dissipation minimizing configurations (as opposed to locating only local minimizers within a rough landscape)<cit.>.By contrast, the problem of minimizing other functions on networks has received relatively little attention. This is likely because, although there is strong evidence that some biological transport networks, such as fungal mycelia and slime mold tubes<cit.>, are adapted to maximize the amount of mixing of the fluids, nutrients and organelles that are transported by the network, microvascular networks have generally been thought to conform to the same principles of dissipation minimization as larger vessels. However, our own work on the embryonic zebrafish vasculature shows that the fine vessels in the trunk are organized to all receive red blood cells at identical rate<cit.>. Red blood cell partitioning is achieved by increasing the resistance of vessels near the head of the fish over vessels near its tail, leading to a large (11 fold) increase in the dissipation within the network. This study therefore suggests that uniformity of flows, rather than minimization of dissipation, underlies the design of the zebrafish trunk microvasculature. However, our ability to determine whether the principle of flow uniformity may rule in other real networks, or to test alternate candidate optimization principles, is limited because, unlike dissipation, there is no existing method for optimizing general functions that can be evaluated over transport networks. The main mathematical challenge that must be overcome to create such an optimization method is to ensure that in addition to minimizing the given function with given constraints, for example on the total material, the optimal network must respect constraints associated with Kirchoff's laws, which are not automatically satisfied at optima if the function of interest is not the energy dissipation within the network.Here we devise a method for minimizing arbitrary functions on networks. The method is described in Sections <ref> and <ref>. It uses gradient descent that can be rigorously shown to locate local minima of a given function, with a heuristic simulated annealing method, that has previously been shown<cit.> to be capable of finding global minima in rough landscapes. As a consistency check, we initially use this method to generate networks that minimize dissipation for a given amount of material, checking first that it is consistent with previous results on optimal networks (in Section <ref>), and second showing how these results can be modified if the non-Newtonian rheology of real blood is incorporated into models (in Section <ref>). Then, inspired by our demonstration of uniform flow in the zebrafish vascular network<cit.>, we go on to minimize a function representing the uniformity of flow within transport networks (Section <ref>), enabling us to calculate the optimal zebrafish trunk vasculature (Section <ref>). Finally (also in Section <ref>) we use our method to solve for hybrid functionals in which a linear combination of uniformity and dissipation are minimized: allowing the relative priority of uniformity and dissipation to be continuously varied, and allowing us to generate diverse optimal networks to compare with experimental observations. § SETUPFirst we mathematically frame the problem of finding optimal networks for general network topology. Consider an undirected graph (𝒱,ℰ) with V vertices k=1,…,V. For any given 2 nodes k,l we write ⟨ k,l⟩ = 1 if there is a edge linking k and l and ⟨ k,l⟩ = 0 if k and l are not linked. Each edge in the network is assigned a conductance κ_kl; the flow Q_kl in the link is then determined by Q_kl = (p_k-p_l)κ_kl, where p_k and p_l are respectively the pressures at the vertices k and l. In typical microvascular networks vessel diameters are on the order of 10 μm, and blood flow velocities are on the order of 1 mm/s, so the Reynolds number, which represents the relative importance of inertia to viscous stresses, is Re = UL/ν≈ 4 × 10^-3, using the viscosity of whole blood ν≈ 2.74mm^2/s. Since Re ≪ 1 inertial effects may be neglected, and by default the conductances of individual vessels will be obtained from he Hagen-Poiseuille's law<cit.>: κ = π r^4/8μℓ where κ is the conductance, μ is the blood viscosity, ℓ is the vessel length, and r is the vessel radius. In ascribing a well-defined pressure to each vertex within the graph, and applying the Hagen-Poiseuille law to compute edge flows from pressures, we assume that there are unidirectional flows within each vessel, ignoring the entrance and exit effects that occur when vessels branch or merge. At moderate Reynolds numbers, entrance effects can strongly affect the flow through vessels, for example by leading to phase separation, whereby red blood cells divide in different ratios at a junction than whole blood<cit.>. However, these effects contribute quite weakly for the low Reynolds number flows being modeled in this paper, for example in our previous studies of the zebrafish trunk vascular network, we found that total variation in hematocrit from vessel to vessel was no more than 2-fold. Moreover, we expect the entrance and exit effects to penetrate a distance comparable to the vessel diameter. Since typical microvascular vessels have diameters on the order of 5-10 μm and lengths on the order of hundreds of μm, we therefore expect entrance and exit effects to contribute negligibly to the total resistance of the vessel.The networks we consider consist of vertices and predescribed edges where conductance may be positive (or zero if required by the algorithm) along with two kinds of boundary conditions on vertices (Fig. <ref>). At any vertice in the network we can either impose Kirchoff's first law (conservation of flux)∑_l : ⟨ k,l⟩ = 1 Q_kl = ∑_l : ⟨ k,l⟩ = 1κ_kl(p_k-p_l) = q_k ∀ 1≤ k≤ V ,where q_k is the total flow of blood entering the network (or leaving it if q_k<0) at vertex k, or we impose p_k = p̅_k (i.e. pressure is specified). We say a node is inif pressure is specified, or inif Kirchhoff's first law is imposed, with possible inflow or outflow. This system of V linear equations forms a discretized Poisson equation with Neumann and Dirichlet boundary conditions imposed on selected nodes, and the flow is uniquely solvable if and only if each connected component of the network (connected by edges with positive conductances) either has at least one Dirichlet vertex or ∑_k∈ q_k = 0 with sum restricted to the component<cit.>. The general problem that this paper will address is how to tune the conductances within the network to minimize a predetermined objective functional f({p_k},{κ_kl}), where {p_k} means the set of all p_k's and {κ_kl} denotes the set of all κ_kl's. Previous works (see Table 1) have shown how to generate networks that minimize the total viscous dissipation occurring within the network: f({p_k},{κ_kl}) =∑_k>l, ⟨ k,l⟩ = 1κ_kl(p_k-p_l)^2.However, the pressures {p_k} and conductances {κ_kl} are coupled through Equations (<ref>). Since the relationship between {p_k} and {κ_kl} is holonomic, we may incorporate it into a functional via Lagrange multipliers. The functional that we want to minimize in this paper will take the form:Θ= f({p_k},{κ_kl}) + λ[ ∑_k>l,⟨ k,l⟩ = 1( aκ_kl(p_k-p_l)^2+ κ_kl^γ d^1+γ_kl) - K ]- ∑_k μ_k(∑_l,⟨ k,l⟩=1κ_kl(p_k-p_l)-q_k).which has V_N +1 Lagrange multipliers: a set {μ_k|k∈} enforcing Kirchoff's first law on Neumann vertices (the setwith || = V_N), and a single multiplier λ that constrains the amount of energy that the organism can invest in pushing blood through the network and in maintaining the vessels that make up the network. The transport constraint is made up of two terms: ∑κ_kl(p_k-p_l)^2 represents the total viscous dissipation within the network, while ∑κ_kl^γ d_kl^γ+1 represents the total cost of maintaining the network (the material constraint), with d_kl being the vessel length. The exponent γ can be altered to embody different models for the cost of maintaining a network. In our default model (Equation <ref>) conductance of an edge is proportional to the fourth power of its radius, so if the cost of maintaining a particular vessel is proportional to its surface area (and thus to its radius), then we expect γ=1/4, while if the cost is proportional to volume then γ=1/2. In general we need γ≤ 1 to produce well posed optimization problems (otherwise, the cost of building a vessel can be indefinitely reduced by subdividing the vessel into finer parallel vessels). Although in (<ref>) we initially adopt the same material cost function definition as was used in previous work<cit.>, we will go on to modify the cost function to incorporate networks in which vessels have different lengths, or in which the non-Newtonian rheology of real blood is modeled. Throughout, we incorporate a parameter a>0 that represents the relative importance of network maintenance and dissipation to the cost of maintaining the network. When presenting optimal networks, we will discuss the effect of varying a (as well as asymptotic limits in which a→ 0) upon the network geometry. Since Murray's work on dissipation-minimizing networks<cit.> is equivalent to minimizing this constraint function, we will adopt the shorthand of calling the network cost term the Murray constraint.Table 1 gives a systematic description of previous work on minimizing functionals across networks, as well as outlining the new results that will be presented here on the optimization of (<ref>).§ OPTIMIZATION OF GENERAL FUNCTIONS ON A NETWORK BY GRADIENT DESCENT At any local minimum of Θ, each of the partial derivatives of (<ref>) must vanish. In order to locate such points, we adopt a gradient descent approach, in which κ_kl are treated as adiabatically changing variables. That is: ∂Θ/∂κ_kl is calculated, and an optimal perturbation of the form δκ_kl = -α∂Θ/∂κ_kl is applied to ensure Θ decreases each time the conductances in the network are updated. At the same time, the other variables in the system, namely {p_k,μ_k,λ}, are assumed to vary much more rapidly, to remain at a local equilibrium, so that:∂Θ/∂ p_k =∂Θ/∂μ_k =∂Θ/∂λ=0 .Our ability to perform gradient descent therefore hinges on our ability to solve the system of 2V_N +1 equations (<ref>) for each set of conductances {κ_kl} that the network passes through on its way to the local minimum. Fortunately it turns out that only one nonlinear equation in a single unknown variable needs to be solved for to solve all of the conditions (<ref>); the other equations are linear and can be solved with relatively low computational cost.Because we will consider multiple variants of the Murray constraint, in what follows we will write the summand that enforces the Murray constraint in the general form: λ g({p_k},{κ_kl}). Then the condition that ∂Θ/∂μ_k=0, k∈, merely enforces the system of mass conservation statements at each Neumann-vertex in the network (<ref>). These equations represent a discretized form of the Poisson equation and can be solved by inverting a sparse V_N × V_N matrix with O(E,V_N) entries<cit.>. That is, we write:Dp = fwhere f_k = q_k is the prescribed inflow at Neumann vertices and f_k = p̃_k, the prescribed pressure at Dirichlet vertices. -D is a form of graph Laplacian:D_kl≐{[ ∑_l,⟨ k,l⟩ =1κ_kl k=l, k∉; -κ_kl ⟨ k,l⟩ =1, k∉;κ1 k=l, k∈; 0 otherwise ].where κ1 = 1. (For any κ1≠ 0 D is full rank; we will make use of other positive constant values for κ1 later.)To solve for {μ_k}, we consider the system of equations ∂Θ/∂ p_k = 0, k∈:0 = (∂ f/∂ p_k +λ∂ g/∂ p_k)- ∑_l, ⟨ k,l⟩ =1 (μ_k-μ_l)κ_kl.If λ, {p_k} and {κ_kl} are all known then these equations again take the form of a discrete Poisson equation, however, just as with the solution of the pressure equation, these equations themselves do not admit unique solutions unless a reference value of μ_k is established. If ≠ϕ, i.e. if pressure is specified at least one vertex within (,E) then μ_k = 0∀ k∈ and the μ_k equations admit a unique solution; otherwise μ_k's are determined up to a constant (see <ref>). For some forms of target function f and constraint function g, we will show that μ_k's for the minimizer are directly related to the pressures, with no need to solve the Poisson equation by a separate matrix inversion. However, to use Equation (<ref>) to solve for μ_k it is still necessary to know the Lagrange multiplier that enforces the Murray constraint (i.e. λ). The simplest way to derive λ is to dictate that the variational of the constraint function should vanish when κ_kl is updated since the constraint function should remain constant when its variational under changes in conductances, i.e.:0 =∑_k∉∂ g/∂ p_kδ p_k +∑_k>l, ⟨ k,l⟩ =1∂ g/∂κ_klδκ_kl (we set δ p_k = 0 if k∈) where δκ_kl = -α∂Θ/∂κ_kl = -α( ∂ f/∂κ_kl+λ∂ g/∂κ_kl - κ_kl (μ_k-μ_l)(p_k-p_l)).At this point {δ p_k} and {μ_k} are undetermined. The lagrange multipliers {μ_k} can be solved in terms of the still unknown λ from (<ref>) (see <ref>). The {μ_k} are linear functions of λ since (<ref>) is a linear system. To obtain δ p_k for each k∈ we calculate the variational in Kirchhoff's first law: ∑_l, ⟨ l,k⟩ = 1δκ_kl(p_k-p_l) + κ_kl(δ p_k - δ p_l) =0. When written in matrix form, the matrix multiplying {δ p_k} is again the negative of the graph Laplacian, -D. Thus {δ p_k} can be solved in terms of λ so long as the original matrix system is solvable for {p_k}. Since {μ_k} are linear in λ, {δ p_k} are also linear in λ, which implies that the right hand side of Equation (<ref>) is linear in λ. Therefore λ can be solved in closed form from Equation (<ref>), and the optimal variation δκ_kl can be determined from equation (<ref>).With {p_k}, {μ_k}, and λ solvable given {κ_kl} we can perform gradient descent using Equation (<ref>) and numerically approach a minimizer. However our descent method has the following limitations: 1. For finite step sizes α, conductances may drop below 0 when perturbed according to Equation (<ref>). 2. The method only conserves the Murray function up to terms of O(δκ). To avoid negative conductances we truncate at a small positive value ϵ at each step, i.e. set: κn+1/2_kl = max{κn_kl - α∂Θ/∂κ_kl,ϵ}.To ensure that the constraint is exactly obeyed we then project the conductances {κ_kln+1/2} onto the constraint manifold g({p_k},{κ_kl}) = 0, via a projection function: κn+1_kl = h(κn+1/2_kl) ∀⟨ k,l⟩ =1, k>l. Throughout this work we consider three possible projection functions: One choice is to project according to the normal of the constraint surface: κn+1_kl = κn+1/2_kl - β∂ g/∂κ_kl({p_kn+1/2},{κ_kln+1/2}),∀⟨ k,l⟩ =1,k>l The value of β must be chosen numerically to ensure that g({pn+1_k}, {κ_kln+1}) = 0 exactly. This entails recomputing the pressure distribution {pn+1_k} for each β value, and secant search on β to obtain the root. Another approach we have followed is varying the parameter λ. This method has comparable complexity to projection on {κn+1/2_kl}; since the {μ_k} depend linearly on λ via Equation (<ref>), {κn+1_kl} depends linearly on the parameter λ. However, just as with the projection method, we must still recompute the {pn+1_k} for each trial set of {κn+1_kl}. Moreover, for some target functions f or constraint functions g, it is difficult to derive closed-form expressions for λ (i.e. to calculate the partial derivatives ∂ f/∂ p_k and ∂ g/∂ p_k). In this case λ may only be computed numerically, by solving g({p_kn+1(λ)},{κ_kln+1(λ)}) = 0. A third approach that we have adopted is to simply scale the conductances: κn+1_kl = βκn+1/2_kl,∀⟨ k,l⟩ =1,k>l where β is chosen to satisfy the Murray constraint. This method produces theoretically suboptimal corrections on the conductances, but it is typically easy to compute a value of β that satisfies the Murray constraint. In particular, under certain boundary conditions, e.g. p_k = p̅,∀ k∈ within each connected component of the network meaning that all pressure vertices within a single connected component have the same imposed pressures, a rescaling of the conductances throughout the network leaves the fluxes on each edge unaffected. In this case, the dissipation decreases in inverse proportion to β, while the maintenance cost increases proportionately to β^γ. § MINIMIZING DISSIPATION §.§ Single source, single sink networksAs a first test for our optimization method we recompute dissipation minimizing networks; that is we set a=0, so our constraint function only reflects the total material cost of the network, and set the target function equal to ∑_k>l,⟨ k,l⟩ =1κ_kl(p_k-p_l)^2 so that our algorithm finds the minimal dissipation among all networks built using a given quantity of material. Our base network is a square grid (Fig. <ref>A). In addition to allowing for simple vertex indexing, this architecture resembles the regular capillary bed networks observed, for example in the rat gut<cit.>. We impose an inflow boundary condition on the upper left corner and a fixed zero pressure on the lower right corner. The dissipation-minimizing network is a single geodesic (i.e. path) between source and sink, allowing us to benchmark our optimization method's ability to find known global optima. To test our gradient descent method we form the function: Θ = ∑_⟨ k,l⟩ =1, k>lκ_kl (p_k-p_l)^2 + λ(∑_⟨ k,l⟩ =1, k>lκ_kl^γ - K^γ) - ∑_k∉μ_k (∑_l,⟨ k,l⟩ =1κ_kl(p_k-p_l) - q_k).Here we ignore d_kl since we assume all the vessels have the same length which may be scaled to 1 by choice of units. The adiabatic variation of p_k and μ_k is derived from ∂Θ/∂ p_k = ∑_l,⟨ k,l⟩ =1 2κ_kl(p_k-p_l) - ∑_l,⟨ k,l⟩ =1κ_kl(μ_k-μ_l), k∉ and the fixed pressure boundary condition on pressure nodes allows us to specify that: μ_i = 0∀ i∈ The μ_k are therefore solving a variant of the Kirchhoff flux conservation equations: D μ = 2 D pwith D as defined in Equation <ref>.This system can be solved for μ_k under the same conditions as the presure equations being solvable (see <ref>). In particular if, as here, the only pressure boundary conditions imposed at vertices inare of the form p=0, then μ_k = 2p_k , ∀ k ∈, i.e. μ_k's exactly represent the pressures for a stationary network. Now we calculate the derivatives with respect to the conductances:∂Θ/∂κ_kl = (p_k-p_l)^2 + λγκ_kl^γ-1 - (μ_k-μ_l)(p_k-p_l) = λγκ^γ-1_kl - (p_k-p_l)^2.In general we determine λ from Equations (<ref>,<ref>,<ref>). However the constraint function g is independent of {p_k} in this case, so Equation (<ref>) becomes0 = ∑_k>l,⟨ k,l⟩ =1∂ g/∂κ_klδκ_kland we can solve λ directly in terms of {p_k},{κ_kl}:λ = ∑_⟨ k,l⟩ =1,k>lκ^γ-1_kl(p_k-p_l)^2/∑_⟨ k,l⟩ =1,k>lγκ^2γ-2_klAs described in Section <ref> we project {κ_kl} along ∂ g/∂κ_kl = γκ_kl^γ-1 after each step of the algorithm. At each step of the algorithm, we solve for the pressures p_k from the conductances {κ_kl}, then the μ_k, and then descend according to Eqn. (<ref>). Assuming that γ < 1, our algorithm deletes edges and concentrates conductance on a single linked path of edges that connects source with sink (Fig. <ref>B, C). Any linked path that follows one of the equivalent shortest paths from source to sink will minimize dissipation and accordingly different distributions of random initial conductances converge to different optimal networks. Convergence is linear (Fig. <ref>D). §.§ Minimizing dissipation with distributed sinks The ability of the optimization algorithm to identify shortest distance paths between source and sink is a useful sanity check, but a real test of the algorithm requires that we evaluate its ability to produce known branching tree structures<cit.> when the network distributes blood between a single source and multiple, dispersed sinks. We simulate such a network by splitting the grid representing the capillary network in half along the diagonal. The source continues to be one corner of the square, and we space out a number of sinks, with equal output fluxes, along the diagonal (Fig. <ref>A). To make the pressure equation solvable we set pressure at the top-most (source) vertex in the network to p_1 = 0. Sink nodes each have prescribed outflows.Initially we assume the Hagen-Poiseuille law holds in each edge, so Θ takes the form specified in Equation (<ref>); and we follow the same method for updating conductances as in <ref>. Optimal networks take the form of hierarchical branching trees (i.e. loopless networks<cit.>) (Fig. <ref>B) in which thicker vessels bifurcate into narrower vessels, and thence into even narrower vessels similar to Bohn et al.<cit.>. We can quantitatively test for the ability of our algorithm to produce locally optimal networks by checking that the networks that it converges to obey Murray's law<cit.> which states that the flow in each vessel in an optimal dissipation network is proportional to the cube power of the radius of the vessel. Since the total flows through each level y = constant must be equal, Murray's law implies that the sum of the cube of the radii of vessels passing through each level should be equal. To test for local optimality, we calculate a Murray exponent by finding the exponent a that minimizes the variance on ∑ r^a_i where sums are taken over each edge in the same level of the network (Fig. <ref>C). The Murray exponents are tightly clustered around 3 (3.01± 0.03), which agree will with the theoretical value. Although our algorithm always converged to a locally optimal transport network, different initial configurations ultimately converged to different optima, consistent with previous results showing that the dissipation function landscape is rough with many local optima. To map out this landscape we measure the total length of the network for different local optima. Total length can be a measure of whether the branch points are concentrated near the source (i.e. small |y|, producing longer networks) or near the sinks (i.e. large negative y, producing shorter networks). The total length has a large variation among optimal networks and also correlates strongly with the dissipation (Fig. <ref>D, r = 0.98). This suggests that while a network with larger total length could be a local minimum, the dissipation can be decreased by a topological change that decreases the number of links, though this requires moving away from the local minimum. This suggests that the roughness of the dissipation landscape is high and a strategy of global optimization such as combining gradient descent with simulated annealing must be implemented to find the global minimal dissipation network (see Katifori et al.<cit.> and below).Although the assumption that each blood vessel obeys the Hagen-Poiseuille law is a useful default model, the non-Newtonian nature of blood means that in vessels of different diameters, blood may have very different apparent viscosity. In particular the finest vessels in a cardiovascular network are typically comparable in size to the red blood cells they transport. Red blood cells therefore occlude fine vessels, increasing the effective resistance of these vessels. At the same time, in larger vessels, red blood cells tend to self-organize to flow in the center of the vessel, leaving low viscosity layers of plasma adjacent to the vessel walls, reducing resistance to flow in those vessels. It is usual to incorporate these effects into models of vessel conductance by continuing to assume the Hagen-Poiseuille law (Eqn. (<ref>)) κ = π D^4/128(D,ϕ) ℓ, where D,ℓ are the diameter and the length of the vessel and the effective viscosity, , is as a function of vessel diameter and of the concentration (i.e. volume fraction) of blood cells, ϕ <cit.>. Assuming that red blood cells are partitioned in the same ratio as the ratio of whole blood fluxes at points at which vessels divide, we may assume that the red blood cell concentration is constant through the network. This assumption excludes the effect of the Zweifach-Fung effect, in which the finite size of red blood cells reduces their probability of entering fine vessels, so that larger vessels tend to also contain higher concentrations of red blood cells<cit.>. However our own studies of the zebrafish microvasculature show that hematocrit varies only weakly between micro-vessels while conductance variation between similar vessels (such as between different trunk intersegmental vessels) may exceed a factor of 39. Accordingly we incorporatean empirical model for the dependence of viscosity upon vessel diameter only. Pries and Secomb<cit.> measured apparent viscosity of red cell suspensions by analyzing flow of rodent blood through glass capillaries and found that the effective viscosity could be fit empirically by a function: (D) = [ 220exp(-1.3D)+3.2 -2.44exp(-0.06D^0.645)] η_0. Here the vessel diameter, D, is measured in microns, and η_0 is the plasma viscosity, which is comparable to water η_0 ≈ 1 cP). The functional dependence ofupon vessel diameter, D, is shown in Fig. <ref>A.We expect Equation (<ref>) to present a good fit only for blood suspensions where the cell radius and hematocrit are comparable to the experiments of Pries and Secomb. It does not apply therefore to the zebrafish network which we study in Section <ref>. However our algorithm is flexible enough to be able to include different functions in place of Equation (<ref>): We expect qualitatively similar conclusions to hold for different models for the Fahraeus-Lindqvist effect. Incorporating the Fahraeus-Lindqvist effect requires that we rewrite the material constraint since we can no longer simply obtain the radius, and thus volume, of a vessel from its length and conductance. Instead we write: g({κ_kl}) =∑_k>l, ⟨ k,l⟩ =1 d_kl D(κ_kl,d_kl)^2 - K where D(κ,d) maps from the conductance and length of a vessel to its diameter D (we neglect the factor π/4 since we can absorb it into K). We continute to assume that the vessels all have the same length ℓ = 1 so we can write D≡ D(κ). Numerically we find that κ(D) is an increasing function so the inverse function D(κ) exists. The change in cost function does not affect ∂Θ/∂ p_k, so μ_k = 2p_k∀ 1≤ k≤ V still holds. However the conductance derivatives now change to: ∂Θ/∂κ_kl = 2λ D_kl D'(κ_kl) - (p_k - p_l)^2 = 2λD_kl/κ'(D_kl) - (p_k-p_l)^2 where D_kl are the diameters corresponding to κ_kl according to Equation (<ref>). λ can be solved solely from Equation (<ref>): λ = ∑_⟨ k,l⟩=1,k>l(p_k-p_l)^2/κ'(D_kl) D_kl/∑_⟨ k,l⟩=1,k>l2/κ'(D_kl)^2 D^2_kl. The projection works in the same manner: n_kl = 2 D_kl D'(κ_kl) = 2 D_kl/κ'(D_kl). For these networks we found a much larger number of local optima than when flow convergence was assumed to be Newtonian. To deal with these optima and accelerate convergence we adopt one part of the simulated annealing method of Katifori et al.<cit.>. Specifically, when the change in conductance (max{|κn+1-κn|}) becomes too small (in practice we adopt a thrshold of 10^-3, then we multiply all conductances (above threshold ϵ) in the network by a multiplicative noise. Then among all the local minimum visited we select the network with the smallest dissipation. The morphology of non-Newtonian minimally dissipative networks qualitatively resembles Newtonian ones in the sense that they are trees (Fig. <ref>B, C). A strong correlation between dissipation and the total length of the network is again observed (Fig. <ref>D, r=0.99). Here the material of an edge is no longer a certain power of conductance, which is the basis for the original derivation of Murray's law<cit.>. Therefore we expect that the Murray's exponent, defined again by minimization of variance in ∑_i r^a_i, might be far from the theoretical value for Newtonian minimially dissipative networks. However we find that the here the Murray's exponents (2.95 ± 0.081) are quite close to 3, the theoretical value for Newtonian networks, and the sum ∑_i r^a_i is well approximated by constant with the optimized exponent a (Fig. <ref>E, F). It has been proven for Newtonian flow<cit.> under general boundary conditions<cit.> that optimal networks are simply connected. However this proof hinges on the fact that Newtonian flows within a network minimize dissipation (or a related quantity called the complementary dissipation<cit.>). This result does not directly translate to the non-Newtonian flows, including the one described by Equation (<ref>). Our numerical result supports that minimally dissipative networks with the Fahraeus-Lindqvist effect are trees and satisfy Murray's law, but further theoretical work will be needed to confirm that this model for the Fahraeus-Lindqvist effect always produces simply connected optimal networks, or to show that optimal networks are generally simply connected even when other non-Newtonian features of blood (such as the Zweifach-Fung effect) are incorporated.§ OPTIMIZING UNIFORMITY OF FLOW §.§ Optimizing uniformity of flow with material constraint Analyzing minimal dissipation on networks allowed us to compare the performance of the algorithm described in this paper with previous work. We now turn to other target functions that have not been extensively studied. At the level of micro-vessels it is likely that oxygen perfusion rather than transport efficiency is the dominant principle underlying network organization. Indeed our own studies of the embryonic zebrafish trunk vasculature<cit.> showed that red blood cells are uniformly partitioned among different trunk microvessels, and that the "cost" of uniform perfusion (in the sense of the increase in dissipation over a uniform network that did not uniformly perfuse the trunk) was an 11-fold increase in dissipation. We therefore frame this question more generally, i.e. ask what organization of vessels achieves a given amount of flow Q̅ on all links or equivalently, how the flow variation f({p_k},{κ_kl}) = ∑_⟨ k,l⟩ = 1, k>l1/2(Q_kl - Q̅)^2 may be minimized by optimal choice of conductances κ_kl. We can expand the function f and abandon the constant term: f({p_k},{κ_kl}) = ∑_k>l, ⟨ k,l⟩ = 1(1/2Q^2_kl - Q̅Q_kl). Under the assumption that the total flow on all edges is conserved, i.e.: ∑_⟨ k,l⟩ =1,k>l Q_kl = C the function f can be reduced tof({p_k},{κ_kl}) = ∑_⟨ k,l⟩ = 1, k>l1/2Q^2_kl by ignoring constants. The assumption (<ref>) is valid in networks provided that the network may be divided into levels: that is a series of control surfaces may be constructed between source and sink, with no pair of control surfaces intersecting and each edge intersected by one control surface (Fig. <ref>). Then since the total flow across each control surface is the same, the total flow over all network edges is ∑_k>l,⟨ k,l⟩ =1 Q_kl = SF where F is the total sink strength and S is the number of control surfaces. Both symmetric branching trees and quadrilateral grids (such as the one shown in Fig. <ref>) are examples of networks having this property, and both can be used as simplified models of microvascular transport networks<cit.>. Without any constraint the function to be optimized can now be written as Θ = ∑_⟨ k,l⟩ =1,k>l1/2(p_k-p_l)^2 κ_kl^2 - ∑_k∈μ_k (∑_l,⟨ k,l⟩ =1κ_kl(p_k-p_l)-q_k ). Here we show that the optimal networks optimizing (<ref>) have the same flow as the network with uniform conductances, although many different sets of conductances lead to the same set of flow. A stationary network of the functional (<ref>) in which p_k = 0∀ k∈ has the same set of flows as a uniform conductance network with the same support on edges. That is, suppose we let κ_kl, Q_kl be the conductances and flows on the stationary network, and κ'_kl, Q'_kl be those on the uniform conductance network, i.e. κ'_kl = {[ 1; 0; ]..Then Q_kl = Q'_kl∀⟨ k,l⟩ =1.The assumption that all pressure vertices have pressure zero is really an assumption that all pressure vertices have the same pressure: In the latter case since a constant shift in all pressures does not change the flows. To find the critical points of Θ we calculate the derivatives: ∂Θ/∂ p_k = ∑_l,⟨ k,l⟩=1κ^2_kl(p_k-p_l) - ∑_l,⟨ k,l⟩=1 (μ_k - μ_l)κ_kl, k∉∂Θ/∂κ_kl = κ_kl(p_k-p_l)^2 - (μ_k - μ_l)(p_k-p_l) along with μ_i = 0∀ i∈ by assumption. Now we show that a uniform distribution of conductances would result in a critical point ({p_k},{μ_k},{κ_kl}), by rewriting the equation ∂Θ/∂ p_k = 0 (<ref>) into the matrix form:Dμ = D2p. Here D_kl is in Equation (<ref>) and -D2 is another graph Laplacian: D_kl2≐{[ ∑_l,⟨ k,l⟩ =1κ_kl^2 k=l, k∉; -κ_kl^2 ⟨ k,l⟩ =1, k∉;κ2 k=l, k∈; 0 otherwise ].in which the matrix is made full-rank if κ2>0 (similarly to the κ1 constant in D). The κ1 entries in D_kl enforce μ_k =0 at each k∈. The entries in D2 are not needed since p_k=0 at each k∈, but we add values here to emphasize the symmetry between {μ_k} and {p_k}. Now consider uniform conductances, i.e. κ_kl = a>0∀⟨ k,l⟩ =1. We can set κ1 = a and κ2 = a^2. Then we have D = aD2 and since D is invertible (see <ref>) μ = D^-1 D2 p = ap. Now this set of μ_k's and p_k's then also satisfies ∂Θ/∂κ_kl = 0 because ∂Θ/∂κ_kl = a(p_k-p_l)^2 - a(p_k-p_l)^2= 0. Thus the network with uniform conductances along with pressures solved from the Kirchhoff's first law is indeed a critical point. Now we show that any interior critical point, i.e. satisfying κ_kl>0∀⟨ k,l⟩ =1, has the same flows as the uniform conductance network. We will see that for any such network the μ_k's represent the pressures of the uniform conductance network. Since all the conductances are positive we have ∂Θ/∂κ_kl = 0∀⟨ k,l⟩=1. Assume for now p_k - p_l ≠ 0∀⟨ k,l⟩ =1. Then from Equation (<ref>) we obtain that the {μ_k} obey a system of equations κ_kl(p_k-p_l) - (μ_k-μ_l) = 0,∀⟨ k,l⟩ =1 which may be rewritten as μ_k - μ_l = κ_kl(p_k - p_l) = Q_kl,∀⟨ k,l⟩ = 1.Kirchhoff's first law in terms of μ_k's then reads ∑_l,⟨ k,l⟩ =1 (μ_k - μ_l) = q_k ∀ k∈,μ_k = 0 ∀ k ∈. In matrix form the equations can be written as Dμ = F where F_k = q_k if k∈ and is zero otherwise, and D is defined as for network made up of unit conductances: D_kl≐{[ ∑_l,⟨ k,l⟩ =1 1 k=l, k∉;-1 ⟨ k,l⟩ =1, k∉; 1 k=l, k∈; 0o.w. ].Because D is invertible we can solve for μ_k's from Eqn. (<ref>,<ref>). The {μ_k}'s represent the pressures that would occur at each vertex if all conductances in the network were set equal to 1, creating uniform conductance network. Since the flows Q_kl = μ_k - μ_l are determined by μ_k's we conclude that the locally optimal networks would have flows the same as in the network of uniform conductances. To derive (<ref>) from (<ref>) we had to assume that p_k≠_l whenever ⟨ k,l⟩=1. Consider the case where in the optimal network p_k - p_l = 0 for some ⟨ k,l ⟩ =1. For these (k,l)'s Eqn. (<ref>) no longer holds and we have to set ∂Θ/∂ p_k = 0 in Eqn. (<ref>) to obtain extra information. We claim that μ_k = μ_l if p_k - p_l = 0. This can be seen from a loop current argument similar to that used in <ref> to prove existence and uniqueness of the {μ_k}. Specifically, suppose for contradiction that μ_k_1≠μ_k_2 for some pair of vertices with p_k_1-p_k_2 = 0 and without loss of generosity let μ_k_1 > μ_k_2. If k_1 and k_2 ∈ then μ_k_1 = μ_k_2 = 0; so at least one of the two vertices does not lie in . If k_2∉ then ∂Θ/∂ p_k_2 = 0 implies: ∑_l,⟨ k_2,l⟩ =1κ^2_k_2 l(p_k_2-p_l) = ∑_l,⟨ k_2,l⟩ =1κ_k_2 l(μ_k_2-μ_l). Since Eqn. (<ref>) holds when p_k-p_l≠ 0 we have 0 = ∑_l,⟨ k_2,l⟩ =1, p_l = p_k_2κ^2_k_2 l(p_k_2 -p_l) = ∑_l,⟨ k_2,l⟩ =1, p_k_2= p_lκ_k_2 l(μ_k_2-μ_l). Since κ_kl>0∀⟨ k,l⟩ =1 and the sum includes the negative summand κ_k_2 k_1(μ_k_2-μ_k_1) we can find l for which μ_l < μ_k_2 and p_l = p_k_2. We let k_3 = l and repeat the process to find a neighbor of k_3 such that p_l = p_k_3 but μ_l < μ_k_3. We then can keep repeating this process until we reach a vertex k_N ∈ (no vertex may be visited more than once). We have imposed μ_k_N=0. Now we trace through increasing μ_k's starting from k_2 and k_1 and we get k'_1,...,k'_N' such that μ_k'_n < μ_k'_n+1 ∀ n=1,...,N'-1 and μ_k'_1>μ_k_1. By the same reasoning we have k_N'∈ and we reach a contradiction since 0 = μ_k'_N' > μ_k'_N'-1 >⋯ > μ_k'_1 > μ_k_1>⋯>μ_k_N = 0. Therefore μ_k = μ_l when p_k = p_l and Eqn. (<ref>) actually holds for all ⟨ k,l⟩ = 1. Again we conclude that the flows of a locally optimal network with non-zero conductances are the same as the flows in the uniform conductance network. Finally we discuss the boundary case where κ_kl = 0 for some ⟨ k,l⟩ = 1, and we denote this set of links by I. To avoid ill-posedness of pressures we require that that the matrix D is invertible. In this case we do not have Eqn. (<ref>) for κ_kl = 0 because ∂Θ/∂κ_kl need not be zero on these edges. However since there is no flow through links with κ_kl = 0 we can write down Kirchhoff's first law as Dμ = 0,where -D is again the graph Laplacian, but with zero conductance edges removed and other edges with conductance 1: D_kl = {[ ∑_l,⟨ k,l⟩ =1, (k,l)∉ I 1k=l,k∉;-1 ⟨ k,l⟩ =1, (k,l)∉ I; 1 k=l, k∈; 0 otherwise; ]..We can safely remove the zero conductance links from the network because the difference μ_k - μ_l no longer represents the flow Q_kl, and that we know Q_kl = 0 for these links. By assumption we can solve for μ from Eqn. (<ref>) so {μ_k} represent the pressures within the uniform conductance network, but with links κ_kl = 0 removed from the network.Finally we numerically calculate the optimal network for uniformizing flow to verify the theoretical prediction. At each step we can solve for μ_k from Equation (<ref>) and we can calculate the gradient from Eqn. (<ref>). Note that here we have neither Murray nor material constraint, so a numerical projection is not required. The numerical optimal networks have highly heterogeneous conductances within each optimal network (Fig. <ref>A, B), but, as the theory predicted, the flow distribution agrees with the network with uniform conductance (Fig. <ref>C).§.§ Optimal network for uniformizing flows with Murray constraintSo far we have followed previous work<cit.> by calculating all of our optimal networks under constraints on the total material. However both material investment and transport costs (i.e. dissipation) may contribute to the total cost of a particular network. We modify our cost function, g, to include both costs. In this case g({p_k},{κ_kl}) = ∑ (aκ_kl(p_k-p_l)^2 + κ^γ_kl)-K depends on both pressure and conductance, and the full mechanism for keeping g constant during the gradient descent needs to be used. To calculate the optimal network by this method we need an explicit formula for λ. The details are somewhat involved, and we place them in <ref>. Are optimal networks under Murray's constraint morphologically different from those only under material constraint? It is difficult to answer this question for general target functions because it requires us to understand how the constraint surface intersects with the target functions. However for target functions that only depend on flows such as the flow uniformity target function the scaling on conductances can give us additional information. Suppose we find an optimal network under the material constraint. We calculate the total material cost K of this network. Then calculate the optimal network in which Murray's constraint is imposed with allowed total energy K including both material costs and dissipation. Denote by κ_kl the conductances in the network under Murray constraint, and by κ'_kl the conductances in the optimal network under material constraint. If a is sufficiently close to zero then the target function of Murray network will be lower or equal to that of material network. The reasoning is that although ∑κ'^γ_kl + aQ_kl^2/κ'_kl = K does not hold, we can try to solve for a multiplicative scaling β>0 that satisfies ∑ (βκ'_kl)^γ + aQ_kl^2/βκ'_kl = K. Notice that Q_kl does not change under the scaling for this class of networks, so the value of target function is unaffected by scaling conductances. Now if a>0 is small enough we expect to be able to find a solution β and {βκ'_kl} is an admissible network in the sense that it obeys the Murray constraint. Thus the optimal network obeying the Murray constraint must have equal or smaller target function value than the optimal network obeying only the material constraint. By reversing this argument we can see that the optimal networks for small enough a>0 actually agree with those with a=0. The question is how large a has to be so that the Murray network is truly constrained by the total energy cost so that optimal networks under the Murray constraint and under the material constraint diverge. To approach the question we numerically obtained the optimal networks for uniform flow on the topology of capillary bed (Fig. <ref>A) with 0≤ a ≤ 50 and fixed total energy cost. The Murray networks look qualitatively similar to network with only material constraints (Fig. <ref>A), and have the same values of target function the same as analytical lower bound (for a uniform conductance network) (Fig. <ref>B). This result suggests that there could be a wide range of a for which the Murray constraint and material constraints result in identical optimal networks. However the Murray constraint does have an effect on the relative strength of dissipation and material cost. We observe that increasing a decreases material costs (Fig. <ref>C). The trend is unintuitive since a represents the relative costs of dissipation and material. We might therefore expect at larger values of a, the network would invest more in material to reduce dissipation. However if we study the curve of ∑ (βκ'_kl)^γ + aQ_kl^2/βκ'_kl drawn as a function of β, the function is U-shaped and diverges if β→ 0 or if β→∞. When a increases the total energy increases, and the network has to adjust itself to a low energy state. If the network is on the left side of the U this means increasing β, which increases the material cost to realize the constraint. In contrast when the network is on the right side of the curve, decreasing β will be the only way to lower the total energy, which explains the trends depicted in Fig. <ref>B. We will further dissect the role of a in Section <ref>.§ OPTIMAL NETWORKS ON ZEBRAFISH EMBRYO TRUNK VASCULATUREZebrafish are model organisms for studying vertebrate biology. In their embryonic state they are transparent, allowing the microvessels to be seen under the zebrafish's skin. Accordingly the embryonic zebrafish cardiovascular network is widely used to study vascular network growth and the effects of damage on the network<cit.>. Blood flows into the trunk of the zebrafish through the dorsal aorta and then passes into minute vessels called intersegmental (Se) vessels. Blood then returns to the heart via the cardinal vein. These vessels are arranged just like rungs (Se) and parallels (cardinal vein and dorsal aorta) of a ladder (Fig. <ref>A). Most gas exchange in the network is assumed to occur in the Se vessels. As the zebrafish develops further minute vessels form between the Se vessels, converting the trunk into a dense reticulated network<cit.>. We focus on the mechanisms underlying flow distribution in the main fine vessels (Fig. <ref>A). Our previous study of the zebrafish microvasculature<cit.> showed that if each vessel has the same radius then most red blood cells would return to the heart via the highest conductance path, i.e. along the closest Se vessel to the heart, which effectively acts as a short circuit for the network. Our analysis also revealed tradeoffs between preventing short circuits and increasing the dissipation within the network; that is, more flow would pass through distant Se vessels if the conductance of distant Se vessels is increased. But this distribution of conductances has higher dissipation than a network in which all Se vessels have the same conductance. Moreover, although the observed distribution of conductances does not create exactly uniform flows across all Se vessels, creating more uniform distributions of flow would further increase the dissipation within the network. The optimization method described in this paper arose as a way to create a mathematically formal version of the problem: with a given total energy available, how uniformly can flows be divided between intersegmental vessels, and how close is the real embryonic zebrafish network to this constrained optimum?Since the zebrafish trunk network is symmetric we can just consider half of the network consisting of the aorta and intersegmental arteries, designated by vertices v_1,...,v_2n+1 and edges e_1,...,e_2n with n being the number of Se vessels (Fig. <ref>A). Due to the symmetry of the zebrafish trunk vasculature we fix the pressures at v_n+1,...,v_2n+1. We assume the heart pumps a constant volume of blood into the trunk in every time interval so we apply a fixed inflow, F, boundary condition on v_1. First we show how far the network is from minimizing dissipation. If we assign a cost function based only on the total material in the network (i.e. set a=0 and γ=1/2 in Eqn. (<ref>)), then minimizing dissipation eliminates all but the first Se vessel (Fig. <ref>B). Conversely if we instead impose uniform flow at each of the verticesv_n+2,...,v_2n+1 and seek a distribution of conductances that minimizes dissipation, although we see a more realistic distribution of conductances (identical conductances in each Se vessel and tapering aorta (Fig. <ref>C)), in this optimal network the pressures where the Se vessels meet the cardinal vein decrease with distance from the heart (Fig. <ref>D), so that blood flows away from the heart within the cardinal vein which is unphysical.We then explore an alternate organizing principle. Specifically we make uniform flow within Se vessels as our target function. Consider the functionalf({p_k},{κ_kl}) = ∑_i=1^n 1/2(Q_2i -Q̅)^2,where Q̅ is a predetermined flow for all the capillaries (in the following arguments edge-defined quantities such as Q_i are indexed with the edges, and vertex-defined quantities such as p_i are indexed with the vertices). Using this indexing scheme, the function to be optimized becomes:Θ =∑_i=1^n 1/2κ^2_2i p^2_i - ∑_i=1^n Q̅κ_2ip_i - ∑_i=2^n-1μ_i[κ_2i-3(p_i-p_i-1) + κ_2i-1(p_i-p_i+1) + p_iκ_2i]- μ_1[κ_1 (p_1-p_2) + p_1κ_2 - F] - μ_n[κ_2n-3(p_n-p_n-1) + p_n κ_2n-1 + κ_2np_n].Just as in Section <ref> we do not need to introduce a Lagrange multiplier enforcing the material constraint because the target function only depends on flows, and we can scale all conductances to realize any material constraint without affecting the target function. We put the details of the calculation in <ref>. Instead of concentrating all the materials on the first capillary or tapering the aorta, the uniform flow network has constant conductance along the aorta and conductances on the Se vessels that increase exponentially with distance from the heart (Fig. <ref>A). Previously<cit.> we showed that if each Se vessel is assigned the same conductance, then blood flows will decrease exponentially with the index of the Se vessel. To counter this effect and to achieve uniform flow the conductance of Se vessels has to increase from head to tail. Indeed the optimal distribution of conductances matches closely to the experimental data we measured<cit.> (Fig. <ref>B), further suggesting that uniformity might be prioritized over dissipation within zebrafish cardiovascular network.The real zebrafish network agrees well with the optimal set of conductances predicted for a network that uniformizes fluxes across Se vessels. But the agreement is not exact. Is the difference between the two optimal and real networks evidence that the real network has other constraints or target functions that are not modeled by Equation (<ref>)? When given two potential target functions or constraints that may explain the measured geometry of a real transport network, our optimization method provides tools to measure the relative weight the network gives to the two principles. For the zebrafish network, we perform network optimization using the Murray constraint, varying the parameter a to see the extent to which material or transport costs influence the network organization. The gradient descent method with Murray constraint follows <ref> with the target function (and therefore the formula for μ) modified. Specifically χ now becomesχ_kl = (κ_kl(p_k-p_l)-Q̅)(p_k-p_l)I_kl - ∇ (D^-1ζ)_kl∇ p_kl (k,l)∈ℰwhere I_kl = 1 if and only if the edge kl is an intersegmental vessel (ℰ≐{(k,l): ⟨ k,l⟩ =1, k<l}) andζ_k ≐{[ ∑_l,⟨ k,l⟩ = 1 (κ_kl(p_k-p_l)-Q̅)I_klκ_klk∉; 0k∈; ].Finally once λ has been solved for, the expression of μ_k is calculated fromμ = 2aλ p + D^-1ζ(For complete derivation see <ref>). Based on our analysis (in Section <ref>) of uniform partitioning of flows in networks with the Murray constraint, we expect that the optimal zebrafish network will be essentially independent of a over some finite interval of a values, starting at 0. Indeed we find that for small a the target function remains vanishing and the dissipation increases as a increases up to a critical value. However, the arguments given in Section <ref> are silent on how the network changes as a is increased, in particular what happens once a exceeds the critical value, once a exceeds the threshold where it is no longer possible to rescale the conductances in a network that obeys a material constraint into a network that obeys the Murray constraint. We find that a critical value of a_c = 33.3 the network undergoes a phase transition where the target function switches from constant to monotonic increasing and the dissipation decreases (Fig. <ref>A). At the phase transition the conductances of intersegmental vessels transition from solution shown in Fig. <ref>B to becoming non-monotonic with the conductance increasing between vessels near the head and then decreasing at the tail (Fig. <ref>B). Above the critical value of a, the optimal network no longer keep flows uniform between intersegmental vessels (Fig. <ref>C). Put another way, as the parameter a is changed, rather than smoothly interpolating between networks that optimize uniformity and networks that optimize transport, the network optimizes uniformity over a large interval of values of a, and then shifts suddenly to a network that is far from realizing a uniform distribution of fluxes. § DISCUSSION AND CONCLUSION In this work we proposed an algorithm that is able to find locally optimal networks for general target functions under general constraints. We tested that our algorithm is able to reproduce networks that agree with previously calculated optimal transport networks. Motivated by our previous work on zebrafish microvasculature<cit.>, we then studied optimal networks that uniformize network flow and derived an analytical result confirmed by the numerical solutions. To study the tradeoffs between different target functions for a network we introduced a constraint that accounts for both the material cost and dissipation. Finally we applied our algorithm to the zebrafish trunk vasculature and showed that the numerical optimal network agrees with the experimental data. Moreover our results expose a phase transition that occurs as the relative size of transport and material costs is increased. Surprisingly, optimal networks do not continuously interpolate between optimizing uniformity and optimizing dissipation, but instead are initially invariant under changes in the cost of dissipation, and then undergo a sudden phase transition-like reconfiguration when this cost exceeds a certain threshold.Although this result would need to be replicated for other combination of target functions, it offers a surprising biological insight; namely, the departure of real zebrafish networks from the optimum for creating uniform distributions of fluxes cannot be explained from the point of view of the network needing to balance tradeoffs between multiple target functions, and is therefore more likely due to another cause; for example variability (or noise) during vessel formation. More generally adherence to a single target function supports a continued focus on single target functions when studying biological networks, since no two functions will likely shape the network simultaneously.Our algorithm treats the conductances of all edges as independent variables, so the number of degrees of freedom over which optimization is performed is the number of vessels. But the number of vessels in real biological networks may be so large as to defeat direct application of the algorithm. For example, in the mouse brain vascular network there are ∼ 10^4 capillaries in a volume of 2mm^3 <cit.>. More degrees of freedom will also lead to a multiplication of local optima. While parallelization and coupling to global optimization methods for navigation rough landscapes (e.g. simulated annealing) could be potential solutions, another approach is to treat the brain as a multiscale network. Large vessels play different roles from small vessels (such as capillaries). This property may be exloited by numerical methods that treat different scales in different ways.There are many other biological relevant functions to which our algorithm could be applied, for example damage resistance<cit.> and mixing<cit.>. Moreover, our model of oxygen perfusion (which we assume to be uniform, so long as fluxes are uniform between fine vessels) is unlikely to be quantitatively correct for more complex networks. Specifically red blood cells will have lower oxygenation levels the more capillaries they travel through. The history of red blood cell passage through the network will therefore influence their oxygenation. Most optimization problems in this work are constrained either by material or total energy, and it is not clear whether imposing network cost limits as a penalty function rather than as a constraint will give the same result or not. In Murray's original paper the Murray's law was derived by minimizing the total energy formed as a sum of material and transport costs<cit.>. However recent works on minimal dissipation networks impose the material cost as a constraint and minimize dissipation under this constraint. The two approaches carry different physical meanings, and it is not clear which approach is a better model for real biological systems, or whether, indeed, they produce equivalent networks. We are currently studying the conditions under which the two problems are equivalent, i.e. produce equivalent classes of optimal networks<cit.>.In conclusion we proposed a gradient descent algorithm that finds optimal networks with general target functions and constraints. We create this algorithm to reveal the biological organizing principles of microvascular networks. The recent explosion in data streams for microvasculature geometry and flow<cit.>, has created an unmet need for quantitative tools for testing hypotheses on the optimization principles underlying real transport networks. As our zebrafish study shows, our algorithm allows comparison between biological networks and optimal networks achieving different biological functions. While further work will be needed to resolve computational challenges and make rigorous mathematical formulation, our work provides a way to test hypothetical optimal trategies for microvasculature organization, with long term use when understanding microvascular damage, defects and recovery.§ ACKNOWLEDGMENTS This research was funded by grants from the NSF (under grant DMS-1351860). MR. SSC was also supported by the National Institutes of Health, under a Ruth L. Kirschstein National Research Service Award (T32-GM008185). The contents of this paper are solely the responsibility of the authors and do not necessarily represent the official views of the NIH. MR also thanks Eleni Katifori and Karen Alim for useful discussions, and the American Institute of Mathematics for hosting him during one part of the development of this paper. § SOLVABILITY OF {Μ_K} Here we prove that {μ_k} in Equation (<ref>) are solvable under a general configuration of flow (i.e. Neumann) and pressure (i.e. Dirichlet) boundary conditions (BCs). We assume that κ_kl >0∀⟨ k,l⟩ = 1 (since κ_kl = 0 is the same as ⟨ k,l⟩ = 0) and that the network is connected. It suffices to show that the matrix D is invertible. However this is the same matrix in the linear system for solving {p_k} with the specified BCs, so we only have to show that there exists a unique flow given any flow and pressure BCs, which is a well-known<cit.>. However since our derivation makes use of multiple invertibility results for different matrices D, D2 and so on, we provide a proof in order to highlight under what conditions invertibility is allowed. The problem is equivalent to showing thatDp = 0 ⇒ p =0. The solution p for Eqn. (<ref>) corresponds to a network where we do not have any flows into the system except possibly at nodes with pressure BCs, denoted by . The goal is to show that p_k = 0∀ k. Suppose for contradiction that ∃ i∉ s.t. p_i ≠ 0 (since we already have p_j = 0∀ j∈). Then we would have Q_kl≠ 0 for some ⟨ k,l⟩ =1 since the network is connected, and WLOG let Q_kl>0. Now we can trace this flow throughout the network in the following procedure:* Given that Q_k_n-1 k_n >0 first check if k_n ∈, and stop if this is the case.* Consider all nodes l s.t. ⟨ k_n,l⟩ = 1. According to Kirchhoff's first law there must be an l s.t. Q_k_n l >0. Since the network is finite we can pick e.g. the smallest l satisfying these conditions and let k_n+1 = l.* Repeat the procedure until k_N ∈ for some N and stop.If we start with k_1 = k, k_2 = l we can initiate the process since the first condition is satisfied. This procedure has to stop eventually because the network is finite and that k_1,...,k_n are all distinct for any given n>1. To see this suppose k_n = k_m with m>n. Then we would have p_n>p_n+1>⋯ > p_m = p_n, a contradiction. Thus we would end up with a chain of distinct nodes k_1,k_2,...,k_N with ⟨ k_n,k_n+1⟩ =1, Q_k_n k_n+1 > 0∀ n=1,...,N-1, and N∈. Now we repeat the same procedure just with k'_1 = l, k'_2 = k to trace the flows upstream, and we would end up with another chain k'_1,k'_2,...,k'_N' with ⟨ k'_n,k'_n+1⟩ =1, Q_k'_n k'_n+1 < 0∀ n=1,...,N'-1, and N' ∈. Notice that there is no repetition in the set {k_1,...,k_N,k'_1,...,k'_N'}since k_n = k'_m would lead to the same contradiction since pressures must be ordered.§ EXPLICIT FORMULA FOR Λ FOR UNIFORM FLOW NETWORKS WITH MURRAY CONSTRAINT We introduce several notations to be used later. Suppose {b_ij} is a set of quantities defined on the edges of the network. For any real constant c we define the matrix for the graph Laplacian with specified boundary conditions asM_bc = {[ ∑_l,⟨ k,l⟩ =1 b_klk=l, k∉;-b_kl⟨ k,l⟩ =1;ck=l, k∈;0otherwise ]..We also abbreviate M_b = M_b1. In the notation of Equations (<ref>) D = M_κ and D2 = M_κ^2. For a quantity v that is defined on the vertices of the network (such as pressure) we define the graph difference vector ∇ v ∈^E as ∇ v_kl = v_k-v_l(k,l)∈,where ℰ denotes the set of ordered pairs of edges so that each edge only appear once in ℰ. Now we can derive the formula for λ: δκ is given by the explicit formula. From ∂Θ/∂ p_k = 0 we obtain μ = D^-1D2 p + 2λ a p (recall here we have f = ∑_k>l, ⟨ k,l⟩ =11/2κ^2_kl(p_k-p_l)^2, g = ∑_k>l, ⟨ k,l⟩ =1 aκ_kl(p_k-p_l)^2 + κ^γ_kl - K^γ), and so:∂Θ/∂κ_kl = λ[γκ^γ-1_kl - a(∇ p_kl)^2] + κ_kl (∇ p_kl)^2- ∇(D^-1D2p)_kl∇ p_kl.We determine λ from the variational: 0=dg= ∑_k>l,⟨ k,l⟩ =1γκ_kl^γ-1δκ_kl + aδκ_kl∇ p_kl^2 + 2a κ_kl∇δ p_kl∇ p_kl =∑_k>l,⟨ k,l⟩ =1 -α (γκ_kl^γ-1+a∇ p_kl^2){λ[γκ^γ-1_kl - a∇ p_kl^2]+ κ_kl∇ p_kl^2- ∇(D^-1D2p)_kl∇ p_kl} + 2a κ_kl∇δ p_kl∇ p_kl.This formula depends on δ p; the change in p produced by the change κ↦κ+δκ. If we assume p_i = 0∀ i∈ we can write Equation (<ref>) in matrix form as M_δκp + Dδ p = 0 so δ p = -D^-1 M_δκ p.(Equation (<ref>) can be modified by adding a non-zero vector on the right hand side, if inhomogeneous pressure boundary conditions are applied.) Thus if we define auxiliary variables: β≐γκ^γ-1 - a∇ p^2, χ≐κ∇ p^2 - ∇(D^-1D2 p)∇ p, so that δκ = -α(λβ+χ), then: 0 = -α{λ∑_k>l,⟨ k,l⟩ =1(γκ_kl^γ-1 + a∇ p_kl^2)β_kl + ∑_k>l,⟨ k,l⟩ =1(γκ_kl^γ-1 + a∇ p_kl^2)χ_kl} - 2a ∑_k>l,⟨ k,l⟩ =1κ_kl∇ p_kl∇ (D^-1 M_-α{λβ + χ} p)_kl, 0 =λ∑_k>l,⟨ k,l⟩ =1γ^2 κ_kl^2γ-2 - a^2∇ p_kl^4 - 2aκ_kl∇ p_kl∇(D^-1M0_β p)_kl +∑_k>l,⟨ k,l⟩ =1 (γκ_kl^γ-1 + a∇ p_kl^2)χ_kl - 2aκ_kl∇ p_kl∇ (D^-1M-1/α_χp)_kl. Finally we can write down the formula for λ as λ = -∑_k>l,⟨ k,l⟩ =1 (γκ_kl^γ-1 + a∇ p_kl^2)χ_kl - 2a κ_kl∇ p_kl∇ (D^-1M-1/α_χp)_kl/∑_k>l,⟨ k,l⟩ =1γ^2 κ_kl^2γ-2 - a^2 ∇ p_kl^4 - 2a κ_kl∇ p_kl∇(D^-1M0_β p)_kl. The value of λ in Eqn. (<ref>) ensures that g remains constant up to O(δκ_kl) terms. However, we must also adjust {κ_kl} at each step to exactly maintain the constraint following the method given in Section <ref>. In previous applications since g was a function of κ alone this additional projection step did not require perturbation of pressures. Now both the change in κ_kl and the change in flow must be considered when adjusting conductances. We calculate here the additional terms created by involvement of pressures. To project along the constraint surface normal we need to calculate the normal vector: n_kl= ∂/∂κ_kl{∑_i>j,⟨ i,j⟩(κ^γ_ij + aκ_ij(p_i-p_j)^2 ) - K^γ} = γκ^γ-1_kl + a(p_k-p_l)^2 + ∑_⟨ i,j⟩,i>j 2aκ_ij(∂ p_i/∂κ_kl - ∂ p_j/∂κ_kl)(p_i-p_j). To obtain ∂ p_i/∂κ_kl we differentiate Kirchhoff's first law with respect to κ_kl: ∑_j κ_ij (∂ p_i/∂κ_kl - ∂ p_j/∂κ_kl) + (δ_ikδ_jl-δ_ilδ_jk)(p_i-p_j) = 0 or: ∑_j κ_ij (∂ p_i/∂κ_kl-∂ p_j/∂κ_kl) = -(p_k-p_l)(δ_il+δ_ik). Notice that ∂ p_i/∂κ_kl = 0 ∀ i∈ since these p_i are fixed by the boundary conditions. Then we can solve for ∂ p_i/∂κ_kl, 1≤ i ≤ V by solving the linear system (solvability was discussed in <ref>) and calculate the normal vector. § GRADIENT DESCENT METHOD FOR ZEBRAFISH TRUNK NETWORK UNIFORMIZING FLOWS IN INTERSEGMENTAL VESSELSFor performing gradient descent method for zebrafish trunk network uniformizing flows in Se vessels we calculate the partial derivatives of Θ:∂Θ/∂ p_i = {[ κ^2_2i p_i - Q̅κ_2i - (κ_2i-1 + κ_2i-3 + κ_2i)μ_i + κ_2i-1μ_i+1 + κ_2i-3μ_i-1i≠ 1,n; κ^2_2 p_1 - Q̅κ_2 - (κ_1+κ_2)μ_1 + μ_2κ_1 i=1; κ^2_2n p_n -Q̅κ_2n - (κ_2n-3 + κ_2n-1 + κ_2n) μ_n + κ_2n-3μ_n-1 i=n ].. ∂Θ/∂κ_i = {[κ_i p^2_i/2 - Q̅p_i/2 - μ_i/2p_i/2 i|2 = 0; -(μ_i+1/2 - μ_i+3/2)(p_i+1/2 - p_i+3/2) i|2 =1, i≠ 2n-1;-μ_n p_ni=2n-1; ].. Then we impose the physical BCs, i.e. fixed inflow into the network and zero pressure on the ends of the main aorta and the capillaries, and perform gradient descent to find the optimal network.§ EXPLICIT FORMULA FOR Λ FOR UNIFORM FLOW NETWORKS WITH MURRAY CONSTRAINT ON ZEBRAFISH TRUNK VASCULAR NETWORKHere we carry out the calculation of {μ_k}, χ for λ calculation on zebrafish trunk vascular network topology, following <ref>. The only difference lies in the target function: f = ∑_(k,l)∈ℰ1/2(κ_kl(p_k-p_l)-Q̅)^2 I_klwhere ℰ = {(k,l): ⟨ k,l⟩ =1, k<l} under the zebrafish trunk topology and our index convention (Fig. <ref>A), and I is defined as in Equation (<ref>). Again from ∂Θ/∂ p_k = 0 we get μ = 2aλ p + D^-1ζwhere ζ is defined as in Equation (<ref>). Then the gradient of Θ can be calculated as ∂Θ/∂κ_kl = (κ_kl(p_k-p_l)-Q̅)(p_k-p_l)I_kl - aλ (∇ p^2)_kl + λγκ^γ-1_kl - ∇ (D^-1ζ)_kl∇ p_kl≐λβ_kl + χ_kl∀ (k,l)∈ Ewhere β_kl = γκ^γ-1_kl as in <ref>, but χ_kl = (κ_kl(p_k-p_l)-Q̅)(p_k-p_l)I_kl - ∇ (D^-1ζ)_kl∇ p_kl is different. Notice that if we set Q̅ = 0, I_kl = 1∀⟨ k,l⟩ =1 then f is the same as in <ref> and the expression of χ agrees with that in <ref>. Since the expression of β does not change we can simply plug χ into Equation (<ref>) to obtain λ, and use Equation (<ref>) to obtain {μ_k} for the gradient descent.10acheson1990elementary David J Acheson. Elementary fluid dynamics. Oxford University Press, 1990.albers2002transient Gregory W Albers, Louis R Caplan, J Donald Easton, Pierre B Fayad, JP Mohr, Jeffrey L Saver, and David G Sherman. Transient ischemic attack—proposal for a new definition. 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"authors": [
"Shyr-Shea Chang",
"Marcus Roper"
],
"categories": [
"q-bio.QM"
],
"primary_category": "q-bio.QM",
"published": "20170927053316",
"title": "A Gradient Descent Method for Optimization of Model Microvascular Networks"
} |
Touch-based object localization in cluttered environments Huy Nguyen, Quang-Cuong Pham School of Mechanical and Aerospace EngineeringNanyang Technological University,SingaporeEmail: [email protected], [email protected] 30, 2023 ====================================================================================================================================================================================================== Touch-based object localization is an important component of autonomous robotic systems that are to perform dexterous tasks in real-world environments. When the objects to locate are placed within clutters, this touch-based procedure tends to generate outlier measurements which, in turn, can lead to a significant loss in localization precision. Our first contribution is to address this problem by applying the RANdom SAmple Consensus (RANSAC) method to a Bayesian estimation framework. As RANSAC requires repeatedly applying the (computationally intensive) Bayesian updating step, it is crucial to improve that step in order to achieve practical running times. Our second contribution is therefore a fast method to find the most probable object face that corresponds to a given touch measurement, which yields a significant acceleration of the Bayesian updating step. Experiments show that our overall algorithm provides accurate localization in practical times, even when the measurements are corrupted by outliers.§ INTRODUCTIONAccurate object localization is essential for robots to autonomously operate in cluttered, real-world environments. Yet, localization by visual sensors alone might not provide a sufficient precision for many dexterous grasping or manipulation tasks. Consider for instance the assembly a chair <cit.>, where one sub-task consists in inserting wooden pins into the holes on a wooden stick. While object localization by commercial 3D cameras can provide at best 1-2 mm precision, the tightness of the insertion task here requires sub-millimeter precision.One principled approach to address this problem consists in refining the pose estimate by physically interacting with the object: the robot would touch the object of interest (without moving it) at multiple positions, see Fig. <ref>. Contact positions and normals are recorded by forward kinematics (some specialized tactile sensors allow detecting the surface normal at contact <cit.>). The estimation of the object pose from a given a number of such measurements is called the tactile (or touch-based) localization problem.Starting from the 1980's, a large amount of literature has been devoted to this problem and efficient methods have been developed, see<cit.> and our Section <ref>.One major difficulty with these approaches is that, when the target object is placed within a cluttered environment, the robot might touch a different object and/or obstacles instead of the target object, generating thereby an outlier measurement. Existing methods are inherently fragile with respect to such outliers.Here we address the problem of outliers classification by transposing the well-known RANdom SAmple Consensus (RANSAC) method into a Bayesian estimation framework. The algorithm consists in a series of hypothesize-and-verify iterations to select the “best” set of measurements. As these iterations involve (computationally intensive) Bayesian updates, it is crucial to improve these updates in order to achieve practical running times. Our second contribution is therefore a fast method to find the most probable object face that corresponds to a given touch measurement, which yields a significant acceleration of the Bayesian updating step.The remainder of the paper is organized as follows. Section <ref> reviews the related literature and provides the necessary mathematical background. Section <ref> and <ref> present our contributions in detail. Section <ref> reports experimental results, which show that our overall algorithm provides accurate localization in practical times, even when the measurements are corrupted by outliers. Finally, Section <ref> concludes and sketches some future research directions. § RELATED WORKS AND PROBLEM SETTING §.§ Related worksMost previous works on touch-based localization are devoted to reducing the computational complexity of problem, which scales exponentially with the number of DOFs and the size of the initial uncertainty region. Based on Bayesian methods, many variants of particle filters have been proposed and proven to well suit the problem <cit.>. In particular, many approaches are capable of achieving high DOFs localization with large initial uncertainty in a timely fashion. In <cit.>, Petrovskaya introduced the Scaling Series method, which achieved 6-DOF localization with large initial uncertainty of 400mm in position and 360 degrees in orientation. In <cit.>, Vezzani proposed the Memory Unscented Particle Filter that combines a modified particle filter and the unscented Kalman filter. In these works, very often, measurements are obtained through a data collection procedure where the robot's end effector, equipped with a tactile or a force/torque sensor, approaches the object from several different directions. Though these actions can be generated randomly <cit.> or be chosen to maximize the expected information gain <cit.>, there is no guarantee that the set of measurements does not contain extreme erroneous measurements, or outliers, which may result from sensor failures or the presence of other objects in the environments. These outliers, however, will shift the distribution of the object states far from the correct state, leading to a significant loss in localization precision. To mitigate the effect of outlier measurements, one can try to determine whether the received measurement is an outlier. Subsequently, the updating step of the filtering is only performed on a relevant subset of the data. The idea of solving this correspondence problem by classifying measurements into inliers and outliers is not new. There have been many important works on 6-DOF object localization by the computer vision community <cit.>. However, to our knowledge, 6-DOF touch-based estimation in cluttered environments has not been addressed in prior art.§.§ Bayesian estimationWe start out with a quick summary of the problem: one needs to determine the pose ∈SE(3) of an objectof known shape based on a set of tactile measurements .The object is typically represented as a polygonal mesh. The measurements =_0,...,_n are obtained by touching the object with the robot's end effector. Each measurement _k := (^_k,^_k) consists of the acquired contact position ^_k and contact normal ^_k.Note that we consider here the case when the measurement data sets fully contrain the problem. In other words, we assume enough data has been collected in order to sufficiently disambiguate the object pose.Hereafter, the tactile localization problem is cast into the Bayesian framework and addressed as a nonlinear filtering problem.The uncertain knowledge of the object is represented by a probability distribution. The object to be located is assumed to be static during the measurement collection. This assumption is commonly made <cit.> and is realistic: for instance, the object is heavy or is fixed on a support preventing possible movements, or the contact is very slight. Hence, starting with P(_0) – the prior distribution over the state– the goal is to recursively updating the following conditional probability P(_t+1|) = η P(|_t) P(_t). Here P(_t+1|) is known as the posterior, which represent our uncertain belief about the stateafter having incorporated the measurement . On the right-hand side, the first factor P(|_t) is the total measurement probability, which encodes the likelihood of the measurement given the state (measurement model). The second factor P(_t) is the prior, which represents our belief aboutbefore obtaining the measurements . The factor η is a normalizing factor independent of the state _t and needs not be computed.As mentioned before, many variants of particle filters have been proposed and proven to well suit the nonlinear and multi-modal nature of the problem. This paper builds upon these algorithms and provides an automated method to deal with outlier measurements. To illustrate its performance, we apply our method to the Scaling Series algorithm <cit.>, which is able to solve the 6-DOF localization problem efficiently and reliably. The main idea of the Scaling Series approach is to combine Bayesian Monte-Carlo and annealing techniques. It performs multiple iterations over the data, gradually scaling precision from low to high. The number of particles at each iteration is automatically selected on the basis of the complexity of the annealed posterior. §.§ The measurement model The total measurement probability is commonly computed based on the proximity measurement model, where the measurements are considered independent of each other and where both the position and normal components and corrupted by Gaussian noise. For each measurement, the probability is computed based on the distance between the measurement and the object. This model was first introduced authors <cit.> and became popular in the literature owing to its computational efficiency.Assume that the target object is represented as a polygonal mesh, containing a set of faces _i and their corresponding normal vectors _i. Suppose that the object is at state , then the distance between a measurement _k and the face i of the object is defined by(̣_k,^_i) := √((̣^_k,^_i)^2/σ_^2 + (̣^_k,^_i)^2/σ_^2),where (̣^_k,^_i) is the shortest Euclidean distance from ^_k to any point on the face ^_i, (̣^_k,^_i) is the usual angle between two 3D vectors, and σ_, σ_ are the Gaussian noise variances of the position and normal measurement components respectively. Next, the distance between the measurement _k and the object is defined as(̣_k,^) := min_i(̣_k,^_i). For the whole set of measurements , the total measurement error is defined asu(,) := ∑_k (̣_k,^)^2. Finally, the total measurement probability can be computed as followsP(|_t) = η_exp(-1/2u(,_t)^2),where η_ is a constant and will be taken into account during the normalization.Notice that the considered proximity measurement model assumes in some sense that the closest point on the object causes the measurement. Alternatively, one can consider the contribution from all points to the probability of the measurement <cit.>. Though such an approach might be more informative, it is much more computationally intensive.§ ACCELERATING BAYESIAN UPDATES BY EFFICIENT FACE SELECTION§.§ Outline of the algorithm We note that, to compute the likelihood of a measurement, one needs to look for the face that is the most likely to cause the contact and normal measurement,that minimizes the distance (̣_k,^_i).Hence the running time of the updating step depends linearly on the number of faces in the mesh model. A key to improve the speed of the updating step then consists in accelerating the face selection process.Here we propose to do so by pruning out faces based on a pre-computed offline angle dictionary as follows.Offline stage: We compute the angles α_i between a reference vector n⃗_ ( the z-axis in the object reference frame) and the normal vectors n⃗_i of all faces _i. The faces are then sorted according to the value of this angle.Measurement evaluation stage: We first compute the angle α__k between the measurement normal and the reference unit vector. Then, a binary search is used to find, in the list of the α_i's, the two angles α_L and α_R that best approximate α__k from below and from above, respectively. Next, a face _i is added to the subset for evaluation if its associated angle α_i satisfies following conditionα_L - δ_α< α_i < α_H + δ_α,where δ_α is a problem-specific threshold. Fig. <ref> illustrates the face selection algorithm on a partial sphere mesh. In this case, the number of faces considered in the measurement likelihood evaluation has been reduced down to the number of faces in a ring-like region (shown in red). Finally, we compute the distance (̣_k,_i) for all the faces in the subset and choose the face with the lowest distance as representative of the object in (<ref>). §.§ Algorithm parameters One can see that if δ_α is too large, the algorithm will be too conservative and select a larger number of faces than needed, resulting in a longer running time. However, if δ_α is too small, the algorithm might not be able to return a good subset of faces for evaluation. A reasonable choice is to set δ_α = σ_ in order to prune out all faces whose normals are farther than about one-standard-deviation from the measured normal.Another factor that affects the performance of the algorithm is the choice of the reference vector n⃗_. A good choice for n⃗_ would induce an “even” distribution of the α_i, which can be quantified by the Shannon entropy as follows. The range [0,π] is divided into N equal segments. Then the α_i are grouped into N bins depending on their values. The Shannon entropy of the distribution is then given by S := -∑_c=1^N p_clog(p_c),where p_c:= # (bin c)/#faces. Fig. <ref> illustrates the computation of the Shannon entropy for two different reference vectors on the mesh model of the back of a chair.Finally, to choose the best reference vector, we sample random unit reference vectors, compute the Shannon entropy they induce, and choose the one with the highest Shannon entropy.The efficient face selection technique described above can be applied equally well in many Bayesian estimations where normal and contact point measurements are available. The main idea is that the normals are discriminative and that the calculations of distances between normals are very fast, as compared to the calculations of the distances between points and faces. § OUTLIER CLASSIFICATION USING RANSAC §.§ Outline of the algorithm The main challenges of the object localization in cluttered environments are: (i) the correspondance problem (whether the measurement belongs to the object), and (ii) computational complexity. In particular, in cluttered environments, the measurement set is usually contaminated by extreme erroneous measurements, or outliers, which may result from sensor failure or the presence of other objects in the environments. When a particle filter receives an outlier measurement, the weight update will shift most of the weights to a few particles that are far from the correct state. This would lead to a significant loss in localization precision. Thus, to mitigate the effect of outlier measurements, one can try to determine via statistical methods whether the received measurement is an outlier or not. Then the updating step of the filtering is only performed on a relevant set of data ( the dataset containing only inliers). Here we adapt the popular Random Sample Consensus (RANSAC) to simultaneously classify the data into inliers (points consistent with the relation) and outliers (points not consistent with the relation). The algorithm consists of a series of hypothesize-and-verify steps and is presented in pseudo-code in Alg.<ref>. We start by randomly sampling a subset of m measurements called the hypothetical inliers. Based on the elements of this sample subset we find a best estimation (hypothesis) using a particle filter. As stated previsouly, we employ in this article the Scaling Series method <cit.> for its ability to deal with 6-DOF localization with large initial uncertainty in a timely fashion. Once a model has been hypothesized from this minimal subset, the remaining data points are examined to determine which agree with the hypothesis (line 6). This can be achieved by evaluating the Mahalanobis distance as in Eqn. <ref> and <ref>. The points that fit the estimated model well are considered as part of the consensus set. The estimated model is reasonably good if sufficiently many points have been classified as part of the consensus set. Subsequently, the model may be improved by re-estimating it using all members of the consensus set, and a measure of how good the model is can be estimated following Eqn. <ref> (lines 9, 10). These measures are stored and used to select the best hypothesis. This process is then repeated util a termination criterion is met. §.§ Algorithm parameters The standard termination criterion for RANSAC is based on the minimum number of samples required to ensure, with some level of confidence, that at least one of the selected minimal subsets is outlier-free. Let K be the number of iterations, K can be chosen as suggested by [cite]:K := log(1-p)/log(1-w^m),where p is the probability we expect the algorithm to select only inliers from the input data set, w is the probability of choosing an inlier each time a single point is selected. Though this gives an estimate of the required number of iterations, it could result the very large number of iterations when the proportion of inliers is relatively small. Therefore, this should be taken only as the upper limit of iterations. In our algorithm, we terminate the process when the “model goodness” falls below a certain threshold. Here we define the model goodness as follows:G := 1/I∑^I_id_i,where I is the total number of measurements in the current consensus set, d_i is the distance between the measurement i in the set and the object at estimated pose (as defined in Eqn. <ref>,<ref>). In other words, given the measurement consensus set, the proposed model goodness is the average measurement distance between each measurement in the set and the estimated object. Since it encapsulates all the distance errors, G provides a good numerical evaluation of the localization. As a further benefit, it can be computed easily on-line at each iteration and could therefore be monitored to understand when to stop the algorithm. Nevertheless, it is worth noticing that when the measurement errors are too large (indicating by the values of σ_ and σ_), (<ref>) can be non-informative ( G might be low even if it is associated to a wrong object pose). In that case, using the maximum number of iterations K is suggested.Since the presented algorithm requires to perform the Bayesian update steps many times (as part of EstimatePose), it is critical to use the efficient face selection technique presented in Section <ref> in order to achieve practical running times.§ EXPERIMENTSIn order to validate the performance of our proposed methods, a Python implementation has been tested via simulations on differents objects and collection of measurements. All experiments were run on a machine with a 3.40 GHz processor, 4GB RAM. Our implementation is open-source and can be found online at <https://goo.gl/uKaH10>. §.§ Efficient face selectionHere we evaluate the proposed face selection algorithm. As mentioned earlier, the proposed improvement procedure for the measurement likelihood evaluation can be applied equally well in the context of particle filter and many of its variants. Here, for the sake of comparison, we apply our procedure to the Scaling Series algorithm <cit.> and compare it with the vanilla version. The simulation setup consisted of 3 objects: a rectangle box, the back of a chair, and a simplified mesh of a cash register (Fig. <ref>). The performance of the proposed algorithm was assessed in terms of both reliability and execution time. Reliability was measured in terms of the number of successes in all the trials. A trial was considered as a fail if the estimated pose was far from the real pose. Let =(,) and =(,) be the estimated and the real poses, respectively. We defined the distance metrics for rotation and translation as follows:(̣,) = √(log(^-1^2), (̣,) = √( - ^2),where . denoted the Euclidean norm (refer to <cit.> for more details). In our experiment, the thresholds of 0.005 mm in (̣,) and 0.05 rad in (̣,) were used to indicate whether or not a trial was successful. For each object, we ran both methods over 50 trials. The initial uncertainties for all objects were 50 mm along x,y,z and 0.5 rad in rotations about x,y,z. At each trial, we randomly selected a pose from the uncertainty region and used it as the ground truth. The measurements set used in the simulation tests were then drawn by sampling random points on 3D model faces. These measurements were perturbed by Gaussian noises with variances σ_ = 2 mm and σ_ = 0.09 rad. For each trial, we drew a sufficient number of measurements to fully constrain the object estimation and used this set of measurements for both methods. The parameters for the Scaling Series algorithm were chosen as suggested in <cit.> and δ_α = 0.09 rad was used as the threshold of face selection algorithm. After the algorithm terminated at each trial, the pose was estimated by computing the mean of the resulting distribution.Tables <ref> shows execution times for both methods. For all objects, it shows that the proposed algorithm greatly improves the execution time. The improvements were more significant for complex objects ( with a larger number of faces). For example, the time difference ranges from 3.7 times for the box to 6.5 times for the cash register model. This could be expected since our algorithm focuses the computational resources only on a relevant subset of faces during the measurement likelihood evaluation. Note that both methods displayed the same level of reliability since all trials succeeded in both cases. In this experiment, we did not increase the initial uncertainty to keep the running time of the Python implementation moderate. However, with larger initial uncertainty, the required number of particles and measurement evaluation steps will considerably increase. Hence the improvement on the running time will be even more significant.§.§ Object localization in a cluttered environmentIn this experiment, we consider a scene where the object to be located (the box in red) is placed near two other objects (the back of a chair and a wood stick), see Figure <ref>.To sample measurements, we simulated, by a ray tracing method, approaching actions by a manipulator end-effector from random directions. In practice, even when prior information of the box is known and the approaching actions are planned carefully, a number of outliers still appear because of uncertainties arising during the execution. In our experiment, the numer of outliers ranges from 1 to 5 over a total of 15 measurements.While some of these outliers might have been discarded by consideringthe initial probability of the object, others were very difficult to be differentiated from the inliers. Note that, in our case, no information about the other objects was fed into the algorithm, which brings about the need for an outlier classification procedure.The initial uncertainty about the object was assumed to be 10 mm along x, y, z and 0.3 rad in rotations about x, y, z. The measurements were generated using the same parameters as last experiment. The parameters for the Scaling Series algorithm were chosen as suggested in <cit.> and δ_α = 0.09 rad was used as the threshold of face selection algorithm.We performed 50 trials and recorded the execution time, together with the average distances between the estimated poses and the real ones. Over all trials, the proposed algorithm succeed in locating the object of interest with an average execution time 21.3±17.2 seconds. We also performed the Scaling Series method (without outlier classification) over the same data set. In this case, outliers always shift the estimated pose distribution away from the real pose and cause significantly larger errors, see Table <ref>.§ CONCLUSIONThis paper was concerned with the touch-based localization problem in cluttered environments, where outlier measurements can lead to significant loss in precision in existing approaches. Our main contributions consist in applying RANSAC to a Bayesian estimation framework and in proposing a novel face selection procedure to improve the speed of the measurement likelihood evaluation in the Bayesian updating steps. Experiments showed that our algorithm could provide, in a timely fashion, accurate and reliable localization in cluttered environments, in the presence of outliers. Future work includes experiments with real systems and further improvements of the hypothesize-and-verify scheme.§ ACKNOWLEDGMENTThis work was supported in part by NTUitive Gap Fund NGF-2016-01-028 and SMART Innovation Grant NG000074-ENG.IEEEtran | http://arxiv.org/abs/1709.09317v1 | {
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"Huy Nguyen",
"Quang-Cuong Pham"
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"published": "20170927031904",
"title": "Touch-based object localization in cluttered environments"
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[Corresponding author: ][email protected] COMP Centre of Excellence, Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, FinlandCOMP Centre of Excellence, Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, FinlandDepartamento de Física, Universidade Federal do Rio Grande do Norte, Natal, RN, 59078-900, BrazilCOMP Centre of Excellence, Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, FinlandDepartment of Physics, Oakland University, Rochester, Michigan 48309, USADepartment of Chemistry, University of California at Davis, One Shields Avenue, Davis, California 95616, USACOMP Centre of Excellence, Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, FinlandCOMP Centre of Excellence, Department of Applied Physics, Aalto University School of Science, P.O. Box 11000, FI-00076 Aalto, Espoo, Finland Department of Mathematical Sciences and Department of Physics, Loughborough University, Loughborough, Leicestershire LE11 3TU, United KingdomGrain boundaries in graphene are inherent in wafer-scale samples prepared by chemical vapor deposition. They can strongly influence the mechanical properties and electronic and heat transport in graphene. In this work, we employ extensive molecular dynamics simulations to study thermal transport in large suspended polycrystalline graphene samples. Samples of different controlled grain sizes are prepared by a recently developed efficient multiscale approach based on the phase field crystal model. In contrast to previous works, our results show that the scaling of the thermal conductivity with the grain size implies bimodal behaviour with two effective Kapitza lengths. The scaling is dominated by the out-of-plane (flexural) phonons with a Kapitza length that is an order of magnitude larger than that of the in-plane phonons. We also show that in order to get quantitative agreement with the most recent experiments, quantum corrections need to be appliedto both the Kapitza conductance of grain boundaries and the thermal conductivity of pristine graphene and the corresponding Kapitza lengths must be renormalized accordingly. Bimodal grain-size scaling of thermal transport in polycrystalline graphene from large-scale molecular dynamics simulations Tapio Ala-Nissila December 30, 2023 ===========================================================================================================================Chemical vapor deposition, currently the only practical approach to grow wafer-scale graphene necessary for industrial applications, produces polycrystalline graphene containing grain boundaries <cit.> acting as extended defects that may influence electrical and thermal transport <cit.>. The influence of grain boundaries on heat conduction in graphene has been theoretically studied using variousmethods, including molecular dynamics (MD) simulations <cit.>, Landauer-Büttiker formalism <cit.>, and Boltzmann transport formalism <cit.>. Although it is well known that graphene samples prepared by chemical vapor deposition <cit.> have smaller thermal conductivity than those prepared by micromechanical exfoliation <cit.>, experimental measurements of the Kapitza conductances of individual grain boundaries in bicrystalline graphene samples <cit.> and thermal conductivities of polycrystalline graphene samples with controlled grain sizes <cit.> have only been attempted recently. A central issue for polycrystalline samples is how the thermal conductivity scales with the grain size d. Previous MD studies have focused on individual grain boundaries <cit.> or have only considered relatively small grain sizes, typically of a few nanometers <cit.>. However, due to the broad distribution of phonon mean free paths that extend well beyond one micron in pristine graphene, accurate determination of the scaling properties requires considering relatively large grains and sample sizes.Recently, an efficient multiscale approach <cit.> for modelling large polycrystalline graphene samples has been developed within the phase field crystal framework <cit.>. This method, combined with a newly developed highly efficient MD code for thermal conductivity calculations <cit.>, allows for direct atomistic simulations of the heat transport properties of large-scale realistic polycrystalline graphene samples (cf. Fig.<ref>). In this Letter, we compare thermal conductivity values obtained by MD simulations from samples up to 192 nm in linear size to recent experimental data <cit.>, resolving the discrepancy between the small Kapitza conductance (3.8 GW m^-2 K^-1) as extracted from the experimental data <cit.> and the larger values predicted from nonequilibrium MD <cit.> (15 - 47 GWm^-2K^-1) and Landauer-Büttiker <cit.> (≈ 8 GWm^-2K^-1)simulations. Further, we show that to obtain quantitative agreement with experiments quantum corrections need to be applied to classical simulation data. We use the equilibrium Green-Kubo method <cit.> within classical MD simulations to calculate the spatially resolved components <cit.>, κ^in (for in-plane phonons), κ^out (for out-of-plane, or flexural phonons), and κ^cross (a cross term), of the thermal conductivity in polycrystalline graphene, as explained in Methods. Its running components with different grain sizes d are shown in Fig. <ref>. The first main result here is that time scale (which roughly corresponds to an average phonon relaxation time) for κ^out is reduced from ≈ 1 ns in pristine graphene (see Fig. 2 of Ref. ) to ≈ 10 ps in polycrystalline graphene. Thus in polycrystalline graphene κ^out < κ^in in contrast to pristine graphene. This shows that the influence of grain boundaries is much stronger on the out-of-plane than the in-plane component. Another remarkable difference is that the cross term κ^cross does not converge to zero as in pristine graphene, although its contribution is still relatively small. This is due to enhanced coupling between the out-of-plane and in-plane phonon modes in the presence of larger surface corrugation. To study the scaling of κ with d, we first plot κ^in, κ^out, and the total κ^tot = κ^in + κ^out against d in Fig. <ref> (we have included the small cross term κ^cross into κ^in as they have similar time scales). More details on the thermal conductivity data are presented in Supporting Information Figure S1 and Table S1. Previously, the scaling of κ^tot with d has been modelled by the following simple formula both in theoretical <cit.> and experimental works <cit.>: 1/κ^tot(d) = 1/κ^tot_0 + 1/(G^totd), where κ^tot_0 is the total thermal conductivity of pristine graphene (i.e., polycrystalline graphene in limit of infinite d) and G^tot is the Kapitza conductance (or grain boundary conductance), which here characterises the average influence of the different grain boundaries on the heat flux across them.In our previous MD simulations for pristine graphene <cit.> we have obtained κ^tot_0 = 2 900 ± 100 Wm^-1K^-1. Thus the only unknown quantity in Eq. (<ref>) is G^tot, which can be treated as a fitting parameter. In Fig. <ref>(a) it can been seen that the fit is not adequate. The reason is that the in-plane and out-of-plane components have very different properties, resulting in a nonlinear behavior of 1/κ^tot(d) with respect to 1/d. Following Ref.we conclude that this bimodal behavior must be taken into account by separating the two components as 1/κ^in(d) = 1/κ^in_0 + 1/(G^ind);1/κ^out(d) = 1/κ^out_0 + 1/(G^outd), with κ^tot(d) = κ^in(d) + κ^out(d), and κ^in_0 ≈ 850 Wm^-1K^-1 and κ^out_0 ≈ 2 050 Wm^-1K^-1 from Ref. . As it can be seen in Fig. <ref>, Eqs. (<ref>) - (<ref>) give accurate fits yielding G^in≈ 21 GWm^-2K^-1 and G^out≈ 5GWm^-2K^-1. The total Kapitza conductance G^tot = G^in+G^out≈ 26 GWm^-2K^-1 lies well within the range predicted by nonequilibrium MD simulations <cit.>.The scaling parameter L^i = κ_0^i/G^i (i=in, out) in the equations above has the dimension of a length and it defines the Kapitza length <cit.>. In terms of the Kapitza length, the conductivity ratios κ^i(d)/κ^i_0can be written as κ^i(d)/κ^i_0 = 1/(1+L^i/d). This shows that when the grain size equals the Kapitza length, κ^i(d) reaches half of κ^i_0. The Kapitza lengths for the in-plane and out-of-plane components from our MD data are L^in≈ 40 nm and L^out≈ 400 nm, differing by an order of magnitude which reflects the difference in the scaling of the corresponding conductivity components with d (cf. Fig. <ref>).In Fig. <ref> we compare our MD data for the scaling of the components of κ(d)/κ_0 vs. d with previous theoretical predictions <cit.>. They are closer to our results for the in-plane component indicating that κ^out(d) was not properly accounted for due to either an incorrect definition of the heat current or unconverged size scaling. In fact, the calculations in Ref. were based on the heat current formula in LAMMPS <cit.> which is incorrect for many-body potentials <cit.>. Indeed, κ^tot_0 in Ref. was estimated to be about 720 Wm^-1K^-1, which is even smaller than our κ^in_0. The calculations in Ref. were based on the approach-to-equilibrium MD method <cit.> with a fixed sample size of 200 nm long and 20 nm wide, which also significantly underestimates the contribution from the out-of-plane component. The Kapitza lengths for the data from Refs.andcan be estimated to be ≈ 15 nm and 25 nm, respectively, in stark contrast with our results.In Fig. <ref> we also plot the data from the most recent experimental measurements (triangles) <cit.>. Although our new data are much closer to the experiments than the previous theoretical results, there is still a quantitative difference. One reason for the discrepancy is the use of classical statistics in view of the high Debye temperature (≈ 2000 K) of graphene <cit.>. Using classical statistics can grossly overestimate the Kapitza conductance. Indeed, quantum mechanical calculations based on the Landauer-Büttiker formalism <cit.> predict graphene grain boundary conductance several times smaller than that from classical non-equilibrium MD simulations <cit.>. While there is no rigorous way to include all the quantum effects within the present calculations, we can gauge their importance by applying the mode-by-mode quantum correction in Ref. to the spectral conductance. In Fig. <ref> we show our data for the in-plane and out-of-plane components of the spectral conductance g^i(ω) (i=in, out) for the 24 × 24 nm^2 polycrystalline system, calculated using the spectral decomposition method in Ref. .The mode-to-mode quantum corrections can be incorporated by multiplying g^i(ω) (i=in, out) by the factor x^2e^x/(e^x-1)^2 (x = ħω / k_B T), which yields the dotted and dot-dashed lines in Fig. <ref>. The influence of these corrections is significant; the integrated conductance is reduced by a factor of three for the in-plane component and by a factor of two for the out-of-plane component. Therefore, our estimates for G^in and G^out are modified to 7 GWm^-2K^-1 and 2.5 GWm^-2K^-1, respectively. The corresponding Kapitza lengths are changed to L^in≈ 120 nm and L^out≈ 800 nm, respectively. The results for the modified conductivity ratio κ^tot(d)/κ^tot_0 are plotted in Fig. <ref> with squares. The agreement with the experiments is better, but still not at a quantitative level.To fully resolve the discrepancy between our data and the experiments we need to revisit the case of pristine graphene. Classical statistics can underestimate the thermal conductivity in the pristine case, too, by overestimating the phonon-phonon scattering rates of the low-frequency phonon modes <cit.>, the major heat carriers in pristine graphene. This explains why the thermal conductivity of pristine graphene calculated by our previous MD simulations (2 900 Wm^-1K^-1) is significantly smaller than the prediction from lattice dynamics calculations <cit.> (about 5 000 Wm^-1K^-1)and the most recent experiments on high-quality monocrystalline graphene <cit.> (5 200 Wm^-1K^-1) . Unfortunately, unlike in the case of grain boundary conductance, there is so far no feasible quantum correction method for classical MD thermal conductivity calculations in the diffusive regime where phonon-phonon scattering dominates. In view of the fact that the differences between the results from classical MD and quantum mechanical lattice dynamics methods mainly concern the out-of-plane component <cit.>, we can resolve this issue here by scaling κ_0^out such that κ_0^tot equals the experimental reference value <cit.> of 5 200 Wm^-1K^-1. Combining this with the mode-to-mode quantum corrections to the Kapitza conductance above, the final revised Kapitza lengths for the two components now become L^in≈ 0.12 μm and L^out≈ 2 μm, respectively, differing by more than an order of magnitude. With quantum corrections to both G and κ, the scaling of κ(d)/κ_0 (circles in Fig. <ref>) finally agrees with the experiments at a fully quantitative level.Finally, we note that in the experimental work <cit.>, the grain-size scaling was interpreted in terms of a single Kapitza conductance of 3.8 GWm^-2K^-1, which is much smaller than that from Landauer-Büttiker calculations <cit.> (8 GWm^-2K^-1). In contrast, our bimodal grain size scaling with the two quantum corrections gives G^tot = G^in + G^out=9.5 GWm^-2K^-1, which is a reasonable value considering that the harmonic approximation used in Ref. can somewhat underestimate the Kapitza conductance at room temperature. In summary, by using high-accuracy MD simulations of large polycrystalline graphene samples generated by a multiscale modeling approach, we have demonstrated that the inverse thermal conductivity does not scale linearly with respect to the inverse grain size but shows a bimodal behavior. The Kapitza lengths for the grain boundaries associated with the in-plane and out-of-plane phonon branches differ by more than one order of magnitude. While the grain-size scaling is dominated by the out-of-plane phonons with a much larger Kapitza length, the in-plane phonons contribute more to the Kapitza conductance. We have also demonstrated that in order to obtain quantitative agreement with the most recent experiments of heat conduction in polycrystalline and pristine graphene samples, quantum corrections to both the Kapitza conductance of grain boundaries and the thermal conductivity of pristine graphene must be included and the corresponding Kapitza lengths must be renormalized accordingly. We note that we have only considered suspended graphene samples in this work. For supported graphene, heat transport by the out-of-plane phonons will be significantly suppressed. Whether or not the bimodal scaling will survive in the presence of a substrate is an interesting question which requires further study. Finally, we point out that our samples were generated by the phase field crystal method, which has been shown to reproduce realistic grain size distributions in the asymptotic limit in two dimensions <cit.>. Such samples may not correspond to those observed in some experiments <cit.>, but additional non-uniformity and anisotropy should influence the results only quantitatively, not qualitatively. The concept of an effective grain size is analogous to that of an effective phonon mean free path, which, in spite of being a relatively crude estimate, captures the essential physics. Because the in-plane and out-of-plane phonons have drastically distinct transport properties, we expect that the bimodal scaling would survive even if the influence of additional non-uniformity and anisotropy were taken into account. This argument is further supported by the fact that while the Kapitza conductance of individual grain boundaries depends on the angle of misorientation, for angles between about 20 and 40 degrees, this dependence is rather weak. We have carried out a comprehensive study of the Kapitza conductance for grain boundaries of different orientations and the results will be published elsewhere. Methods. Realistic polycrystalline samples are constructed with the phase field crystal model by starting with small random crystallites that grow in a disordered density field. We grow the polycrystalline samples akin to chemical vapor deposition by assuming non-conserved dynamics. The relaxed density field is converted into a discrete set of atomic coordinates suited for the initialization of MD simulations <cit.>. To investigate the scaling properties, we construct polycrystalline samples with various characteristic grain sizes, d, defined as d=(A/n)^1/2, where A is the total planar area, and n is the number of grains comprising it. We considersystems of four sizes: 24 × 24 nm^2, 48 × 48 nm^2, 96 × 96 nm^2, and 192 × 192 nm^2. Each case was initializedwith 16 randomly placed and oriented crystallites. The final number of grains in a sample is typically smaller than 16 and the effective grain sizes, averaged over a few realizations for each sample size, are found to be 8 nm, 14 nm, 26 nm, and 50 nm, respectively. Figure <ref> shows a typical structure of our smallest 24 × 24 nm^2 polycrystalline sample after MD relaxation.We use the Green-Kubo method <cit.>, with equilibrium MD, to calculate the thermal conductivities. In this method, one can calculate the running thermal conductivity tensorκ_μν(t) (μ andν can be x or y for two-dimensional materials) as a function of the correlation time t as κ_μν(t) = ∫_0^t⟨ J_μ(0)J_ν(t') ⟩ dt', where ⟨ J_μ(0)J_ν(t') ⟩ is the heat current autocorrelation function evaluated as the time average ⟨ ⟩ of the product of two heat currents separated by t'. A decomposition of the heat current, J_μ=J_μ^in+J_μ^out, which is essential for two-dimensional materials, was introduced in Ref. . We stress that the decomposition is in terms of the velocity, and the out-of-plane heat current J_μ^out is not a heat current perpendicular to the graphene basal plane (taken as the xy plane), but a heat current in the basal plane (μ direction) contributed by the out-of-plane vibrational modes.With this decomposition, the thermal conductivity decomposes into three terms, κ^in_μν(t), κ^out_μν(t), and κ^cross_μν(t), associated with the autocorrelation functions ⟨ J^in_μ(0)J^in_ν(t')⟩, ⟨ J^out_μ(0)J^out_ν(t')⟩, and 2⟨ J^in_μ(0)J^out_ν(t')⟩, respectively. As our systems can be assumed statistically isotropic, the conductivity in the basal plane can be treated as a scalar κ = (κ_xx+κ_yy)/2.We perform the equilibrium MD simulations using an efficient GPUMD code <cit.>. The Tersoff potential <cit.> optimized for graphene <cit.> is used here. The velocity-Verlet method <cit.> is used for time integration, with a time step of 1 fs for all the systems. All the simulations are performed at 300 K and a weak coupling thermostat (Berendsen) <cit.> is used to control temperature and pressure during the equilibration stage (which lasts 2 ns for all the systems). The production stage (in which heat current is recorded) lasts 5 ns and 10 independent simulations are performed for each sample. Periodic boundary conditions are applied on the xy plane.Although the magnitude of the out-of-plane deformation can exceed 1 nm in some cases, we assume a uniform thickness of 0.335 nm for the monolayer when reporting the effective three-dimensional thermal conductivity values.The spectral conductance g^i(ω) (i=in, out) is calculated using a spectral decomposition method <cit.> in the framework of nonequilibrium MD simulations. The system is divided into a number of blocks along the transport direction, with the two outermost blocks being taken as heat source and sink, maintained at 320 K and 280 K, respectively, using the Nosé-Hoover chain thermostat <cit.>. The transverse direction is treated as periodic and the two edges in the transport direction are fixed. After achieving steady state, we calculate the correlation function K^i_A → B(t) (i=in, out) defined in Ref. . Then the spectral conductance is calculated as <cit.> g^i(ω) =∫_-∞^+∞ dt e^iω t[ 2K^i_A → B(t)/(SΔ T) ], where S is the cross-sectional area and Δ T is the temperature difference between the source and the sink. Supporting informationThe following files are availablefree of charge. * samples.zip: All the polycrystalline graphene samples created by the phase field crystal method.* supp.pdf: Detailed results on the thermal conductivity of the polycrystalline graphene samples.NotesThe authors declare no competing financial interests. This research has been supported in small part by the Academy of Finland through its Centres of Excellence Program (Project No. 251748). We acknowledge the computational resources provided by Aalto Science-IT project and Finland's IT Center for Science (CSC). Z.F. acknowledges the support from the National Natural Science Foundation of China (Grant No. 11404033). P.H. acknowledges financial support from the Foundation for Aalto University Science and Technology, and from the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters. L.F.C.P. acknowledges financial support from the Brazilian government agency CAPES for project “Physical properties of nanostructured materials” (Grant No. 3195/2014) via its Science Without Borders program. K.R.E. acknowledges financial support from the US National Science Foundation under Grant No. DMR-1506634. | http://arxiv.org/abs/1709.09498v1 | {
"authors": [
"Zheyong Fan",
"Petri Hirvonen",
"Luiz Pereira",
"Mikko Ervasti",
"Ken Elder",
"Davide Donadio",
"Ari Harju",
"Tapio Ala-Nissila"
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"published": "20170927133006",
"title": "Bimodal grain-size scaling of thermal transport in polycrystalline graphene from large-scale molecular dynamics simulations"
} |
An Isolated Microlens Observed from K2, Spitzer and Earth A. Yonehara December 30, 2023 =========================================================empty§ INTRODUCTION Starting with standard silica fibers, supercontinuum generation in optical fibers is studied extensively in the past decades. Today fiber-based supercontinuum sources are largely used in a variety of applications including optical metrology, fiber communication systems and biomedical imaging <cit.>. Evolution dynamics of supercontinuum generation inside of single-mode or few-mode fibers are well understood and photonic crystal fibers are allowed scientists to tailor key fiber properties to achieve octave spanning supercontinuum sources with a wide range of source parameters <cit.>. Recently graded-index multimode fibers (MMFs) drew attention by featuring unprecedented nonlinear phenomena. Intermodal interactions and spatio-temporal evolution with periodic imaging leads to discovery of unique phenomena such as Kerr self-cleaning <cit.>, spatio-temporal solitons <cit.>, ultrabroadband dispersive waves <cit.>, spatio-temporal instability <cit.>, quasi-phase matched four-wave mixing (FWM) <cit.> and cascaded Raman scattering <cit.>. With exploiting aforementioned nonlinear effects in graded-index MMFs a robust and relatively less sophisticated method to generate supercontinua is presented in the literature<cit.>. These efforts to generate supercontinuum in graded-index MMF by pumping in normal dispersion regime investigated the nonlinear dynamics by choosing low repetition rate (Hz-kHz) sources to achieve high pump peak powers. Hence maximum average powers of the reported supercontinua are in hundreds of mWs range so far even though graded-index MMFs are promoted with high power level handling potentials. Many applications of supercontinuum sources require achieving high average powers for detection purposes. All-fiber lasers are generally preferred to obtain high average and peak powers with high repetition rates. As a result of this capability, fiber based supercontinua are studied extensively in photonic crystal fibers with femtosecond to continuous-wave regimes <cit.>. Hence fiber lasers are suitable supercontinuum source candidates to generate supercontinua in MMFs with high average powers. Here, we study the generation of octave-spanning high power and high repetition rate supercontinuum generation in graded-index MMFs with 62.5 μm core diameter (Thorlabs-GIF625). Pump pulses with MHz repetition rates, ∼30 kW peak power and 70 ps pulse duration are launched to graded-index MMF. Spectrally flat supercontinua with 1.88 W and 3.5 W output average powers are generated in 20 m fiberwith 1 MHz and 2 MHz repetition rates respectively. Although our system is pump power limited, our observations regarding power scaling with increasing repetition rate suggest >100 W powers can be achieved in graded-index MMFs with higher repetition rate pump laser systems.§ EXPERIMENTAL RESULTS Fig. <ref> shows a schematic of the experimental setup. A home-built Yb-doped dispersion managed mode-locked fiber laser is employed as a pump pulse generator <cit.>. Chirped pulses are first amplified by a preamplifier to ∼70 mW average power. AOM is employed to change fundamental repetition rate of the pulse train before the main amplifier. Due to the loss of AOM and the change of repetition rate from ∼40 MHz to 1 MHz average power drops to ∼ 2mW after the AOM. Thus another preamplifier is placed before the double clad main amplifier. At the end of the main amplifier, we obtain 70 ps pulses centered around 1045 nm with ∼20 nm bandwidth and adjustable repetition rates (kHz-MHz) which corresponds to ∼30 kW peak power. We collimate the output of the system with a biconvex lens and high power free-space isolator is used to prevent back reflections. We use a half-wave plate to change the polarization of the pump pulse to achieve best conversion condition in graded-index MMF. To excite the fiber we use a biconvex lens with 2 cm focal length which creates ∼ 20 μm beam waist size at the facet of the MMF. Optimum coupling condition is achieved with a three-axis translation stage which enables free space coupling efficiency greater than 80%. By using pump pulses with 1 MHz repetition rate generation of supercontinuum is demonstrated in 20 m and 10 m graded-index MMFs with 2.19 W and 1.88 W output average power as presented in Fig. <ref>(b-c). Pump pulse spectrum contains low unabsorbed pump around ∼ 980 nm but at the end of graded-index MMF, supercontinuum become dominated in that region as well. Spectral measurements are performed with an optical spectrum analyzer covering a range from 600 nm to 1700 nm. Due to the lack of a suitable measurement tool, no information above 1700 nm obtained thus end of generated supercontinua on the longer wavelengths is unknown. By comparing results of 10 m and 20 m fiber we notice increasing the propagation length helps to generate conversions for shorter wavelengths and broader spectrum can be achieved but due to the losses of fiber average output power starts to drop. Evolution of supercontinuum for 20 m fiber is investigated in detail by studying the effect of pump launch power [Fig. <ref>]. For low powers, we observe small wavelength conversion from pump to 780 nm and 1550 nm. These frequency shifts are matching with intermodal FWM and spatiotemporal instability but with increasing launch power SRS becomes dominant nonlinear mechanism. When 0.82 W output power is reached, generation of cascaded SRS with ∼ 13 THz frequency shifts is observed as reported by Pourbeyram et al. <cit.> [Fig. <ref>(b)]. When SRS peaks reach zero dispersion wavelength (ZDW) of the fiber (∼ 1330 nm), broad spectral formation is emerges from the noise. This behavior can be explained by complex parametric phenomena including collision-based spectral broadening and the reduction in SRS gain near ZDW <cit.>. After we reached 1.89 W output power we observed a sudden drop at average output power and wavelength generation at shorter wavelengths starts to emerge. Evolution of supercontinuum at visible wavelengths can be explained with the coupling of Raman and parametric gain which also takes place in supercontinuum generation for picosecond pulses in photonic crystal fibers <cit.>. This results in the generation of anti-Stokes wavelengths even without proper phase-matching <cit.>. In the end, more than octave spanning spectrally flat supercontinuum achieved with 1.88 W average output power for 1 MHz pump repetition rate. Overall spectral intensity deviation of the continuum is calculated as 52%. On the other hand, above the pump wavelength (between 1060-1700 nm), spectral intensity deviation is incredibly small as 24%.We study supercontinuum generation in 20 m graded-index multimode fiber with 50 μm core diameter to demonstrate versatility of this low-cost supercontinuum generation method. For 1 MHz repetition rate and same output average power, spectral difference is presented in Fig. <ref>(a). Spectral features resembles the supercontinuum generated in graded-index multimode fiber with 62.5 μm core diameter. To demonstrate power scalability of the supercontinuum generation method in graded-index MMF, while peak power of the pump pulse remains same as we sweep repetition rate of pump pulses in kHz region [Fig. <ref>(b)]. In the experiments we set the pump peak power as 25 kW and increase average output power from 350 mW to 1.4 W. Here we focus on MHz repetition rates and as shown in Fig. <ref>(a), with increasing pump pulse repetition rate from 1 MHz to 2 MHz by preserving pump peak power ultra-broad supercontinua could be reproduced. By doubling pump pulse repetition rate from 1 MHz to 2 MHz, while keeping peak powers constant, we achieve 3.96 W and 3.50 W average powers for supercontinua generated in 10 m and 20 m graded-index MMF, respectively. Our scaling experiments are pump power limited but these results indicate that by increasing average power and repetition rate, a hundred watts could be obtained with standard graded-index fiber while octave-spanning supercontinuum features remain.Next, we measure near field spatial distribution of supercontinuum using a beam profiler operating up to 1200 nm. Due to the device limitation, we could measure the beam profile from 730 nm to 1200 nm Fig. <ref>(b). Additionally with a long-pass filter, we measure beam profile from 1100 nm to 1200 nm Fig. <ref>(c), as well. Even though fiber preferred in the experiments supports hundreds of modes, Gaussian-like spatial profile with high-order modes in the background is observed. Raman or Kerr beam cleaning could be the reason of observed spatial distributions <cit.>.§.§ Numerical Results In order to develop better understanding of supercontinuum evolution inside graded-index MMF we perform numerical simulations with 1+1D generalized nonlinear Schrödinger equation: ∂ A/∂ z + ( ∑_n≥ 2β _ni^n-1/n!∂^n /∂ t^n )A=iγ (z) (1+∂/∂ t )( (1-f_R)A | A|^2+f_RA∫_0^∞h_R(t') | A(z,t-t')|^2 )with a periodic nonlinear coefficient <cit.>. For numerical integration with high accuracy in simulations, we prefer the fourth-order Runge-Kutta in the Interaction Picture (RK4IP) method <cit.>. In simulations, we include Raman process (f_R), shock terms and high-order dispersion coefficients up to β _7. SRS is included in the equation via use of a response function <cit.>. Even though our simulations start from quantum noise, we average the simulations over 20 sets of initial conditions to simulate experimental observations more accurately. To decrease computation time, we consider pump pulse duration 10 times smaller than the experiments while pulse energy remains the same. It allows us to decrease propagation length 10 times. Therefore simulations are performed with 7 ps pulse duration, ∼300 kW peak power at 1040 nm central wavelength and we set n_0 as 1.470, n_2 as 2.7x10^-20 m^2/W, relative index difference as 0.01, time window width as 70 ps with 2 fs resolution.Spectral and temporal evolution obtained from numerical simulations for 1 m graded-index MMF with 62.5 μm core diameter are presented in Fig. <ref>. Obtained results are equivalent to experimental supercontinuum generation after propagation of 10 m fiber. Detailed analysis on the evolution of the pump pulse inside the fiber reveals that simulations also present strong cascaded SRS generation with relatively low parametric wavelength conversions in the spectral domain. After SRS peaks reach to ZDW, generation of new wavelengths at anomalous dispersion resembles development of Raman soliton components. In the literature, this phenomenon is explained as the transformation of SRS peaks above ZDW to ultrashort solitons <cit.>. These solitons experience self-frequency shift and more uniform spectra can be formed. Preceding soliton dynamics lead to temporal breakup seen in the simulations after 0.5 m propagation and with the help of spectrally broadened SRS peaks, a supercontinuum starts to appear. Spectral flatness and intensity distribution behavior matches well with experiments even though loss terms are not included in the simulations. § CONCLUSION In summary, we demonstrated the generation of high average power spectrally flat octave-spanning supercontinua using a graded-index MMF pumped with an all-fiber laser system. The highest supercontinuum output power of 3.96 W is achieved in graded-index MMF with 62.5 μm core diameter using picosecond pulses at MHz repetition rate. Numerical simulations reveal that unique cascaded SRS observed in graded-index MMF plays a significant role in the spectral evolution. We have shown that this recently discovered, low-cost graded-index multimode fiber based supercontinuum source could benefit from multimode features of the fiber and high power level average powers are feasible to achieve. Adaptability of the high power and high repetition rate supercontinuum generation method is studied as well. Further power scaling based on fiber pump lasers enable high repetition rate all-fiber supercontinuum systems with >100W average powers. Therefore noise and coherency measurements of the supercontinua generated with graded-index MMF studies become feasible. § ACKNOWLEDGMENTSThe authors thank TUBITAK, TUBA-GEBIP, BAGEP, METU Prof. Dr. Mustafa Parlar Foundation and FABED for support and Ç. Şenel for insightful helpful suggestions. | http://arxiv.org/abs/1709.09158v1 | {
"authors": [
"Uğur Teğin",
"Bülend Ortaç"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170926174936",
"title": "High power high repetition rate supercontinuum generation in graded-index multimode fibers"
} |
[email protected]é de Bordeaux-CNRS-CEA, Centre Lasers Intenses et Applications, UMR 5107, 33405 Talence, France Université de Bordeaux-CNRS-CEA, Centre Lasers Intenses et Applications, UMR 5107, 33405 Talence, France CEA/CESTA, 15 Avenue des Sablières, CS 60001 33116 Le Barp cedex France CEA/CESTA, 15 Avenue des Sablières, CS 60001 33116 Le Barp cedex France CEA/CESTA, 15 Avenue des Sablières, CS 60001 33116 Le Barp cedex France Université de Bordeaux-CNRS-CEA, Centre Lasers Intenses et Applications, UMR 5107, 33405 Talence, France The absorbed laser energy of a femtosecond laser pulse in a transparent material induces a warm dense matter region which relaxation may lead to structural modifications in the surrounding cold matter. The modeling of the thermo-elasto-plastic material response is addressed to predict such modifications. It has been developed in a 2D plane geometry and implemented in a hydrodynamic lagrangian code. The particular case of a tightly focused laser beam in the bulk of fused silica is considered as a first application of the proposed general model. It is shown that the warm dense matter relaxation, influenced by the elasto-plastic behavior of the surrounding cold matter, generates both a strong shock and rarefaction waves. Permanent deformations appear in the surrounding solid matter if the induced stress becomes larger than the yield strength. This interaction results in the formation of a sub-micrometric cavity surrounded by an overdense area. This approach also allows one to predict regions where cracks may form. The present modeling can be used to design nano-structures induced by short laser pulses. Thermo-elasto-plastic simulations of femtosecond laser-induced structural modifications: application to cavity formation in fused silica Guillaume Duchateau December 30, 2023 ========================================================================================================================================§ INTRODUCTIONFemtosecond laser pulses are widely used to structure transparent dielectric materials by modifying their local properties, resulting in the formation of designed structureson the surface or in the bulk <cit.>.This leads to various industrial and technological applications as optical data storage, waveguide or grating writing <cit.>. In the case of femtosecond laser pulses tightly focused in the bulk, relatively high intensities, above 10^14 W/cm^2, can be reached. Such intensities enable to ionize most of the transparent dielectric materials (even those exhibiting large optical bandgaps) by inducing nonlinear electronic responses.Because the laser beam deposits its energy on a timescale smaller than hydrodynamic characteristic times (of the order of tens picoseconds), there is a decorrelation between the energy absorption processes andthe evolution of macroscopic matter properties. A warm dense matter with both a high temperature and pressure is then created in the focal volume of characteristic dimensions of the order of a few tenths of micrometer. This area is surrounded by a non-modified cold material. The relaxation of this heated matter generates a strong shock and rarefaction waves which propagate through the solid.For energy densities above the material damage threshold, a significant structural modification may appear by compression and traction of the material. In particular, several studies and experimental observations have shownthat a single ultrashort laser pulse could form a cavity in the bulk material <cit.>.Important efforts and significant progresses have been performed to understand the physical processes at play during the laser-material interaction including the solid phase response <cit.>. Especially, the recent work by Najafi et al.<cit.> investigates the birefringence induced by the axisymmetric stress due to the interaction of an ultrashort laser with a transparent material. In the particular case of tightly focused beams theoretical and numerical studies have been carried out by only accounting for the hydrodynamic behavior <cit.> (fluid behavior) or by including the solid mechanical properties <cit.>. In the latter study, despite only small material density variations have been considered, it has been shown that the behavior of thesolid is very important and play a significant role in the formation of the desired structures. Based on these considerations, it appears that theoretical and numerical developments are still desired to model and understand significant laser induced material modifications by tightly focused beam.The aim of this paper is to model these laser-induced modifications and understand the transient mechanisms leading to a cavity formation. The solid behavior is introduced in the hydrodynamic Euler's equations. Generally, materials exhibit a viscoelastic behavior mixing both the elastic properties of solids and the viscosity one of liquids <cit.>. For the present applications, only small material deformations are assumed since experimental observations show that the overdense region around the cavity is significantly smaller than the size of the total modified area <cit.>. The material response thus mainly exhibits an elastic behavior where the viscosity influence may be neglected. However, due to these deformations, when the induced mechanical stress exceeds the elastic limit, permanent deformations are produced. Within this framework, the solid material response then consists of two regimes of deformation: an elastic (reversible) regime and a plastic (irreversible) regime.In the present work, to describe the elasto-plastic (EP) response of the solid, the physical model proposed by Wilkins <cit.> is used (Section <ref>). It has been implemented in the hydrodynamic CHIC code <cit.>to study the cavity formation by a tightly focused laser beam (Section <ref>). The simulations are performed for fused silica by assuming an instantaneous energy deposition. Throughout the paper, the EP predictions are compared to fluid simulations. First, 1D simulations are considered in order to exhibit the influence of the EP behavior. Especially, a difference between the shock waves (shape and velocity) clearly appears, leading to a significant discrepancy between the density profiles. Results of 2D simulations demonstrate the role of the solid response for the stabilization of the laser-induced nano-structure, providing caracteristic deformations in a good agreement with experimental observations.Finally, a prediction of the critical zoneswhere potential fractures may appear is proposed. Conclusions and perspectives are drawn in Section <ref>.§ THEORETICAL MODELING Following references [Wilkins1964] and [Maire2013], the relaxation of the heated matter in the surrounding colder material can be described by a more general form of the hydrodynamic conservation laws,where the pressure P is substituted by the Cauchy stress tensor <cit.> σ̅̅̅. Within their lagrangian form, this set of equations reads: ρd/dt(1/ρ)-∇.V=0ρdV/dt-∇.σ̅̅̅=0ρdE/dt-∇.(σ̅̅̅.V)=0where ρ, V and E are the density, the velocity and the specific total energy, respectively. The Cauchy stress tensor consists of two terms:σ̅̅̅=-PI_d+S̅̅̅The first term of <ref> corresponds to the spherical part of the stress tensor, representing the fluid behavior of matter, which only changes the volume of the material when a pressure is applied. I_d is the identity matrix. The second term, S̅̅̅, is the deviatoric part ofthe stress tensor accounting for the solid behavior. It includes both longitudinal and shear stresses. The time evolution of the deviatoric stress is given by <cit.>:dS̅̅̅/dt = 2μ(D̅̅̅_0-D̅̅̅^p)-(S̅̅̅W̅̅̅-W̅̅̅S̅̅̅)where μ is the shear modulus characterizing how the material deforms under the influence of a shear stress, and W̅̅̅ is the antisymmetric part of the velocity gradientW̅̅̅=1/2[∇.V-(∇.V)^t]D̅̅̅_0 is the deviatoric part of the strain rate tensor D̅̅̅,D̅̅̅_0=D̅̅̅-1/3Tr(D̅̅̅)I_dwith D̅̅̅ defined as the symmetric part of the velocity gradient:D̅̅̅=1/2[∇.V+(∇.V)^t]The plastic strain rate D̅̅̅^p in (<ref>) is determined through the equation:D̅̅̅^p=χ(N̅̅̅^p:D̅̅̅^p)N̅̅̅^pwhere the symbol : denotes the inner product of tensors defined as A̅̅̅:B̅̅̅=Tr(A̅̅̅^tB̅̅̅). D̅̅̅^p represents the rate of deformation due to the formation and motion of dislocations in the material. N̅̅̅^p is the plastic flow directionN̅̅̅^p=S̅̅̅/|S̅̅̅|and χ is the switching parameter defined byχ = 0iff<1or iff=1and(N̅̅̅^p:D̅̅̅^p)≤0 1iff=1and(N̅̅̅^p:D̅̅̅^p)>0where f is the yield function. The first case corresponds to different regimes except the plastic domain: the elastic regime (f<1), elastic unloading (f=1 and (N̅̅̅^p:D̅̅̅^p)<0)and neutral loading (f=1 and (N̅̅̅^p:D̅̅̅^p)=0). The second case corresponds to the plastic regime.The evolution of the deviatoric stress S̅̅̅ is thus subjected to the yield function allowing to determine the deformation regime of the solid submitted to a mechanical stress. In the present study the yield function is defined by the von Mises yield criterion <cit.>, which is widely used. In this case, the yield function reads:f=σ_eq/Y where Y defined the yield strength of the material which represents its elastic limit, andσ_eq is a local effective stress, called the equivalent stress. σ_eq =√(3/2Tr(S̅̅̅.S̅̅̅)) Within the plastic regime, aphenomenon of hardening may take place increasing the flow resistance (the yield strength) <cit.>. However, in this study, a perfect plastic behavior is considered corresponding to the idealcase where the yield strength remains constant. This assumption is correct for a brittle material <cit.> like fused silica which is studied in Section <ref>.Within the elastic regime, the flow velocity can be decomposed into a longitudinal and a transverse components.Each direction may support waves which velocities c_l and c_t, respectively, read <cit.>:c_l^2 = c_h^2 + 4/3μ/ρandc_t^2 = μ/ρwhere c_h is the standard hydrodynamic sound velocity. Note that the longitudinal sound velocity (Eq. (<ref>)) is larger in a solid than in a fluid due to the lattice structure. Within the plastic regime, only compression waves can propagate since the solid structure is removed for large stresses, i.e. c_l = c_h and c_t = 0.Finally, a thermal softening is introduced in the model to account for the solid-liquid phase transitions. This is a general model where a polynomial g(ε), which is a function of the specific internal energy ε,weights the shear modulus and the yield strength. It is equal to unity and zero for the solid and liquid phases, respectively. This polynomial accounts for a smooth transition between the two phases (see Appendix A for more details).Based on the previously presented model, a second order cell-centered Lagrangian numerical scheme is implemented in the hydrodynamic CHIC code <cit.>. This scheme possesses several efficient numerical features <cit.>. It is developed in a planar geometry allowing one to study more accurately physical phenomena than in a 1D geometry as presented hereafter.§ RESULTS AND DISCUSSION The cavity generation in the bulk of fused silica is considered. Such a structure may be induced by tightly focusing a femtosecond laser pulse into the material. During the laser-matter interaction, nonlinear processes lead to a significant absorption ofthe laser energy. The latter is then confined in a volume with characteristic dimensions smaller than the laser wavelength. For absorbed energies of the order of 10 nJ,the induced high energy densities create high enough temperatures and pressures, the latter exceeding the bulk modulus of the solid, to form a void with observed dimensions between 0.2 μm and 0.5 μm.In the present work, a gaussian energy deposition is assumed in order to simplify the study of the dynamics of cavity formation.The energy deposition radius <cit.> is set to 0.13 μm at half maximum. This radius is chosen to remain constant even for higher energy of the laser beam where the volume of absorption may be larger in experiments.This work is motivated by the fact that the main aim of the present study is to demonstrate the importance of the solid response. A more detailed study, including an initial energy deposition evaluated by the solving of the Maxwell's equations, will be provided elsewhere. The absorbedenergy density is chosen (value provided hereafter) such that the simulation leads to cavity sizes similar to the experimental observations; 0.4 μm is chosen in particular.It is noteworthy that the regime leading to such cavity radii, as shown hereafter, exhibitsthe influence of most of the possible physical mechanisms for such a system.An equation of state (EOS) is required to obtained the pressure and the temperature from the knowledge of the density and the specific internal energy. For that purpose, the SESAME table 7386 <cit.> for fused silica is used. The initial density is equal to 2.2 g/cm^3. The yield strength and the shear modulus for initial standard conditions are Y_0=7.1 GPa, μ_0=22.6 GPa, respectively.Within the present hypothesis of an ideal plastic behavior, Y_0 remains constant whatever the deformation. This assumption is correct for SiO_2 since it is a brittle material <cit.>. Despite fused silica can present non ideal plastic deformations at the micron scale <cit.>, the plasticity domain remains relatively short.The thermal conductivity λ_th=1.381 W.m^-1.K^-1 is also assumed to be constant. Indeed, within the nanosecond timescale (t=1 ns) corresponding to the simulation time, with a specific heat capacity C of 1000 J.kg^-1.K^-1, the characteristic length of diffusion l is about 0.025 μm following the relation l=√(λ t/ρ C). This is smaller than the characteristic lengths of deformations studied here, making the influence of the thermal conductivity negligible for the present study. Moreover, we have checked that higher values of the thermal conductivity do not change the results on this timescale.Note that the solid-solid phase transitions are also neglected. As mentioned in [Michel2006], for stresses above 35 GPa, a change in slope of the Hugoniot curve is observed due to acrystallization toward the stichovite phase of SiO_2. However, the time required for the matter to undergo this phase transition is of the order of 1 to 10 ns depending on the strength of the shock <cit.>.For the present studies, this timescale is significantly longer than the time for which the matter experiences the required conditions when the shock passes through.Moreover, experimental observations have never revealed, to our knowledge, the stishovite phase in fused silica after an interaction with a femtosecond laser, further supporting the present assumption. §.§ Comparison of the fluid model and the elasto-plastic model in the 1D case Let us first considered a 1D simulation to compare the EP model to the fluid description in order to exhibit the influence of the solid response. The initial absorbed energy density is set to 90 kJ/cm^3 (0.9 nJ) in the center of the 1D domain. The full cartesian mesh size is set to 20 μm with initial cell length set to 20 nm.For various times, Fig. <ref> presents the spatial evolution of the pressure (fluid model) and the stress σ_xx (EP model). We remind that the stress is a generalization ofthe pressure within the EP model, the comparison thus makes sense. The stress is defined as positive in compression for the sake of clarity. The propagation of a shock, induced by the relaxation of the heated matter, is observed in both cases but with different shapes and velocities. In the fluid model case, the shock propagates with a straight front without change in its shape as usually. Only its amplitude decreases in course of propagation. In the case of the EP model, the shock propagates similarly as in the pure fluid model up to 100 ps. However, from 100 ps, a change is observed in the shape of the shock wave and disappears after 300 ps.This behavior is due to the surrounding matter which is in a solid state. The matter is submitted to strong stresses and the EP behavior split the shock into two waves: a plastic wave and an elastic precursor. The first, inducing permanent deformations in the material, propagates with the hydrodynamic speed of sound c_h, itself depending on the local density value. The second, inducing reversible deformations, propagates with the augmented longitudinal speed of sound c_l (Eq. (<ref>)). Its amplitude is equal to the Hugoniot elastic limit (HEL) Y_HELwhich is around 12 GPa as provided by both the SESAME table and the mechanical properties of the material, following the relation <cit.>: Y_HEL=1-ν/1-2νY_0 with ν the Poisson's ratio (around 0.3-0.4 for glass).Note that the HEL is slightly different from the yield strength. It defines the threshold in longitudinal stress where there is a transition between a pure elastic behavior to an elastic-plastic one <cit.>. The elastic wave then is faster than the plastic wave explaining the observed splitting. Finally, after 300 ps, only an acoustic wave propagates because σ_xx<Y_HEL (σ_eq<Y), permanent deformations in compression then are not further allowed. With this particular two-wave structure, the shock front first compresses the matter elastically and then plastically. The previously compressed material is then decompressed by the back of the wave, and finally is stretched (sign change in the deviatoric stress) after the wave.If this traction is strong enough, it may deform plastically the matter a second time. This scenario is illustrated by Fig. <ref> which shows the stress profile at 400 ps. The second plastification is demonstrated by the plotted plasticity threshold curve that provides the value of σ_eq/Y for each position.These differences between the fluid and EP behaviors have consequences on the density evolution as shown in Fig. <ref>, where density profiles are presented for various times. First, the formation of a cavity (from a density decreased by a factor two) is observed in both cases. The compression of the material by the shock wave induces then a strong relaxation (decompression) of the previously compressed matter behind it. This leads to rarefaction waves which digs the material.The major difference appears in the surrounding matter where an overdense zone is formed in the case of the EP model. With a strong shock front, it is possible to increase enough the internal energy of matter to liquefy it locally.This phenomenon can be observed from 100ps to 200 ps in the case of the EP model. The shock front always experiences the matter in the solid phase. It induces an internal energy in excess of the energy of melting, inducing a liquid phase which is experienced by the back of the shock. Since the speed of sound in the liquid phase is smaller than in the solid, the back of the shock is decelerated leading to the overdense region.The evolution as a function of time of the cavity radius is plotted in Fig. <ref>. This radius is defined as the distance from the center of the cavity, where the density is the lowest,to the position where the density is equal to half of the initial density. We have checked that a slightly different criterion leads to similar conclusions. For both models the cavity radius increases with respect to time. Up to roughly 200 ps, the cavity expansion is supported by the presence of the shock, leading to an almost constant speed of expansion for both models. When the shock has traveled sufficiently far, the flow velocity decreases in the central region, explaining the observed slower expansion of the cavity. In the case of the EP model, the cavity grows slightly slower than in the fluid case due to an additionnal force derived from the deviatoric part of the stress tensor opposed to the matter displacements in the solid phase.To summarize, this 1D study has first shown that the shock divides into two parts due to the EP response. Secondly, the associated rarefaction wave creates a cavity which is surrounded by an overdense area. For the studied timescale, the cavity radius increases monotically with respect to time without exhibiting any significant saturation whereas a finite size is expected based on experimental observations. An additional spatial dimension,by adding shear stress, is then expected to modify the cavity expansion dynamics as shown hereafter. §.§ 2D simulation of cavity formation Let us now consider the 2D case. The full cartesian mesh size is set to 18 μm×18 μm with initial cell size set to 50 nm×50 nm (simulations with smaller initial cells size werealready performed to verify the numerical convergence). All parameters are similar to the previous section except the energy densitywhich is set to 0.6 MJ/cm^3 (6 nJ) in the center of the 2D domain to get a final cavity radius around 0.4 μm. Note that this absorbedenergy density is different from the 1D case due to the additional spatial dimension. With this energy deposition, the EOS predicts initial pressure and temperature around 0.3 TPa and 10^5 K, respectively, in the energy deposition area. Figure <ref> presents the evolutions of (a) σ_xx, (b) σ_xy and (c) the density profiles for various times.As in the 1D case, the shock first propagates with a straight front in the liquid phase (Fig. <ref>(a), roughly before 100 ps). After 100 ps (the temperature in the center of the heated matter is around several 10^4K), the shock front experiences the solid state but induces a phase transitionto the liquid state due to its large amplitude. Since the sound velocity in the liquid is smaller, the back of the shock is decelerated leading to a change in the stress shape(as explained in the 1D case) with a split between the back and the front of the shock. The elastic precursor is not really visible for this energy deposition but its signature can be seen around σ_xx=Y_HEL=12 GPa between 100 and 200 ps where the front shape slightly changes. Figure <ref>(b) exhibits an accumulation of the shear stress σ_xy during 200 ps around the cavity and following the front of the compression wave. A decreasing of the shear stress is visible behind the wave (negative during 150 ps) due to the relaxation. After 200 ps, the shear stress still increases around the cavity but decreases while the shock transforms into an acoustic wave. Between this two regions, residual shear stresses (plateau) is created,signature of the induced permanent deformations. Figure <ref>(c) shows that a cavity is formeddue to the strong relaxation behind the shock. As in the 1D case, an overdense region is induced in the vicinity of the cavity for the same reasons. The final thickness (at 1.4 ns) of the overdense region, depending on the plastic deformation, is between 0.3 and 0.4 μm (the temperature at 1.4 ns in the center of the cavity is still high, around 10^4 K).Note this thickness may slightly change depending on the choice of the yield criterion(because the plasticity region change) <cit.>. However, all the main trends presented above should be similar. For the present purpose of laser structuration of materials, the von Mises yield criterion is well adapted. Other criteria and models could be more suited for advanced studies, as the accurate description of the hardening in the densification process of SiO_2 for instance <cit.>.Evolutions of the cavity radius as a function of time for various initial energy densities, predicted by the EP model, are plotted in Fig. <ref>. An evolution forthe largest energy density is also simulated with the fluid approach for the sake of comparison. Within the EP model, the evolution of the cavity radius consists mainly of three steps: an expansion, a contraction (shrinking), and a stabilization. The first step is due to the formation of a strong shock as in the 1D case. During the expansion, shear stress is accumulated (Fig. <ref>(b)), inducing an additional force opposing to the matter displacement.Ultimately, the expansion stops when the shock transforms into a pure elastic wave. The cavity radius then reaches a maximum value. It is worth noting that this behavior takes place on a shorter timescale than the 1D case due to theinfluence of the shear stress which is more significant in the 2D simulations. Then, corresponding to the above-mentioned second step, the cavity radius decreases due to the relaxation of the elastic deformations closing partially the cavity.Finally, the cavity stabilizes (third step) due to the induced permanent deformations.That is this final radius which has to be compared to experimental observations in post-mortem analysis.The evolution of the previous trends with respect to the absorbed energy density is now discussed. During the first step, the larger the absorbed energy density, the stronger the shock, and the faster the expansion of the cavity. That leads to a maximum cavity radius which increases and is reached for longer times. For the second step, the larger the energy absorption, the longer the elastic relaxation timeto reach the stabilization of the cavity. Indeed, elastic deformations are more important. The final radius increases as a function of the absorbed energy density; its behavior is discussed in more details hereafter.The final radius can also be estimated from the reached maximum value by subtracting the total elastic deformations. Within this procedure, we take into account the possible phase changes which remove the solid deformations. As shown byFig. <ref> (horizontal dashed lines), results of these predictions are in a good agreement with the final radius as predicted by the full EP modeling. This result supports the analysis performed in the previous paragraph.Figure <ref> also presents the evolution of the cavity radius in the case of the fluid model for an absorbed energy density of 2MJ/cm^3. The radius increases monotonically without exhibiting any decrease, neither stabilization as observed with the EP model, demonstrating the importance of the latter. However, a final cavity radius may be determined by comparing the front shock pressure P_shock to the Hugoniot elastic limit, i.e. it is obtained when P_shock=Y_HEL. In the present case, that takes place at 480 ps, leading to an estimation around 0.9 μm. This value corresponds approximatively to the maximal radius given by the EP model. However, the fluid model is not able to exhibit the elastic relaxation (second step) as the EP model which predicts asmaller final cavity radius of 0.8 μm or so. Figure <ref> presents the evolution of the maximal and the final cavity radius as a function of the absorbed energy within the EP and fluid models.With an absorbed energy in a range from 2 to 20 nJ (0.2 to 2 MJ/cm^3), the final cavity radius is about 0.2 to 0.8 μm within the EP model. It increases with the absorbed energy as a square root (checked by a fit), i.e. r_cavity∝√(E_abs), due to the 2D geometry (for a 3D geometry, r_cavity∝√(E_abs)). The difference between the maximal cavity radius and the final cavity radius increases with the absorbed energy. This is due to the increase of the region size where the matter has been compressed. The elastic relaxation then leads to a larger difference between transient and final radius. Within the fluid model,as mentionned above, the final cavity radius corresponds to the maximal radius reached by the EP model, due to the absence of elastic behavior. Thus the discrepancies exhibited by the fluid model in both the final value and the temporal behavior of the cavity radius clearly shows the importance of the EP behavior. For a more realistic energy deposition, especially for a larger absorption volume possibly corresponding to a higher energy of the laser beam, a change in the temporal evolution is expected, but not necessarily in the final cavity size. Indeed, the energy density should be lower due to the increase of the absorption volume leading to a lower pressure and a lower temperature. Thus, the shock wave should be weaker and should induce a smaller increase of the cavity radius accompanied by a weaker plastic deformation. This should lead to a lower final cavity size. However, the larger absorption zone should lead to a larger final cavity size. Thus, both influences are in competition and should lead to comparable final cavity sizes. The improvement of the EP approach is also its ability to predict the potential crack formation as shown hereafter. §.§ Predictions of potential cracks formationThis modeling permits to determine different critical areas where potential cracks (fractures) may appear.In the case of ideal and homogeneous materials without any defects, the resistance corresponds to the pressure capable of separating atoms <cit.>, leading to very large value of the material resistance. In the realistic case, the resistance (ability to support stress) is significantly decreased by the presence of defects which initiate the formation of crack. Effective macroscopic values of the intrinsic mechanical limitsof materials can then be defined. They consist of the resistancein compression L_c, traction L_t, and shear <cit.> L_t/2. L_t is set to a characteristic value <cit.> (microscopic scale) of 8 GPa and L_c is set to 10 GPa (resistance in compressionis generally larger than resistance in tractionfor brittle materials <cit.>).The present EP simulations allows us to perform the laser induced stress in the material. From it, principal stresses in compression-traction and maximum shear stress can be defined. If the laser induced principal stress or maximum shear stress exceeds the associated previous limits, then the material may break, i.e.cracks may appear. Details are provided in Appendix B.Within the conditions of the previous section, an evaluation of this critical area has been performed. In the present case of isotropic energy deposition and material mechanical properties, the symmetry of the system leads to conclude that stress in compression is the main component. Figure <ref>(a) shows its spatial distribution at 200 ps for an energy deposition of 0.6 MJ/cm^3.At this time, the stress is maximal, i.e. the crack formation is the most probable, and located around the cavity. As expected, the stress is distributed isotropically with a maximum value for a radius slightly larger than 1 μm which is roughly twice the cavity radius (0.4 μm). This induced stress leads to a critical area as shown in Fig. <ref>(b). It is distributed similarly as the previously discussed maximum stress, exhibiting a thin ring shape which the probability of meeting and activating of defects (inducing a crack) depends on its size and the density of defects. The density of activated defects can be defined as a function of their spatial distribution (Weibull modulus), their size and the stress applied to the material. The probability of crack initiation can then beevaluated through the Weibull distribution <cit.>. Furthermore a model of mechanical fracturation <cit.> could be investigated in order to charaterize the fracture length as a function of the cavity radius.However, such a study is out of the scope of the present work. In order to exhibit the influence of the shear stress, an inhomogeneous energy density deposition is now considered.The total energy is set to 12 nJ (1.2 MJ/cm^3 in average) which is twice the previous energy, and distributed within two main spots separated by a few tenths of micrometers, (see Fig.<ref>(a)). Figure <ref>(b) shows the induced 2D density profile at 250 ps. Two cavities, separated by a slightly denser region, corresponding to the main energy deposition spots are created. As in the previous homogeneous case, a shock wave launching is associated to each cavity formation. In the horizontal direction outwards the cavities, the shock waves mainly propagate as in the homogeneous case. However, for the inward direction, the shock waves collide, leading to a daughter wave in the perpendicular direction. That results in two symmetric overdense regions in the vertical direction as depicted by Fig. <ref>(b).Figure <ref> shows the principal stress in compression (Fig. <ref>(a)) and the maximum shear stress (Fig. <ref>(c)) together with their associated critical areas (Figs. <ref>(b) and (d), respectively) at 250 ps.As discussed for the density map, the daughter shock wave creates the maximum compressive stress in the vertical direction, resulting in both the upper and lower regions of maximal stress. That leads to critical areas exhibiting a similar appearance as depicted by Fig. <ref>(b). The overall clearly shows the anisotropic distribution of stress. The latter then gives rise to an elongation in the vertical direction leading to the apparition of shear stress on the left and right sides of the double-cavity structure, as shown by Fig. <ref>(c). The critical sheared region also exhibits the same anisotropy with a main axis oriented vertically. The potential formation of cracks is thus expected to take place in a preferential direction due to the initial anisotropic energy deposition. We also expect a crack propagation in a preferential direction. These numerical results are in agreement with a recent experimental study <cit.> devoted to a similar physical system. The observations have shown the same trends, i.e. an elongation of the created cavities and preferential cracks directions.§ CONCLUSION A theoretical and numerical modeling has been presented to analyze and understand the formation of structural modifications in transparent materials induced by a femtosecond laser pulse tightly focused in the bulk. The standard fluid description has been augmented by the solid response through the elasto-plastic behavior. This approach has been implemented in the 2D hydrodynamic CHIC code. Numerical simulations have been performed for fused silica irradiated within such conditions that a cavity may form. It has been shown that the creation of a shock wave and its subsequent propagation lead to a cavity formation and surrounding overdense region. Due to the elasto-plastic behavior, the shock exhibits a two-wave structure: an elastic precursor inducing reversible deformations and a plastic wave leading to permanent deformations.The induced shear stress contributes significantly to the dynamics of the cavity formation, ultimately leading to stop its expansion. The maximum cavity size is reached when the shock transforms into a pure elastic wave.The relaxation of the elastic deformations then closes partially the cavity. The permanent deformations are responsible for the final cavity stabilization and the final overdense area.By taking the same order of magnitude of the absorbed energy deposition as experimentally obtained <cit.>, the present approach, as it is, permits to account for the morphology of the observed cavities.Such behaviors cannot be obtained by the fluid approach. In particular, the latter cannot lead to any cavity stabilization which size could be compared to experimental observations. To overcome this flaw, by comparing the hydrodynamic pressure to the Hugoniot elastic limit, an ad hoc criterion renders it possible to give the order of magnitude of the final cavity radius, however without being able to describe the surrounding matter. It turns out that a reliable comparison to experimental data is only possible by accounting for the elasto-plastic influence.By evaluating the induced mechanical stress, the present approach is also able to predict the regions where potential fractures may appear during the interaction. With an isotropic energy deposition, potential fractures are mainly induced by a compressive stress. Thanks to the present 2D approach, anisotropic conditions can be considered. In addition to the compression, an anisotropic energy deposition is shown to possibly create fractures from shear stresses.Since the present approach is general, simulations for other materials could also be performed by adapting the material properties as the equation of state, the mechanical constants, etc. This thermo-elasto-plastic tool could thus be used to design laser-induced nano-structures in material within conditions preventing from formation of fractures (which are detrimental to the structure functionalities).Finally, the simulation of the laser propagation, including the electrons dynamics, will be performed to obtain a more realistic energy deposition. A 3D modelling should also be developed to perform more quantitative predictions which may account accurately for experimental observations. Since the main physical mechanisms are expected to be the same as those exhibited in the present work, the present modeling and conclusions may be used as a baseline for future developments in the field of the laser-matter interaction.§ ACKNOWLEDGMENTSVladimir Tikhonchuk is acknowledged for fruitful discussions. The CEA and the Région Aquitaine (MOTIF project) are acknowledged for supporting this work. Jocelain Trela is also akcnowledged for the help brought on plots of 2D maps. § THERMAL SOFTENING The solid-liquid phase change is modelized by a thermal softening model. A polynomial g(ε) weights the shear modulus and the yield strength. It is equal to unity and zero for the solid and liquid phases, respectively, depending on the internal energy ε. A third order polynomial is chosen to ensure a smooth transition between the two states. More precisely, it is defined as follows. If ε<ε_solid, then g(ε)=1. If ε>ε_solid and ε<ε_melting, then:g(ε)=1-3(ε-ε_solid/ε_melting-ε_solid)^2+2(ε-ε_solid/ε_melting-ε_solid)^3And if ε>ε_melting, then g(ε)=0. ε_solid is the internal energy of the solid at 300 K (ε_solid=0.3 kJ/g) and ε_melting is the internal energy of melting (ε_melting=2.15 kJ/g, corresponding to a softening temperature around 2000 K <cit.>).§ STRESSES ANALYSIS In order to perform the analysis of plane stresses, the knowledge of principal stresses in every points of the material is required. The matrix elements of σ̅̅̅ in the 2D cartesian directions of coordinates system aredefined as σ_x, σ_y and τ_xy, the longitudinal stresses and the shear stress, respectively. They are the projection on the two axis of applied stresses. Knowing these values in these particular directions,other projections of stressesdefined as σ_x1, σ_y1 and τ_x1y1 can be calculated in any directions making an angle θ with the abscissa axis <cit.>.σ_x1 = σ_x+σ_y/2+σ_x-σ_y/2cos2θ+τ_xysin2θ τ_x1y1 = -σ_x-σ_y/2sin2θ+τ_xycos2θ σ_y1 = σ_x+σ_y/2-σ_x-σ_y/2cos2θ-τ_xysin2θA particular angle θ_p exists, where σ_x1 is maximized and simultaneously σ_y1 is minimized, and τ_x1y1 is canceled. This is theprincipal coordinate system which amounts to diagonalize the stress tensor. In this case, the matter is in a traction/compression state. It is thus interesting to use this frame in order to compare principal stresses to traction/compression limits. σ_I is defined as the maximum (minimum) stress andσ_II as the minimum (maximum) stress in traction (compression). By convention σ_I>σ_II.σ_I =σ_m+R σ_II = σ_m-Rwithσ_m = σ_x+σ_y/2andR = √((σ_x-σ_y/2)^2+τ_xy^2)The angle θ_p can be performed through the relation:tan2θ_p = 2τ_xy/σ_x-σ_ySimilarly, a frame rotation with an angle θ_s can be defined, at ±45^∘ relative to the principal coordinates system (Mohr's circle <cit.>), where the shear stressis maximized whereas normal stresses are equal to the mean stress σ_m. This frame is useful to know sites where themaximum shear stress may create dislocations or cleavages. The relations read:τ_max = Randθ_s = θ_p ± 45^∘Finally, to determine critical sites, where stresses exceed intrinsic mechanical limits of material, Mohr's criterion is used <cit.>. One can define a compression limit L_c and atraction limit L_t, especially in breakable materials where they are differents. Thus, thanks to the principal basis, σ_I and σ_II can be compared to these limits deducing critical points. If σ_Iand σ_II have the same sign, the largest one must not exceed L_c or L_t depending on the stress state. In the case of a traction state, stresses are positives, σ_Irepresents the maximum stress and σ_II the minimum stress in traction. Within this condition, a point is critical if σ_I/L_t≥1In the opposite case of a compression state, a point is critical if |σ_II|/L_c≥1If σ_I and σ_II have opposite signs, that corresponds to torsion. A point is critical if:|σ_I-σ_II|/L_t≥1or|σ_I-σ_II|/L_c≥1which is illustrated in Fig.<ref>.48 fxundefined [1]ifx#1fnum [1]#1firstoftwosecondoftwo fx [1]#1firstoftwosecondoftwonoop [0]secondoftworef[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0]rl [1]href #1 @bib@innerbibempty[Davis et al.(1996)Davis, Miura, Sugimoto, and Hirao]Davis1996 author author K. M. Davis, author K. Miura, author N. Sugimoto,andauthor K. Hirao, @noopjournal journal Opt. 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"authors": [
"Romain Beuton",
"Benoît Chimier",
"Jérôme Breil",
"David Hébert",
"Pierre-Henri Maire",
"Guillaume Duchateau"
],
"categories": [
"physics.app-ph"
],
"primary_category": "physics.app-ph",
"published": "20170927171618",
"title": "Thermo-elasto-plastic simulations of femtosecond laser-induced structural modifications: application to cavity formation in fused silica"
} |
=24pt [ C. E. Pugh1 V. M. Nakariakov1,2 A.-M. Broomhall1,3 A. V. Bogomolov4 I. N. Myagkova4 Received September 15, 1996; accepted March 16, 1997 ===================================================================================================================================================== The small-cluster exact-diagonalization calculations and the projectorquantum Monte Carlo method are used to examine the competing effectsof geometrical frustration and interaction on ferromagnetismin the Hubbard model on the Shastry-Sutherland lattice.It is shown that the geometrical frustration stabilizes the ferromagnetic state at high electron concentrations (n ≳ 7/4), where strong correlations between ferromagnetism and the shapeof the noninteracting density of states are observed. In particular, it is found that ferromagnetism is stabilizedonly for these values of frustration parameters, which leadto the single peaked noninterating density of states at the band edge.Once, two or more peaks appear in the noninteracting density of statesat the band egde the ferromagnetic state is suppressed. This opens a new route towards the understanding of ferromagnetism in strongly correlated systems. § INTRODUCTIONSince its introduction in 1963, the Hubbard model <cit.> hasbecome, one of the most popular models of correlated electronson a lattice. It has been used in the literature to study a great varietyof many-body effects in metals, of which ferromagnetism, metal-insulatortransitions, charge-density waves and superconductivity are the most common examples. Of all these cooperative phenomena, the problem of ferromagnetism in the Hubbard model has the longest history. Although the model was originally introduced to describe the itinerant ferromagnetismin narrow-band metals like Fe, Co, Ni and others, it soon turnedout that the single-band Hubbard model is not the canonical modelfor ferromagnetism. Indeed, the existence of saturated ferromagnetism has been proven rigorously only for very special limits. The first well-known example is the Nagaoka limit that corresponds to the infinite-U Hubbard model with one hole in a half-filledband <cit.>. Another example, where saturated ferromagnetism has been shown to exist, is the case of the one-dimensional Hubbard model with nearest and next-nearest-neighbor hopping at low electron densities <cit.>. Furthermore, several examples of the fully polarized ground statehave been found on special lattices as are the bipartite latticeswith sublattices containing a different number of sites <cit.>, the fcc-type lattices <cit.>, the lattices with long-range electronhopping <cit.>, the flat bands <cit.> and the nearly flat-band systems <cit.>. This indicates that the lattice structure, which dictates the shape of the density of states (DOS), plays an important role in stabilizingthe ferromagnetic state.Motivated by these results, in the current paper we focus our attentionon the special type of lattice, the so-called Shastry-Sutherlandlattice (SSL). The SSL represents one of the simplest systems withgeometrical frustration, so that putting the electrons on this latticeone can examine simultaneously both, the effect of interaction as well asthe effect of geometrical frustration on ground-state properties of theHubbard model. This lattice was first introduced by Shastryand Sutherland <cit.> as an interesting example ofa frustrated quantum spin system with an exact ground state.It can be described as a square lattice withthe nearest-neighbor links t_1 and the next-nearest neighbors links t_2 in every second square (see Fig. 1a).The SSL attracted much attention after its experimental realization in the SrCu_2(BO_3)_2 compound <cit.>. The observation of a fascinating sequence of magnetization plateaus (atm/m_s=1/2, 1/3, 1/4 and 1/8 ofthesaturatedmagnetization m_s)in this material <cit.> stimulated further theoretical and experimentalstudies of the SSL. Some time later, many other Shastry-Sutherland magnets have been discovered <cit.>. In particular, this concerns an entire group of rare-earth metal tetraborides RB_4 (R=La-Lu). These materials exhibit similar sequences of fractional magnetization plateaus as observed in the SrCu_2(BO_3)_2 compound.For example, for TbB_4 the magnetization plateau has been foundat m/m_s=2/9, 1/3, 4/9, 1/2and 7/9 <cit.> and for TmB_4at m/m_s=1/11, 1/9, 1/7 and 1/2 <cit.>. To describe some ofthe above mentioned plateaus correctly, it was necessearry to generalizethe Shastry-Sutherland model by including couplings between the thirdand even between the forth nearest neighbors <cit.>.The SSL with the first, second and third nearest-neighbor links isshown in Fig. 1b and this is just the lattice that will be used in ournext numerical calculations.Thus our starting Hamiltonian, corresponding to the one band Hubbard model on the SSL, can be written as H= -t_1∑_⟨ ij ⟩_1, σc^+_iσc_jσ -t_2∑_⟨ ij ⟩_2, σc^+_iσc_jσ -t_3∑_⟨ ij ⟩_3, σc^+_iσc_jσ +U∑_in_i↑n_i↓, where c^+_iσ and c_iσ are the creation and annihilation operatorsfor an electron of spin σ at site i and n_iσ is the corresponding number operator (N=N_↑+N_↓=∑_iσ n_iσ). The first three terms of (1) are the kinetic energies corresponding to the quantum-mechanical hopping of electrons between the first, second and third nearest neighbors and the last term is the Hubbard on-site repulsion between two electrons with opposite spins. We set t_1=1 as the energy unit and thus t_2 (t_3) can be seen as a measure of the frustration strength. To identify the nature of the ground state of the Hubbard modelon the SSL we have used the small-cluster-exact-diagonalization(Lanczos) method <cit.> and the projector quantum Monte-Carlomethod <cit.>.In both cases the numerical calculations proceed in the folloving steps.Firstly, the ground-state energy of the model E_g(S_z) iscalculated in all different spin sectors S_z=N_↑-N_↓as a function of model parameters t_2,t_3 and U. Then the resulting behaviors of E_g(S_z)are used directly to identify the regions in the parametric space of the model, where the fully polarized statehas the lowest energy. § RESULTS AND DISCUSSIONTo reveal possible stability regions of the ferromagnetic state in the Hubbard model on the SSL, let us first examine the effects of the geometrical frustration, represented by nonzero values of t_2 and t_3, on the behavior of the non-interacting DOS.The previous numerical studies of the standard one and two-dimensionalHubbard model with next-nearest <cit.> as well aslong-range <cit.> hopping showed that just this quantitycould be used as a good indicator for the emergence of ferromagnetism in the interacting systems. Indeed, in both models the strong correlation between ferromagnetism and the anomalies in the noninteracting DOS are observed. In the first model the ferromagentic state is foundat low electron concentrations and the noninteracting DOS is strongly enhanced at the low-energy band edge, while in the second one the ferromagnetic phase is stabilized at the high electron concentrations and the spectral weight is enhanced at the high-energy band edge. This leads to the scenario according to which the large spectral weight in the noninteracting DOS that appears at the low (high) energy band edges allows for a small kinetic-energy loss for a state with total spin S0 in reference to onewith S=0. At some finite value of interaction U, the Coulomb repulsion paid for the low-spin states overcomes this energy loss and the high-spin state becomes energetically favored. The key point in this picture is the assumption that the shape of the DOS is only weakly modified as the interaction U is switched on, at least within its low (high) energy sector.The noninteracting DOS of the U=0 Hubbard model on the SSLof size L=200 × 200, obtained by exact diagonalizationof H (for U=0) is shown in Fig. 2. The left panels correspondto the situation when t_2 >0 and t_3=0, while the right panels correspondto the situation when both t_2 and t_3 are finite.One can see that once the frustration parameter t_2 is nonzero, the spectral weight starts to shift to the upper band edge and the noninteracting DOS becomes strongly asymmetric. Thus taking into account the above mentioned scenario, there is a real chance that the interacting system could be ferromagnetic in the limit of high electron concentrations. To verify this conjecture we haveperformed exhaustive numerical studies of the model Hamiltonian (1) for a wide range of the model parameters U,t_2 and n at t_3=0.Typical results of our PQMC calculations obtained on finite cluster ofL=6×6 sites, in two different concentration limits(n≤ 1 and n>1) are shown in Fig. 3. There is plotted the difference Δ E=E_f-E_min between the ferromagnetic state E_f, which can be calculated exactly and the lowest ground-state energyfrom E_g(S_z) as a function of the frustration parameter t_2.According to this definitionthe ferromagnetic state corresponds to Δ E=0. It is seen that for electron concentrations below the half filled band case n=1, Δ E is the increasing function of t_2, and thus there is no signof stabilization of the ferromagnetic state for n≤ 1, in accordance with the above mentionedscenario. The situation looks more promising in the oppositelimit n>1. In this case, Δ E is considerably reduced with increasing t_2, however, this reduction is still insufficient to reach the ferromagnetic state Δ E=0 for physicallyreasonable values of t_2 (t_2<1.6) that correspond to the situation in the real materials. To exclude the finite-size effect, we have alsoperformed the same calculations on the larger cluster of L=8 × 8 sites, but again no signs of stabilization the ferromagnetic state have been observed (see inset to Fig. 4b).For this reason we have turned our attention to the case t_2>0 and t_3>0. The noninteracting DOS corresponding to this case is displayed in Fig. 1 (the right panels). These panels clearly demonstrate that with the increasing value of thefrustration parameter t_3, still a more spectral weight is shiftedto the upper band edge. A special situation arises at t_3=0.6, when the spectral weight is strongly peaked at the upper band edge. In this case the nonintercting DOS is practically identical withone corresponding to noninteracting electrons with long-range hopping <cit.>. Since the long-range hopping supports ferromagnetism in the standard Hubbard model for electron concentrations above the half-filled bandcase <cit.>, we expect that this could be true alsofor the Hubbard model on the SSL, at least for some values of frustrationparameters t_2 and t_3. Therefore, we have decided to perform numericalstudies of the model for a wide range of t_3 values at fixed t_2, Uand n (t_2=1, U=1, n=7/4). To minimize the finite-size effects, thenumerical calculations have been done on two different finite clustersof L=6× 6 and L=8× 8 sites. The results of our calculations for Δ E as a function of t_3 are displayed in Fig. 4a. In accordance with the above mentioned assumptions we find a relativelywide region of t_3 values around t_3=0.6, where the ferromagneticstate is stable. It is seen that the finite-size effects on the stability region of the ferromagnetic phase are negligible and thus these results can be satisfactorily extrapolated to the thermodynamic limit L=→∞. Moreover, the same calculations performedfor different values of the Hubbard interaction U showed that correlationeffects(nonzero U) further stabilize the ferromagnetic state and leadto the emergence of macroscopic ferromagnetic domain in the t_3-Uphase diagram (see inset to Fig. 4a). This confirms the crucial role of the Hubbard interaction U in the mechanism of stabilizationof ferromagnetism on the geometrically frustrated lattice. In Fig. 4b we have also plotted the comprehensive phase diagrams of the model in the t_3-n as well as t_3-t_2 plane,which clearly demonstrate that the ferromagnetic state is robustwith respect to doping (n ≳ 7/4) and frustration. To check the convergence of PQMC results we have performed the samecalculations by the Lanczos exact diagonalization method. Of course,on such a large cluster, consisting of L=6 × 6 sites, we wereable to examine (due to high memory requirements) only several electronfillings near the fully occupied band (N=2L). The exact diagonalizationand PQMC results for the width of the ferromagnetic phaseobtained on finite cluster of L=6 × 6 sites, for three different electron fillings from the high concentration limit(N=66,67,68), are displayed in the inset to Fig. 4b andthey show a nice convergence of PQMC results.Let us finally turn our attention to the question of possible connection between ferromagnetism and the noninteracting DOS that has been discussed at the beginning of the paper. Figs. 4a and 4bshow, that for each finite U and n sufficiently large (n ≳ 7/4), there exists a finite intervalof t_3 values, around t_3∼ 0.6, where the ferromagnetic state isthe ground state of the model. To examine a possible connectionbetween ferromagnetism and the noninteracting DOS, we havecalculated numerically the noniteracting DOS for several differentvalues of t_3 from this interval and its vicinity. The results obtained for U=1, n=7/4 andt_2=1 are displayed in Fig. 5.Comparing these results with the ones presented in Fig. 4a forthe stability region of the ferromagnetic phase at the same values of U,n and t_3, one can see that there is an obvious correlationbetween the shape of the noninteracting DOS and ferromagnetism.Indeed, the ferromagnetic state is stabilized only for these valuesof frustration parameters t_2,t_3, which lead to the single peakednoninterating DOS at the band edge. Once, two or more peaks appearin the noninteracting DOS at the band egde (by changing t_2 or t_3),ferromagnetism is suppressed. In summary, the small-cluster exact-diagonalization calculationsand the PQMC method were used to examine possible mechanisms leadingto the stabilization of ferromagnetism in strongly correlated systemswith geometrical frustration. Modelling such systems by the Hubbard modelon the SSL, we have found that the combined effects of geometricalfrustration and interaction strongly support the formation of theferromagnetic phase at high electron densities. The effects of geometricalfrustration transform to the mechanism of stabilization of ferromagnetismvia the behaviour of the noninteracting DOS, the shape of which is determineduniquely by the values of frustration parameters t_2 and t_3. We havefound that it is just the shape of the noninteracting DOS near the band edge(the single peaked DOS) that plays the central role in the stabilizationof the ferromagneticstate. Since the same signs have been observed alsoin some other works (e.g., the Hubbard model with nearest and next-nearestneighbor hopping, or the Hubbard model with long range hopping), it seemsthat such a behaviour of the noninteracting DOS near the band edge shouldbe used like the universal indicator for the emergence of ferromagnetismin the interacting systems. This work was supported by the Slovak Research and Development Agency (APVV) under Grant APVV-0097-12 andERDF EU Grant under the contract No. ITMS26210120002 and ITMS26220120005.99 Hubbard J. Hubbard, Proc. R. Soc. London A 276, 238 (1963). Nagaoka Y. Nagaoka, Phys. Rev. 147, 392 (1966). M_H E. Müller-Hartmann, J. Low. Temp. Phys. 99, 342 (1995). Lieb E. H. Lieb, Phys. Rev. Lett.62, 1201 (1989). Ulmke M. Ulmke, Eur. Phys. J. B 1, 301 (1998). Pandey S. Pandey and A. Singh, Phys. Rev. B 75, 064412 (2007). UH1 P. Pieri, Mod. Phys. Lett. B 10, 1277 (1996).UH2 M. Salerno, Z. Phys. B 99, 469 (1996).Fark1 P. Farkašovský, Phys. Rev. B 66, 012404 (2002). Fark2 P. Farkašovský and Hana Čencariková,Cent. Eur. J. Phys. 11, 119 (2013). Fark3 P. Farkašovský, EPL 110, 47007 (2015).FB2 A. Mielke, J. Phys. A 25, 4335 (1992). FB3 H. Tasaki, Phys. Rev. Lett. 69, 1608 (1992). FB4 A. Mielke and H. Tasaki, Commun. Math. Phys. 158, 341 (1993). FB5 H. Katsura, I. Maruyama, A. Tanaka and H. Tasaki, EPL 91, 57007 (2010).NFB1 H. Tasaki, Phys. Rev. Lett. 73, 1158 (1994). NFB2 A. Mielke, Phys. Rev. Lett. 82, 4312 (1999). NFB3 A. Tanaka and H. Ueda, Phys. Rev. Lett. 90, 067204 (2003). NFB4 M. Maksymenko, A Honecker, R. Moessner, J. Richter and O. Derzhko,Phys. Rev. Lett. 109, 096404 (2012). ShastryB. S. Shastry, B. Sutherland, Physica Band C 108, 1069 (1981). Kageyama2 H. Kageyama, K. Yoshimura, R. Stern, N. V. Mushnikov, K. Onizuka, M. Kato, K. Kosuge, C. P. Slichter, T. Goto, and Y. Ueda, Phys. Rev. Lett. 82, 3168 (1999). Kodama K. Kodama, M. Takigawa, M. Horvatic, C. Berthier, H. Kageyama, Y. Ueda, S. Miyahara, F. Becca, and F. Mila, Science 298, 395 (2002). Yoshii S. Yoshii, T. Yamamoto, M. Hagiwara, S. Michimura, A. Shigekawa, F. Iga, T. Takabatake, and K. Kindo, Phys. Rev. Lett. 101, 087202 (2008). Gabani K. Siemensmeyer, E. Wulf, H. J. Mikeska, K. Flachbart, S. Gabani, S. Matas, P. Priputen, A. Efdokimova, and N. Shitsevalova, Phys. Rev. Lett. 101, 177201 (2008). pssb K. Čenčariková and P. Farkašovský, Physica Status Solidi b 252, 333 (2015). Lanczos E. Dagotto, Rev. Mod. Phys. 66, 763 (1994). PQMC M. Imada and Y. Hatsugai, J. Phys. Soc. Jpn. 58, 3752 (1989). | http://arxiv.org/abs/1709.09461v1 | {
"authors": [
"Pavol Farkasovsky"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170927114957",
"title": "Effects of geometrical frustration on ferromagnetism in the Hubbard model on the Shastry-Sutherland lattice"
} |
Some New Results on Charged Compact Boson Stars [ December 30, 2023 =============================================== [1]Corresponding author: [email protected] (Ingo Nitschke) [2]Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany [3]Institut für Theoretische Physik II - Soft Matter, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany [4]Dresden Center for Computational Materials Science (DCMS), 01062 Dresden, GermanyWe consider a thin film limit of a Landau-de Gennes Q-tensor model. In the limiting process we observe a continuous transition where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. Main properties of the thin film model, like uniaxiality and parameter phase space, are preserved in the limiting process. For the derived surface Landau-de Gennes model, we consider an L^2-gradient flow. The resulting tensor-valued surface partial differential equation is numerically solved to demonstrate realizations of the tight coupling of elastic and bulk free energy with geometric properties. nematic liquid crystals, thin film limit, surface equation§ INTRODUCTIONWe are concerned with nematic liquid crystals whose molecular orientation is subjected to a tangential anchoring on a curved surface. Such surface nematics offer a non trivial interplay between the geometry and the topology of the surface and the tangential anchoring constraint which can lead to the formation of topological defects. An understanding of this interplay and the resulting type and position of the defects is highly desirable.As an application, nematic shells have been proposed as switchable capsules optimal for a steered drug delivery <cit.>. The defect structure thereby essentially determines where the shells can be opened in a minimal destructive way. Moreover, nematic shells are possible candidates to form supramolecular building blocks for tetrahedral crystals with important implications for photonics <cit.>.Besides such equilibrium structures, defects also play a fundamental role in active systems. In <cit.> the spatiotemporal patterns that emerge when an active nematic film of microtubules and molecular motors is encapsulated within a lipid vesicle is analyzed. The combination of activity, topological constraints, and geometric properties produces a myriad of dynamical states. Understanding these relations offers a way to design biomimetic materials, with topological constraints used to control the non-equilibrium dynamics of active matter.Defects in nematic shells are intensively studied on a sphere <cit.> and under more complicated constraints, see, , <cit.>. However, most of these studies use particle methods. Despite the interest in such methods a continuum description would be more essential for predicting and understanding the macroscopic relation between position and type of the defects and geometric properties of the surface. For bulk nematic liquid crystals the Landau-de Gennes Q-tensor theory <cit.> is a well established field theoretical description. For a mathematical review we refer to <cit.>. However, its surface formulation is still under debate. Surface models have been postulated by analogue derivations on the surface <cit.>, by considering the limit of vanishing thickness for bulk Q-tensors models <cit.> or via a discrete-to-continuum limit <cit.>. The derived models differ in details and strongly depend on the made assumptions in the derivation.Our approach aims to derive a surface Q-tensor model by dimensional reduction via a thin film limit of a general bulk Landau-de Gennes model. In contrast to previous work we only make assumptions on the boundary of the thin film where we admit only states conforming to critical points of the free energy.In the limiting process we observe a continuous transformation where the normal and tangential parts of the Q-tensor decouple and various intrinsic and extrinsic contributions emerge. The obtained surface Landau-de Gennes energy is compared with previous models <cit.> and an L^2-gradient flow is considered. The resulting tensor-valued surface partial differential equation is solved numerically on an ellipsoid.The paper is structured as follows. In Section 2 we present the main results, including the surface Landau-de Gennes energy, a formulation for the evolution problem, and numerical results to illustrate the mentioned interplay between the geometry, the topology of the surface, and the positions and type of the defects. Section 3 establishes the notation essential for the derivation of the thin film limit, which is derived in Section 4 for the energy and the L^2-gradient flow. A discussion of mathematical and physical implications of the derived model and a comparison with previously postulated thin film models is provided in Section 5. Conclusions are drawn in Section 6 and details of the analysis are given in the Appendix.§ MAIN RESULTSWe consider Q-tensor fields on Riemannian manifolds ℳ defined by 𝒬(ℳ):= {∈^(2)(ℳ) := 0,= ^T}. We assume ℳ as well as n-tensor bundles ^(n)(ℳ) to be sufficiently smooth and consider two types of manifolds ℳ, a surfacewithout boundaries and a thin film _h:=×[-h/2,h/2] of thickness h. We have 𝒬() ⊂𝒬() and we can tie a surface Q-tensor ∈𝒬() with a restricted bulk Q-tensor ∈𝒬() by the orthogonal projections = Id - ⊗, with identity Id and surface normalanda Q-tensor projection defined in (<ref>), ,= ( |_) = |_ + 1/2( |_)For Q-tensors ∈𝒬(_h) we consider the elastic and bulk free energy ℱ^_h = ℱ^_h_el + ℱ^_h_bulk withℱ^_h_el[]:= 1/2∫__h L_1∇^2+ L_2^2+ L_3⟨∇ , ( ∇)^T_(2 3)⟩+ L_6⟨( ∇) , ∇⟩,ℱ^_h_bulk[]:= ∫__h a^2 + 2/3b^3 + c^4,see, , <cit.>, with elastic parameters L_i and thermotropic parameters a,b, and c. For simplicity we restrict our analysis to achiral liquid crystals, i.e. L_4 = 0, see the general form in <cit.>.Moreover, let == 0 and = β be essential anchoring conditions at ∂_h, where β is considered to be constant. Consequently, we obtain the natural anchoring conditions( ( L_1 + β L_6) (∇) + L_3(∇)^T_(2 3)) =0 ∂_hwhich ensure vanishing boundary integrals in the first variation δℱ^_h. Foras in (<ref>) we obtain in the thin film limit 1/hℱ^_h[] = ℱ^[] + 𝒪(h^2) the corresponding surface free energy ℱ^ = ℱ^_el + ℱ^_bulk with ℱ^_el[]:= 1/2∫_L'_1∇^2 + L_6⟨( ∇) , ∇⟩ + M_1^2+ M_2⟨,⟩^2+ M_3^2⟨,⟩+ M_4⟨,⟩ + C_0 ,ℱ^_bulk[]:= ∫_ a'^2 + c^4 + C_1 ,and shape operator =-(∇). In contrast to (<ref>), all operators are defined by the Levi-Civita connection and inner products are considered at the surface. All parameter functions L'_1,M_1,M_2,M_3,M_4,C_0,C_1, and a' can be related to the thin film parameters L_i, the surface quantities(mean curvature) and(Gaussian curvature), and β, see (<ref>). The L^2-gradient flow ∂_t= - ∇_L^2ℱ^ reads∂_t =L'_1Δ^dG+ L_6(( ∇∇): + ( ∇)· - 1/2( ∇)^T_(1 3):∇ + 1/4∇^2)-( M_1 + M_3⟨,⟩+ 2a' +2c ^2) -( M_2⟨,⟩ + M_3/2^2 + M_4/2)( - 1/2)on × [0,T] with the div-Grad (Bochner) Laplacian Δ^dG. The same evolution equation also follows as the thin film limit of the corresponding L^2-gradient flow ∂_t= - ∇_L^2ℱ^_h for (<ref>).To numerically solve the tensor-valued surface partial differential equation (<ref>) we use a similar approach as considered in <cit.>. We reformulate the equation in ^3 euclidean coordinates and penalize all normal contributions ·, to enforce tangentiality of the tensor. This leads to a coupled nonlinear system of scalar-valued surface partial differential equations for the components of , which can be solved using surface finite elements <cit.>. <ref> shows the evolution on a spheroidal ellipsoid. The initial configuration is set as in <cit.>(Fig. 1), with three nodes and a saddle defect, which are placed along an equatorial plane. In accordance with the Poincaré-Hopf theorem the topological charges of these defects add up to the Euler characteristic of the surface, 1+1+1-1 = 2. After some rearrangement all four defects split into pairs of + 1/2 and - 1/2 defects, which move away from each other perpendicular to the initial equatorial plane. Equally charged defects repel each other and oppositely charged defects attract each other. This leads to an annihilation of two pairs of +1/2 and -1/2 defects. According to the geometric properties of the ellipsoid the remaining four +1/2 defects arrange pairwise in the vicinity of the high curvature regions, with each pair perpendicular to each other. This deformed tetrahedral configuration is known to be the minimal energy state, see <cit.> for a sphere and <cit.> for ellipsoids. We further observe the principle director to be aligned with the minimal curvature lines in the final configuration. This alignment is a consequence of the extrinsic contributions in (<ref>), where our model differs from previous studies. Another remarkable feature of the derived surface Landau-de Gennes model is the possibility of coexisting isotropic and nematic phases. Such coexistence is know in three-dimensional models and results from the presence of the ^3 term in (<ref>). Such a term is absent in two-dimensional models in flat space. This difference in the three- and two-dimensional model typically changes the phase transition type. In our model the dependency of M_1 on curvature, see (<ref>), allows to locally modify the double-well potential in (<ref>) and thus allows for coexisting states due to changing geometric properties of the surface.§ NOTATIONAL CONVENTION AND THIN FILM CALCULUSFor notational compactness of tensor algebra we use the Ricci calculus, where lowercase indices i,j,k,… denote components in a surface coordinate system and uppercase indices I,J,K,… denote components in the extended three dimensional thin film coordinate system. Brackets [] and {} are used to switch between components and object representation, , for a 2-tensorwe write []_ij=t_ij for the components and {t_ij}= for the object. Most of the tensor formulations in this paper are invariantcoordinate transformations, thus a co- and contravariant distinction in the object representation is not necessary. However, if such a distinction is needed, we use the notation of musical isomorphisms ♯ and ♭ for raising and lowering indices, respectively. These are extended to tensors in a natural way, , for a 2-tensor ={t^i_j}∈^1_1 we write ^♭^♯ = {t_i^j} = {t^i_j}^-1∈_1^1 with metric tensorin . Finally, a tensor product denotes a contraction [s]_ij:=s_i^kt_k_j and the Frobenius norm of a rank-n tensorwill be denoted by _, , _^2=⟨, ⟩_ with ⟨s, ⟩_ := s_i_1⋯ i_nt^i_1⋯ i_n, that has to be understoodthe corresponding metricfor raising and lowering the indices[The suffixwill be omitted, if it is clear which metric the scalar product refers to.].The first, second, and third fundamental form are denoted by g_ij = ⟨∂_i, ∂_j⟩ (metric tensor), []_ij = -⟨∂_i, ∂_j⟩ (covariant shape operator), and []_ij = ⟨∂_i, ∂_j⟩, respectively. With this, curvature quantities can be derived: = det^♯ (Gaussian curvature) and = tr=^i_i (mean curvature). The Kronecker delta will be denoted by = {δ^i_j} and the Christoffel symbols (of second kind) will be denoted by ijk=1/2g^kl(∂_i g_jl + ∂_j g_il - ∂_l g_ij) at the surface and _IJ^K=1/2G^KL(∂_I G_JL + ∂_J G_IL - ∂_L g_IJ) in the thin film, whereis the metric tensor of the thin film , , G_IJ = δ_IJ and _IJ^K = 0 in the euclidean case.The surfaceand the thin filmas Riemannian manifolds are equipped with different metric compatible Levi-Civita connections ∇. We use “;” in the thin film and “|” at the surface to point out the difference for covariant derivatives in index notation, ,[ ∇]_IJK = Q_IJ;K = ∂_KQ_IJ - _KI^LQ_LJ - _KJ^LQ_IKin[ ∇]_ijk = q_ij|k = ∂_kq_ij - Γ_ki^lq_lj - Γ_kj^lq_ikin We define the coordinate in normal directionof the surfaceby ξ∈[ -h/2, h/2]. The local surface coordinates are (u,v) defined on every chart in the atlas of ,the immersion : (u,v) ↦^3 parametrize the surface. Adding these up, we obtain a parametrization : (u,v,ξ) ↦^3 of the thin film , defined by (u,v,ξ) := (u,v) + ξ(u,v) This means, the lowercase indices i,j,k,… are in {u,v} and the uppercase indices I,J,K,… are in {u,v,ξ}. The canonical choice of basis vectors in the tangential bundles are ∂_i∈ and ∂_I∈. Therefore, the metric tensors are defined by g_ij = ∂_i·∂_j and G_ij = ∂_I·∂_J. Consequently, it holds G_iξ = G_ξ i = 0, G_ξξ = 1, and by (<ref>), we get for the inverse metric tensor G^iξ = G^ξ i = 0, G^ξξ=1. The pure tangential components of the thin film metric and its inverse can be expressed as a second order surface tensor polynomial in ξ and a second order expansionG_ij = g_ij - 2ξ B_ij + ξ^2[ ]_ijand G^ij = g^ij + 2ξ B^ij + (ξ^2)respectively. Consequently, there is no need for rescaling while lowering or rising the normal coordinate index ξ, for an arbitrary thin film tensorit holds*W*^…_…^ξ_^…_… = G^ξ I*W*^…_…^_I^…_…= *W*^…_…^_ξ^…_…Moreover, a contraction of two arbitrary thin film tensorandrestricted to the surface results in a contraction of the tangential partthe surface metric and a product of the normal part, ,*W*^…_…^_I^…_…*W*^…_…^I_^…_… = G^IJ*W*^…_…^_I^…_…*W*^…_…^_J^…_… = g^ij*W*^…_…^_i^…_…*W*^…_…^_j^…_…+ *W*^…_…^_ξ^…_…*W*^…_…^_ξ^…_…= *W*^…_…^_i^…_…*W*^…_…^i_^…_…+ *W*^…_…^_ξ^…_…*W*^…_…^_ξ^…_… To deal with covariant derivatives, we have to take the Christoffel symbols into account. It is sufficient to expand _IJ^K first order in normal direction, since we only use first order derivatives and no partial derivatives of the symbols are necessary. Hence, (<ref>) result in_ij^k = Γ_ij^k + (ξ)_ij^ξ = B_ij + (ξ) _ξξ^K = _Iξ^ξ = _ξ I^ξ = 0and_iξ^k = _ξ i^k = -B_ij + (ξ)The volume elementcan be split up into a surface and a normal part by (<ref>), ,= √() dudv = (1 - ξ + ξ^2)√() dudv= (1 - ξ + ξ^2) § THIN FILM LIMITThin film limits require a reduction of degrees of freedom.We deal with this issue by setting Dirichlet boundary conditions for the normal parts ofand postulate a priori a minimum of the free energy on the inner and outer boundary of the thin film. This is achieved by considering natural boundary condition of the weak Euler-Lagrange equation. In this setting we restrict the density of ℱ^_h to the surface and integrate in normal direction to obtain the surface energy ℱ^. In the same way, we also show the consistency of the thin film and surface L^2-gradient flows. The next subsection considers the reformulation of the surface Landau-de Gennes energy to obtain the formulation in (<ref>), which allows a distinction of extrinsic and intrinsic contributions. Finally, we present a strong formulation of the derived equation of motion. §.§.§ Derivation of thin film limitsThe free energy (<ref>) in the thin filmin index notation readsℱ^_h_el[]= 1/2∫_ L_1Q_IJ;KQ^IJ;K + L_2Q_I^J_;JQ^IK_;K + L_3Q_IJ;KQ^IK;J + L_6Q^KLQ_IJ;KQ^IJ_;L ℱ^_h_bulk[]= ∫__h a Q_IJQ^JI + 2/3b Q_IJQ^JKQ_K^I + c Q_IJQ^JKQ_KLQ^LIWith respect to arbitrary thin film Q-tensors ∈𝒬(_h), the corresponding first variationsδℱ^_h_el( , )=∫__hΨ_IJ;K( L_1Q^IJ;K + L_3Q^IK;J + L_6Q^KLQ^IJ_;L)+ L_2Ψ_I^J_;JQ^IK_;K+ L_6/2Ψ_IJQ_KL^;IQ^KL;J δℱ^_h_bulk( , )= 2∫__h( ( a + cQ_KLQ^KL)Q^IJ + b Q^IKQ_K^J) Ψ_IJwhich are used to find local minimizers of the functional ℱ^_h = ℱ^_h_el +ℱ^_h_bulk, using the L^2-gradient flow∫_⟨∂_t, ⟩ = -δℱ^_h( , ) = -∫_⟨∇_L^2ℱ^_h, ⟩for all ∈𝒬(_h). However, integration by parts of (<ref>) givesδℱ^_h( , )= ∫_⟨∇_L2ℱ^_h, ⟩+ ∫_∂ L_2Q_I^J_;JΨ^I_ξ+( L_1 Q_IJ;ξ + L_3Q_Iξ;J + L_6Q^ξ KQ_IJ;K)Ψ^IJdAwhere dA is the volume form of the boundary surfaces. For the choice of essential boundary conditions, we require thathas to have two directors in the boundary tangential bundle and the remaining director has to be the boundary normal, , for ∈∂ a pure covariant representation ofat the boundary is= S_1^♭⊗^♭ + S_2^♭⊗^♭ - 1/3( S_1+S_2)with scalar order parameter S_1 and S_2. Hence, it holds Q_iξ = Q_ξ i = 0 and Q_ξξ= 1/3 (2S_2 - S_1). For simplicity, we set the pure normal part ofconstant, , Q_ξξ= β∈ at ∂. Therefore,has to be in 𝒬_0(_h) := {∈𝒬(_h): Ψ_Iξ = Ψ_ξ I = 0at ∂}, and we consider the natural boundary conditions 0 = ( L_1 + L_6β)Q_ij;ξ + L_3Q_iξ;j at ∂ so that the boundary integral in (<ref>) vanishs. Here, our analysis differs from previous results, which deal with a global determination of the normal derivatives in the whole bulk ofby parallel transport ∇_ξ = 0, or by ∂_ξ = 0, see <cit.>.With <ref> we can relate the anchoring conditions to surface identitiesQ_ξξ = β + (h^2)∂_ξQ_ξξ = (h^2)( L_1 + L_6β)Q_ij;ξ + L_3Q_iξ;j = (h^2) Q_iξ = Q_ξ i = (h^2)∂_ξQ_i ξ = ∂_ξQ_ξ i = (h^2)Evaluating ∈𝒬_0(_h) at the surface results in Ψ_Iξ = Ψ_ξ I =∂_ξΨ_I ξ = ∂_ξΨ_ξ I =(h^2) The restricted Q-tensor {Q_ij}∈^(2) is not a Q-tensor, because _{Q_ij} = _ - Q_ξξ = - Q_ξξ. We thus project {Q_ij} to 𝒬() with the orthogonal projection:^(2) →𝒬()↦1/2(+ ^T - (_))and define ∈𝒬() by:= {Q_ij}= {Q_ij} + β/2 + (h^2)For ∈𝒬_0(_h) the tangential part is already a Q-tensor up to (h^2). Therefore we define ψ_ij:= Ψ_ij + 1/2ψ_ξξg_ij = Ψ_ij + (h^2) where ∈𝒬(). With (<ref>), (<ref>), (<ref>), (<ref>), (<ref>), and the tensor shift ω :=- ω/2β we can determine all covariant derivatives restricted to the surface byQ_ξξ;ξ = ∂_ξQ_ξξ = (h^2)Q_iξ ;ξ = Q_ξ i;ξ = ∂_ξQ_i ξ - _ξ i^KQ_Kξ = (h^2) Q_ξξ;k = ∂_kQ_ξξ - 2_k ξ^LQ_Lξ = (h^2) Q_iξ ;k = Q_ξ i;k = ∂_kQ_i ξ - _k i^lQ_lξ - _k i^ξQ_ξξ - _k ξ^lQ_il= -β B_ik + ( q_il - β/2g_il) B^l_k + (h^2)= [ 3]_ik + (h^2)Q_ij;ξ = -L_3/L_1 + L_6βQ_iξ;j + (h^2)= -L_3/L_1 + L_6β[ 3]_ik + (h^2) Q_ij;k = ∂_kQ_i j - _ki^lQ_lj - _ki^ξQ_ξ j - _kj^lQ_il - _kj^ξQ_i ξ=∂_kQ_i j - Γ_ki^lQ_lj - Γ_kj^lQ_il + (h^2) = [ 1]_ij|k + (h^2)= q_ij|k + (h^2)Analogously, for the components of the covariant derivative ∇, we obtain Ψ_Iξ;ξ = Ψ_ξ I;ξ = Ψ_ξξ;I = (h^2), Ψ_iξ;k = Ψ_ξ i ;k = [ ]_ik + (h^2) and Ψ_ij;k = ψ_ij|k + (h^2) Note, in absence of natural boundary conditions for , the covariant normal derivatives Ψ_ij;ξ of the tangential components stay undetermined. However, as we will see, the thin film limit of the L^2-gradient flow (<ref>) does not depend on these derivatives. Adding up the three terms in (<ref>) with factors L_1, L_3, and L_6, factoring ∇ out, restricting to the surface and considering (<ref>) and (<ref>), results inL_1∇^2_ + L_3⟨∇ , ( ∇)^T_(2 3)⟩_ + L_6⟨( ∇) , ∇⟩_= Q_IJ;K( L_1Q^IJ;K + L_3Q^IK;J + L_6Q^KLQ^IJ_;L)= Q_ij;k( L_1Q^ij;k + L_3Q^ik;j + L_6Q^klQ^ij_;l)+ Q_ξ j;k( L_1Q^ξ j;k + L_3Q^ξ k;j + L_6Q^klQ^ξ j_;l)+ Q_iξ ;k( L_1Q^iξ ;k + L_3Q^ik;ξ + L_6Q^klQ^iξ_;l)+ Q_ij;ξ( L_1Q^ij;ξ + L_3Q^iξ ;j + L_6Q^ξξQ^ij_;ξ)+ (h^2)= (L_1-β/2L_6) ∇^2_ + L_3⟨∇ , ( ∇)^T_(2 3)⟩_ + L_6⟨( ∇) , ∇⟩_ + ( 2L_1 - L_3^2/L_1+L_6β)3^2_+ L_3_( 3)^2 + 2L_6⟨31, 3⟩_+ (h^2)With ^3 = 0 we obtain for the remaining terms^2_ = Q_I^J_;JQ^IK_;K= Q_i^j_;jQ^ik_;k + Q_ξ^j_;jQ_ξ^k_;k +(h^2) = _^2 + ( _(3) )^2 +(h^2)_^2 = Q_IJQ^JI= Q_ijQ^ji + (Q_ξξ)^2 +(h^2)= _(- β/2)^2 + β^2 +(h^2)= _^2 + 3/2β^2 +(h^2) _^3 = Q_IJQ^JKQ_K^I = Q_ijQ^jkQ_k^i + (Q_ξξ)^3 +(h^2)= _(- β/2)^3 + β^3 +(h^2) = 3/2β( β^2/2 - _^2) +(h^2)_^4 = 1/2(_^2)^2 = _^4 + 3/2β^2_^2 + 9/8β^4 +(h^2)Adding all these up, we can define ℱ^ := ℱ^_el + ℱ^_bulk byℱ^_el[]:= 1/2∫_(L_1-β/2L_6) ∇^2+ L_2^2+ L_3⟨∇ , ( ∇)^T_(2 3)⟩+ L_6⟨( ∇) , ∇⟩ + ( 2L_1 - L_3^2/L_1+L_6β)3^2+ L_2( (3) )^2+ L_3( 3)^2 + 2L_6⟨31, 3⟩ ℱ^_bulk[]:= ∫_1/2(2a - 2bβ + 3cβ^2)^2 + c ^4 + β^2/8(12a + 4bβ + 9cβ^2)and by the rectangle rule and (<ref>), we obtain for h→ 01/hℱ^ = 1/h∫_F^ = 1/h∫_-h/2^h/2∫_( 1- ξ + ξ^2)F^ = ∫_ F^ + (h^2)= ℱ^ + (h^2) ⟶ℱ^ Consequently, the energies ℱ^ and ℱ^ are consistentthe thickness h. To show a similar asymptotic behavior for the L^2-gradient flows, we investigate the first variation δℱ^ = δℱ^_el + δℱ^_bulk in (<ref>) and compare with the first variation δℱ^= δℱ^_el + δℱ^_bulk, whereδℱ^_el(,)=∫_(L_1-β/2L_6) ⟨∇, ∇⟩ + L_2⟨,⟩ + L_3⟨∇ , ( ∇)^T_(2 3)⟩ + L_6( ⟨( ∇) , ∇⟩ + 1/2⟨( ∇) , ∇⟩) + ( 2L_1 - L_3^2/L_1+L_6β)⟨3, ⟩ + L_2⟨3, ⟩⟨, ⟩ + L_3⟨3, ⟩ + L_6( 2⟨31, ⟩+⟨( 3)^2,⟩) δℱ^_bulk(,)=∫_ (2a - 2bβ + 3cβ^2)⟨, ⟩ + 2c^2⟨, ⟩Proceeding as before we restrict the terms under the integral of δℱ^ in (<ref>) to the surface. For δℱ^_el we obtainΨ_IJ;K( L_1Q^IJ;K + L_3Q^IK;J + L_6Q^KLQ^IJ_;L)= Ψ_ij;k( L_1Q^ij;k + L_3Q^ik;j + L_6Q^klQ^ij_;l)+Ψ_iξ;k( 2L_1Q^iξ;k + L_3( Q^ik;ξ + Q^kξ;i) + 2L_6Q^klQ^iξ_;l) +(h^2)= ψ_ij|k( ( L_1 - β/2L_6)q^ij|k + L_3q^ik|j + L_6q^klq^ij_|l)+ [ ]_ik(( 2L_1 - L_3^2/L_1+L_6β) [3]^ik + L_3[3]^ki)+2L_6[ ]_ik[31]^ik+(h^2) Ψ_I^J_;JQ^IK_;K = ψ_i^j_|jq^ik_|k +[ ]^j_j[ 3]^k_k +(h^2)1/2Ψ_IJQ_KL^;IQ^KL;J = 1/2Ψ_ij( Q_kl^;iQ^kl;j + 2Q_kξ^;iQ^kξ;j) +(h^2)= 1/2ψ_ijq_kl^|iq^kl|j + ψ_ij[ (3)^2]^ij +(h^2)and for δℱ^_bulk2(a+cQ_KLQ^KL)Q^IJΨ_IJ = ( 2a + 2cq_klq^kl + 3cβ^2)( q^ijψ_ij - β/2ψ^i_i)+(h^2) = ( 2a + 3cβ^2) ⟨, ⟩ + 2c^2⟨, ⟩ +(h^2)2bQ^IKQ_K^JΨ_IJ = 2bQ^ikQ_k^jΨ_ij +(h^2) = 2b( [ ^2]^ij - β q^ij + β^2/4g^ij)ψ_ij +(h^2) = -2bβ q^ijψ_ij + b( ^2 + β^2/2)ψ^i_i+(h^2)= -2b ⟨, ⟩ +(h^2)where we used <ref>, , 2^2=(^2), particularly. In summary, we see that ⟨∇_L^2ℱ^_h, ⟩ =⟨∇_L^2ℱ^, ⟩ + (h^2) is valid. Moreover, as ∂_t = 0 for a stationary surface, we obtain [ ∂_t]_ij = [ ∂_t]_ij + (h^2). Finally, as in (<ref>), we argue with the rectangle rule in normal direction and observe1/h∫_⟨∇_L2ℱ^_h + ∂_t, ⟩ =∫_⟨∇_L^2ℱ^ + ∂_t, ⟩ + (h^2)§.§.§ Surface energyTo have a better distinction between extrinsic terms, , ⟨, ⟩, and terms depending only on scalar curvaturesandin the surface energy (<ref>), we use <ref> and obtain the substitutions( ( 3) )^2 = ⟨, ⟩^2 - 3β⟨, ⟩ +9/4β^2^2 ( 3)^2 = ⟨, ⟩^2 + ^2 -3β⟨, ⟩ + 9/4β^2( ^2 - 2) 3^2 = 1/2(^2 - 2)^2-3β⟨, ⟩ + 9/4β^2( ^2 - 2) ⟨31 , 3⟩ = 1/2^2⟨, ⟩ -β( 3 ⟨, ⟩^2 + 1/4( ^2 + 10))^2+ 15/4β^2⟨, ⟩ - 9/8β^3( ^2 - 2)at the surface . Terms with invariant measurement of the gradient ∇ differ only in zero order quantities for a closed surface, see <ref>. Adding all these up, we obtain (<ref>) and therefore in index notationℱ^_el[] =1/2∫_ L'_1 q_ij|kq^ij|k + L_6 q^klq_ij|kq^ij_|l + M_1 q_ijq^ij + M_2 B^ijB^klq_ijq_kl + M_3 B^ijq_ijq^klq_kl + M_4B^ijq_ij + C_0ℱ^_bulk[]:= ∫_ a' q_ijq^ij + c q_ijq^jkq_klq^li + C_1with coefficient functionsL'_1 := L_1 + 1/2( L_2 + L_3 - L_6β)M_2 := L_2 + L_3 - 6L_6βM_3 := L_6M_1 := 1/2( -L_6( ^2+10)β +( 2L_1 - L_3^2/L_1+L_6β)( ^2-2) +( L_2 + L_3))M_4 := -3( 2L_1 + L_2 + L_3 - 5/2L_6β - L_3^2/L_1+L_6β)βC_0 := 9/4( ( 2L_1 + L_3 - L_6β - L_3^2/L_1+L_6β)( ^2-2) +L_2^2)β^2a':= 1/2(2a - 2bβ + 3cβ^2) andC_1 := β^2/8(12a + 4bβ + 9cβ^2)§.§.§ Surface equation of motionTo obtain the strong form of the surface L^2-gradient flow ∂_t= -∇_L^2ℱ^ we have to ensure∫_⟨∂_t, ⟩ = -∫_⟨∇_L^2ℱ^, ⟩, ∀∈𝒬(), the L^2 inner product over the space of Q-tensors and thus ∇_L^2ℱ^∈𝒬(). While for the first variations δ in direction ψ1/2δ∫_∇^2= ∫_⟨ -∇ , ⟩1/2δ∫_^2= ∫_⟨ , ⟩1/2δ∫_^4= ∫_⟨( ^2) , ⟩the left argument of the inner product is already in 𝒬(), we have to applydefined in (<ref>) for the remaining terms, ,1/2δ∫_⟨( ∇) , ∇⟩= ∫_( -( q_ij|kq^kl)_|l + 1/2q_kl|iq^kl_|j)ψ^ij = ∫_( - q_ij|k|lq^kl - q_ij|kq^kl_|l + 1/2[{q_kl|iq^kl_|j}]_ij)ψ^ij = ∫_( - q_ij|k|lq^kl - q_ij|kq^kl_|l +1/2q_kl|iq^kl_|j -1/4q_kl|mq^kl|mg_ij)ψ^ij = ∫_⟨( -∇∇): - ( ∇)· +1/2( ∇)^T_(1 3):∇ - 1/4∇^2, ⟩1/2δ∫_⟨,⟩^2= ∫_⟨⟨,⟩ , ⟩ = ∫_⟨⟨,⟩ , ⟩ = ∫_⟨⟨,⟩( - 1/2) , ⟩1/2δ∫_^2⟨,⟩= ∫_⟨1/2^2 + ⟨,⟩ , ⟩ = ∫_⟨1/2^2 + ⟨,⟩ , ⟩ = ∫_⟨1/2^2( - 1/2) + ⟨,⟩ , ⟩1/2δ∫_⟨,⟩= ∫_⟨1/2, ⟩ = ∫_⟨1/2, ⟩ = ∫_⟨1/2( - 1/2), ⟩Finally, with [ Δ^dG]_ij := q_ij^|k_|k, the div-Grad (Bochner) Laplace operator, we get the equation of motion (<ref>), which reads in index notation∂_tq_ij = L'_1q_ij^|k_|k+ L_6(- q_ij|k|lq^kl - q_ij|kq^kl_|l +1/2q_kl|iq^kl_|j -1/4q_kl|mq^kl|mg_ij)- ( M_1 + M_3B_klq^kl+ 2a' +2c q_klq^kl)q_ij - ( M_2 B_klq^kl + M_3/2q_klq^kl + M_4/2)(B_ij - 1/2 g_ij) After establishing weak consistences for the energies and the L^2-gradient flows in the thin film and at the surface in (<ref>) and (<ref>), we also have pointwise consistence for the evolution equation in the Q-tensor space restricted to the surface for sufficient regularity, ,[ ( ∂_t + ∇_L^2ℱ^[ ] )] -(∂_t + ∇_L^2ℱ^[ ]) _ = (h^2) boundary conditions forat ∂ and initial condition |_t=0 = |_(,t=0) + |_(,t=0). This means, the order of performing the limit h→ 0 and formulating the local dynamic equation,∇_L^2 flow, does not matter, , the diagramℱ^rh→ 0 [swap]1/h[mapsto]d[swap]∇_L^2flow ℱ^[mapsto]d∇_L^2flow ∂_t = -∇_L^2ℱ^[ ]rh→ 0 [swap],∂_t = -∇_L^2ℱ^[ ]commutes. § DISCUSSIONWe now discuss similarities and differences between the thin film and surface Landau-de Gennes energy and their physical implication. Besides the terms containing the extrinsic quantityand its scalar valued invariants, the surface Q-tensor energy (<ref>) is similar to the thin film Q-tensor energy (<ref>). While we have three scalar invariants for the gradient ∇_ in the thin film controlled by L_1, L_2, and L_3, at the surface we need only one for ∇_ to formulate the distortion of , see <ref>. This behavior seems to be a consequence of reducing the degree of freedoms of Q-tensors. Particularly, 𝒬(_h) is a five dimensional function-vector space, while 𝒬() is only a two dimensional function-vector space with improper rotation endomorphisms in the tangential bundle as basis tensors. Moreover, at the surface we only consider the trace of even powers offor the bulk energy as for r≥ 0 it holds^2(r+1) = ⟨( ^2)^r+1, ⟩ = 2^-(r+1)( ^2)^r+1^2 = 2^-r( ^2)^r+1 ^2r+1 = ⟨( ^2)^r, ⟩ = 2^-r( ^2)^r = 0,see <ref>. This has several consequences. In principle it leads to a change in phase transition type, as coexistence between a nematic and an isotropic phase is not possible without the ^3 term. However, as we will see, our model still allows coexistence. We first show that we can preserve the phase diagram of the thin film bulk energy. To limit complexity we have considered =β to be constant. Similar assumptions have been made in <cit.>. Our approach chooses β such that surface and thin film formulation of bulk energy match. For β=-1/3 S^*, where S^*=1/4c(-b+√(b^2-24ac)) indeed the minima of ℱ^_h_bulk and ℱ^_bulk are equal and are achieved for S=S^*, with S = S_1 = S_2 or S = S_1 if S_2 = 0 or S = S_2 if S_1 = 0. The reconstructed thin film Q-tensor = -β/2 + β⊗ is uniaxial with eigenvalues [2/3S,-1/3S,-1/3S]. <ref>shows the phase diagram. Contrary to the modeling via degenerate states with β=0, see, , <cit.>, the phase diagram of the bulk energy is preserved for β = - 1/3 S^*.With the emergence of defects the assumption β = const becomes questionable and a more precise modeling would require to treat β as a degree of freedom. However, this would lead to an excessive amount of additional coupling terms in the elastic energy and thus makes the complexity of the model infeasible. A detailed derivation and interpretation of the additional terms thus remains an open question.Considering the elastic energy, the surface model provides a set of new terms consisting of combinations of ^2 and ⟨,⟩. These terms interact with the double-well potential a'^2 + c ^4 of the surface bulk energy. By this interaction the bulk potential can be deformed locally, as e.g. M_1 ^2, depends on geometric properties M_1=M_1(,). So, while the bulk potential itself inhibits isotropic-nematic phase coexistence, a global phase coexistence can emerge on surfaces by local variance of geometric properties, see <ref>.The term⟨,⟩ imposes restrictions on energetic favorable ordering. This term can be expressed in terms of principal directorofby ⟨,⟩ = -1/2 illustrating a geometric forcing towards the ordering along lines of minimal curvature. Such forcing does eliminate the rotational invariance of the four +1/2 defect configuration on an ellipsoid as demonstrated in <ref>. The same effect has also been observed in surface Frank-Oseen modell for surface polar liquid crystals <cit.>.Combining these effects provides a wide range of intriguing mechanisms coupling geometry and ordering with significant impacts on minimum energy states and dynamics. A more detailed elaboration of these interactions as well as a detailed description of the used numerical approach will be given elsewhere.As a complementary result, we point out that the surface model for degenerate states in <cit.> can be reproduced by our model by choosing β=0, k=L_1= 1/√(2)L_2=-1/√(2)L_3, k_24 = -√(2)k and defining 2a=A, 2c=C. A one-to-one comparison with the models derived in <cit.> is more complicated, as in contrast to our approach, which only uses the Levi-Civita connections ∇, other surface derivatives are introduced in <cit.>, which make these models depending on the chosen coordinate system. A detailed comparison of numerical simulations might allow to point out similarities and differences.§ CONCLUSIONWe have rigorously derived a surface Q-tensor model by performing the thin film limit. Instead of making assumptions on the Q-tensor field in the thin film we have prescribed a set of boundary conditions for the thin film. By requiring the normal components ofto be compatible with the minimum of the bulk energy we were able to transfer main features of the thin film model, like uniaxiality or parameter-phase space, to the surface model. Nonetheless, these features break down in areas of defects. It still remains an open question how to treat defect areas properly in surface Q-tensor models.The proposed approach to derive thin film limits is general and can also be used for other tensorial problems, e.g. in elasticity. Note that for deriving thin film limits containing higher order derivatives, also higher order expansions for thin film metric quantities are needed,_ij^k = Γ_ij^k + ξΘ_ij^k + (ξ^2) with Θ_ij^k := B_i^k_|j + B_j^k_|i - B_ij^|k for thepure tangential components of Christoffel symbols to express second order covariant derivatives like the Laplace operator Δ in the thin film. Our analysis also indicates that the surface evolution equation can be derived directly without a detour of a global energy minimization problem. However, there is no general theory regarding sufficient prerequisites of this analysis, and we can not ensure, that, for example, every well posed tensorial thin film problem results in a well posed tensorial surface problem.Even with the made approximations in the modeling approach the numerical results provide new insights on the tight coupling of topology, geometry, and energetic minimal states as well as dynamics. In a next step the derived coupling terms should be investigated systematically and the model should be validated versus experimental data. Various extensions of the proposed model, like coupling to hydrodynamics and/or activity open up a wide array of possible physical applications in material science or biophysics. For recent work on hydrodynamics on surfaces we refer to <cit.>. Also investigations on energy minimization and dynamics on moving domains seem now feasible. However, to deal with these problems numerically requires a more detailed investigation of the regularity. In contrast to our assumption for the tensor fields to be sufficiently smooth, which was made for simplicity, tensorial Sobolev spaces should be investigated, see e.g. <cit.>. AcknowledgementsHL and AV acknowledge financial support from DFG through Lo481/20 and Vo899/19, respectively. We further acknowledge computing resources provided by JSC under grant HDR06 and ZIH/TU Dresden. § APPENDIXFor all surface q-Tensors ∈𝒬() holds∫_^2 = ∫_1/2∇^2 + ^2 ∫_⟨∇,( ∇)^T_(2 3)⟩ = ∫_1/2∇^2 - ^2With the surface Levi-Civita tensor ≅ defined byE_ij := (∂_i, ∂_j) = √() ε_ijwith Levi-Civita symbols ε_ij, we use the 2-tensor curl[ ]_i := [ -∇ : ]_i = -E_jkq_i^jkand observe[ -·]_i = E_ilE_jkq^lj|k= ( g_ijg_lk - g_ikg_lj)q^lj|k= q^l_i|l - q_j^j_|i= q_i^l_|l= [ ]_iMoreover, in this case, -· is isomorph to theHodge-star operator * on differential 1-forms and therefore it can be seen as a length preserving pointwise counterclock quarter turn, that is why =-· = holds for the norm. We remark, that ∈^(2) is compatible with ∇ and hence, we calculate∫_^2 = 1/2∫_^2 + ^2 = -1/2∫_( q_i^k_|k|l + E_kjE_lmq_i^k|j|m) q^il= -1/2∫_( q_i^k_|k|l + q_il^|j_|j - q_i^k_|l|k) q^ilThe Riemannian curvature tensor has only one independent component on surfaces and is given by R = ⊗∈^(4). Hence, for changing the order of covariant derivatives, holdsq_i^k_|k|l -q_i^k_|l|k = R^j_iklq_j^k - R^k_jklq_i^j = ( ( δ^j_kg_il -δ^j_lg_ik)q_j^k- ( δ^k_kg_jl -δ^k_lg_jk)q_i^j) = -2 q_ilFinally, we get∫_^2 = -1/2∫_( q_il^|j_|j -2q_il) q^il= ∫_1/2∇^2 + ^2 ∫_⟨∇,( ∇)^T_(2 3)⟩ = -∫_q_i^k_|l|k q^il= -∫_( q_i^k_|k|l + 2 q_il) q^il= ∫_^2 -2^2= ∫_1/2∇^2 - ^2For all 2-tensors ∈^(2) at surfaceholds^2 =( ) + 1/2( ^2-( )^2)With the surface Levi-Civita tensordefined in (<ref>), the quarter turn in the row and column space of a 2-tensor ∈^(2) is[ ]_ij = E_ikE_ljt^kl= ( g_ilg_kj - g_ijg_kl)t^kl= t_ji - t^k_kg_ij= [ ^T - ( )]_ijParticularly, (<ref>) is also valid for the square of , ,^2 = ( ^2)^T - ( ^2)On the other hand side, with =-, (<ref>) and (^T)^2 = (^2)^T, we calculate^2 = -( )^2= -( ^T - ( ))^2= -(^T)^2 + 2( )^T - ( )^2= -( ^2)^T + 2( ) + ( )^2Averaging identities (<ref>) and(<ref>) results in^2 = ( ) + 1/2(( )^2 -^2)Finally, we obtain(<ref>) by a quarter turnwith in the row and column space of (<ref>).For all full covariant 2-tensors ∈^0_2 on surfaceholds( )^2 - ^2 = 2 /= 2^♯wheremeans the determinant of the matrix proxy. We can interpretas its matrix proxy with components t_ij due to the stipulation of the height of the indices. Hence, the determinant can be calculated applying the Levi-Civita symbols ε_ij∈{-1,0,1}, ,= 1/2∑_i,j,k,lε_ijε_klt_ikt_jlWith the Levi-Civita tensor defined in (<ref>) we obtain the transformation propertyE^ij = 1/E_ij = 1/√()ε_ijTherefore, (<ref>) results in= /2E^ijE^klt_ikt_jl= /2( g^ikg^jl - g^ilg^jk)t_ikt_jl= /2( t_i^it_j^j - t_i^jt_j^i)Additionally, we observe^♯ = ( ·^-1)= /For shape operatorand Q-tensor ∈𝒬() the following identities are valid. ^2 = ^2 = ^2 - 2 ^2 =-⟨^2 , ⟩ = ⟨, ⟩ ^2 = 1/2(^2) ^2 = 1/2(^2)( ^2 - 2) ()^2 = ⟨, ⟩^2 + ^2The proofs here are very straightforward with all the spadework above. (<ref>) is a consequence of <ref> for ∈^(2) and hence, we obtain also (<ref>) with <ref>. Since∈𝒬() is trace-free, we follow from (<ref>), that ⟨^2 , ⟩ = ⟨, ⟩ - 2 and therefore (<ref>). Again,is a Q-tensor and thus <ref> results in (<ref>). The shape operatoris self-adjoint, so with (<ref>) we can calculate^2 = ⟨, ⟩ = ⟨^2,^2⟩ = 1/2(^2)⟨^2, ⟩ = 1/2(^2)^2and get (<ref>) with (<ref>). We note that ⟨, ⟩^2 = ( )^2. Therefore,<ref> results in (<ref>), because( ( ) )^2 - ()^2 = 2( )^♯= 2( ^♯)( ^♯)= ( ( )^2 - ^2)= -^2 For the inverse thin film metric ^-1 holdsG^ij = ( g^ik + ∑_ = 1^∞ξ^[ ^]^ik)( δ_k^j + ∑_ = 1^∞ξ^[ ^]_k^j) = [(+ ∑_ = 1^∞ξ^^)^2]^ijG^ξξ = 1 G^iξ = G^ξ i = 0First we define the pure tangential components of the thin film metric tensor as _t := { G_ij}. With = {δ^i_j} the Kronecker delta, we can write down in usual matrix notation·^-1 = [ _tO;O1 ]·[{ G^ij}{ G^iξ}; { G^ξ i} G^ξξ ] =[ O; O 1 ]Thus, we obtainG^ξξ = 1 G^iξ = G^ξ i = 0{ G^ij} = _t^-1=(- ξ)^-2= (- ξ)^-1·(- ξ^♯)^-1For h small enough, so that ξ≤ h < 1 and exponent with a dot indicate matrix (endomorphism) power, we can use the Neumann series(- ξ^♯)^-1 =+ ∑_ = 1^∞ξ^( ^♯)^·and therefore the assertion, becausewith ^= ( ·^-1)^·· we get( ^♯)^· = ( ·^-1)^· = ^·^-1 = ( ^)^♯and(- ξ)^-1 = ( (- ξ^♯)·)^-1 = ^-1·(- ξ^♯)^-1 For the determinant of the thin shell film tensorholds= ( 1 - ξ + ξ^2)^2The mixed components are zero, so we get= G_ξξ_t = _tNow, we define √(_t^♯) := ( - ξ)^♯ as a square root of _t^♯, because_t^♯ = ( (-ξ)^2)^♯ = ( (-ξ)^♯(-ξ))^♯ = (-ξ)^♯(-ξ)^♯ = ( √(_t^♯))^2Hence, we can calculate= _t = _t^♯ =_t^♯ =√(_t^♯)^2For the determinant of √(_t^♯), we regard that ^♯= is the Kronecker delta, so we obtain√(_t^♯) = ( ^♯ - ξ^♯)= ( 1 - ξB_u^u)( 1 - ξB_v^v) - ξ^2B_u^vB_v^u= 1 - ξ( B_u^u + B_v^v) + ξ^2( B_u^uB_v^v - B_u^vB_v^u)= 1 - ξ + ξ^2^♯=1 - ξ + ξ^2 Letbe an arbitrary n-tensor in the thin film (with sufficient regularity), which vanish at the boundaries, , ∈{∈^(n): =0at ∂}, holds|_ = ∂_ξ|_ = (h^2)We denote the boundary at ξ=h/2 by Υ^+ and Υ^- at ξ=-h/2, Υ^+∪Υ^- = ∂. Taylor expansions at the surface result in0= |_Υ^±= |_±h/2∂_ξ|_ + h^2/8∂_ξ^2|_ + (h^3)And we yield0= |_Υ^+ + |_Υ^- = 2|_ + (h^2)0= |_Υ^+ - |_Υ^- = h∂_ξ|_ + (h^3)20ptsiam | http://arxiv.org/abs/1709.09436v1 | {
"authors": [
"Ingo Nitschke",
"Michael Nestler",
"Simon Praetorius",
"Hartmut Löwen",
"Axel Voigt"
],
"categories": [
"cond-mat.soft",
"math-ph",
"math.AP",
"math.MP",
"35Q82, 53Z05, 58J99, 58Z05, 82D30, 76A15"
],
"primary_category": "cond-mat.soft",
"published": "20170927103538",
"title": "Nematic liquid crystals on curved surfaces - a thin film limit"
} |
Joint Detection and Recounting of Abnormal Events by Learning Deep Generic Knowledge^*Ryota Hinami^1, 2, Tao Mei^3, and Shin'ichi Satoh^2, 1^1The University of Tokyo,^2National Institute of Infomatics, ^3Microsoft Research Asia [email protected], [email protected], [email protected] ============================================================================================================================================================================================================== In this paper, we investigate the existence of solution for systems of Fokker-Planck equations coupled through a common nonlinear congestion. Two different kinds of congestion are considered: a porous media congestion or soft congestion and the hard congestion given by the constraint ρ_1+ρ_2 ⩽ 1. We show that these systems can be seen as gradient flows in a Wasserstein product space and then we obtain a constructive method to prove the existence of solutions. Therefore it is natural to apply it for numerical purposes and some numerical simulations are included. Keywords:Wasserstein gradient flows, Jordan-Kinderlehrer-Otto scheme, crowd motion, nonlinear cross-diffusion systems.MS Classification: 35K40, 49J40, 49J45.§ INTRODUCTIONThe modelling of crowd behaviour has become a very active field of applied mathematics in recent years. These models permit to understand many phenomena such as cell migration, tumor growth, etc. Several models already exist to tackle this problem. The first one, microscopic, consists in seeing a population as a high number of individuals which satisfy ODEs, see for instance <cit.> and the second is macroscopic and consists in describing a population by a density ρ satisfying a PDE, where ρ(t,x) represents the density of individuals in x at time t. In the latter framework, different methods to handle the congestion effect have been proposed. The first one consists in saying that the motion has to be slower when the density is very high, see for example <cit.> for adifferent approach with applications to crowd dynamics. Another way of modelling the congestion effect is to use a threshold: the density evolves as we would expect until it touches a maximal level and then the motion has to be adapted in these regions (to not increase the density there), see for example <cit.> for crowd motion model and <cit.> for application to dendritic growth. For a comparison between microscopic and macroscopic models, we refer to <cit.>. In <cit.>, Mészáros and Santambrogio proposed a model for hard congestion where individuals are subject to a Brownian diffusion. This corresponds to modified a Fokker-Planck equation with an L^∞ constraint on the density.Since in macroscopic models, we have mass conservation, the theory of optimal transportation is a very natural tool to attack them. In <cit.>, the authors investigated a model of room evacuation. They showed that if the desired velocity field of the individuals is given by a gradient, say V=∇ D, where D is the distance to a given target, then the problem has a gradient flow structure in the Wasserstein space and the velocity field has to be adapted by a pressure field to handle congestion effect. More recently in <cit.>, a splitting scheme has been introduced to handle velocity fields which are not necessarily gradient field. The scheme consists in combining steps where the density follows the unconstrained Fokker-Planck equation and Wasserstein projections onto the set of densities which cannot exceed 1.A natural variant of the model of <cit.>, consists in considering two (or more) populations, each of whom is subject to an advection term coming from different potential gradients but coupled through the constraint that the total density cannot exceed a given threshold, say 1, and then subject to a common pressure field.Note that variant problems withtotal density equal to 1 are treated in <cit.> and for more general cross-diffusion systems, we refer, for instance, to <cit.>. For a linear diffusion (corresponding to a Brownian noise on each species), the two-species crowd dynamic is expressed by the PDEs{[ ∂_t ρ_1 -Δρ_1- (ρ_1 (∇ V_1+∇ p))=0,; ∂_t ρ_2 -Δρ_2- (ρ_2 (∇ V_2+∇ p))=0,;p⩾ 0, ρ_1+ρ_2 ⩽ 1, p(1-ρ_1-ρ_2)=0,; ρ_1(0, ·) = ρ_1,0,ρ_2(0, ·) = ρ_2,0 ]. on Ω a convex compact subset of ^n with smooth boundary such that |Ω | >2. The assumption (<ref>) is made to ensure that the subset𝒦 := { (ρ_1, ρ_2) ∈ (Ω)^2 :ρ_1 +ρ_2 ⩽ 1 a.e.}is neither empty nor trivial. We put no-flux boundary conditions to preserve the mass in Ω,(∇ρ_1 + ρ_1 (∇ V_1+∇ p) ) ·ν =0and(∇ρ_2 + ρ_2 (∇ V_2+∇ p) )·ν =0a.e. on ^+ ×∂Ω, where ν is the outward unit normal to ∂Ω. In this paper, we show that this system is the gradient flow for the Wasserstein product distance of the energy _∞(ρ_1, ρ_2):={[ ∑_i=1^2∫_Ω (ρ_i log(ρ_i)+ V_i ρ_i)+∫_Ωχ_[0,1] (ρ_1(x)+ρ_2(x))xif ρ_i log(ρ_i) ∈ L^1(Ω),;+∞, otherwise, ]. where χ_[0,1] is the indicator function of [0,1],χ_[0,1](z):={[0 ifz ∈ [0,1],; +∞ otherwise. ]. In addition, for a different energy of the form _m(ρ_1, ρ_2):= {[ ∑_i=1^2 ∫_Ω (ρ_ilog(ρ_i)+V_i ρ_i )+∫_Ω1/m-1(ρ_1(x)+ρ_2(x))^m dxif ρ_i log(ρ_i),(ρ_1+ρ_2)^m∈ L^1(Ω),;+∞otherwise, ]. for m > 1,the gradient flow of _m leads to the following nonlinear system {[ ∂_t ρ_1= Δρ_1 + (ρ_1 ∇( V_1 +m/m-1(ρ_1+ρ_2)^m-1)); ∂_t ρ_2= Δρ_2 + (ρ_2 ∇( V_2 +m/m-1(ρ_1+ρ_2)^m-1)); ρ_1(0, ·) = ρ_1,0,ρ_2(0, ·) = ρ_2,0 ]. with no flux boundary conditions. Then for a given small time step h>0, the JKO scheme for this energy reads,(ρ_1^k+1, ρ_2^k+1)=_(ρ_1,ρ_2){∑_i=1^2 1/2h W_2^2(ρ_i, ρ_i^k)+_m(ρ_1, ρ_2) }which, in the particular case of the linear diffusion crowd motion problem with two species, takes the form (ρ_1^k+1, ρ_2^k+1)=_ ρ_1+ρ_2 ⩽ 1{∑_i=1^2 (1/2h W_2^2(ρ_i, ρ_i^k)+∫_Ω (ρ_i log(ρ_i)+ V_i ρ_i) ) }.We want to mention that the results in this paper have been obtained in the authors's PhD thesis, <cit.>, back in 2016. Note that recently, in <cit.>, Kim and Mészáros studied problems (<ref>) and (<ref>) without individual diffusions. They prove existence of weak solution in dimension 1 for segregated initial conditions and ordered drifts. In any dimension, they prove existence of very weak solutions. The difficulty is to handle the cross diffusive term which needs to have strong compactness in ρ_1,ρ_2 and ρ_1 +ρ_2. Here, this difficulty is overcome by assuming that individuals of each populations are subject to a Brownian diffusion. This allows us to obtain separated estimates on ρ_i and ρ_1+ρ_2. In <cit.>, Laurençot and Matioc give a similar result inand m=2. In this paper, we extend this result on Ω⊂^n and with m∈ [1,+∞]. Furthermore, taking advantage of the gradient flow structure, we give numerical simulations implemented by the augmented Lagrangian scheme introduced in <cit.>. We want to point out that uniqueness of systems (<ref>) and (<ref>) is still an open question due to the lack of geodesic convexity of the common energy and we do not adress this problem in this paper. We refer to <cit.> for further discussions on this subject. This paper is organized as follows. In section <ref>, we introduce our assumptions and we state our main results. In section <ref>, we prove the existence of a weak solution for system (<ref>). The key ingredient is the flow interchange argument (see <cit.> for example) which gives separated estimates on the gradient of ρ_1+ρ_2 and on the gradient of ρ_i. Section <ref> provides the proof of existence of a weak solution for system (<ref>). In this section we use again the flow interchange argument to obtain stronger estimates. In section <ref>, we focus on the particular case where ∇ V_1= ∇ V_2. In this case, we are able to show the convergence when m → +∞ of a solution to (<ref>) to a solution to (<ref>) and we prove a L^1-contraction theorem. In the final section <ref>, numerical simulations are presented. § PRELIMINARIES AND MAIN RESULTSThroughout the paper, Ω is a smooth convex bounded subset of . We start to recall some results from the optimal tranportation theory and then we will state our main results. §.§ Wasserstein spaceFor a detailed exposition, we refer to reference textbooks <cit.>. We denote ℳ^+(Ω) the set of nonnegative finite Radon measures on Ω, (Ω) the space of probability measures on Ω, and (Ω), the subset of (Ω) of probability measures on Ω absolutely continuous with respect to the Lebesgue measure.For all ρ,μ∈(Ω), we denote Π(ρ,μ), the set of probability measures on Ω×Ω having ρ and μ as first and second marginals, respectively, and an element of Π(ρ,μ) is called a transport plan between ρ and μ. Then for all ρ,μ∈(Ω), we denote by W_2(ρ,μ) the Wasserstein distance between ρ and μ, defined as W_2^2(ρ,μ)= min{∬_Ω×Ω |x-y|^2dγ(x,y) :γ∈Π(ρ,μ)}. Since this optimal transportation problem is a linear problem under linear constraints, it admits a dual formulation given by W_2^2(ρ,μ)=sup{∫_Ωφ(x)dρ(x) +∫_Ωψ(y) dμ(y) : φ, ψ∈𝒞(Ω)s.t. φ(x) +ψ(y) ⩽ |x-y|^2 }. Optimal solutions to the dual problem are called Kantorovich potentials between ρ and μ. If ρ∈(Ω), a well-known result proved by Brenier, <cit.>, states that the optimal transport plan, γ, is unique and is induced by an optimal transport map, T, i.eγ is of the form(Id × T)_#ρ, where T_#ρ =μ and T is the gradient of a convex function. Moreover, the optimal transport map is given by T=Id -∇φ where (φ , ψ) is a pair of Kantorovich potentials between ρ and μ. It is well known that (Ω) endowed with the Wasserstein distance defines a metric space and since Ω is compact, W_2 metrizes the narrow convergence of probability measures. §.§ Assumptions and main results For i ∈{1,2}, we define _i :(Ω) → the potential energy associated to V_i ∈ W^1,∞(Ω) as_i(ρ) := ∫_Ω V_i(x) dρ(x). We introduce the Entropydefined, for all probabilty measures ρ, as (ρ) := {[ ∫_Ω H(ρ(x) ) dx if ρ≪_|Ω,;+∞otherwise, ].,H(z):=zlog(z)for all z∈ℝ^+. Finally, for m ∈ [1,+∞), we define _m:^+(Ω) →∪{ +∞} as _m(ρ) := {[ ∫_Ω F_m(ρ(x) ) dx if ρ≪_|Ω,;+∞otherwise, ]., F_m(z):= {[ z log zifm=1,; z^m/m-1 ifm >1. ]. for all z∈ℝ^+, and, for m=+∞, _∞:^+(Ω) →∪{ +∞} is defined by _∞(ρ) := {[ 0 if ρ_∞⩽ 1 ,;+∞otherwise. ].[Weak solution] * We say that (ρ_1,ρ_2) : [0, +∞) →(Ω)^2 is a weak solution to (<ref>) if for all i ∈{1,2} and for all T<+∞, ρ_i ∈𝒞^0,1/2([0,T],(Ω)) ∩ L^2-1/m((0,T),W^1,2-1/m(Ω))∩ L^2m-1((0,T) ×Ω), ρ_i ∇ F_m'(ρ_1 + ρ_2) ∈ L^2-1/m((0,T)×Ω) and for all ϕ∈𝒞_c^∞([0,+∞) ×), ∫_0^+∞∫_Ω[ ρ_i ∂_t ϕ - (ρ_i ∇ V_i +ρ_i ∇ F_m'(ρ_1 + ρ_2)+ ∇ρ_i)·∇ϕ] dx dt = -∫_Ωϕ(0,x) ρ_i,0(x) dx. * We say that (ρ_1,ρ_2,p) : [0, +∞) →(Ω)^2 × H^1(Ω) is a weak solution to (<ref>) if for all i ∈{1,2} and for all T<+∞, ρ_i ∈𝒞^0,1/2([0,T],(Ω)) ∩ L^2((0,T),H^1(Ω)), p ∈ L^2((0,T),H^1(Ω)) with p ⩾ 0, ρ_1 + ρ_2 ⩽ 1 and p(1-ρ_1-ρ_2)=0 a.e. in [0,T] ×Ω. In addition, for all ϕ∈𝒞_c^∞([0,+∞) ×), ∫_0^+∞∫_Ω[ ρ_i ∂_t ϕ - (ρ_i ∇ V_i +ρ_i ∇ p + ∇ρ_i)·∇ϕ] dx dt = -∫_Ωϕ(0,x) ρ_i,0(x) dx.The main results of this paper are Assume that ρ_1,0,ρ_2,0∈(Ω) satisfy(ρ_1,0)+(ρ_2,0)+_m(ρ_1,0+ρ_2,0) <+∞,then (<ref>) admits at least one weak solution.andAssume that Ω satisfies (<ref>). If (ρ_1,0,ρ_2,0)∈ satisfies(ρ_1,0)+(ρ_2,0) <+∞ , then there exists at least one weak solution to (<ref>). * These models can be generalized to more than two species. Moreover, instead of assuming that individuals of different populations take the same space, we can generalize to densities evolving under the constraints on α_1 ρ_1 + α_2 ρ_2, for α_1, α_2 >0. Then system (<ref>) becomes ∂_t ρ_i= (ρ_i ∇ V_i) +Δρ_i +α_i(ρ_i∇ F'_m(α_1ρ_1+α_2ρ_2)), i=1, 2.and system with hard congestion becomes {[ ∂_t ρ_1 -Δρ_1- (ρ_1 (∇ V_1+∇ p))=0,; ∂_t ρ_2 -Δρ_2- (ρ_2 (∇ V_2+∇ p))=0,; p⩾ 0, α_1ρ_1+α_2 ρ_2 ⩽ 1, p(1-α_1 ρ_1-α_2 ρ_2)=0. ]. * These results can be generalized to more general velocities. Indeed, using the semi-implicit scheme introduced by DiFrancesco and Fagioli in <cit.> and developped in <cit.> or the splitting method introduced in <cit.>, we can treat vector fields depending on the densities and which come not necessarily from a potential. These extensions allow to treat nonlocal interactions between different species, of the form V_i[ρ_1,ρ_2]=K_i,1∗ρ_1 + K_i,2∗ρ_2 where K_i,j∈ W^1,∞, which are subject to a common congestion effect . * To simplify the exposition, during the whole paper, we deal with linear self-diffusion terms but it is possible to extend Theorems <ref> and <ref> to nonlinear self-diffusions. In particular, we can deal with porous medium diffusion of the form Δρ_i^q_i. This can be done replacing the Entropy (ρ_i) by the functional _q_i(ρ_i). In the analysis, the individual estimates found in Proposition <ref> and in Proposition <ref> become L^2((0,T),H^1(Ω)) estimates onρ_1,h^q_1 /2 and ρ_2,h^q_2 /2 (see for example <cit.>) without modifying the joint estimate. In addition, discret solutions are not globally supported anymore, i.e. Lemma <ref> and Lemma <ref> do not hold, but Proposition <ref> and Proposition <ref> can be recovered, see for example <cit.> for m <+∞ and <cit.> in the case m=∞.§ COUPLING THROUGH COMMON SOFT CONGESTIONIn this section, we prove Theorem <ref> using the implicit JKO scheme, firstly introduced by Jordan, Kinderlherer and Otto in <cit.>. Given a time step h>0, we construct by induction two sequences ρ_1,h^k and ρ_2,h^k with the following scheme: ρ_i,h^0=ρ_i,0 and for all k ⩾ 0,(ρ_1,h^k+1,ρ_2,h^k+1) ∈_(ρ_1,ρ_2) ∈(Ω)^2{∑_i=1^2 ( W_2^2(ρ_i,ρ_i,h^k)+2h((ρ_i) +_i(ρ_i) ) ) +2h _m( ρ_1+ ρ_2)}.These sequences are well-defined by standard compactness and l.s.c argument. Then we define the piecewise constant interpolations ρ_i,h:^+ →(Ω) byρ_i,h(t):= ρ_i,h^k+1, ift ∈ (kh,(k+1)h].In the first part of this section, we study the convergence of these sequences and then we give the proof of Theorem <ref>.§.§ Estimates and convergencesWe start retrieving classical estimates coming from the JKO scheme, <cit.>, and then, we develop stronger estimates using the flow interchange argument, <cit.>. First, the minimization scheme givesFor all T <+∞ and for all i ∈{ 1 ,2 }, there exists a constant C <+∞ such that for all k ∈ℕ and for all h with kh⩽ T and let N=⌊T/h⌋, we have(ρ_i,h^k) ⩽ C, _m(ρ_1,h^k+ρ_2,h^k) ⩽ C,∑_k=0^N-1 W_2^2(ρ_i,h^k,ρ_i,h^k+1) ⩽ Ch. These results are obtained easily taking ρ_i=ρ_i,h^k as competitors in (<ref>), see <cit.>. Notice that estimate (<ref>) does not depend on m. This Remark will be useful in section <ref> to show that a solution to (<ref>) converges to a solution to (<ref>). In the next proposition, stronger estimates are obtained in order to pass to the limit in the nonlinear diffusive term. The main argument to prove this proposition is the flow interchange argument, introduced in <cit.>. First we recall the definition of a κ-flow. A semigroup 𝔖_Ψ:^+ ×𝒫^ac(Ω) →𝒫^ac(Ω) is a κ-flow for the functional Ψ: 𝒫^ac(Ω)→∪{ +∞} with respect toW_2 if, for all ρ∈𝒫^ac(Ω), the curve s ↦𝔖 _Ψ^s[ρ] is absolutely continuous on ^+ and satisfies the evolution variational inequality (EVI) 1/2d^+/dσ|_σ=s W_2^2(𝔖 _Ψ^s[ρ], ρ̃) +κ/2W_2^2(𝔖 _Ψ^s[ρ], ρ̃) ⩽Ψ(ρ̃)-Ψ(𝔖 _Ψ^s[ρ]),for all s>0 and for all ρ̃∈𝒫^ac(Ω) such that Ψ(ρ̃) < +∞, where d^+/dtf(t) := lim sup_s→ 0^+f(t+s) -f(t)/s.In <cit.>, the authors showed that the fact a functional admits a κ-flow is equivalent to κ-displacement convexity.For all T>0, there exists a constant C_T>0 such that,ρ_1,h^1/2_L^2((0,T),H^1(Ω))^2 +ρ_2,h^1/2_L^2((0,T),H^1(Ω))^2 +1/m (ρ_1,h+ρ_2,h)^m/2_L^2((0,T),H^1(Ω))^2 ⩽ C_T. We use the flow interchange argument, introduced in <cit.>, to find a stronger estimate as in <cit.>. In other words, we perturb ρ_1,h^k and ρ_2,h^k by the heat flow. Let η_i be the solution to{[∂_t η_i = Δη_i in(0,T) ×Ω,;∇η_i ·ν =0in(0,T) ×∂Ω,; η_i_|t=0=ρ_i,h^k. ].Since the Entropy is geodesically convex then the heat flow is a 0-flow of the Entropy , and satisfies the Evolution Variational Inequality, (<ref>), see <cit.>,1/2d^+/dσ_|σ=s W_2^2(η_i(s),ρ) ⩽(ρ) - (η_i(s)),for all s>0 and ρ∈(Ω). Taking (η_1(s),η_2(s)) as a competitor in the minimization (<ref>), we get∑_i=1^21/2d^+/d sW_2^2(η_i(s),ρ_i,h^k-1)_|s=0 +hd^+/d s( ∑_i=1^2 ((η_i(s)) +_i(η_i(s)) ) +_m(η_1(s)+ η_2(s)))_|s=0⩾ 0. Since η_i(s) is a smooth positive function for s>0, the following computations are justified∂_s (∑_i=1^2( (η_i(s)) . +_i(η_i(s)) ) + _m(η_1(s)+ η_2(s) ) ) =∑_i=1^2 (∫_ΩΔη_i(s) ((1+log(η_i(s))) +V_i ) + ∫_ΩΔ (η_1(s)+η_2(s)) F_m'(η_1(s)+η_2(s))=- ∑_i=1^2 ( ∫_Ω|∇η_i(s)|^2/η_i(s) + ∫_Ω∇ V_i ·∇η_i(s) )- ∫_Ω |∇ (η_1(s)+η_2(s))|^2F_m”(η_1(s)+η_2(s)).In addition, Young's inequality gives-∫_Ω∇ V_i(s) ·∇η_i⩽ ∫_Ω |∇ V_i||∇η_i(s)|⩽1/2∫_Ω |∇ V_i|^2η_i(s) +1/2∫_Ω|∇η_i(s)|^2/η_i(s)Then, we have ∂_s (∑_i=1^2 ((η_i(s)) +_i(η_i(s)) ) + _m(η_1(s)+ η_2(s)) )⩽∑_i=1^2 ( - 1/2∫_Ω|∇η_i(s)|^2/η_i(s) +1/2∫_Ω |∇ V_i|^2 η_i(s))- ∫_Ω |∇ (η_1(s)+η_2(s))|^2F_m”(η_1(s)+η_2(s)). By definition of F_m, for m⩾ 1, F_m”(z) =m z^m-2 for all z⩾0 and, since V_i ∈ W^1,∞(Ω),∂_s (∑_i=1^2 ((η_i(s))+_i(η_i(s)) ) + _m(η_1(s)+ η_2(s)) )⩽ C- 1/2∑_i=1^2∫_Ω |∇η_i(s)^1/2|^2 - 4/m∫_Ω |∇ (η_1(s)+η_2(s))^m/2|^2.By a lower semi-continuity argument,1/2∑_i=1^2∫_Ω |∇ (ρ_i,h^k)^1/2|^2 + 4/m∫_Ω |∇ (ρ_1,h^k+ρ_2,h^k)^m/2|^2⩽ C -d^+/d s( ∑_i=1^2 ((η_i(s)) +_i(η_i(s)) ) +_m(η_1(s)+ η_2(s)))_|s=0 .Combining with (<ref>) and (<ref>), we obtainh ∑_i=1^2∫_Ω |∇ (ρ_i,h^k)^1/2|^2 + 4h/m∫_Ω |∇ (ρ_1,h^k+ρ_2,h^k)^m/2|^2 ⩽∑_i=1^2 ( (ρ_i,h^k-1) -(ρ_i,h^k) ) +Ch.Then summing over k, we obtainρ_1,h^1/2_L^2((0,T),H^1(Ω))^2 +ρ_2,h^1/2_L^2((0,T),H^1(Ω))^2 +1/m (ρ_1,h+ρ_2,h)^m/2_L^2((0,T),H^1(Ω))^2 ⩽ C_T,where we use the fact that ρ_i,h^1/2_L^2((0,T) ×Ω)^2 =T and 1/m (ρ_1,h+ρ_2,h)^m/2_L^2((0,T)×Ω)^2 ⩽ CT by (<ref>).The bound on ρ_i,h^1/2_L^2((0,T),H^1(Ω)) does not depend on m. However, if we multiply the Entropyby a small parameter >0 in the JKO scheme (<ref>), individual bounds blow up asgoes to 0. Now we can deduce the following convergences.For all T<+∞, there exist ρ_1 and ρ_2 in 𝒞^0,1/2([0,T],(Ω)) such that, up to a subsequence, * ρ_i,h converges to ρ_i in L^∞([0,T],(Ω)), * ρ_i,h converges strongly to ρ_i in L^1((0,T) ×Ω),* (ρ_1,h+ρ_2,h)^m/2 converges strongly to (ρ_1 +ρ_2)^m/2 and ∇ (ρ_1,h+ρ_2,h)^m/2 converges weakly to ∇ (ρ_1 +ρ_2)^m/2 in L^2((0,T) ×Ω).* The first convergence is classical. We use the refined version of Ascoli-Arzelà's Theorem, <cit.>, and we immediately deduce that there exists a subsequence such that, for i=1,2, ρ_i,h converges to ρ_i ∈𝒞^1/2([0,T],(Ω)) in L^∞([0,T],(Ω)). The next two strong convergence results are obtained applying an extension of the Aubin-Lions Lemma proved by Rossi and Savaré in <cit.>. In the sequel, we work with the convergent subsequence obtained in the first step.* Let : L^1(Ω) → (-∞, +∞] and g : L^1(Ω)× L^1(Ω)→ [0, +∞] defined by(ρ):={[ρ^1/2_H^1(Ω) if ρ∈(Ω)and ρ^1/2∈ H^1(Ω);+∞otherwise, ].and g(ρ,μ):={[ W_2(ρ,μ) if ρ,μ∈(Ω); +∞ otherwise, ].is l.s.c and its sublevels are relatively compact in L^1(Ω) (see <cit.>) and g is a pseudo-distance. According to (<ref>) and (<ref>), we have sup_h⩽ 1∫_0^T (ρ_i,h(t) ) dt <+∞,and lim_τ↘ 0sup_h⩽ 1∫_0^T-τ g(ρ_i,h(t+τ), ρ_i,h(t)) dt =0,then applying Rossi-Savaré's Theorem, there exists a subsequence, not-relabeled, such that for i=1,2, ρ_i,h converges in measure with respect to t in L^1(Ω) to ρ_i. Moreover by Lebesgue's dominated convergence Theorem, ρ_i,h converges to ρ_i strongly in L^1((0,T)×Ω) .* With the same argument, we get a strong convergence on a nonlinear quantity of ρ_1,h+ρ_2,h. Let define by(ρ):={[ρ^m/2_H^1(Ω) if ρ∈(Ω)and ρ^m/2∈ H^1(Ω);+∞otherwise, ].and g defined as before. We want to apply Theorem 2 of <cit.> in L^m(Ω) over the sequence ρ_1,h+ρ_2,h/2. By (<ref>), we obtainsup_h⩽ 1∫_0^T (ρ_1,h(t)+ρ_2,h(t)/2) dt <+∞.Since, it is well-known that for all ρ_1,ρ_2,μ_1,μ_2 ∈(Ω),W_2^2(ρ_1+ρ_2/2,μ_1+μ_2/2) ⩽1/2 W_2^2(ρ_1,μ_1) + 1/2 W_2^2(ρ_2,μ_2), by (<ref>), we obtainlim_τ↘ 0sup_h⩽ 1∫_0^T-τ g(ρ_1,h+ρ_2,h/2(t+τ), ρ_1,h+ρ_2,h/2(t)) dt =0.Theorem 2 in <cit.> and Lebesgue's dominated convergence Theorem imply that ρ_1,h+ρ_2,h converges strongly to ρ_1+ρ_2 in L^m((0,T)×Ω). In addition, Krasnoselskii's Theorem, <cit.>, implies that (ρ_1,h+ρ_2,h)^m/2 converges to (ρ_1+ρ_2)^m/2 in L^2((0,T)×Ω). To conclude, ∇ (ρ_1,h+ρ_2,h)^m/2 is bounded in L^2((0,T)×Ω), thanks to (<ref>), then ∇ (ρ_1,h+ρ_2,h)^m/2 weakly converges to ∇ (ρ_1+ρ_2)^m/2 in L^2((0,T)×Ω).It is possible to obtain a strong convergence result in L^1((0,T)×Ω) for the pressure F'_m(ρ_1,h +ρ_2,h). Indeed, since ρ_1,h +ρ_2,h strongly converges in L^m((0,T)×Ω), then up to a subsequence, F_m'(ρ_1,h +ρ_2,h) → F_m'(ρ_1 +ρ_2) a.e. In addition using De La Vallée Poussin's Theorem, we show that (F'_m(ρ_1,h +ρ_2,h))_h is uniformly integrable. We conclude applying Vitali's convergence Theorem. Notice that we can drop one individual diffusion. Assume that we drop the individual Entropy in the JKO scheme (<ref>) for one of the two densities, for instance ρ_2. The difficulty is to obtain a strong convergence for the sequence (ρ_2,h)_h. Proposition <ref> gives the strong convergence of ρ_1,h and ρ_1,h +ρ_2,h in L^1((0,T) ×Ω) and L^m((0,T)×Ω) respectively, and then pointwise on (0,T) ×Ω. Consequently, ρ_2,h= (ρ_1,h+ρ_2,h) - ρ_1,h converges pointwise on (0,T) ×Ω. Moreover,∫_0^T ∫_Ωρ_2,h(t,x)^mdxdt ⩽∫_0^T ∫_Ω (ρ_1,h(t,x)+ρ_2,h(t,x))^mdxdt ⩽ C_T.Then Vitali's convergence Theorem implies that ρ_2,h strongly converges to ρ_2 in L^1((0,T) ×Ω).§.§ Existence of weak solutions to (<ref>) In this section, we start by giving the optimality conditions for (<ref>). Instead of using horizontal perturbations, ρ_i, = Φ__#ρ_i,h^k+1, as introduced in <cit.> by Jordan, Kinderlherer and Otto, we will perturb ρ_i,h^k+1 with vertical perturbations introduced in <cit.>, and revisited in <cit.>, which consist in taking ρ_i,= (1-) ρ_i,h^k+1 + _i, for any _i ∈ L^∞(Ω). Before giving the optimality conditions for (<ref>), we state the following Lemma.For all k ⩾ 1, ρ_i,h^k>0 a.e. and log(ρ_i,h^k)∈ L^1(Ω).The proof is the same as <cit.>.This Lemma ensures the uniqueness (up to a constant) of the Kantorovich potential in the transport from ρ_i,h^k+1 to ρ_i,h^k and then, we can easily compute the first variation of W_2(·,ρ_i,h^k) according to <cit.>. For i ∈{1,2}, ρ_i,h^k+1 satisfies∇ V_i +∇log(ρ_i,h^k+1) + ∇ F_m'(ρ_1,h^k+1+ρ_2,h^k+1) + ∇φ_i,h^k+1/h = 0 ρ_i,h^k+1-a.e,where φ_i,h^k+1 is the (unique) Kantorovich potential from ρ_i,h^k+1 to ρ_i,h^k. The proof is a straightforward adaptation of classical result, see for instance <cit.>. A classical consequence of the previous Proposition is that ρ_1,h and ρ_2,h are solutions to a discrete approximation of system (<ref>).Let h>0, for all T>0, let N such that N=⌊T/h⌋. Then for all (ϕ_1, ϕ_2) ∈𝒞^∞_c ([0,T)×)^2 and for all i ∈{1,2},∫_0^T∫_Ωρ_i,h(t,x)∂_t ϕ_i(t,x)dxdt + ∫_Ωρ_i,0(x) ϕ_i(0,x) dx= h∑_k=0^N-1∫_Ω∇ V_i(x) ·∇ϕ_i (t_k,x) ρ_i,h^k+1(x) dx +h∑_k=0^N-1∫_Ω∇ρ_i,h^k+1(x) ·∇ϕ_i (t_k,x) dx+h∑_k=0^N-1∫_Ω∇ F_m'(ρ_1,h^k+1 + ρ_2,h^k+1) ·∇ϕ_i(t_k,x) ρ_i,h^k+1(x) dx+∑_k=0^N-1∫_Ω×Ωℛ[ϕ_i(t_k,·)](x,y) dγ_i,h^k (x,y)where t_k=hk (t_N :=T) and γ_i,h^k is the optimal transport plan in W_2(ρ_i,h^k,ρ_i,h^k+1). Moreover, ℛ is defined such that, for all ϕ∈𝒞^∞_c([0,T) ×),|ℛ[ϕ](x,y)| ⩽1/2D^2 ϕ_L^∞ ([0,T) ×) |x- y|^2.We multiply by ρ_i,h^k+1 and take the L^2-inner product between the l.h.s. of (<ref>) and ∇ϕ_i(t_k, ·), for any ϕ_i ∈𝒞^∞_c ([0,T)×) and the proof is the same as in <cit.>, for example. Another consequence of (<ref>) is an improvment of the regularity of ρ_i,h.For all T>0 and i=1,2, we have * (ρ_1,h +ρ_2,h)^1/2∇ F_m'(ρ_1,h +ρ_2,h) is bounded in L^2((0,T)×Ω), * ρ_i,h, ρ_1,h + ρ_2,h are boundedin L^2m-1(((0,T)×Ω), * ∇ F_m'(ρ_1,h +ρ_2,h) ρ_i,h is bounded in L^2-1/m((0,T) ×Ω) and ρ_i,h is bounded in L^2-1/m((0,T), W^1,2-1/m(Ω)). The first item is a direct consequence of (<ref>), using Proposition <ref>, see for example <cit.>, and by Poincaré-Wirtinger inequality, we prove the second item. Now we will prove the third item. The first part is straightforward applying Hölder's inequality,∇ F_m'(ρ_1,h +ρ_2,h) ρ_i,h_L^2-1/m⩽∇ F_m'(ρ_1,h +ρ_2,h) ρ_i,h^1/2_L^2^1-1/2mρ_i,h_L^2m-1^1/2m < +∞. According to (<ref>), we obtain a.e.|∇ρ_i,h^k+1 |^2-1/m⩽ C(| ∇φ_i,h^k+1ρ_i,h^k+1/h|^2-1/m + (|∇ V_i|ρ_i,h^k+1)^2-1/m + (|∇ F_m'(ρ_1,h^k+1+ρ_2,h^k+1)|ρ_i,h^k+1)^2-1/m).We have already seen that ∇ F_m'(ρ_1,h +ρ_2,h) ρ_i,h is bounded in L^2-1/m((0,T) ×Ω) and since ρ_i,h∈ L^1 ∩ L^2m-1 ((0,T) ×Ω), ∇ V_iρ_i,h_L^2-1/m⩽ C. To deal with the last term, notice that by Hölder's inequality,∫_Ω| ∇φ_i,h^k+1ρ_i,h^k+1/h|^2-1/m⩽1/h^2-1/m W_2(ρ_i,h^k, ρ_i,h^k+1)^2-1/mρ_i,h^k+1_L^2m-1^(2m-1)/2m,and then,h∑_k=0^N-1∫_Ω|∇φ_i,h^k+1ρ_i,h^k+1/h|^2-1/m ⩽C h^1/m -1 N^1/2m( ∑_k=0^N-1 W_2^2(ρ_i,h^k, ρ_i,h^k+1) )^(2m-1)/2m⩽C T^1/2m( ∑_k=0^N-1 W_2^2(ρ_i,h^k, ρ_i,h^k+1)/h)^(2m-1)/2m⩽C_T,by (<ref>) where T=Nh. Then ∇ρ_i,h is bounded in L^2-1/m((0,T) ×Ω) and we conclude the proof with Poincaré-Wirtinger inequality.Now we are able to prove Theorem <ref>.We have to pass to the limit in all terms in Proposition <ref> as h ↘ 0. The remainder term converges to 0 using the total square distance estimate (<ref>) and the linear term converges to ∫_0^T ∫_Ωρ_i ∂_t ϕ_i - ∫_0^T ∫_Ω∇ V_i ·∇ϕ_i ρ_i,when h goes to 0 thanks to Proposition <ref>. Furthermore, since ∇ρ_i,h is bounded in L^2-1/m((0,T)×Ω), because of Proposition <ref> and the fact that ρ_i,h strongly converges to ρ_i in L^1( (0,T) ×Ω), we conclude that ∇ρ_i,h converges weakly to ∇ρ_i in L^2-1/m((0,T)×Ω). This implies that the individual diffusion term converges to ∫_0^T ∫_Ω∇ϕ_i ·∇ρ_idxdt . It remains to study the convergence of the nonlinear cross diffusion term. First, we remark that ∇ F_m'(ρ_1,h^k+1 +ρ_2,h^k+1) can be rewritten as∇ F_m'(ρ_1,h^k+1 +ρ_2,h^k+1)=2(ρ_1,h^k+1 +ρ_2,h^k+1)^m/2/ρ_1,h^k+1 +ρ_2,h^k+1∇ (ρ_1,h^k+1 +ρ_2,h^k+1)^m/2.Then ∇ F_m'(ρ_1,h^k+1 +ρ_2,h^k+1)ρ_i,h^k+1 = 2 G_1-m/2(ρ_1,h^k+1,ρ_2,h^k+1)∇ (ρ_1,h^k+1 +ρ_2,h^k+1)^m/2,where G_α:^+ ×^+ → is the continuous function (for α <1) defined byG_α(x,y):= {[x/(x+y)^α if x>0, y ⩾ 0,;0 otherwise. ].As m⩾ 1, 1-m/2 <1 so G_1-m/2 is continuous and since, up to a subsequence, ρ_i,h converges to ρ_i a.e., we obtain that G_1-m/2(ρ_1,h,ρ_2,h) converges to G_1-m/2(ρ_1,ρ_2) a.e. in (0,T)×Ω. In addition, | G_1-m/2(ρ_1,h,ρ_2,h) | = | (ρ_1,h+ρ_2,h)^m/2ρ_1,h/ρ_1,h+ρ_2,h| ⩽ (ρ_1,h+ρ_2,h)^m/2. Up to a subsequence, ρ_i,h and ρ_1,h+ρ_2,h converge a.e. in (0,T) ×Ω, and, since (ρ_1,h+ρ_2,h)^m/2 converges to(ρ_1+ρ_2)^m/2 in L^2((0,T)×Ω), there exists a function g ∈ L^2((0,T)×Ω) such that,|(ρ_1,h+ρ_2,h)^m/2 | ⩽ g.Then Lebesgue's dominated convergence Theorem implies that G_1-m/2(ρ_1,h,ρ_2,h) converges strongly in L^2((0,T)×Ω) to G_1-m/2(ρ_1,ρ_2). Moreover, ∇ (ρ_1,h^k+1 +ρ_2,h^k+1)^m/2 converges weakly in L^2((0,T)×Ω), by Proposition <ref>, then ∇ F_m'(ρ_1,h +ρ_2,h)ρ_i,h converges weakly in L^1((0,T)×Ω) to ∇ F_m'(ρ_1 +ρ_2)ρ_i and h∑_k=0^N-1∫_Ω∇ F_m'(ρ_1,h^k+1 + ρ_2,h^k+1) ·∇ϕ_i(t_k,x) ρ_i,h^k+1(x) dx →∫_0^T ∫_Ω∇ F_m'(ρ_1 +ρ_2) ·∇ϕ_i ρ_idxdt.In addition, by Proposition <ref>, we obtain that ∇ F_m'(ρ_1 +ρ_2) ρ_i ∈ L^2-1/m((0,T)×Ω), which concludes the proof. § COUPLING BY HARD CONGESTIONIn this section we prove the existence of a weak solution to (<ref>), i.e. Theorem <ref>. This system can be seen as gradient flow in a Wasserstein product space. Using the Jordan-Kinderlherer-Otto scheme, we construct two sequences defined in the following way: let h>0 be a time step, we construct a sequence ( ρ_1,h^k,ρ_2,h^k) with ( ρ_1,h^0,ρ_2,h^0)= ( ρ_1,0,ρ_2,0) and ( ρ_1,h^k+1,ρ_2,h^k+1) is a solution toinf_(ρ_1,ρ_2) ∈∑_i=1^2[ 1/2h W_2^2(ρ_i,ρ_i,h^k) + (ρ_i) + _i(ρ_i) ],where :={ (ρ_1,ρ_2) ∈(Ω)^2 :ρ_1 +ρ_2 ⩽ 1 } and |Ω|>2. The direct method shows that these sequences are well-defined. As before, we define the piecewise constant interpolations ρ_i,h:^+ →(Ω) byρ_i,h(t):= ρ_i,h^k+1, ift ∈ (kh,(k+1)h].§.§ Estimates and convergencesIn the following proposition, we list the classical estimates coming from the Wasserstein gradient flow theory.Let T>0. Then there exists C>0 such that for i ∈{1,2} and for all k⩾ 0 such that k⩽ N:=⌊T/h⌋,ρ_1,h^k + ρ_2,h^k ⩽ 1,(ρ_i,h^k) ⩽ C, ∑_k=0^N-1 W_2^2(ρ_i,h^k,ρ_i,h^k+1)⩽ Ch.As in the previous section, we need stronger estimates in order to handle the very degenerate cross diffusion term, (ρ_i∇ p). For all T>0, there exists a constant C_T>0 such thatρ_1,h^1/2_L^2((0,T),H^1(Ω)) +ρ_2,h^1/2_L^2((0,T),H^1(Ω))⩽ C_T. We apply the flow interchange technique as previously, Proposition <ref>. Keeping the same notations as in the previous section, we denote by η_i the heat flow with initial condition ρ_i,h^k. Since the heat flow decreases the L^∞-norm, (η_1(s),η_2(s)), defined in (<ref>), is admissible for the minimization problem (<ref>), for all s ⩾ 0. Then the same computations as in Proposition <ref> give the result.Consequently, we deduce the following convergences.For all T>0, there exist ρ_1 and ρ_2 in 𝒞^0,1/2([0,T],(Ω)) such that, up to a subsequence, * ρ_i,h converges to ρ_i in L^∞([0,T],(Ω)),* ρ_i,h converges strongly to ρ_i in L^p((0,T) ×Ω), for all p ∈ [1, +∞) and ∇ρ_i,h converges narrowly to ∇ρ_i. The total square distance estimate (<ref>) and the refined version of Ascoli-Arzelà's Theorem, <cit.>, implies that ρ_i,h converges to ρ_i ∈𝒞^1/2([0,T],(Ω)) in L^∞([0,T],(Ω)). As in Proposition <ref>, applying <cit.>, we obtain that ρ_i,h converges strongly to ρ_i in L^1((0,T) ×Ω). And noticing that ρ_i,h,ρ_i ⩽ 1 a.e., we deduce that the strong convergence holds in L^p((0,T) ×Ω), for all p ∈ [1, +∞). To conclude, we remark that ∇ρ_i,h = 2 ρ_i,h^1/2∇ρ_i,h^1/2, ρ_i,h^1/2 strongly converges to ρ_i^1/2 in L^2((0,T)×Ω) and ∇ρ_i,h^1/2 weakly converges to ∇ρ_i^1/2 in L^2((0,T)×Ω).We end this section by a lemma implying the uniqueness of the pair of Kantorovich potentials from ρ_i,h^k+1 to ρ_i,h^k and then the existence of the first variation of W_2^2(·,ρ_i,h^k) (Propositions 7.18 and 7.17 from <cit.>). Minimizers of (<ref>) satisfy ρ_i,h^k >0 a.e. and log(ρ_i,h^k) ∈ L^1(Ω). The proof is the same as in <cit.>. Indeed we can use a constant perturbationbecause (, ) is admissible in (<ref>) (+ = 2/|Ω| ⩽ 1 by (<ref>)).§.§ Pressure field associated to the constraint In this section, we introduce a discrete pressure associated to the constraint ρ_1,h^k+1 +ρ_2,h^k+1⩽ 1. This common pressure is obtained arguing as in <cit.> in the case of one population.Let (ρ_1,h^k+1,ρ_2,h^k+1) be the unique solution to (<ref>). Then for all (ρ_1,ρ_2) ∈,∫_Ωψ_1,h^k+1 (ρ_1 - ρ_1,h^k+1) + ∫_Ωψ_2,h^k+1 (ρ_2 - ρ_2,h^k+1) ⩾ 0,where ψ_i,h^k+1 = φ_i,h^k+1/h +V_i + 1 +log(ρ_i,h^k+1) and φ_i,h^k+1 is the optimal (up to a constant) Kantorovich potential in W_2(ρ_i,h^k+1,ρ_i,h^k).The proof of this result is the same as Lemma 3.1 in <cit.>.Notice that (<ref>) can be rewritten as∫_Ωψ_1,h^k f_1 + ∫_Ωψ_2,h^k f_2 ⩾ 0,for all functions f_1 ,f_2 ∈ L^∞(Ω) such that f_1 +f_2 ⩽1-ρ_1,h^k -ρ_2,h^k/,f_i ⩾-ρ_i,h^k/ and ∫_Ω f_i =0,for all 0<≪ 1. In the next proposition, we introduce the common discrete pressure.There exists p_h^k ⩾ 0 such that for all, k ⩾ 1,p_h^k(1-ρ_1,h^k - ρ_2,h^k)=0a.e.In addition, p_h^k satisfies ∇ p_h^k = - ∇φ_i,h^k/h -∇ V_i - ∇log(ρ_i,h^k)a.e,for i=1,2.Let S:={ρ_1,h^k +ρ_2,h^k =1 } be the set where the constraint is saturated. Firstly, we choose f_2=0 on Ω and f_1=0 on S in Remark <ref>. Then we have∫_S^cψ_1,h^k f_1⩾ 0,for all f_1∈ L^∞(Ω). This implies that there exists a constant C_1 such that ψ_1,h^k=C_1 a.e. on S^c. Applying the same argument with f_1 =0 on Ω and f_2 =0 on S, we find a constant C_2 such that ψ_2,h^k=C_2 a.e. on S^c. And since f_1 and f_2 satisfy (<ref>), we have∫_Ω (ψ_1,h^k -C_1 )f_1+∫_Ω (ψ_2,h^k -C_2 )f_2⩾ 0.Now, choosing f_1 =f and f_2=-f on S and by symmetry (f_1=-f and f_2=f), we find∫_S ((ψ_1,h^k -C_1 ) -(ψ_2,h^k -C_2 ) ) f =0,for all f ∈ L^∞(Ω). We conclude that (ψ_1,h^k -C_1 )=(ψ_2,h^k -C_2 ) =:ψ_h^k a.e. on S and consequently∫_S ψ_h^k(f_1+f_2) ⩾ 0.On the other hand, since f_1+f_2 ⩽ 0 on S, ψ_h^k ⩽ 0 a.e on S, then we define p_h^k byp_h^k := (C_1 - ψ_1,h^k)_+ = (C_2 - ψ_2,h^k)_+.By definition, we have p_h^k(1-ρ_1,h^k -ρ_2,h^k)=0 a.e. and since ψ_i,h^k is differentiable a.e., the proof is completed. Now, we define the piecewise interpolation p_h:^+ → L^1(Ω) byp_h(t):= p_h^k+1, ift ∈ (kh,(k+1)h].Notice that p_h(t) ⩾ 0 and for all t⩾ 0, p_h(t)(1-ρ_1,h(t)-ρ_2,h(t))=0 a.e. Therefore, we immediately deduce the following estimate on the pressure. For all T>0, p_h is bounded in L^2((0,T) , H^1( Ω)). First, we prove that ∇ p_h is bounded in L^2((0,T)×Ω) and then we will conclude using Poincaré's inequality. By definition of p_h^k+1, we have∫_Ω |∇ p_h^k+1|^2(ρ_1,h^k+1 +ρ_2,h^k+1)=∑_i=1^2 ∫_Ω |∇ψ_i,h^k+1|^2ρ_i,h^k+1⩽C ∑_i=1^2 ( ∫_Ω| ∇ϕ_i,h^k+1/h|^2ρ_i,h^k+1 + ∫_Ω |∇ V_i |^2 ρ_i,h^k+1 + ∫_Ω|∇ρ_i,h^k+1|^2/ρ_i,h^k+1)⩽ C ∑_i=1^2( 1/h^2 W_2^2(ρ_i,h^k,ρ_i,h^k+1) +C + (ρ_i,h^k+1)^1/2_H^1(Ω)),where the last line is obtained using the fact that ∇ V_i ∈ L^∞(Ω). Summing the previous inequalities over k and by (<ref>) and (<ref>), we obtain that∫_0^T ∫_Ω | ∇ p_h(t)|^2 (ρ_1,h(t) + ρ_2,h(t)) ⩽ C.Since p_h(t) =0 a.e. on {ρ_1,h(t) + ρ_2,h(t) <1}, we deduce∫_0^T ∫_Ω | ∇ p_h(t)|^2 = ∫_0^T ∫_Ω | ∇ p_h(t)|^2 (ρ_1,h(t) + ρ_2,h(t))⩽ C.We conclude with the same argument as <cit.>. Using Poincaré's inequality, since |{p_h(t) =0 } | ⩾ |{ρ_1,h(t) + ρ_2,h(t) <1 } | ⩾ |Ω| -2 >0, by (<ref>), we obtain that p_h is bounded in L^2((0,T), H^1(Ω)). Using Proposition <ref>, the regularity of ρ_i can be improved. For all T>0 and i=1,2, ρ_i,h is bounded in L^2((0,T) , H^1( Ω)).By (<ref>) combined with ρ_i,h^k+1⩽ 1, we obtain that|∇ρ_i,h^k+1 |^2 ⩽ C(|∇φ_i,h^k+1|^2/h^2ρ_i,h^k+1 +|∇ V_i|^2ρ_i,h^k+1 +|∇ p_h^k+1 |^2 )a.e.Since, by Proposition <ref>, ∇ p_h is bounded in L^2((0,T) ×Ω) and h∑_k=0^N-1∫_Ω|∇φ_i,h^k+1|^2/h^2ρ_i,h^k+1⩽ C,because of (<ref>), we have∇ρ_i,h_L^2((0,T)×Ω)⩽ C.The proof is concluded noticing thatρ_i,h_L^2((0,T) ×Ω)⩽ρ_i,h_L^∞((0,T) ×Ω)^1/2ρ_i,h_L^1((0,T) ×Ω)^1/2⩽ T^1/2.To analyse the pressure field p_h, we recall the following lemma, <cit.>,<cit.>Let (p_h)_h>0 be a bounded sequence in L^2([0,T],H^1(Ω)) and (ρ_h)_h>0 a sequence of piecewise constant curves valued in (Ω) which satisfiy W_2(ρ_h(t),ρ_h(s)) ⩽ C√(t-s-h) for all s<t ∈ [0,T] and ρ_h ⩽ C for a fixed constant C. Suppose thatp_h ⩾ 0,p_h(1-ρ_h)=0, ρ_h ⩽ 1,and that p_h ⇀ pweakly inL^2([0,T],H^1(Ω)) andρ_h →ρ uniformly in (Ω).Then p(1-ρ) =0. Consequently, one hasThere exists p ∈ L^2([0,T],H^1(Ω)) such that p_h converges weakly in L^2([0,T],H^1(Ω)) to p, where p satisfiesp ⩾ 0,p(1-ρ_1 -ρ_2)=0, ρ_1 +ρ_2 ⩽ 1 a.e. in[0,T]×Ω.In addition, ρ_i,h∇ p_h narrowly converges to ρ_i∇ p. We apply Lemma <ref> to ρ_h:=ρ_1,h + ρ_2,h and p_h. According to Proposition <ref>, p_h weakly converges in L^2((0,T), H^1(Ω)) to p such that p ⩾ 0,p(1-ρ_1-ρ_2)=0, ρ_1+ρ_2⩽ 1.Moreover, using the estimate on p_h, we know that ∇ p_h weakly converges to ∇ p in L^2((0,T)×Ω). Then since ρ_i,h strongly converges to ρ_i in L^2((0,T) ×Ω) (Proposition <ref>), by strong-weak convergence, we obtain that ρ_i,h∇ p_h narrowly converges to ρ_i∇ p. §.§ Existence of weak solutions to (<ref>)Arguing as in Proposition <ref>, (ρ_1,h,ρ_2,h) is solution to a discrete approximation of system (<ref>).Let h>0, for all T>0, let N such that N=⌊T/h⌋. Then for all (ϕ_1, ϕ_2) ∈𝒞^∞_c ([0,T)×)^2 and for all i ∈{1,2},∫_0^T ∫_Ω ρ_i,h(t,x)∂_t ϕ_i(t,x)dxdt + ∫_Ωρ_i,0(x) ϕ_i(0,x) dx= h∑_k=0^N-1∫_Ω∇ V_i(x) ·∇ϕ_i (t_k,x) ρ_i,h^k+1(x) dx+h∑_k=0^N-1∫_Ω∇ρ_i,h^k+1(x) ·∇ϕ_i (t_k,x) dx+h∑_k=0^N-1∫_Ω∇ p_h^k+1·∇ϕ_i(t_k,x) ρ_i,h^k+1(x) dx +∑_k=0^N-1∫_Ω×Ωℛ[ϕ_i(t_k,·)](x,y) dγ_i,h^k (x,y)where t_k=hk (t_N :=T) and γ_i,h^k is the optimal transport plan in W_2(ρ_i,h^k,ρ_i,h^k+1). Moreover, ℛ is defined such that, for all ϕ∈𝒞^∞_c([0,T) ×),|ℛ[ϕ](x,y)| ⩽1/2D^2 ϕ_L^∞ ([0,T) ×) |x- y|^2. Combining Propositions <ref>, <ref>, <ref> and <ref>, the rest of the proof of Theorem <ref> is identical to the proof of Theorem <ref> in the previous section.As in Remark <ref>, it is possible to drop one diffusion. Say we drop the individual Entropy for the second species, ρ_2. The difficulty is to pass to the limit in the nonlinear term ρ_2,h∇ p_h. This term can be rewritten as (ρ_1,h +ρ_2,h) ∇ p_h - ρ_1,h∇ p_h.Taking advantage of the definition of p_h, we deduce that (ρ_1,h +ρ_2,h) ∇ p_h = ∇ p_h a.e and then converges weakly to∇ p=(ρ_1 + ρ_2) ∇ p in L^2((0,T) ×Ω). Moreover, since ρ_1,h strongly converges in L^2((0,T) ×Ω) by Proposition <ref> and ∇ p_h converges weakly in L^2((0,T) ×Ω) we can pass to the limit in the second term by strong-weak convergence. Then we deduce that ρ_2,h∇ p_h weakly converges to ρ_2∇ p. § SYSTEMS WITH A COMMON DRIFT In this section, we focus on the special case where ∇ V_1= ∇ V_2=: ∇ V ∈ L^∞(Ω). Although this asumption is very restrictive, it allows us to obtain better estimates on solutions (Proposition <ref> and Proposition <ref>) which are hard to get in the general case due to the lack of convexity of _m(ρ_1 +ρ_2), see Remark <ref>. Therefore, in this case, we will be able to prove the convergence of a solution to (<ref>) to a solution to (<ref>), when m goes to +∞. Moreover, under some regularity we give a L^1-contraction result for systems (<ref>) and (<ref>).It is well-known in the Wasserstein gradient flow theory that the λ-geodesic convexity of the functional implies a W_2-contraction of the flow. Unfortunately, as mentioned in <cit.>, in general, (ρ_1,ρ_2) ∈(Ω)^2 ↦_m(ρ_1+ρ_2) is not displacement convex. Indeed, for m=2, we can rewrite the functional as_2(ρ_1+ρ_2)= _2(ρ_1) +_2(ρ_2) +2∫_Ωρ_1 ρ_2.Let ρ_2 be a fixed density, we study the displacement convexity of ρ↦_2(ρ) +2∫_Ωρ_2 ρ. We know, see <cit.>, that ρ∈(Ω) ↦_2(ρ) is displacement convex but ρ↦∫_Ωρ_2 ρ is displacment convex if ρ_2 is λ-convex. To overcome this lack of convexity, we need to obtain a stronger estimate, independent on m, on ∇F_m'(ρ_1,m +ρ_2,m), where (ρ_1,m,ρ_2,m) is a solution to (<ref>). In the case of a common drift, this estimate can be found observing that ρ_m:=ρ_1,m+ρ_2,m is the Wasserstein gradient flow of ++ _m and then, solves ∂_t μ - Δμ -(μ∇ V) - (μ∇ F_m'(μ))=0,with initial condition μ_|t=0=ρ_1,0+ρ_2,0.Let (ρ_1,m,ρ_2,m) be a solution to (<ref>) in L^2((0,T),H^1(Ω)) with ∇ V_1= ∇ V_2=: ∇ V ∈ L^∞(Ω). Then ρ_m:=ρ_1,m+ρ_2,m is unique and F_m'(ρ_m) is bounded independently of m in L^2((0,T),H^1(Ω)), for all T<+∞.As we remark above, ρ_m is solution to (<ref>).By geodesic convexity ofand _m, we know that solution to (<ref>) is unique (see <cit.>).To conclude, we reason as in <cit.>. The proof is based on the flow interchange technique with the (smooth) solution to{[∂_t η =Δη^m-1 +Δη in(0,T)×Ω,; (∇η^m-1 +∇η) ·ν =0in(0,T)×∂Ω,;η_|t=0=ρ_h,m^k,].where ρ_h,m^k is constructed using the JKO scheme. We obtain, whengoes to 0 and using a lower semi-continuity argument, ∇ F'_m(ρ_m) _L^2((0,T),H^1(Ω))⩽ C_T, for all T>0, where C_T is a constant independent on m. The L^1-estimate of F_m'(ρ_m) and the Poincaré-Wirtinger inequality conclude the proof.Now, we show that (ρ_1,m,ρ_2,m) converges to a solution to (<ref>), (ρ_1,∞,ρ_2,∞), as m↗+∞.Assume that the initial data satisfy ρ_1,0 + ρ_2,0⩽ 1. Up to a subsequence, as m→ +∞, a solution to (<ref>), (ρ_1,m,ρ_2,m), converges strongly in L^2((0,T)×Ω) to (ρ_1,∞,ρ_2,∞) and p_m:=F_m'(ρ_1,m+ρ_2,m) converges weakly in L^2((0,T),H^1(Ω)) to p_∞, where (ρ_1,∞,ρ_2,∞,p_∞) is a solution to (<ref>).First we prove the convergence of ρ_i,m. We start noticing that the estimate (<ref>) does not depend on m and then by Remark <ref>, we haveρ_i,m^1/2_L^2((0,T),H^1(Ω))⩽ C_TandW_2(ρ_i,m(t),ρ_i,m(s))⩽ C_T|t-s|^1/2,for all t,s ⩽ T and where C_T is a contant independent on m. Then using the Rossi-Savaré Theorem we obtain that ρ_i,m converges to ρ_i,∞ in L^1((0,T)×Ω). In fact, ρ_i,m converges strongly to ρ_i,∞ in L^2((0,T)×Ω). Indeed, for m≫ 2, ρ_i,m_L^m((0,T) ×Ω) is uniformly bounded in m so (ρ_i,m^2)_m is uniformly integrable. Then, ρ_i,m converges weakly in L^2((0,T) ×Ω)to ρ_i,∞ and Vitali's convergence Theorem implies that ρ_i,m_L^2((0,T) ×Ω) =ρ_i,m^2 ^1/2_L^1((0,T) ×Ω)→ρ_i,∞^2 ^1/2_L^1((0,T) ×Ω)=ρ_i,∞_L^2((0,T) ×Ω). Furthermore,p_m converges weakly in L^2((0,T),H^1(Ω)) to p_∞, Proposition <ref>, and obviously p_∞⩾ 0. Consequently, we can pass to the limit in the weak formulation of the system (<ref>) to obtain the weak formulation of sytem (<ref>).To conclude the proof, it remains to prove that ρ_1,∞+ρ_2,∞⩽ 1and p_∞(1-ρ_1,∞ - ρ_2,∞) = 0a.e. We start to show that ρ_1,∞+ρ_2,∞⩽ 1. The argument is the same as in <cit.>. The estimate (<ref>) does not depend on m so we have ∫_0^T ∫_Ω (ρ_1,m+ρ_2,m-1)_+^2dxdt ⩽2C/m→ 0, when m→ +∞, which implies that ρ_1,∞+ρ_2,∞⩽ 1 a.e.To obtain the second part of the claim, we start proving ∫_0^T ∫_Ω p_m(1-ρ_1,m-ρ_2,m) φ dxdt →∫_0^T ∫_Ω p_∞ (1-ρ_1,∞-ρ_2,∞)φ dxdt,for all φ∈𝒞_c^∞((0,T)×Ω). With the same argument as before, ρ_1,m+ρ_2,m→ρ_1,∞+ρ_2,∞ strongly in L^2((0,T)×Ω) and p_m ⇀ p_∞ weakly in L^2((0,T)×Ω), then by strong-weak convergence, we obtain the result.Now, we show that∫_0^T ∫_Ω p_m(1-ρ_1,m-ρ_2,m)φ dxdt → 0,for all nonnegative φ∈𝒞_c^∞((0,T)×Ω). We start splitting the integral,∫_0^T ∫_Ω p_m(1-ρ_1,m-ρ_2,m)φ dxdt= ∬_{ρ_1,m+ρ_2,m⩽ 1 }p_m(1-ρ_1,m-ρ_2,m)φ dxdt + ∬_{ρ_1,m+ρ_2,m⩾ 1 }p_m(1-ρ_1,m-ρ_2,m)φ dxdt.Remark that, since ρ_1,m+ρ_2,m→ρ_1,∞+ρ_2,∞ strongly in L^1((0,T)×Ω), up to a subsequence, ρ_1,m(t,x)+ρ_2,m(t,x) →ρ_1,∞(t,x)+ρ_2,∞(t,x) (t,x)-a.e. Let (t,x) ∈ [0,T] ×Ω be a point where the convergence a.e. holds. If ρ_1,∞(t,x)+ρ_2,∞(t,x)<1, then ρ_1,m(t,x)+ρ_2,m(t,x) ⩽ (1-), for large m and p_m(t,x) ⩽m/m-1(1-)^m-1→ 0, therefore p_m(t,x)(1-ρ_1,m(t,x)-ρ_2,m(t,x))→ 0. On the other hand, if ρ_1,∞(t,x)+ρ_2,∞(t,x)=1 and, for large m, ρ_1,m(t,x)+ρ_2,m(t,x)⩽ 1, then 1-ρ_1,m(t,x)-ρ_2,m(t,x)→ 0 and p_m(t,x) ⩽m/m-1 remains bounded. Thus,p_m(t,x)(1-ρ_1,m(t,x)-ρ_2,m(t,x))→ 0 a.e. and since on {ρ_1,m+ρ_2,m⩽ 1 }, ρ_1,m+ρ_2,m is bounded by 1 and p_m ⩽m/m-1⩽ 2, by Lebesgue convergence Theorem, we obtain ∬_{ρ_1,m+ρ_2,m⩽ 1 }p_m(1-ρ_1,m-ρ_2,m)φ dxdt → 0. The convergence of the second term is obtained by applying Cauchy-Schwarz inequality, (<ref>) and Proposition <ref>,| ∬_{ρ_1,m+ρ_2,m⩾ 1 }p_m(1-ρ_1,m-ρ_2,m)φ dxdt | ⩽ p_m _L^2((0,T) ×Ω)C/m^1/2→ 0,when m ↗ +∞. Then, for all φ∈𝒞_c^∞((0,T)×Ω),∫_0^T ∫_Ω p_∞ (1-ρ_1,∞-ρ_2,∞)φ dxdt=0.Since p_∞ (1-ρ_1,∞-ρ_2,∞) ⩾ 0, we conclude that p_∞ (1-ρ_1,∞-ρ_2,∞)=0 a.e. in (0,T) ×Ω.To end this section, we give a L^1-contraction result for m ∈ [1,+∞] under some regularity on solutions but first we establish maximum principle for m ∈ [1, +∞).Assume that ρ_i,0 + ρ_2,0⩽ M_0. For all m∈ [1, +∞) and T <+∞, there exists a constant M_T>0 such that ρ_1,m + ρ_2,m_L^∞((0,T)×Ω)⩽ M_T. In addition, we have ∇ρ_i,m , ∇ F_m'(ρ_1,m + ρ_2,m) ∈ L^2((0,T) ×Ω).It is well known that the solution μ to (<ref>) satisfies a maximum principle, see for instance <cit.>. Then by uniqueness of the solution, there exists M_T such that ρ_1,m + ρ_2,m_L^∞((0,T)×Ω)⩽ M_T. We obtain then | ∇ρ_i,m | ⩽ 2M_T^1/2 | ∇ρ_i,m^1/2 |and|ρ_i,m∇ F_m'(ρ_1,m + ρ_2,m) | ⩽ M_T|∇ F_m'(ρ_1,m + ρ_2,m) |. Since, ∇ρ_i,m^1/2 and ∇ F_m'(ρ_1,m + ρ_2,m) are in L^2((0,T) ×Ω) (Proposition <ref>), the proof is concluded. In the sepcial case of a common drift, by Proposition <ref>, we can improve the regularity of solutions to (<ref>) inDefinition <ref> if we start with L^∞ initial conditions. Then, as in <cit.>, we notice that, by density, we can consider test functions in W^1,1((0,T),L^1(Ω)) ∩ L^2((0,T),H^1(Ω)) in Definition <ref> for system (<ref>) and system (<ref>). Let (ρ_1,m^1,ρ_2,m^1) and (ρ_1,m^2,ρ_2,m^2) be two solutions to (<ref>) (or (<ref>) if m=+∞) with intial conditions (ρ_1,0^1 ,ρ_2,0^1) and (ρ_1,0^2 , ρ_2,0^2), respectively. Assume there exists M_0 >0 such thatρ_1,0^1 + ρ_2,0^1_L^∞( Ω),ρ_1,0^2 + ρ_2,0^2_L^∞( Ω)⩽ M_0.If ∂_t ρ_i,m^1,∂_tρ_i,m^2 ∈ L^1((0,T)×Ω), then ρ_i,m^1(t,·) - ρ_i,m^2(t,·) _L^1(Ω)⩽ρ_i,0^1 - ρ_i,0^2 _L^1(Ω).First if m<+∞, since ρ_1,m+ρ_2,m solves (<ref>), then it is unique and according to Proposition <ref>, p_m:=F_m'(ρ_1,m+ρ_2,m) is in L^2((0,T),H^1(Ω)). Moreover, we have already shown in Theorem <ref> that the pressure p_∞ associated to the constraint ρ_1,∞+ρ_2,∞⩽ 1is in L^2((0,T), H^1(Ω)) and, according to <cit.>, (ρ_1,∞+ρ_2,∞,p_∞) is unique. Then, for m ∈ [1,+∞], ρ_1,m^i solves ∂_t ρ_1,m^i -Δρ_1,m^i -(ρ_1,m^i (∇ V + ∇ p_m)) =0. Now, by the same argument as <cit.>, we prove the L^1-contraction. We prove the result for i=1 and the argument is the same for i=2. We note Ω_T:=(0,T) ×Ω. Define the smooth function, for z ∈, f(z) = e^-1/ze^-1/(1-z) if z ∈ (0,1) and 0 otherwise and M:=f_L^∞. Then for δ>0, define the smooth function ϕ_δ byϕ_δ(z):= 1/Z∫_0^z/δ f(ξ) dξ,whereZ := ∫_0^1 f(ξ) dξ.Consider ζ_δ := ϕ_δ (ρ_1,m^1-ρ_1,m^2).By definition, ζ_δ∈ W^1,1((0,T), L^1(Ω))∩ L^2((0,T), H^1(Ω)) ∩ L^∞ (Ω_T). Then taking ζ_δ as an admissible test function in Definition <ref>, see Remark <ref>, we obtain ∬_Ω_T∂_t(ρ_1,m^1-ρ_1,m^2)ζ_δ=-∬_Ω_T((ρ_1,m^1-ρ_1,m^2)(∇ V+∇ p_m) ·∇ζ_δ+∇ ( ρ_1,m^1-ρ_1,m^2) ·∇ζ_δ)dxdt.We introduce Ω_T^δ:= Ω_T ∩{0 < ρ_1,m^1-ρ_1,m^2 < δ}. Then by definition of ζ_δ∬_Ω_T ∂_t(ρ_1,m^1-ρ_1,m^2)ζ_δ= -1/Zδ∬_Ω_T^δ (ρ_1,m^1-ρ_1,m^2)(∇ V +∇ p_m) ·∇( ρ_1,m^1-ρ_1,m^2)f(ρ_1,m^1 - ρ_1,m^2/δ) dxdt- 1/Zδ∬_Ω_T^δ |∇ ( ρ_1,m^1-ρ_1,m^2)|^2 f(ρ_1,m^1 - ρ_1,m^2/δ) dxdt. Young's inequality gives∬_Ω_T ∂_t(ρ_1,m^1-ρ_1,m^2)ζ_δ⩽M/2Zδ∬_Ω_T^δ(ρ_1,m^1-ρ_1,m^2)^2|∇ V +∇ p_m|^2 dxdt-1/2Zδ∬_Ω_T^δ|∇ ( ρ_1,m^1-ρ_1,m^2)|^2f(ρ_1,m^1 - ρ_1,m^2/δ) dxdt⩽M/2Z∇ V +∇ p_m_L^2(Ω_T)^2 δ. Then, when δ↘ 0, by Fatou's Lemma, ∬_Ω_T ∩{ρ_1,m^1-ρ_1,m^2 ⩾ 0}∂_t(ρ_1,m^1-ρ_1,m^2)⩽ 0. Reversing the roles of ρ_1,m^1 and ρ_1,m^2, we have∬_Ω_T∂_t(|ρ_1,m^1-ρ_1,m^2|) ⩽ 0,which concludes the proof.§ NUMERICAL SIMULATIONS To end this paper, we use the algorithm introduced in <cit.> to present numerical simulations in dimension 2 on the square Ω=[-1/2, 1/2]^2. Simulations are carried out using a 50 × 50 discretization in space with a time step h=0.01. The first system we study is the transport equation with common porous media congestion, without individual diffusions,∂_t ρ_i -α_i(ρ_i ∇ F_m'(α_1ρ_1+α_2ρ_2)) -(ρ_i ∇ V_i)=0, i =1,2,which, at least formally, is the gradient flow in Wasserstein space for the energy(ρ_1,ρ_2):= ∫_Ω V_1ρ_1 + ∫_Ω V_2ρ_2 + ∫_Ω F_m(α_1ρ_1+α_2ρ_2).Arguing as in <cit.>,setting ϕ=(ϕ_1, ϕ_2), (Dϕ_1, Dϕ_2):=(∂_t ϕ_1, ∇ϕ_1, ∂_t ϕ_2, ∇ϕ_2),q=(q_1, q_2)=(a_1, b_1, c_1, a_2, b_2, c_2),σ=(σ_1, σ_2)=((μ_1, m_1, _1), (μ_2, m_2, _2)) and defining the convex set K:={(a,b) ∈^n+1: a+ 1/2 |b|^2⩽ 0}, one can rewrite one step of the JKO scheme, (<ref>), withasa saddle-point problem for the augmented Lagrangian L_r(ϕ, q , σ) =∑_i=1^2 ∫_Ωϕ_i(0, x) ρ_i,h^k(x)x+∑_i=1^2 ∫_0^1∫_Ωχ_K(a_i(t,x), b_i(t,x)) x t+ ∑_i=1^2 ∫_0^1∫_Ω((μ_i, m_i)·(Dϕ_i -(a_i, b_i) ) +r/2|D ϕ_i -(a_i, b_i)|^2 ) x t+∑_i=1^2 ∫_Ω(r/2|ϕ_i(1,x)+c_i(x)|^2x - (ϕ_i(1,x)+c_i(x)) _i(x))x +h^*(c_1/h, c_2/h),where ^* is the Legendre tranform ofextended by +∞ on (-∞,0]. A saddle point of L_r satisfies μ_i(1,·)=μ̃_i and the solution to one JKO step is ρ_i,h^k+1=μ̃_i. Then, we use the augmented Lagrangian algorithm, ALG2-JKO, introduced in <cit.> to compute numerically (ρ_1,h^k+1,ρ_2,h^k+1) and we refer to <cit.> for a detailed exposition.Figure <ref> represents two populations crossing each other subject to porous media congestion with α_1=α_2=1 and m=50. Initial conditions are given by ρ_1,0 = 1_[-0.45,-0.15]^2 and ρ_2,0 = 1_[0.15,0.45]^2.The motion is imposed by potentials V_1(x,y)= 4 (x,y) - (0.3, 0.3) ^2 and V_2(x,y)= 4 (x,y) + (0.3, 0.3) ^2.We remark that the two populations have the same behaviour and when they cross each other, the density has to spread. In Figure <ref>, we study the same behaviour but subject to the porous medium constraint on ρ_1+2ρ_2. We can see that the population where the constraint plays a higher role, ρ_2, has to deviate in order to let pass ρ_1 through.Although the theory is not fully understood for system (<ref>) (see discusions in <cit.>), we notice that in Figures <ref> and <ref>, it seems that the unique discrete solutions behave numerically stable. In the two populations crowd motion model with linear diffusion, we saw that we can find a solution as the gradient flow of(ρ_1,ρ_2):= ∫_Ω (V_1 +log(ρ_1))ρ_1 + ∫_Ω (V_2+log(ρ_2))ρ_2 + _∞(α_1 ρ_1 +α_2 ρ_2).In this context, we use the same initial datas and potentials as previously. The small parameter =0.01 in the simulations is taken to reduce the effect of the diffusion.In Figure <ref>, we see two populations which cross each other. When they start to cross each other at time t=0.05, we remark that the density of ρ_1 and ρ_2 decrease and the sum is saturated. In this situation, individuals of both populations take the same space. Now assume that an individual of the second population takes twice the space than an individual of the first population. Then if we study the one population model (without interaction), populations ρ_1 and ρ_2 are subject to constraints ρ_1(x) ⩽ 1 and ρ_2(x) ⩽1/2. In our case, where populations interact each other,ρ_1 and ρ_2 are subject to the common constraint ρ_1(x) +2ρ_2(x) ⩽ 1. Notice that when ρ_1(x) =0 or ρ_2(x) =0, we recover the expected behaviour, ρ_2(x) ⩽1/2 and ρ_1(x) ⩽ 1. In Figure <ref>, we represent two populations crossing each other subject to this constraint. Immediately, the second population sprawls to saturate the constraints ρ_2(x) ⩽ 1/2 and then when they start crossing the density of ρ_1 and ρ_2 decrease and we have ρ_1(x) +2ρ_2(x)=1. In Figures <ref> and <ref>, the same situations as in Figures <ref> and <ref> are presented adding an obstacle in the middle of Ω. This can be done using a potential with very high value in this area. §.§ AcknowledgementsThe author gratefully thanks G. Carlier for suggesting this problem and for fruitful discussions about this work. plain | http://arxiv.org/abs/1709.09109v2 | {
"authors": [
"Maxime Laborde"
],
"categories": [
"math.AP"
],
"primary_category": "math.AP",
"published": "20170926161224",
"title": "On cross-diffusion systemsfor two populations subject to a common congestion effect"
} |
Hybrid Bi-LSTM-CRF for Russian NER Neural Networks and Deep Learning Lab, Moscow Institute of Physics and Technology, Russia{burtcev.ms, arkhipov.mu}@mipt.ru Faculty of Information Technology, Vietnam Maritime University, Viet Nam [email protected] of a Hybrid Bi-LSTM-CRF model to the task of Russian Named Entity Recognition Anh L. T.1, 2, Arkhipov M. Y.1, Burtsev M. S.1 December 30, 2023 ========================================================================================= Named Entity Recognition (NER) is one of the most common tasks of the natural language processing. The purpose of NER is to find and classify tokens in text documents into predefined categories called tags, such as person names, quantity expressions, percentage expressions, names of locations, organizations, as well as expression of time, currency and others. Although there is a number of approaches have been proposed for this task in Russian language, it still has a substantial potential for the better solutions. In this work, we studied several deep neural network models starting from vanilla Bi-directional Long Short Term Memory (Bi-LSTM) then supplementing it with Conditional Random Fields (CRF) as well as highway networks and finally adding external word embeddings. All models were evaluated across three datasets: Gareev's dataset, Person-1000 and FactRuEval-2016. We found that extension ofBi-LSTM model with CRF significantly increased the quality of predictions. Encoding input tokens withexternal word embeddings reduced training time and allowed to achieve state of the art for the Russian NER task.§ INTRODUCTION There are two main approaches to address the named entity recognition (NER) problem <cit.>. The first one is based on handcrafted rules, and the other relies on statistical learning. The rule based methods are primarily focused on engineering a grammar and syntactic extraction of patterns related to the structure of language. In this case, a laborious tagging of a large amount of examples is not required. The downsides of fixed rules are poor ability to generalize and inability to learn from examples. As a result, this type of NER systems is costly to develop and maintain. Learning based systems automatically extract patterns relevant to the NER task from the training set of examples, so they don't require deep language specific knowledge. This makes possible to apply the same NER system to different languages without significant changes in architecture. NER task can be considered as a sequence labeling problem. At the moment one of the most common methods to address problems with sequential structure is Recurrent Neural Networks (RNNs) due their ability to store in memory and relate to each other different parts of a sequence.Thus, RNNs is a natural choice to deal with the NER problem. Up to now, a series of neural models were suggested for NER. To our knowledge on the moment of writing this article the best results for a number of languages such as English, German, Dutch and Spanishwere achieved with a hybrid model combining bi-directional long short-term memory network with conditional random fields (Bi-LSTM + CRF) <cit.>. In our study we extended the original work by applying Bi-LSTM + CRF model to NER task in Russian language. We also implemented and experimented with a series of extensions of the NeuroNER model <cit.>. NeuroNER is a different implementation of the same Bi-LSTM + CRF model. However, the realizations of the models might differ in such details as initialization and LSTM cell structure. To reduce training time and improve results, we used the FastText[An open-source library for learning text representations and text classifiers. URL: https://fasttext.cc/] model trained on Lenta corpus[A Russian public corpus for some tasks of natural language processing. URL: https://github.com/yutkin/lenta.ru-news-dataset] to obtain external word embeddings. We studied the following models: * Bi-LSTM (char and word); * Bi-LSTM (char and word) + CRF; * Bi-LSTM (char and word) + CRF + external word embeddings;* Default NeuroNER + char level highway network;* Default NeuroNER + word level highway Bi-LSTM;* Default NeuroNER + char level highway network + word level highway Bi-LSTM.To test all models we used three datasets: * Gareev's dataset <cit.>; * FactRuEval 2016[The dataset for NER and Fact Extraction task given at The International Conference on Computational Linguistics and Intellectual Technologies - Moscow 2016]; * Persons-1000 <cit.>.Our study shows that Bi-LSTM + CRF + external word embeddings model achieves state-of-the-art results for Russian NER task.§ NEURONAL NER MODELS In this section we briefly outline fundamental concepts of recurrent neural networks such as LSTM and Bi-LSTM models. We also describe a hybrid architecture which combines Bi-LSTM with a CRF layer for NER task as well as some extensions of this baseline architecture. §.§ Long Short-Term Memory Recurrent Neural Networks Recurrent neural networks have been employed to tackle a variety of tasks including natural language processing problems due to its ability to use the previous information from a sequence for calculation of current output. However, it was found <cit.> that in spite theoretical possibility to learn a long-term dependency in practice RNN models don't perform as expected and suffer from gradient descent issues. For this reason, a special architecture of RNN called Long Short-Term Memory (LSTM) has been developed to deal with the vanishing gradient problem <cit.>. LSTM replaces hidden units in RNN architecture with units called memory blocks which contain 4 components: input gate, output gate, forget gate and memory cell. Formulas for these components are listed below:i_t = σ(W_ixx_t + W_ihh_t-1 + b_i),f_t = σ(W_fxx_t + W_fhh_t-1 + b_f),c_n = g(W_cxx_t + W_chh_t-1 + b_c),c_t = f_t ∘ c_t-1 + i_t ∘ c_n,h_t = o_t ∘ g(c_t),o_t = σ(W_oxx_t + W_ohh_t-1 + b_o),where σ, g denote the sigmoid and tanh functions, respectively; ∘ is an element-wise product; W terms denotes weight matrices; b are bias vectors; and i, f, o, c denote input gate, forget gate, output gate and cell activation vectors, respectively. §.§ Bi-LSTM Correct recognition of named entity in a sentence depends on the context of the word. Both preceding and following words matter to predict a tag. Bi-directional recurrent neuronal networks <cit.> were designed to encode every element in a sequence taking into account left and right contexts which makes it one of the best choices for NER task. Bi-directional model calculation consists of two steps: (1) the forward layer computes representation of the left context, and (2) the backward layer computes representation of the right context. Outputs of these steps are then concatenated to produce a complete representation of an element of the input sequence. Bi-directional LSTM encoders have been demonstrated to be useful in many NLP tasks such as machine translation, question answering, and especially for NER problem. §.§ CRF model for NER task Conditional Random Field is a probabilistic model for structured prediction which has been successfully applied in variety of fields, such as computer vision, bioinformatics, natural language processing. CRF can be used independently to solve NER task (<cit.>, <cit.>). The CRF model is trained to predict a vector 𝐲 = { y_0, y_1, .., y_T } of tags given a sentence 𝐱 = { x_0, x_1, .., x_T }. To do this, a conditional probability is computed:p(𝐲|𝐱) = e^Score(𝐱, 𝐲)/∑_𝐲^' e^Score(𝐱, 𝐲^'),where Score is computed by the formula below <cit.>:Score(𝐱, 𝐲) = ∑_i=0^T A_y_i, y_i+1 + ∑_i=1^TP_i, y_i,where A_y_i, y_i+1 denotes the emission probability which represents the score of transition from tag i to tag j, P_i, j is transition probability which represents the score of the j^th tag of the word i^th.In the training stage, log probability of correct tag sequence log(p(𝐲|𝐱)) is maximized.§.§ Combined Bi-LSTM and CRF model Russian is a morphologically and grammatically rich language. Thus, we expected that a combination of CRF model with a Bi-LSTM neural network encoding <cit.> should increase the accuracy of the tagging decisions. The architecture of the model is presented on the figure <ref> .In the combined model characters of each word in a sentence are fed into a Bi-LSTM network in order to capture character-level features of words. Then these character-level vector representations are concatenated with word embedding vectors and fed into another Bi-LSTM network. This network calculates a sequence of scores that represent likelihoods of tags for each word in the sentence. To improve accuracy of the prediction a CRF layer is trained to enforce constraints dependent on the order of tags. For example, in the IOB scheme (I – Inside, O – Other, B – Begin) tag I never appears at the beginning of a sentence, or “O I B O” is an invalid sequence of tags.Full set of parameters for this model consists of parameters of Bi-LSTM layers (weight matrices, biases, word embedding matrix) and transition matrix of CRF layer. All these parameters are tuned during training stage by back propagation algorithm with stochastic gradient descent. Dropout is applied to avoid over-fitting and improve the system performance. §.§ Neuro NER Extensions NeuroNER is an open-source software package for solving NER tasks. The neural network architecture of NeuroNER is similar to the architecture proposed in the previous section. Inspired by success of character aware networks approach <cit.> we extended NeuroNER model with a highway layer on top of the Bi-LSTM character embedding layer. This extension is depicted on figure <ref>. Dense layer makes character embedding network deeper. The carry gate presented by sigmoid layer provides a possibility to choose between dense and shortcut connections dynamically. A highway network can be described by the following equation: 𝐲 = H(𝐱, 𝐖_𝐇) · G(𝐱, 𝐖_𝐆) + 𝐱· (1 - G(𝐱, 𝐖_𝐆))where 𝐱 is the input of the network, H(𝐱, 𝐖_𝐇) is the processing function, G(𝐱, 𝐖_𝐆) is the gating function. The dimensionality of 𝐱, 𝐲, G(𝐱, 𝐖_𝐆), andH(𝐱, 𝐖_𝐇) must be the same.Another extension of NeuroNER we implemented is a Bi-LSTM highway network <cit.>. The architecture of this network is quite similar to the character-aware highway network. However, the carry gate is conditioned on the input of the LSTM cell. The gate provides an ability to dynamically balance between raw embeddings and context dependent LSTM representation of the input. The scheme of our implementation of the highway LSTM is depicted in figure <ref>. § EXPERIMENTS§.§ Datasets Currently, there are a few Russian datasets created for the purpose of developing and testing NER systems. We trained and evaluated models on the three Russian datasets: * Dataset received from Gareev et al. <cit.> contains 97 documents collected from ten top cited "Business" feeds in Yandex "News" web directory. IOB tagging scheme is used in this data sets, and entity types are Person, Organization, Other. * The FactRuEval 2016 corpus <cit.> contains news and analytical texts in Russian. Sources of the dataset are Private Correspondent[http://www.chaskor.ru/] web site and Wikinews[http://ru.wikinews.org]. Topics of the texts are social and political. Tagging scheme is IOB. * Person-1000 <cit.> is a Russian news corpus with marked up person named entities. This corpus contains materials from the Russian on line news services.Statistics on these datasets are provided in the table <ref>.§.§ External Word Embedding News and Lenta are two external word embeddings we used to initialize lookup table for the training step.News[Word embeddings, which are available todownload from http://rusvectores.org] is a Russian word embeddings introduced by Kutuzov, et al. <cit.>. Corpus for this word embedding is a set of Russian news (from September 2013 until November 2016). Here are more details about news: * Corpus size: near 5 billion words* Vocabulary size: 194058* Frequency threshold: 200* Algorithm: Continuous Bag of Words* Vector size: 300Lenta is a publicly available corpus of unannotated Russian news. This corpus consists of 635000 news from Russian online news resource lenta.ru. The size of the corpus is around 46 million words. The corpus spans vocabulary of size 376000 words. To train embeddings on this corpus, we use skip-gram algorithm enriched with subword information<cit.>. Parameters of the algorithm were the following:* Vector size: 100* Minimal length of char n-gram: 3* Maximal length of char n-gram: 6* Frequency threshold: 10§.§ ResultsThe purpose of the first experiment was to compare tagging accuracy of three implementations: Bi-LSTM, Bi-LSTM + CRF, Bi-LSTM + CRF + external word embedding news. To do this, we evaluated these implementations on the Gareev's dataset. Parameters of the dataset and hyper-parameters of the models are listed below: * Word embedding dimension: 100* Char embedding dimension: 25* Dimension of hidden layer: 100 (for each LSTM: forward layer and backward layer)* Learning method: SGD, learning rate: 0.005* Dropout: 0.5* Number of sentences: 2136 (for training/ validation/ testing: 1282/ 427/ 427)* Number of words: 25372 (unique words: 7876). 7208 words (account for 91.52% of unique words) was initialized with pre-trained embedding Lenta* epochs: 100We used ConllEval[A Perl script was used to evaluate result of processing CoNLL-2000 shared task: http://www.cnts.ua.ac.be/conll2000/chunking/conlleval.txt] to calculate metrics of performance. The result is shown in the Table <ref>. One can see that adding CRF layer significantly improved prediction. Besides that, using external word embeddings also reduced training time and increased tagging accuracy. Due to absence of lemmatization in our text processing pipeline news embeddings matched only about 15% of words in the corpus, embeddings for other words were just initialized randomly. Therefore, the improvement was not really significant and prediction for Organization type was even lower with news embeddings. To deal with this problem, in the second experiment we decided to use FastText trained on Lenta corpus in order to build an external word embedding. After that, we used this embedding to train on Gareev's dataset one more time using the same configuration with the previous experiment.Table <ref> shows the confusion matrix on the test set. We also experimented on two other datasets: Persons-1000, FactRuEval 2016. The summary of experiments on these datasets are shown in the Table <ref>. We compare Bi-LSTM + CRF + Lenta model and other published results as well as NeuroNER and its extensions on three datasets mentioned in the subsection <ref>. Results are presented in the table <ref>.Bi-LSTM + CRF + Lenta model significantly outperforms other approaches on Gareev's dataset and Persons-1000. However, the result on FactRuEval 2016 dataset is not as high as we expected.§ DISCUSSION Traditional approaches to Russian NER heavily relied on hand-crafted rules and external resources. Thus regular expressions and dictionaries were used in <cit.> to solve the task. The next step was application of statistical learning methods such as conditional random fields (CRF) and support vector machines (SVM) for entity classification. CRF on top of linguistic features considered as a baseline in the study of <cit.>. Mozharova and Loukachevitch <cit.> proposed two-stage CRF algorithm. Here, an input for the CRF of the first stage was a set of hand-crafted linguistic features. Then on the second stage the same input features were combined with a global statistics calculated on the first stage and fed into CRF. Ivanitskiy et al. <cit.> applied SVM classifier to the distributed representations of words and phrases. These representations were obtained by extensive unsupervised pre-training on different news corpora. Simultaneous use of dictionary based features and distributed word representations was presented in <cit.>. Dictionary features were retrieved from Wikidata and word representations were pre-trained on Wikipedia. Then these features were used for classification with SVM.At the moment deep learning methods are seen as the most promising choice for NER. Malykh and Ozerin <cit.> proposed character aware deep LSTM network for solving Russian NER task. A distinctive feature of this work is coupling of language modeling task with named entity classification.In our study we applied current state of the art neural network based model for English NER to known Russian NER datasets. The model consists of three main components such as bi-directional LSTM, CRF and external word embeddings. Our experiments demonstrated that Bi-LSTM alone was slightly worse than CRF based model of <cit.>. Addition of CRF as a next processing step on top of Bi-LSTM layer significantly improves model's performance and allow to outperform the model presented in <cit.>. The difference of Bi-LSTM + CRF model from the model presented in <cit.> is trainable feature representations. Combined training of Bi-LSTM network on the levels of words and characters gave better results then manual feature engineering in <cit.>. Distributed word representations are becoming a standard tool in the field of natural language processing. Such representations are able to capture semantic features of words and significantly improve results for different tasks. When we encoded words with news or Lenta embeddings results were consistently better for all three datasets. Up to now, the prediction accuracy of the Bi-LSTM + CRF + Lenta model outperforms published models on Gareev's dataset and Persons-1000. However, the results of bothBi-LSTM + CRF + Lenta and NeuroNER models on the FactRuEval dataset were better then results reported in <cit.> and <cit.> but not as good as SVM based model reported in <cit.>. In spite the fact that both models we tested have the same structure, performance of NeuroNER <cit.>is a bit lower than Bi-LSTM+CRF model <cit.>. This issue can be explained by different strategies for initialization of parameters.Our extension of the baseline model with a highway network for character embedding provides moderate performance growth in nearly all cases. Implementation of the Bi-LSTM highway network for tokens resulted in a slight increase of performance for Persons-100 and FactRuEval 2016 datasets and a decrease of performance for Gareev's dataset. Simultaneous extension of the NeuroNER with character and token Bi-LSTM highway networks results in the drop of performance in the most of the cases.We think that results of LSTM highway network can be improved by different bias initialization and deeper architectures. In the current work the highway gate bias was initialized with 0 vector. However, bias could be initialized to some negative value. This initialization will force the network to prefer processed path to the raw path. Furthermore, stacking highway LSTM layers might improve results allowing a network dynamically adjust complexity of the processing. Alternatively, character embedding network can be built using convolutional neural networks (CNN) instead of LSTM. A number of authors <cit.> reported promising results with a character level CNN. Another promising extension of presented architecture is an attention mechanism <cit.>. For NER task this mechanism can be used to selectively attend to the different parts of the context for each word giving additional information for the tagging decision. § CONCLUSIONS Named Entity Recognition is an important stage in information extraction tasks. Today, neural network methods for solving NER task in English demonstrate the highest potential. For Russian language there are still a few papers describing application of neural networks to NER. We studied a series of neural models starting from vanilla bi-directional LSTM then supplementing it with conditional random fields, highway networks and finally adding external word embeddings. For the first time in the literature evaluation of models were performed across three Russian NER datasets.Our results demonstrated that (1) basic Bi-LSTM model is not sufficient to outperform existing state of the art NER solutions, (2) addition of CRF layer to the Bi-LSTM model significantly increases it's quality, (3) pre-processing the word level input of the model with external word embeddings allowed toimprove performance further and achieve state-of-the-art for the Russian NER. § ACKNOWLEDGMENTS The statement of author contributions. AL conducted initial literature review, selected a baseline (Bi-LSTM + CRF) model, prepared datasets and run experiments under supervision of MB.AM implemented and studied extensions of the NeuroNER model. AL drafted the first version of the paper. AM added a review of works related to the Russian NER and materials related to the NeuroNER modifications. MB, AL and AM edited and extended the manuscript.This work was supported by National Technology Initiative and PAO Sberbank project ID 0000000007417F630002.30Ms. Maithilee et al. Maithilee L. Patawar,M. A. Potey: Approaches to Named Entity Recognition: A Survey. International Journal of Innovative Research in Computerand Communication Engineering, Vol 3. Issue 12, 12201 – 12208 (2015).Guillaume Lample et al. Guillaume Lample, Miguel Ballesteros, Sandeep Subramanian, Kazuya Kawakami, Chris Dyer: Neural Architectures for Named Entity Recognition. ArXiv preprint arXiv: 1603.01360 (2016). 2017neuroner Dernoncourt F.,Lee, J. Y., Szolovits P.: NeuroNER: an easy-to-use program for named-entity recognition based on neural networks. ArXiv preprint arXiv:1705.05487 (2017). Rinat Gareev et al. Rinat Gareev, Maksim Tkachenko, Valery Solovyev, Andrey Simanovsky, Vladimir Ivanov: Introducing Baselines for Russian Named Entity Recognition. Computational Linguistics and Intelligent Text Processing, 329 – 342 (2013). Trofimov Trofimov, I.V.: Person name recognition in news articles based on the persons-1000/1111-F collections. In: 16th All-Russian Scientific Conference “Digital Libraries: Advanced Methods and Technologies, Digital Collections”, RCDL 2014, pp. 217 – 221 (2014). Mozharova et al. Mozharova V., Loukachevitch N.: Two-stage approach in Russian named entity recognition. In Intelligence, Social Media and Web (ISMW FRUCT), 2016 International FRUCT Conference, 1 – 6 (2016). Ivanitskiy et al. Ivanitskiy Roman, Alexander Shipilo, Liubov Kovriguina: Russian Named Entities Recognition and Classification Using Distributed Word and Phrase Representations. In SIMBig, 150 – 156. (2016). Sysoev et al. Sysoev A. A., Andrianov I. A.: Named Entity Recognition in Russian: the Power of Wiki-Based Approach. dialog-21.ruValentin Malykh et al. Valentin Malykh, Alexey Ozerin: Reproducing Russian NER Baseline Quality without Additional Data. In proceedings of the 3rd International Workshop on Concept Discovery in Unstructured Data, Moscow, Russia, 54 – 59 (2016). Bengio et al. Yoshua Bengio, Patrice Simard, Paolo Frasconi: Learning long-term dependencies with gradient descent is difficult. IEEE Transactions on Neural Networks, Volume 5, Issue 2, 157 – 166 (1994). Sepp Hochreiter et al. Sepp Hochreiter, Jürgen Schmidhuber: Long Short-Term Memory. MIT Press, Vol. 9, No. 8, 1735-1780 (1997). BiRNN Schuster, Mike, and Kuldip K. Paliwal: Bidirectional recurrent neural networks. IEEE Transactions on Signal Processing 45.11, 2673 – 2681 (1997). Wenliang Chen et al. Wenliang Chen, Yujie Zhang, Hitoshi Isahara: Chinese Named Entity Recognition with Conditional Random Fields. In Proceedings of the Fifth SIGHAN Workshop on Chinese Language Processing, 118 – 121 (2006). Kutuzov A. et al. Kutuzov A., Kuzmenko E.: WebVectors: A Toolkit for Building Web Interfaces for Vector Semantic Models. In: Ignatov D. et al. (eds) Analysis of Images, Social Networks and Texts. AIST 2016. Communications in Computer and Information Science, vol 661. Springer, Cham, 155 – 161 (2016). Asif Ekbal et al. Asif Ekbal, Rejwanul Haque, Sivaji Bandyopadhyay: Named Entity Recognition in Bengali: A Conditional Random Field Approach. IJCNLP conference, 589 – 594 (2008).Ashish Vaswani et al. Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, Illia Polosukhin: Attention is all you need. ArXiv preprint arXiv: 1706.03762 (2017) Mikolov et al. T. Mikolov, M. Karafiat, L. Burget, J. Cernocky, S. Khudanpur. 2010: Recurrent neural network based language model. Interspeech, 1045 – 1048 (2010). Starostin A. et al. Starostin A. S., Bocharov V. V., Alexeeva S. V., Bodrova A., Chuchunkov A. S., Dzhumaev S. S., Nikolaeva M. A.. FactRuEval 2016: Evaluation of Named Entity Recognition and Fact Extraction Systems for Russian. In Computational Linguistics and Intellectual Technologies. Proceedings of the Annual International Conference «Dialogue»(2016). No. 15, 702 – 720 (2016).Bojanowski at al. Bojanowski Piotr, Edouard Grave, Armand Joulin, and Tomas Mikolov: Enriching word vectors with subword information. ArXiv preprint arXiv:1607.04606 (2016). Vlasova at al. Vlasova N.A., Suleymanova E.A., Trofimov I.V: Report on Russian corpus for personal name retrieval. In proceedings of computational and cognitive linguistics TEL'2014, Kazan, Russia, 36 – 40 (2014) Rubaylo et al. Rubaylo A. V., Kosenko M. Y.: Software utilities for natural language information retrievial. Almanac of modern science and education, Volume 12 (114), 87 – 92. (2016) Kim et al. Kim, Y., Jernite, Y., Sontag, D., Rush, A. M.: Character-Aware Neural Language Models. Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, 2741 – 2749. (2016) Pundak et al. Pundak Golan, and Tara N. Sainath: Highway-LSTM and Recurrent Highway Networks for Speech Recognition. Proc. Interspeech 2017, ISCA (2017) Tran et al. Tran P. N., Ta V. D., Truong Q. T., Duong Q. V., Nguyen T. T., Phan X. H.: Named Entity Recognition for Vietnamese Spoken Texts and Its Application in Smart Mobile Voice Interaction. In Asian Conference on Intelligent Information and Database Systems. Springer Berlin Heidelberg. 170 – 180. (2016) BengioAttn Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014). | http://arxiv.org/abs/1709.09686v2 | {
"authors": [
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"M. S. Burtsev"
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"published": "20170927181832",
"title": "Application of a Hybrid Bi-LSTM-CRF model to the task of Russian Named Entity Recognition"
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height1.41ex depth-1.27ex width0.34em đd-0.36em height1ex depth-0.86ex width0.4em D-0.73em0.33em 0=1=1 by 40pt Can Two-Way QKD Be Considered Secure?Pavičić M. Pavičić Center of Excellence for Advanced Materials (CEMS), Ruđer Bošković Institute, Research Unit Photonics and Quantum Optics, Zagreb, Croatia and Nanooptics, Department of Physics, Humboldt-Universität zu Berlin, Germany. [email protected] Can Two-Way Direct Communication Protocols BeConsidered Secure? Mladen Pavičić Received: Jul 6, 2017 / Accepted: Sep 11, 2017 ================================================================= We consider attacks on two-way quantum key distribution protocols in which an undetectable eavesdropper copies all messages in the message mode. We show that under the attacks there is no disturbance in the message mode and that the mutual information between the sender and the receiver is always constant and equal to one. It follows that recent proofs of security for two-way protocols cannot be considered complete since they do not cover the considered attacks.03.67.Dd 03.67.Ac 42.50.Ex§ INTRODUCTION Quantum cryptography, in particular quantum key distribution(QKD) protocols, offers us, in contrast to the classical one, provably unbreakable communication based on thequantum physical properties of the information carriers<cit.>. So far, the implementations were mostly based on the BB84protocol <cit.> which is unconditionally secure provided thequantum bit error rate (QBER) is low enough. However,QBER in BB84-like protocols might be high and since we cannotdiscriminate eavesdropper's (Eve's) bit flips from bit flipscaused by losses and imperfections the request of having QBER lowenough for processing the bits is often difficult to satisfy. E.g., 4-state BB84 with more than 11% <cit.>and 6-state BB84 <cit.> with more than 12.6% <cit.> of disturbance (D) have to be aborted (D is defined as the percentage of polarization-flips caused by Eve, maximum being 0.5). Since D cannot be discriminated from the inherent QBER in the line, these levels of total QBER are insecure (mutual information between the sender (Alice) and Eve (I_AE) surpasses the one between Alice and the receiver (Bob) (I_AB): I_AE>I_AB for D>0.11, 0.126, respectively) and therefore cannot be carried out just because Eve might be in the line. In search for more efficient protocols, two-way protocols were proposed and implemented. In particular, entangled photon two-way protocols based on two <cit.> (also called a ping-pong (pp) protocol) and four (Ψ^∓,Φ^∓) <cit.> Bell states, on the one hand and a single photon deterministic Lucamarini-Mancini (LM05) protocol, on the other <cit.>. Several varieties,modifications, and generalisations of the latter protocol are given in <cit.>. Two varieties were implemented in <cit.> and<cit.>. The former pp protocol was implemented by Ostermeyer and Walenta in 2008 <cit.> while theprotocol with four Bell states cannot be implemented with linear optics elements <cit.>. In the aforementioned references various security estimations have been obtained. In <cit.> Lu, Fung, Ma, and Cai provide a security proof of an LM05 deterministic QKD for the kind of attack proposed in <cit.>. Nevertheless, they claim it to be a proof of the unconditional security of LM05. In <cit.> Han, Yin, Li, Chen, Wang, Guo, and Han provide a security proof for a modified pp protocol and prove its security against collective attacks in noisy and lossy channel.All considerations of the security of two-way protocolsassume that Eve attacks each signal twice, once on the way from Bob to Alice, and later on its way back fromAlice to Bob, and that, in doing so, she disturbs the signal in the message mode. However, as we show below, there are other attacks in which an undetectable Eve encodes Bob's signals according to Alice's encoding of a decoy signal sent to her and later on read by Eve. In this paper we show that in the two-way deterministic QKD protocols under a particular intercept and resend attack an undetectable Eve can acquire all messages in the message mode (MM) and that the mutual information between Alice and Bob is constant and equal to one. That means that the security of the protocols cannot be established via standard procedures of evaluating the secret fraction of key lengths.§ METHODSWe analyze the attacks on two different two-way QKD protocols: entangled photon and single photon ones. In particular, we elaborate on the procedure which enables Eve to read off all the messages in the message mode while remaining undetectable.Subsequently, we carry on a security analysis, so as to calculate mutual information between Alice and Eve, as well as between Alice and Bob, as a function of the disturbance that Eve might introduce while eavesdropping. Eventually, we apply the obtained results on the procedure which aims at proving an unconditional security of two-way protocols.§ RESULTS AND DISCUSSION §.§ Entangled Photon Two-Way ProtocolsWe consider an entangled-photon two-way protocol based on two Bell states (pp protocol) <cit.>. Bob prepares entangled photons in one of the Bell statesand sends one of the photons to Alice and keeps the other one in a quantum memory. Alice either returns the photon as is or acts on it so as to put both photons into another Bell state. The Bell states she sends in this way are her messages to Bob. Bob combines the photon he receives from Alice with the one he kept and at a beam splitter (BS) he decodes Alice's messages. Such messages are said to be sent in a message mode (MM). There is also a control mode (CM) in which Alice measures Bob's photon. She announces switching between the modes over a public channel as well as the outcomes of her measurements in CM. We define the Bell basis as a basis consisting of two Bell states|Ψ^∓⟩=1/√(2)(|H⟩_1|V ⟩_2∓|V⟩_1|H⟩_2),where |H⟩_i (|V⟩_i), i=1,2, represent horizontal (vertical) polarized photon states.Photon pairs in the state |Ψ^-⟩ are generated by a down-converted entangled photon source. To send |Ψ^-⟩ state Alice just returns her photon to Bob. To send |Ψ^+⟩ she puts a half-wave plate (HWP(0^∘)) in the path of her photon, as shown in Fig. <ref>(b). The HWP changes the sign of the vertical polarization. At Bob's BS the photons in state |Ψ^-⟩ will split and those in state |Ψ^+⟩ will bunch together. Eve carries out her attack, designed by Nguyen <cit.>, as follows. She first puts Bob's photon in a quantum memory and make use of a copy of Bob's device to send Alice a photon from a down-converted pair in state |Ψ^-⟩ as shown in Fig. <ref>. When Eve receives the photon from Alice she combines it with the other photon from the pair and determines the Bell state in the same way Bob would. She uses this result to generate the same Bell state for Bob by putting the appropriate HWPs in the path of Bob's photon. Thus, Eve is able to copy every single message in MM and therefore sending of messages in MM is equivalent to sending of plain text “secured” by CM. We will come back to this point later on.Here we stress that photons cover four times the distance they cover in BB84. So, if the probability of a photon to be detected over only Bob-Alice distance is p, the probability of being detected over Bob-Alice-Bob distance will be p^4 which with the exponentially increasing losses over distance also exponentially decreases the probability of detecting the disturbance Eve introduces in CM. §.§ Single Photon Two-Way ProtocolsWe start with a brief presentation of the LM05 protocol <cit.>. As shown in Fig. <ref>, Bob prepares a qubit in one of the four states |0⟩,|1⟩ (the Pauli Z eigenstates), |+⟩, or |-⟩ (Pauli X eigenstates) and sends it to his counterpart Alice. In the MM she modifies the qubit state by applying either I, which leaves the qubit unchanged and encodes the logical 0, or by applying i Y= Z X, which flipsthe qubit state and encodes the logical 1. (i Y|0⟩=-|1⟩,i Y|1⟩=|0⟩,i Y|+⟩=|-⟩,i Y|-⟩=-|+⟩.)Alice now sends the qubit back to Bob who measures it in the same basis in which he prepared it and deterministically infers Alice’s operations, i.e., her messages, without basis reconciliation procedure.The attack on LM05 protocol we consider is proposed by Lucamarini in <cit.>. It is shown in Fig. <ref>. Eve delays Bob's photon (qubit) in a fiber spool (a quantum memory) and sends her own decoy photon in one of the four states |0⟩, |1⟩, |+⟩, or |-⟩ to Alice, instead. Alice encodes her message via I or i Y and sends the photon back. Eve measures it in the same basis in which she prepared it, reads off the message, encodes Bob's delayed photon via I, if she read 0, or via i Y, if she read 1, and sends it back to Bob. Eve never learns the states in whichBob sent his photons but thatis irrelevant in the MM since only polarization flipping or notflipping encode messages. Alice also need not know Bob's states <cit.>. This means that, Eve could only be revealed in CM in which Alice carries out a projective measurement of the qubit along a basis randomly chosen between Z and X, prepares a new qubit in the same state as the outcome of the measurement, sends it back to Bob, and reveals this over a classical public channel <cit.>, as shown in Fig. <ref> Here, it should be stressed that photons in LM05 cover twice the distance they cover in BB84. So, if the probability of a photon to be detected over only Bob-Alice distance is p, the probability of being detected over Bob-Alice-Bob distance will be p^2 and Eve would be able to hide herself in CM exponentially better than in BB84. §.§ Security of Two-Way ProtocolsIn a BB84 protocol with more 11% of disturbance the mutual information between Alice and Eve I_AE is higher than the mutual information between Alice and Bob I_AB and one has to abort it. For our attacks, there is no disturbance (D) that Eve induces in MMand the mutual information between Alice and Bob is equal to unity. I_AB=1.Therefore, unlike in BB84, I_AB and I_AE arenot functions of D and that prevents us from proving thesecurity using the standard approach.Also, in a realistic implementation there is no significant D in MM, either. When Bob, e.g., sends a photon in |H⟩ state and Alice does not change it, then Bob will detect |H⟩ with a probability close to 1, with or without Eve, and independently of distance. The only QBER which depends on the fiber length is the one that stems from the dark counts of detectors <cit.>. In a recent implementation of a one-way QKD the total QBER was under 2% over a 250 km distance <cit.>. We can practically completely eliminate the dark counts, and therefore any uncontrolled polarization flips, by making use of superconducting transition edge sensor (TES) photon detectors. The highest efficiency of such detectors is currently over 98% <cit.> and their dark count probability is practically zero.For BB84, and practically all one-way one-photon protocols recently implemented or considered for implementation, the security of the protocols are evaluated via the critical QBER by calculating the secret fraction <cit.>r=lim_N→∞l/n=I_AB - I_AEwhere l is the length of the list making the final key and nis the length of the list making the raw key,I_AB=1+Dlog_2D+(1-D)log_2(1-D) and I_AE=-Dlog_2D-(1-D)log_2(1-D) and their intersection yields D=0.11. Equivalently, r=1+2Dlog_2D+2(1-D)log_2(1-D) goes down to 0 when D reaches 0.11.We do not have such an option for our attacks on two-way protocols since it follows from Eqs. (<ref>) and (<ref>) that r is never negative. Actually it approaches 0 only when Eve is in the line all the time.Since D is not related to MM mode in any way it is on Alice and Bob to decide after which D they would abort their transmission. However, whichever 0≤ D≤ 0.5 they choose I_AB-I_AE shall always be non-negative and they will not have a critical D as in BB84 where the curves I_AB(D) and I_AE(D) intersect for D=0.11 in MM as shown in Fig. <ref>(a). For two-way deterministic protocols, the level of D, which is defined in CM (and not in MM), has no effect on I_AB, i.e., there is no difference whether D=0 or D=0.5, as shown in Fig. <ref>(b); 0≤ D < 0.5 would only mean that Eve is not in the line all the time, but Bob always gets full information from Alice: when Eve is not in the line, because she not in the line, and when Eve is in the line, because she faithfully passes all Alice's messages to Bob.We can assume that Eve snatches only a portion of messages so as to keep QBER in CM at a low level (and have I_AE≤ 1) which would be acceptable to Alice and Bob. With that in mind, we can try to carry out the security evaluation for our attack and verify whether the proofs of unconditional security carried out for other kind of attack on LM05 in <cit.> might apply to it aswell. In the aforementioned security proof <cit.>, which is claimed to be unconditional, the authors first, in Sec. III.A, claim that Eve has to attack the qubits in both the Bob-Alice andAlice-Bob channels in to gain Alice’s key bits and in Sec. III.B, Eq. (1,3) they assume that Eve reads off Bob's qubit and induces a disturbance in the message mode in both Bob-Alice and Alice-Bob channels (error rate e; last paragraph of Sec. III.B and 1st paragraph of Sec. III.F).However, in the considered attacks Eve does not measure Bob's qubits. She just stores them in a quantum memory. She sends her own qubits to Alice and reads off whether she changed them (Y) or not (I). Then she applies Y or I to stored Bob's qubits and sends them back to him. Consequently she does not induce any disturbance in the Alice-Bob channel, either. Also she does not make use of any ancillas as in <cit.>. Therefore, the analysis of getting the key bits carried out in <cit.> is inapplicable to our attack. Hence, since the proof of security presented in <cit.> applies only to the attack considered in it and not to the above Lucamarini's attack, it is not universal, i.e., it cannot be considered unconditional.Let us now consider whether some standard known procedure can beused to establish the security of LM05 protocol. In the protocol, we have neither sifting nor any error rate in the message mode.So, the standard error reconciliation cannot be applied either. The only procedure we are left with to establish the security isthe privacy amplification. When Eve possesses just a fraction of datashe will loose trace of her bits and Alice and Bob's ones will shrink.Eve might be able to recover data by guessing the bits she misses andreintroduces all bits again in the hash function. If unsuccessful herinformation will be partly wiped away. However, Alice and Bob meeta crucial problems with designing their security procedure (e.g.,hash function) which would guarantee that Eve is left with noinformation about the final key. They do not have a critical amountof Eve's bits as in BB84 (11%) which are explicitly included inthe equations of the privacy amplification procedure<cit.>.In a word, the privacy which should be amplified is notwell defined. To design a protocol for such a “blind” privacyamplification is a complex undertaking<cit.> and it is a question whether sending of—in effect—plain text via MM secured by occasionalverification of photon states in CM offers us any advantage overor a better security than the BB84 protocol. In Table <ref> we list the properties of a BB84-like protocolunder an arbitrary attack vs. two-way protocols under the aboveattacks, which seem to indicate that it would be hard to answer the aforementioned question in the positive. § CONCLUSIONTo summarise, we considered deterministic attacks on two kinds of two-way QKD protocols (pp with entangled photons and LM05 with single photons) in which an undetectable Eve can decode all the messages in the message mode (MM) and showed that the mutualinformation between Alice and Bob is not a function of disturbance but is equal to unity no matter whether Eve is in the line or not. Eve induces a disturbance (D) only in the control mode (CM) and therefore the standard approach and protocols for estimating and calculating the security are not available since they all assume the presence of D in MM. As a result, a critical D cannot be determined, the standard error correction procedure cannot be applied for elimination of Eve's information, the efficiency of the privacy amplification is curtailed, and the unconditional security cannot be considered proved. In a way, Alice's sending of the key is equivalent to sending an unencrypted plain text “secured” by an unreliable indicator of Eve's presence and such protocols cannot be considered for implementation at least not before one proves or disproves that a novel kind of security procedures for such deterministic attacks can be designed.We stress that for deciding whether a protocol is unconditionally secure or not it is irrelevant whether Eve can carry out attacks which are more efficient than the attacks considered above, for a chosen D in CM. A proof of unconditional security should cover them all.Financial supports by the Alexander von Humboldt Foundation and the DFG (SFB787) are acknowledged. Supports by the Croatian Science Foundation through project IP-2014-09-7515 and by the Ministry ofScience and Education of Croatia through Center of Excellence CEMSare also acknowledged. spmpsci 10 urlstylelucamarini-mancini-13 Beaudry, N.J., Lucamarini, M., Mancini, S., Renner, R.: Security of two-way quantum key distribution. Phys. Rev. A 88, 062302-1–9 (2013)bb84 Bennett, C.H., Brassard, G.: Quantum cryptography, public key distribution and coin tossing. In: International Conference on Computers, Systems & Signal Processing, Bangalore, India, December 10-12, 1984, pp. 175–179. IEEE, New York (1984)bennett-uncond-sec-ieee-95 Bennett, C.H., Brassard, G., Crépeau, C., Maurer, U.M.: Generalized privacy amplification. IEEE Trans. Inf. Theory 41(6), 1915–1923 (1995)bostrom-felbinger-02 Boström, K., Felbinger, T.: Deterministic secure direct communication using entanglement. Phys. Rev. Lett. 89, 187902–1-4 (2002)bruss Bruß, D.: Optimal eavesdropping in quantum cryptography with six states. Phys. Rev. Lett. 81, 3018–3021 (1998)cai-li-04 Cai, Q., Li, B.: Improving the capacity of the Boström-Felbinger protocol. Phys. Rev. A 69, 054301–1-3 (2004)cere-06 Cerè, A., Lucamarini, M., Di Giuseppe, G., Tombesi, P.: Experimental test of two-way quantum key distribution in the presence of controlled noise. Phys. Rev. Lett. 96, 200501–1-4 (2006)elliot-DARPA Elliott, C., Colvin, A., Pearson, D., Pikalo, O., Schlafer, J., Yeh, H.: Current status of the DARPA Quantum Network. In: E.J. Donkor, A.R. Pirich, H.E. Brandt (eds.) SPIE Quantum Information and Computation III, Proceedings of SPIE, vol. 5815, pp. 138–149. SPIE, Bellingham, Washington (2005)fukuda-11 Fukuda, D., Fujii, G., Numata, T., Amemiya, K., Yoshizawa, A., Tsuchida, H., Fujino, H., Ishii, H., Itatani, T., Inoue, S., Zama, T.: Titanium-based transition-edge photon number resolving detector with 98% detection efficiency with index-matched small-gap fiber coupling. Opt. Express 19, 870–875 (2011)han-14 Han, Y.G., Yin, Z.Q., Li, H.W., Chen, W., Wang, S., Guo, G.C., Han, Z.F.: Security of modified Ping-Pong protocol in noisy and lossy channel. Sci. Rep. 4, 4936–1-4 (2007)henao-15 Henao, C.I., Serra, R.M.: Practical security analysis of two-way quantum-key-distribution protocols based on nonorthogonal states. Phys. Rev. A 92, 052317–1-9 (2015)khir-12 Khir, M.A., Zain, M.M., Bahari, I., Suryadi, Shaari, S.: Implementation of two way quantum key distribution protocol with decoy state. Opt. Commun. 285, 842–845 (2012)korzh-gisin15 Korzh, B., Lim, C.C.W., Houlmann, R., Gisin, N., Li, M.J., Nolan, D., Sanguinetti, B., Thew, R., Zbinden, H.: Provably secure and practical quantum key distribution over 307 km of optical fibre. Nature Phot. 9, 163–168 (2015). Supp. Info.kumar-lucamarini-08 Kumar, R., Lucamarini, M., Giuseppe, G.D., Natali, R., Mancini, G., Tombesi, P.: Two-way quantum key distribution at telecommunication wavelength. Phys. Rev. A 77, 022304–1-10 (2008)nam-apl13 Lamas-Linares, A., Calkins, B., Tomlin, N.A., Gerrits, T., Lita, A.E., Beyer, J., Mirin, R.P., Nam, S.W.: Nanosecond-scale timing jitter for single photon detection in transition edge sensors. Appl. Phys. Lett. 102, 231117–1-4 (2013)lita-nam-10 Lita, A., Calkins, B., Pellouchoud, L., Miller, A.J., Nam, S.W.: Superconducting transition-edge sensors optimized for high-efficiency photon-number resolving detectors. In: SPIE symposium; Advanced Photon Counting Techniques IV, April 5-9, 2010. 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Commun. 281, 4540–4544 (2008)peev-zeilinger-njp09 Peev, M. et al.: The SECOQC quantum key distribution network in Vienna. New J. Phys. 11, 075001–1-37 (2009)pirandola-nat-08 Pirandola, S., Mancini, S., Lloyd, S., Braunstein, S.L.: Continuous-variable quantum cryptography using two-way quantum communication. Nature Phys. 4, 726–730 (2008)sasaki-tokyo-qkd-10 Sasaki, M. et al.: Field test of quantum key distribution in the Tokyo QKD network. Optics Express 19, 10387–10409 (2011)scarani-09 Scarani, V., Bechmann-Pasquinucci, H., Cerf, N.J., Dušek, M., Lütkenhaus, N., Peev, M.: The security of practical quantum key distribution. Rev. Mod. Phys. 81, 1301–1350 (2009)shaari-mancini-15 Shaari, J.S., Mancini, S.: Finite key size analysis of two-way quantum cryptography. Entropy 17, 2723–2740 (2015)stucki-gisin-02 Stucki, D., Gisin, N., Guinnard, O., Ribordy, G., Zbinden, H.: Quantum key distribution over 67 km with a plug&play system. New. J. Phys. 4, 41.1–41.8 (2002)vaidman99 Vaidman, L., Yoran, N.: Methods for reliable teleportation. Phys. Rev. A 59, 116–125 (1999) | http://arxiv.org/abs/1709.09262v1 | {
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A Profit-Maximizing Strategy of Network Resource Management for 5G Tenant Slices Bin Han, Member, IEEE, Di Feng, Lianghai Ji, Student Member, IEEE, and Hans D. Schotten, Member, IEEE B. Han and H. D. Schotten are with Institute for Wireless Communication and Navigation, Department of Electrical and Computer Engineering, University of Kaiserslautern, 67663 Kaiserslautern, Germany. Emails: {binhan,schotten}@eit.uni-kl.de D. Feng is with Universitat Autònoma de Barcelona and Barcelona GSE, Cerdanyola del Vallès, 08193 Spain. Email: [email protected] December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ Supported by the emerging technologies of Network Function Virtualization (NFV) and network slicing, 5G networks allow tenants to rent resources from mobile network operators (MNOs) in order to provide services without possessing an own network infrastructure. The MNOs are therefore facing the problem of deciding if to accept or decline the resource renting requests they receive. This paper builds a stochastic model that describes the MNO's revenue and opportunity cost of accepting a contract, and therewith proposes a strategy that is analytically derived to maximize the expected profit at every decision. Network slicing, multi-tenant network, profit model, network resource management, 5G network optimization§ INTRODUCTIONNetwork slicing was proposed by the Next Generation Mobile Networks (NGMN) Alliance <cit.>, since then it has become one of the hottest topics in the filed of future 5th Generation (5G) mobile communication networks. Generally, the concept of network slicing can be understood as creating and maintaining multiple independent logical networks (slices) on a common physical infrastructure, each slice operates a separate business service. Enabled and supported by the emerging technologies of software defined networks (SDN) and network function virtualization (NFV), network slicing exhibits great potentials, not only in supporting specialized applications with extreme performance requirements, but also in benefiting the mobile network operators (MNOs) with increased revenue <cit.>.As pointed out by Rost et al. <cit.>, a sliced mobile network manages its infrastructure and virtual resources in independent scalable slices, each slice runs a homogeneous service with simple business model. Thus, an MNO can dynamically and flexibly create, terminate and scale its slices to optimize the resource utilization for a better revenue or profit. In a previous paper <cit.>, we have proposed a profit optimization model for sliced mobile networks that run in the traditional business mode: the MNOs with network resources implement the slices and provide all network services directly to their end-users. In this case, an MNO has full a priori knowledge about the service demand and the cost/revenue models of its every slice. It is able to scale the slices according to their profiting efficiencies, in order to achieve the maximal overall profit under the resource constraint. This is a classic multi-objective optimization problem (MOOP), in which the main challenge is to solve the optimum, or at least to find a satisfactory solution, with an affordable computing effort.Unfortunately, this model does not apply to the slices run by tenants such as mobile virtual network operators (MVNOs), which are considered to play an important role in 5G networks <cit.>. Tenants are third-parties that provide services without owning any network infrastructure, e.g. utility/automotive companies and over-the-top service providers such as YouTube. To implement services, they have to be granted by MNOs with network resources, including radio / infrastructure resources and virtualized resource blocks. In legacy networks, every tenant makes its contractual agreement with the MNO(s), to pay a fixed and coarsely estimated annual/monthly fee for these resource sharing concepts. In the context of network slicing, in contrast, the resources are first bundled into slices before granted to tenants upon demand. Depending on the type, size and lifetime of granted slice, the fee is specified. This approach improves the sharing efficiency and the resource utilization rate. However, as such slices are maintained by tenants, the MNO has neither insight into their efficiencies of making revenue at end-users, nor authentication to rescale or terminate them during their lifetime. Instead, the MNO formulates the fee rate for different resource bundles, and chooses if to accept or decline the slice requests from tenants, like discussed in <cit.>. In this case, the MNO cannot jointly optimize all slices in a fully dynamic approach, but only attempt to make the best decision for every received request, which is a problem of decision theory and operations research.In this paper, we will focus on the case of tenant slices, and propose an economic model that evaluates the profit of an MNO to accept a certain slice request from tenant. Based on this model we propose a decision strategy to maximize the expected overall network profit at every decision step. The rest part of the paper is organized as follows: In Sec. <ref> we simplify the problem described above to an approximate business model. Then we build the profit model in Sec. <ref>, starting with an ideal simple caseand then approach to the complex reality step-by-step, on every step we deduce a profit-maximizing decision strategy from the proposed profit model.At the end we close the paper with our conclusion and outlooks in Sec. <ref>. § SIMPLIFIED BUSINESS PROBLEM §.§ Fundamental AssumptionsOur study begins with some basic assumptions and approximations on the business case. First of all, in most countries and regions, the mobile network infrastructure is controlled by only a few or even one single MNO, i.e. the network resource market is never perfectly competitive but highly oligarchy or monopoly. Hence, in this work we consider the case with only one MNO, ignoring the competition between different MNOs.Second, the MNO holds a resource pool, which contains resources of certain types. Self-evidently, resource of every type is limited in amount. To rent these resources to tenants, a list of available contracts is provided by the MNO, every contract defined by a resource bundle, a contract period and a periodical payment. Every resource bundle is specified for a reference slice of certain type and size. We assume that the list of available contracts are predefined and remains consistent.When a tenant requires resources to implement a slice, it selects one from the available contracts, requests to possess the corresponding resource bundle for the contract period. The MNO then decides if to accept or decline the request. Upon acceptance, the contract is confirmed and the tenant periodically pays the quota defined in the contract. If denied, the requested resource bundle will not be dedicated, and can be flexibly exploited for the MNO's own slices to make revenue. We consider that a confirmed contract cannot be terminated or modified within its period. We neglect the priority of contract renewals over new contract establishments, i.e. a tenant obtains no advantage for its future requests from the current contract. We also consider only nonelastic slices and neglect resource multiplex over slices, i.e. no resource can be allocated to multiple contracts simultaneously. Therefore, when accepting the current request, the MNO also loses some opportunity of accepting potential better deals in future.The requests arrive stochastically. Usually, the arriving rate remains on a certain level and the intervals between different arrivals are independent from each other. Hence, it is reasonable to consider the number of arriving requests in a certain period as Poisson distributed. To simplify the model we consider an enough short unit period so that the request arrivals can be approximated as a Bernoulli process.We also assume that the MNO possesses full a priori knowledge about the statistics of arriving requests (resource bundle and contract period), which we consider as consistent. §.§ Model SetupTo normatively describe the simplified business model above, we define the following sets, variables and mappings: * Ψ= [0,1]^N: a general N-dimensional non-negative Euclidean space to measure normalized network resource bundles in reference to the maximal resource pool, where N is the number of resource types. * ψ_t∈Ψ: the normalized measure of idle resource pool available at discrete time t∈ℕ. * Ω_t⊂Ψ: the finite, discrete set of resource bundles defined by all contract options provided by the MNO with its idle resources at time t. Each element in Ω_t is a possible resource bundle requested at time t. The case of no request arrival is considered as a null-request, i.e. ω_null=0^N∈Ω_t,∀ t∈ℕ. We also consider that the entire resource pool is idle at t=0 so that Ω_0 is a predefined set of resource bundles defined by the list of all available contracts, and generally we have a generation function: Ω_t=G(ψ_t)={ω|ω≤ψ_t}∩Ω_0, ∀ t∈ℕ[Note that here with ω≤ψ_t we denote ω is not greater than ψ_t in any of the N dimensions.] * 𝕋⊂ℕ^+: the finite, discrete set of contract periods in all contract options, inf(𝕋)=1 denotes a unit time period. * P ⊂ℝ: the finite, discrete set of payments defined by all contract options. * Ω_0[0,1]: a probability measure on Ω_0 representing the probabilities of upcoming request for different contracts in each unit time period. * Ω_0×𝒫(Ω_0)[0,1]: a probability measure representing the probabilities of upcoming request for different contracts that can be supported by a given idle resource pool[𝒫(Ω_0) denotes the power set of Ω_0]. We consider requests for oversize contracts beyond the support of current idle resource pool as equivalent of null contracts, hence generally: f(ω,Ω) = 0 ω>sup(Ω) g(ω_null)+∑_μ>sup(Ω)g(μ) ω=ω_null g(ω) otherwise * Ψ×Ω_0 Ψ: the decision strategy that determines if to accept the arrived requests. It maps from the current idel resource pool and a received request to the idle resource pool at the next period, i.e. F(ψ_t,ω_t)=ψ_t+1. * Ω_0×𝕋 P: a pricing function that maps the resource bundle and contract period (ω,T) to the corresponding periodical payment, the payment of a null contract is zero i.e. p(ω_null,T)=0,∀ T∈𝕋. * Ω_0ℝ: a function that maps a resource bundle to the corresponding revenue it can generate within one unit time period through its deployment in MNO's own slices. In this work we consider q as a linear function of ω.§ PROFIT MODEL AND DECISION STRATEGYNow we analyze the MNO's profit of accepting a resource request from tenant. In the following part of the paper, we use the term (ψ_t,ω_t,T_t) to represent the request arriving at t. §.§ Two-Step Decision, Non-Expiring ContractWe start with a simple two-step model, where both the MNO and the tenant only act two unit periods, i.e. t∈{0,1}. We also assume that no contract will expire, which means T_t=2-t. Now for the two periods we have two idle resource pools ψ_0,ψ_1. As there is no contract expiry, no resource is released at t=1 so thatψ_1=ψ_0-ω_0 (ψ_0,ω_0,2) accepted; ψ_0 otherwise.Obviously, as the MNO only operates two periods, it should accept any contract request at t=1 as long as its resource pool supports. So the focus is on the decision at t=0. Given any contract request (ψ_0,ω_0,2) arrived at t=0, the expected payoff of accepting this request isΓ_1(ψ_0,ω_0)=(1+β)p(ω_0,2)-C_1(ψ_0,ω_0),where β∈(0,1) is the discount factor to describe the time value of money (TVM) <cit.>, which is determined by the capital market. C_1 is the Opportunity Cost (OC) of this contract: C_1(ψ_0,ω_0)=(1+β)q(ω_0)+β∑_ω∈Ω_0[f(ω,Ω_0)-f(ω,G(ψ_0-ω_0)]p(ω,1).Obviously, to maximize the profit, the MNO is supposed to follow the optimal strategy of binary decision:ψ_1=F_1,opt(ψ_0,ω_0)= ψ_0-ω_0 Γ_1(ψ_0,ω_0)≥0; ψ_0 otherwise,where case 1 denotes acceptance and case 2 for declination. §.§ Multi-Step Decision, Non-Expiring ContractThen we progress towards the multi-step model, where the MNO and the tenant act t_max unit periods. Once again, as of this step we still consider non-expiring contracts, so that the request arriving at t has a contract period of T_t=t_max-t and hence a periodical payment of p(ω_t,t_max-t). Takingit into account that time present value of any future payment x in Δ t periods from current under a discount factor β is xβ^Δ t,we can compute the total present value of all payments for the requested contract (ψ_t,ω_t,t_max-t) asp_t_max(ω_t)=∑_τ=t^t_max-1β^τ-tp(ω_t,t_max-t). Similarly, the present value of exploiting the resource bundle ω_t on the MNO's own slice for t_max-t can be computed asq_t_max(ω_t)=∑_τ=t^t_max-1β^τ-tq(ω_t). Thus, the OC of any request arriving at t isC_t_max(ψ_t,ω_t)=q_t_max(ω_t)+∑_τ=t^t_max-1β^τ-t+1 ×∑_ω∈ G(ψ_τ)[f(ω,Ω_τ)-f(ω,G(ψ_τ-ω_t))]p(ω,t_max-t).Therefore, letΓ_t_max(ψ_t,ω_t)= p_t_max(ω_t)- C_t_max(ψ_t,ω_t),the profit-maximizing decision by the MNO should be ψ_t+1=F_t_max,opt(ψ_t,ω_t)= ψ_t-ω_t Γ_t_max(ψ_t,ω_t)≥0; ψ_t otherwise.Note that as the OC is not promised to be convexabout the decision sequence [ψ_0,…,ψ_t_max-1], (<ref>) is not guaranteed to achieve the global optimum of C_t_max(ψ_t,ω_t), but a local maximum.Besides, according to (<ref>),the OC C_t_max depends on the expected idle resource pool in future ψ_τ>t, which is determined by the target decision strategy F_t_max,opt, whose solution (<ref>) relies on C_t_max. This closed loop encourages to apply an iterative approach, as briefly described in Fig. <ref>. As C_t_max has non-negative terms and bounded partial sums, it converges with increasing t_max. Therefore the sequence (C_t_max^i)_i∈𝕀 is bounded where 𝕀={1,2,…,i_max}. Thus, according to the Bolzano-Weierstrass theorem <cit.> it always has a converging subsequence (C_t_max^j)_j∈𝕁⊆𝕀. By selecting a reasonable i_max we can renew 𝕀=𝕁 in order to construct a converging sequence (C_t_max^i), so that the iterative algorithm converges to a limit as i approaches to i_max.[Herewith we have analytically derived the convergence, but the converging speed must be numerically evaluated, which shall be a follow-up work.] In real world, the MNO and tenants shall be considered as long-term or even eternally operating, i.e. t_max→+∞. According to (<ref>), the sequence (ψ_0, ψ_1,…) monotonically decreases, and G(ψ_t) converges to an empty set:lim_t→+∞G(ψ_t)=∅.Therefore we know that C_t_max(ψ_t,ω_t) is bounded as t_max→+∞. Meanwhile, as p is bounded and β∈(0,1), p_t_max also converges to a bounded value as t_max→+∞, so the proposed approach also applies to the infinite-step case[Nevertheless, as the MNO is usually interested in a short-term or intermediate-term (e.g. monthly or annual) profit, instead of the long-term overall profit till forever, it is still practical to artificially set a finite t_max^*.].Generally, in the case of non-expiring contract, at every step of decision, the impacts of all historical decisions about previous requests are completely reflected in the current idle resource pool without any extra influence in the future. Hence, the non-expiring contract model is a Markov model, and the proposed decision strategy is also Markovian as well. §.§ Multi-Step Decision, Expiring ContractSubsequently we bring the issue of contract expiry into our discussion. Consider flexible contract periods T_t<t_max-t permitted, the resource bundle assigned to it will be released to the MNO's resource pool after the expiry. Thus, the idle resource pool size ψ_t is not monotonically decreasing with t, but jointly determined by all previous requests and decisions during [0,t-1]. Hence, the model becomes non-Markovian and the Markovian decision strategy (<ref>) does not apply as it lacks information about contract periods and previous decisions.As a solution to this, given an arbitrary request (ψ_τ,ω_τ,T_τ) arriving at τ, we denote the indicator functionI_τ(t,T_τ)= 1 τ≤ t≤τ+T_τ, (ψ_τ,ω_τ,T_τ) accepted; 0 otherwiseto represent its validity at time t. Thus,we can represent the resource bundle reserved for it at any time t asω_t^T_τ=ω_tI_τ(t,T_τ) Then we define the scalarω̃_t=∑_τ=0^t-1ω_t^T_τto track and aggregate the resource bundles currently reserved by all previously accepted contracts. Thus, instead of (ψ_t, ω_t, T_t), now we use (ω̃_t, ω_t, T_t) to represent a contract request.the equations (<ref>) and (<ref>) become p_t_max^T_t(ω_t) =∑_τ=t^t+T_t-1β^τ-tp(ω_t,T_t), q_t_max^T_t(ω_t) =∑_τ=t^t+T_t-1β^τ-tq(ω_t),respectively, and thus the OC of accepting the request isC_t_max^T_t(ω̃_t,ω_t)=q_t_max^T_t(ω_t)+∑_τ=t^t_max-1β^τ-t+1×∑_ω∈ G(ψ_τ)[f(ω,Ω_τ)-f(ω,G(ψ_0+ω̃_τ-ω_t^T_t))]p(ω,T_t).Defining the payoff functionΓ^T_t_t_max(ω̃_t,ω_t)= p^T_t_t_max(ω_t)-C^T_t_t_max(ω̃_t,ω_t),the non-Markovian profit-maximizing decision strategy is F_t_max,opt^T_t(ψ_t,ω_t)= ψ_0+ω̃_t+1-ω_t Γ_t_max^T_t(ψ_t,ω_t)≥0; ψ_0+ω̃_t+1 otherwise. In finite-step cases where t_max is finite, the number of possible sequences ((ω_t, T_t))_0≤ t≤ t_max is limited so that C_t_max^T_t is bounded and the iterative approach in Fig. <ref> still applies. However, when t_max→+∞, as (ψ_t)_t∈ℕ is not monotonic about t, (<ref>) fails to hold and no more convergence is guaranteed. In this case, an artificial finite t_max^* is needed, like we did in the footnote <ref>.§ DISCUSSIONSSo far we have closed our study on the MNO's decision strategy under the assumptions in Sec. <ref>. Nevertheless, concerning the strength of our assumptions, some discussions about their feasibilities may be necessary.The first concern can be raised by the monopoly model with only one MNO, because the tenants are may fail to obtain resources to maintain services in this case. Practically, a pre-ordering mechanism can be applied to allow early requests before the due of slice creation. Thus, if a request is declined, the tenant is still able to reattempt with another contract option with better chance of acceptance. Moreover, it is true that in practice there isusually not only one but several MNOs, i.e. the market is actually oligarchy. While providing the tenant more alternative options when its request is rejected by one MNO, this fact does not necessarily conflict with our results under the monopoly assumption, because oligarchy markets differ from monopoly ones only in the supply, demand and pricing mechanism, but not in the decision making logic <cit.>, which we focus on in this paper. Nevertheless, the competition between MNOs in oligarchy markets worths further study.Another doubt may arise about the exclusion of unexpected contract termination and modification. Certainly they are ignored here for model simplification, and can be eventually involved in future work by introducing another random variable with corresponding statistics. Similarly, in this work we have ignored the preference in renewing old expiring contracts over making new ones. For a better approximation to reality we can consider all contracts as non-expiring, and apply the random termination event instead to describe the tenant cancellation.At last, the assumption that MNO possesses full a priori knowledge about the request statistics may be argued. In practice, although the a priori model is hard to obtain, it can be estimated in a Bayesian approach from the a posteriori historical records that every MNO keeps, as long as it remains consistent. Even in case of non-stationarity, short-term consistence can still be approximated with periodical updates.§ CONCLUSIONIn this work, in an operations research perspective we have investigated the 5G network resource management problem of creating tenant slices upon request, aiming at a strategy that maximizes the expected MNO revenue at every binary decision. Different cases have been studied and an iterative algorithm proposed. The convergence of proposed method has been mathematically derived. For future works, numerical experiments are expected to evaluate the converging speed and the revenue gain of the proposed algorithm in both short and long terms, elastic slices shall also be considered to enable resource multiplexing between slices .IEEEtran | http://arxiv.org/abs/1709.09229v1 | {
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[email protected] Statistical Physics Group and Collège Doctoral 𝕃^4 for Physics of Complex Systems, IJL, UMR Université de Lorraine - CNRS 7198, 54506 Vandœuvre les Nancy, France [email protected] Departamento de Física, Universidade Federal Rural de Pernambuco,52171-900, Recife, PE, Brazil [email protected] Centro de Nanociencias y Nanotecnología, Universidad Nacional Autonoma de México, Apdo. Postal 14,22800 Ensenada B.C., Mexico [email protected] Statistical Physics Group, Laboratoire de Physique et Chimie Théoriques, Université de Lorraine, 54506 Vandœuvre les Nancy, [email protected] Statistical Physics Group, Laboratoire de Physique et Chimie Théoriques, Université de Lorraine, 54506 Vandœuvre les Nancy, France We examine the effect of a wiggly cosmic string for bothmassless and massive particle propagation along the string axis. We show that thewave equation that governs the propagation of a scalar field in the neighborhood of a wiggly string is formally equivalent to the quantum wave equation describing the hydrogen atom in two dimensions.We further show that the wiggly string spacetime behaves as a gravitational waveguide in which the quantized wave modes propagate with frequencies that depend on the mass, string energy density,and string tension. We propose an analogy with an optical fiber, defining an effective refractive index likely to mimic the cosmic string effect in the laboratory. The wiggly cosmic string as a waveguide for massless and massive fields Sébastien Fumeron December 30, 2023 =======================================================================§ INTRODUCTION The thermal history of the Universe started 14 billion years ago with an extremely hot and dense quark-gluon plasma that cooled down in the inflation era. As a consequence, it has undergone a succession of phase transitions involving spontaneous symmetry-breaking (SSB) mechanisms. Below an energy scale M_ GUT∼ 10^16 GeV, the strong forces are represented by the three-fold color symmetry, associated with the gauge group SU(3)_ color, whereas the weak and electromagnetic forces are mixed into the electroweak interaction, represented by weak isospin symmetry (gauge group SU(2)_ L) and hypercharge symmetry (gauge group U(1)_ Y) <cit.>.This isthe realm of the particle physics standard model, which has been tested to a very high precision. Above M_ GUT, strong and electroweak interactions unify within a larger gauge symmetry group G where grand unified theories involving supersymmetry (SUSY GUT) have been considered as suitable description for such energy scales <cit.>. As well-known in condensed matter physics (CMP), spontaneoulsy symmetry breaking (SSB) lead to phase transitions which often give rise to the appearance of topological defects. Interestingly, a mechanism originally proposed by Kibble <cit.> to describe the birth and dynamics of a network of defects in a cosmological context revealed to be relevant in the condensed matter realm, for example in liquid crystals. <cit.>. To determine what kind of topological defect emerges for a given SSB transition G→ H, one may study the content of homotopy groups π_k(G/H) of the vacuum manifold ℳ=G/H <cit.>. If π_k(G/H)≠ 0, defects of dimension 2-k are formed: for k=0, defects are 2D (grain boundaries in CMP, domain walls in cosmology), for k=1, defects are line-like (disclinations or dislocations in CMP, cosmic strings in cosmology) and for k=2, defects are point-like (e.g. hedgehogs in CMP, monopoles in cosmology...). Even though the SUSY extension to the standard modelstill has to pass experimental verification (the first run of the LHC found no evidence for supersymmetry), it provides a route to the formation of cosmic strings. In a seminal paper, Jeannerot et al <cit.> examined all possible SSB patterns from the large possible SUSY GUT gauge groups down to the standard model SU(3)_ color× SU(2)_ L× U(1)_ Y and concluded that cosmic string formation was unavoidable. Another possibility for cosmic string generation is brane inflation <cit.>. Cosmic strings, seen as lower-dimensional D-branes which are one-dimensional in the noncompact directions, may have been abundantly produced by brane collision towards the end of the brane inflationary period <cit.>. Despite alltheoretical justifications for the existence of cosmic strings, the observational evidenceis still feeble and mostly indirect. Nevertheless, the search for cosmic strings is very active and it happens in such diverse fronts as the cosmic microwave background <cit.> and gravitational wave bursts <cit.>. As warned by Copeland and Kibble <cit.>, “Both cosmic strings and superstrings are still purely hypothetical objects. There is no direct empirical evidence for their existence, though there have been some intriguing observations that were initially thought to provide such evidence, but are now generally believed to have been false alarms. Nevertheless, there are good theoretical reasons for believing that these exotic objects do exist, and reasonable prospects of detecting their existence within the next few years."In this paper, we study the dynamics of particles in the vicinity of a wiggly cosmic string. Regular straight strings are linear defects for which the geometry is globally that of a cone and therefore, spacetime is locally flat, except on string axis. Indeed, for such objects, the line tension T_0 exactly matches the energy density per unit length μ_0, such that straight strings do not gravitate. Recent data on the Cosmic Microwave Background collected from PLANCK satellite have not confirmed the existence of these objects yet, but they have set upper boundaries on their mass-energy density <cit.> Gμ_0 < 10^-7 (c=1). Refined models for cosmic strings may involve small-scale perturbations such as kinks and wiggles <cit.>. The presence of wiggles generates a far gravitational field contribution whichmay be responsible for an elliptical distortion of the shape of background galaxies <cit.> or for the accretion of dark energy around the defect<cit.>. Averaging the effect of these perturbations along a string increases the linear mass density μ̃ and decreases the string tension T̃, respecting the equation of state <cit.>μ̃ T̃=μ_0^2, leading the wiggly string to exert a gravitational pulling on neighboring objects.In the weak-field approximation, the linearized line element representing the spacetime of a wiggly string oriented along the z-axis is given by<cit.>:ds^2 = -(1+8εln(r/r_0))dt^2 +dr^2+α^2 r^2 dθ^2 + (1-8εln(r/r_0))dz^2.Here, α^2=1-4G(μ̃+T̃), where 4G(μ̃+T̃) ≪ 1meaning that conical deficit angle 4Gπ(μ̃+T̃) associated to the string is very small. The parameter ε is defined as the excess of mass-energy density, 2ε=G(μ̃-T̃). It must be emphasized that G(μ̃+T̃) and ε are two independent parameters: the former accounts for the discrepancy between flat and conical geometries, whereas the latter accounts for the discrepancy between straight and wiggly strings. The constant r_0 denotes the effective string radius <cit.>. In the remainder of this work, propagation of particles will be considered only within the region r_0 < r ≪ r_0 e^1/8ε, in order to avoid the logarithmic divergence at small and large distances from the defect. Hence, h_00=8εln(r/r_0)=O(ε) ≪ 1. In the next sections of this work, the wave equation for propagation along the string axis is numerically solved in the background spacetime given by metric (<ref>). The properties of the radially bound states and the dispersion relations are examined in detail for both massless and massive particles. Then an analogy with light propagation in an optical fiber is performed to design a system likely to mimic the effect of a cosmic wiggly string in laboratory. § MASSLESS PARTICLE PROPAGATION In a plane perpendicular to the wiggly string, light propagates in the same way as in the vicinity of a regular straight string <cit.>, that is without experiencing any gravitational force. On the contrary, when the direction of propagation is not perpendicular to the string, the wiggly string exerts gravitational pulling on light passing by, as the background spacetime is not locally Euclidean.Consequently, the wave equation governing propagation of a scalar field in this background geometry needs to account for its curvature. This is done by using the 4-dimensional Laplace-Beltrami operator in the wave equation1/√(-g)∂_μ(√(-g)g^μν∂_ν)Φ=0,where Φ=Φ(r,θ,z,t) is the scalar wave amplitudeand g= (g_μν) with the metric tensor g_μν coming from metric (<ref>). In terms of the metric (<ref>), this gives-(1-h_00)∂_t^2 Φ+1/r∂_r(r∂_r) Φ+1/α^2r^2∂_θ^2 Φ +(1+h_00) ∂_z^2 Φ=0.As the field is single-valued, Φ has to be periodic in θ:Φ(θ)=Φ(θ+2 π).To solve equation (<ref>) we make the ansatz Φ(r,θ,z,t)=e^il θ e^i(ω t-kz) R(r),where the wave vector k ∈ℝ, l=0,±1,±2... specifies the angular momentum and ω is an angular frequency. Substituting the general solution (<ref>) into equation (<ref>),we get-1/rd/d r(rd R/d r)+l^2/α^2 r^2 R+h_00(ω^2+k^2)R=(ω^2-k^2)R.Defining thedimensionless variables ρ=r/γ, ρ_0=r_0/γ,where γ=[8ε(ω^2+k^2)]^-1/2, then multiplying (<ref>) by γ^2∼ O(ε^-1) and rearranging terms gives the eigenvalue equation: -1/ρd/dρ(ρdR/dρ) + (l^2/α^2ρ^2+lnρ/ρ_0)R =ζ̅Rwithζ̅=1/8εω^2 -k^2/ω^2+k^2. We note that the potential behaving logarithmically, the energy scale cannot be fixed at infinity and we work in the following with ω-dependent length units such that ρ_0=1. Eq. (<ref>) is formally equivalent to the Schrödinger equation that describes the hydrogen atom in the 2D Coulomb potential. Hence, the potential term in Eq. (<ref>), V_ eff=l^2/α^2 ρ^2+lnρ/ρ_0(see figure <ref>) only accommodates for bound states <cit.>.As a consequence, in the geometrical optics limit, trajectories are radially bounded helices around the string, as appears in Fig. <ref>, explicitly showing the gravitational pulling by the string. This is in agreement with Ref. Arazi2000, where geodesics near a Brans-Dicke wiggly cosmic string were also found to be bounded. The minimum and maximum radii are solutions of the transcendental equation ζ̅=V_ eff whereas the pitch is given by the ratio between the angular and effective linear momenta l/k. In the case of l=0the trajectory is rectilinear and parallel to the string. In order to solve equation (<ref>), we used a finite difference method <cit.> and computed numerically the radial part of the waves traveling along the wiggly string with their corresponding eigenvalues. The different states are labeled by quantum numbers n(radial quantum number) and l. In Fig. <ref>, we plot the lowest three eigenvalues ζ̅_̅n̅l̅ of the wave equation (<ref>) and the corresponding radial wave amplitudes R_nl(ρ) for l=0,± 1, ± 2. From Eq. (<ref>) we see that the wave modes that propagate along the wiggly string axis are quantized by n and l. It has been suggested that cosmic structures with a non-vanishing Newtonian potential could generally behave as gravitational waveguide for light and massive particles <cit.> and we examine this proposition in the following.The dispersion relation is given by: ω^2_nl= 1/n^2_1 k^2where n_1=[1-8εζ̅_̅n̅l̅/1+8εζ̅_̅n̅l̅]^1/2is an effective refractive index.From Eq. (<ref>) we see that modes are propagative along the string provided that the following requirement is fulfilled:0<ζ̅_nl <1/4 G(μ̃-T̃). This constraint establishes that the number of wave modes propagating along the wiggly string is large but finite as in an ordinary electromagnetic waveguide. As we would expect, we find that the allowed modes, besides being quantized by n and l, their frequency also depend on both the energy density and the tension of the string.§ PROPAGATION OF MASSIVE PARTICLES Since light propagating along a wiggly string is radially confined, as seen in the previous section, it is interesting to investigate what happens to massive particles under the same circumstances.In order to study this possibility we write the Klein-Gordon equation (ħ=1) in the wiggly string background geometry:[1/√(-g)∂_μ(√(-g)g^μν∂_ν)-m^2]Φ(r,θ,z,t)=0,where now Φ isa complex scalar field describing spinless relativistic particles. Using the ansatz given in Eq. (<ref>)in equation (<ref>), andfollowing the same procedures used above in the case of massless particles propagation, we arrive to an identical eigenvalue equation as (<ref>), but with eigenvalues givenby ℰ̅_nl=1/8εω^2 -k^2-m^2/ω^2 +k^2,thus the wavefunctions R_nl(ρ) and the eigenvalues ℰ̅_nl are numerically identical to the solution of the equation (<ref>) (see figure (<ref>)). In addition, the discussion on the geometric optics limit of the propagating massless field is still valid for massive particles. However, inclusion of the mass term in the dispersion relation now introduces a cutoff:ω_nl^2=1/n^2_2 k^2+ω_c^2where n_2 is an effective refractive index defined byn_2=[1-8εℰ̅_nl/1+8εℰ̅_nl]^1/2,which has an identical generic form than n_1, and ω_c^2=m^2/1-8εℰ_nlis a cutoff frequency. The dispersion relation (<ref>) presents a forbidden band as it occurs for electromagnetic waves propagating in an unmagnetized plasma <cit.>. The wave will propagate along the string when its frequency is larger than the cutoff frequency, ω_c, otherwise solutions appear as evanescent waves. At high frequencies, ω≫ω_c, we recover the massless dispersion relation (<ref>). Moreover,the constraint0<ℰ̅_̅n̅l̅ <1/4 G(μ̃-T̃),like in the massless case, sets a limit to a finite numberof propagating modes. Besides the dependence on the density of energy and tension of the wiggly string, the allowed propagating modes also depend on the mass of the particle.There is obviously a strong similarity between the propagation of both massless and massive scalar fields along a wiggly cosmic string and the propagation of electromagnetic waves in optical waveguides.In the next section, we further explore this analogy by proposing a way of designing an optical fiber that mimics the wiggly string in the context described above. § ANALOGUE OPTICAL WAVEGUIDEThe analogy between 3D gravity and optics is an old topic that started with the pioneering works of Gordon on Fresnel dragging effect in moving dielectrics <cit.>. Recently, artificial optical materials (metamaterials) have been proposed as a wayto mimic aspects of curved spacetime in the laboratory. For instance, by manipulating the effective refractive indexof the medium, Sheng et al. <cit.> were able to reproduce gravitational lensing and trapping of light (see also <cit.>).Incidentally, electronic metamaterials may also be used to simulate peculiar spacetime conditions, like a discontinuousLorentzian to Kleinian metric signature change <cit.> which has also been modeled by optical metamaterials <cit.>. Liquid crystals, as well, have been used to simulate straight cosmic strings<cit.> and the Schwarzschild spacetime <cit.>. On the other hand, it has been proposed that, by spatially varying the dopingconcentration, the refractive index profile of optical fibers can be used to control optical transmission in a designer-specified way <cit.>. Following this standpoint, we investigate the proposition of a graded-index optical fiber that reproduces some of the properties of the scalar field propagation along a wiggly string. In general, the wave equations for electromagnetic waves propagating along a circular fiber are coupled <cit.>. This implies that there is no separation into purely TE or TM modes but, in the specific case of a fiber withan azimuthally symmetric refractive index,if the fields have no dependence on the azimuthal angle,the equations uncouple into separate scalar wave equationsof the form[1/r∂/∂ r(r∂/∂ r)+∂^2/∂ z^2+n^2(r)ω^2]Φ=0,where ω is the angular frequency,n(r) is the optical fiber refractive index andΦ represents any real component of the field. For waves propagating along the optical fiber (z-direction), the ansatz Φ(r,z)= e^-ik z R(r), (where k ∈ℝ) substitutedinto Eq. (<ref>) gives -1/rd/d r(rdR/d r)-n^2(r)ω^2R+k^2R=0.Here, we choose the refractive index to be given by n(r)=(1-Ωlnr/r_0)^1/2 ,with the dimensionless parameter Ω≪ 1 in order to be consistent with the wiggly string parameter ε. The quantityr_0is considered to be much smaller than the radius r_f of the fiber and defines an opaque core radius. This way, by considering propagation in the region r_0 <r<r_f, the logarithmic singularity at r=0 is avoided. Like in the previous sections, we change r to dimensionless units by doingthe change of variablesρ=r/ν, ρ_0=r_0/νand setting ν=Ω^-1/2ω^-1. Then, thedimensionless equation for the optical fiber can be written as: -1/ρd/dρ(ρdR/dρ)+(lnρ/ρ_0)R=β̅_nR,whereβ̅_n=1/Ω(1-k^2/ω^2). The radial amplitudes of the wave and their corresponding eigenvalues in the optical fiber with the refractive index given by Eq. (<ref>) obey equations identical to the ones of massless and massive particles propagating with l=0 in the spacetime of a wiggly string. For a given z, the intensity profiles for the propagating waves described by Eq. (<ref>) are given by 2πρ R_nl^2(ρ). In Fig. <ref>, we plot the intensity distribution for different wave modes, solutions of Eq. (<ref>). The optical fiber modes described by Eq. (<ref>) correspond to the cases where l=0. Before ending this section, we remark that the coupled equations for the electromagnetic field in the circular optical fiber, in the more general case where the field depends on the azimuthal angle but the refractive index remains azimuthally symmetric, give rise to hybrid HE modes (no longer TE or TM) <cit.>. The case of a step-index fiber was studied in Ref. <cit.> which found the field to be of the form R_l (ρ) e^ilθ, where R_l is a Bessel function. Even though Eq. (<ref>) is not a Bessel equation, its symmetries and the shape of the numerical solutions shown in Fig. <ref>, suggest that a solution in terms of an expansion on Bessel functions might be rapidly convergent. The θ-dependence of the scalar fields solutions, for l ≠ 0, is therefore reminiscent of what happens in the circular optical fiber.Also, it would be interesting to compare the coupled electromagnetic vector field equations for a wave propagating along a circular optical fiber with refractive index given by Eq. (<ref>) with the electromagnetic field equations in the wiggly string background.§ CONCLUSIONIn this paper, we examined the effect of wiggly cosmic strings on propagation of massless and massive fields. We found that waves propagating along the string axis experience the small-scale perturbations which make the propagation qualitatively different from that of waves propagating in the background spacetime with a unperturbed cosmic string. The non-vanishing Newtonian potential acts as an inhomogeneous dielectric medium so that the massless particles are radially confined in a vicinity of the defect axis. Therefore, the wiggly string spacetime behaves as a gravitational waveguide in which wave modes are quantized.These latter depend on the string energy density and string tension. The number of allowed modes is finite as in a ordinary optical waveguides. On the other hand, the presence of wiggles cause gravitational pullings on massive objects, making the waveguide effect to be also valid for massive fields propagation. In this case, the frequencies of the waves also depend on the mass of the particle.Finally, we proposed the design of an optical fiber with a non homogeneous refractive index profile likely to mimic the effect of a perturbed cosmic string. The radial solutions with the corresponding eigenvalues were found by using a numerical method. Although we have considered here the propagation of massive and massless scalar fields along a wiggly string, the extension to vector fields like vector bosons or the electromagnetic field can be of interest. In particular, as a perspective for future work we mention the study of propagating electromagnetic waves along the wiggly string and a possible correspondence with an optical fiber. This is more complex than the problem presented here since the vector field equations are coupled and cannot be reduced to scalar wave equations.Another aspect of of the optical fiber/wiggly string analogy is whether apropagatingelectromagnetic wave along the wiggly string could act as a tractor field on particles in the string vicinity. This has been proposed recently in the realm of negative index optical waveguides <cit.>: instead of being pushed by radiative pressure, a polarizable particle in such environment is attracted by the source of radiation. Even though the requirement of a negative refractive index rules out the wiggly string as suchwaveguide, a relatedlinear defect, the hyperbolic disclination <cit.>, seems to be a plausible mediator of this effect. The Kleinian signature of its metric simulates a negative refractive index.Networks of cosmic topological defects have been proposed as models for solid dark matter <cit.>. This suggests that one might explore the optical implications of a network of wiggly strings. For instance, for a periodic array of strings one might expect some of the properties of a photonic crystal, like the appearance of band gaps in the dispersion relation, whichlimit the propagation to the allowed regions of the spectrum. This is presently under investigation and will be the subject of a future publication.F.M. is grateful to U. Lorraine, FACEPE, CNPq and CAPES for financial support. F.A. thanks the Collège Doctoral “𝕃^4 collaboration” (Leipzig, Lorraine, Lviv, Coventry) and the Dionicos programme between U. Lorraine and UNAM for financial support. | http://arxiv.org/abs/1709.09514v1 | {
"authors": [
"Frankbelson dos S. Azevedo",
"Fernando Moraes",
"Francisco Mireles",
"Bertrand Berche",
"Sébastien Fumeron"
],
"categories": [
"gr-qc",
"cond-mat.mes-hall"
],
"primary_category": "gr-qc",
"published": "20170927134630",
"title": "The wiggly cosmic string as a waveguide for massless and massive fields"
} |
.eps,.jpg -10truept myheadings Gratton, Gürol, Simon, Toint: Preconditioning least-squares | http://arxiv.org/abs/1709.09031v1 | {
"authors": [
"Serge Gratton",
"Selime Gürol",
"Ehouarn Simon",
"Philippe L. Toint"
],
"categories": [
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"86A5, 86A10, 90C06, 90C30, 15A12",
"G.1.3; G.1.6"
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"primary_category": "math.NA",
"published": "20170926141655",
"title": "A note on preconditioning weighted linear least squares, with consequences for weakly-constrained variational data assimilation"
} |
Institute for Theoretical Physics, TU Wien, Wiedner Hauptstr. 8-10/136, A-1040 Vienna, Austria We introduce a new entropy formula for Kerr black holes inspired by recent results for 3-dimensional black holes and cosmologies with soft Heisenberg hair. We show that also Kerr–Taub–NUT black holes obey the same formula. New entropy formula for Kerr black holes Hernán A. Gonzá[email protected] Daniel [email protected] Wout [email protected] [email protected] 30, 2023 ====================================================================================================================================================================================§ INTRODUCTIONRecently a new horizon entropy formula has emerged <cit.> S=2π (J_0^+ + J_0^-) eq:k1 that is more universal than the Bekenstein–Hawking <cit.> or Wald's <cit.> entropy formulas, albeit only in three dimensions. The quantities J_0^± are zero-mode charges of 𝔲(1) current algebras that appear to arise generically in near horizon descriptions.Formula (<ref>) was first derived for black holes in 3-dimensional Anti-de Sitter space within Einstein gravity <cit.> inspired by related earlier discussions of near horizon boundary conditions <cit.> and the concept of soft hair <cit.>. During the past 18 months the entropy formula (<ref>) was shown to apply also to flat space cosmologies in Einstein gravity <cit.>, to black holes (flat space cosmologies) in higher spin theories <cit.> (<cit.>) where Bekenstein–Hawking fails, and to higher derivative theories with gravitational Chern–Simons term <cit.> where Wald's formula fails.Besides the entropy itself also Cardyology simplifies <cit.> including log-corrections <cit.>, while semi-classical considerations allow for Hardyology <cit.>, by which we mean a counting of explicitly constructed semi-classical black hole microstates that combinatorially reduces to partitions of large integers. Nearly all results so far apply only to three spacetime dimensions, with the exception of <cit.> that applies Hardyology to extremal Kerr black holes. The phenomenologically most interesting non-extremal Kerr black hole <cit.> was not discussed in detail so far.The main purpose of this proceedings contribution is to make modest progress towards lifting the exciting results in three dimensions to four dimensions by providing an analog of the entropy formula (<ref>) for non-extremal Kerr black holes. Our main result is a new entropy formulaS_Kerr = 4π J_0^+ J_0^- eq:k2 that is equivalent to the Bekenstein–Hawking law or Wald's entropy, but expressed in terms of zero mode charges of 𝔲(1) current algebras that are expected to appear in a suitable near horizon description of non-extremal Kerr.This work is organized as follows. In section <ref> we summarize briefly salient aspects of the black hole entropy (<ref>) in three dimensions. In section <ref> we present our new results for the entropy of non-extremal Kerr black holes. In section <ref> we elaborate on inner horizon entropy, limits of Kerr and the generalization to Kerr–Taub–NUT. In section <ref> we provide a brief outlook.§ BLACK HOLE ENTROPY IN THREE DIMENSIONS In this section we review material that appeared in <cit.>, whose conventions and notations we use. We start with the near horizon expansion of non-extremal black holes (or cosmologies).The vicinity of non-extremal horizons allows an expansion around Rindler spaces^2=-α^2r^2 t^2+r^2+γ^2 φ^2+⋯eq:k3 where the ellipsis denotes terms that vanish as the radial coordinate tends to zero, r→ 0. The angular coordinate is periodic, φ∼φ+2π. The exact solutions of Einstein's equations with negative cosmological constant Λ=-1/ℓ^2 that asymptote to the near horizon metric (<ref>) are given by s^2= -α^2ℓ^2sinh^2(r/ℓ) t^2 + r^2 +2αωℓ^2sinh^2(r/ℓ) tφ+(γ^2cosh^2(r/ℓ)-ω^2ℓ^2sinh^2(r/ℓ)) φ^2. eq:k4 As usual, for defining the theory it is crucial to declare which variations of the metric are allowed, i.e., to impose some consistent set of boundary conditions.The near horizon boundary conditions of <cit.> allow for arbitrary variations of the horizon radius function γ and the rotation function ω but no variations of Rindler acceleration α, i.e., δγ≠ 0 ≠δω and δα=0. For constant Rindler acceleration the equations of motion imply conservation of these functions in time, ∂_tγ=0=∂_tω, which are near horizon analogs of the holographic Ward identities ∂_∓ T^±±=0 for the usual Brown–Henneaux boundary conditions <cit.>.The canonical boundary charges associated with these choices are given by the Fourier modesJ^±_n = 1/16πG ∮ e^±in (γ±ωℓ) eq:k5 where G is the 3-dimensional Newton constant. Their algebra consists of two 𝔲(1) current algebras[J_n^±, J_m^±] = ℓ/8G n δ_n+m, 0[J^+_n, J^-_m] = 0 . eq:k6 The zero mode charges J_0^± commute with all generators J_n^±, which encodes algebraically the “soft hair” property. Since the algebra (<ref>) can be mapped to infinite copies of the Heisenberg algebra frequently the name “soft Heisenberg hair” is used to distinguish from other soft hair constructions.The Bekenstein–Hawking entropy is given byS_BH = 2π ∮γ/4G = 2π (J_0^+ + J_0^-) eq:k7 which coincides with the result announced in (<ref>). While in Einstein gravity the result (<ref>) is perhaps not too surprising, its simplicity is still remarkable and should be contrasted with the usual form of the entropy expressed in terms of Virasoro zero modes L_0^± <cit.>,S_Cardy = 2π (√(c L_0^+/6)+√(c L_0^-/6)) eq:k8 with the Brown–Henneaux central chargec=3ℓ/2G. eq:angelinajolie More suprisingly, formula (<ref>) applies also to flat space, where the usual Cardy-like formula is given by <cit.> (L_0, M_0 are BMS_3 zero modes and c_M is the Barnich–Compère central charge <cit.>)S_BMS_3 = 2π L_0 √(c_M/2M_0) eq:k9 and to situations where Bekenstein–Hawking fails, such as higher spin <cit.> black holes <cit.>, where even for the simplest case of spin-3 black holes the entropy looks fairly complicated <cit.> (L_0^± and c are the same as in the spin-2 case; W_0^± denotes the spin-3 zero mode charges with suitable normalization; for W_0^±=0 the Cardy formula (<ref>) is recovered)S_spin 3=2π (√(cL_0^+/6)cos(1/3 arcsin(W_0^+(L_0^+)^-3/2)) + √(cL_0^-/6)cos(1/3 arcsin(W_0^-(L_0^-)^-3/2))) . eq:k10 Similar remarks and results apply to higher spin flat space <cit.> cosmologies <cit.>.Let us note for later purposes that in the flat space case the algebra (<ref>) appears naturally in an off-diagonal basis, J_n = J_n^+ + J_-n^- and K_n = 1/ℓ (J_n^+ - J_-n^-), such that[J_n, K_m] = 1/4Gn δ_n+m, 0 eq:k31 and the entropy (<ref>) readsS = 2πJ_0 . eq:k32The algebraic reasons behind these surprising results are Sugawara-type of relations between the near horizon generators J_n^± and the usual asymptotic variables, like Virasoro generators, BMS_3 generators or higher spin generators. For example, the Virasoro generators L_n^± emerge uniquely from a Miura-type of transformation (spelled out in <cit.>) that leads to a Sugawara constructionL_n^±= 4G/ℓ ∑_p J_n-p^±J_p^±+ in J_n^±eq:k11 with a precise twist-term that allows to recover the Brown–Henneaux central charge (<ref>). Inserting this central charge together with the vacuum expectation value of the zero-mode generatorL_0^±= 4G/ℓ(J_0^±)^2 eq:k12 into the Cardy formula (<ref>) recovers formula (<ref>). In the spin-3 case the Sugawara-type construction involves tri-linears in the spin-2 and spin-3 generators J_(2,3) n^± <cit.> which in a similar way allows to recover formula (<ref>) from the complicated expression (<ref>) and vice versa. A summary of some of these algebraic constructions is provided in <cit.>.Besides these unexpected simplifications of the results for entropy, at least in three dimensions one can go further and exploit the near horizon theory for a microstate counting (including log corrections), see <cit.> and references therein. Moreover, based on the 𝔲(1) current algebras (<ref>) a specific proposal for all microstates of semi-classical BTZ black holes was provided in <cit.>.While it is not yet clear how these more advanced considerations generalize to higher dimensions, the more basic ones reviewed here show at least that there is a simple and fairly universal entropy law (<ref>) in three dimensions that often considerably simplifies the results and paves the way for a deeper analysis of black hole entropy. We show in the next section that something similar can be achieved for non-extremal Kerr black holes.§ KERR BLACK HOLE ENTROPY§.§ Bekenstein–Hawking entropy Kerr black holes are characterized uniquely by their mass M and angular momentum J = aM, in terms of which their Bekenstein–Hawking entropy reads (we set Newton's constant to unity)S_Kerr = 2π M^2 (1 + √(1 - a^2/M^2)) . eq:k13 §.§ Near horizon symmetry algebra by Donnay et al. To continue with our agenda we need some analog of the near horizon boundary conditions reviewed in section <ref>, which so far do not exist. Thus, we use instead the near horizon boundary conditions introduced in <cit.> (see also <cit.>), which lead to a symmetry algebra whose non-vanishing commutators are displayed below.[L_n^±, L_m^±]= (n-m) L_n+m^± [L_n^+, T_(m, k)]= -m T_(n+m, k) [L_n^-, T_(m, k)]= -k T_(m, n+k) The Witt generators L_n^± produce “superrotations” and the remaining generators T_(n, m) “supertranslations”. Note, however, that the algebra (<ref>) differs from the Barnich–Troessaert generalization <cit.> of BMS <cit.>.§.§ Sugawara deconstruction There are three key observations for our purposes. The first one, made in <cit.>, relates the supertranslation double-zero-mode to the Kerr black hole entropy (<ref>),[ For convenience our normalization of T_(0, 0) differs from the one in <cit.> by a factor of surface gravity and a factor 2.]S_Kerr = 4π T_(0,0) eq:k15 The second one (made in the same paper) is that for the Kerr solution the Witt zero mode generators, up to an imaginary factor, are given by the Kerr angular momentum.L_0^±= ±i/2 J eq:k18 The third one, made in <cit.>, provides a “Sugawara-deconstruction” of the algebra (<ref>) in terms of four 𝔲(1) current algebras (commutators not displayed vanish)[_n^±, _m^±] = - [_n^±, _m^±] = n/2δ_n+m, 0eq:k16 namelyT_(n, m) = (_n^+ + _n^+)(_m^- + _m^-) L_n^±= ∑_p (_n-p^±+ _n-p^±) (_p^±- _p^±) . eq:k17The “deconstruction” above is of course not unique and we can change to a more convenient basis, which we do now. Defining the generatorsJ_n^±:= ^±_n + ^±_n K_n^±:= ^±_n - ^±_neq:k25 yields an algebra consisting of two copies of the 3-dimensional flat space near horizon symmetry algebra (<ref>) (again all commutators not displayed vanish)[J_n^±, K_m^±] = nδ_n+m, 0. eq:k26 The generators (<ref>) are given by the bilinearsT_(n, m) = J_n^+ J_m^- L_n^±= ∑_p J_n-p^±K_p^± . eq:k30 §.§ Determination of zero modes Comparing with <cit.> for the non-extremal Kerr solution the four zero modes of the new generators have to obey three algebraic constraints (determined by their relations to L_0^± and T_(0, 0)). To solve uniquely for these zero modes we need a fourth condition. We imposeJ_0^+ + J_0^- + K_0^+ + K_0^- != 2Meq:k33 based on the rationale that this chirally symmetric sum of zero modes should not see the angular momentum, while dimensional analysis shows it must be linear in mass. The least obvious aspect of the postulate (<ref>) is the factor 2 on the right hand side. This factor is fixed uniquely by demanding that in the Schwarzschild limit of vanishing angular momentum, J→ 0, the zero modes agree with each other pairwise, J_0^+ = J_0^- (the other relation, K_0^+=K_0^-, holds automatically in this limit since both quantities tend to zero). Solving the algebraic equations (<ref>), T_(0, 0)=J_0^+ J_0^- and L_0^± = J_0^± K_0^± establishes uniquely results for the zero modes of the new generators. J_0^±= 1/2 (M + √(M^2-a^2) ±i a) K_0^±= 1/2 (M - √(M^2-a^2) ±i a)eq:k27 §.§ Results for non-extremal Kerr in terms of u(1) zero modes We express now standard Kerr parameters in terms of the zero mode charges (<ref>). The black hole and Cauchy horizon radii r_+ = J_0^+ + J_0^- r_- = K_0^+ + K_0^-eq:k22 as well as the black hole mass M = J_0^+ + K_0^- = J_0^- + K_0^+ eq:k28 are linear in the zero mode charges, while angular momentumJ = -2iJ_0^+ K_0^+ = 2iJ_0^- K_0^-eq:k23 and, equivalently, the Virasoro zero modes (<ref>)L_0^±= J_0^±K_0^±eq:k29 are bilinear in them.The Bekenstein–Hawking entropy (<ref>) is also bilinear in the zero modes. S_Kerr = 4π J_0^+ J_0^-eq:k20 The new entropy formula (<ref>) is the main result of this work. The non-trivial aspect is not just that the entropy can be written as in (<ref>), but rather that the quantities therein, J_0^±, are (linear combinations of) zero modes of 𝔲(1) generators that are related in a Sugawara-like way to the near horizon charges (<ref>) discovered in <cit.>. As compared to the 3-dimensional flat space result (<ref>) the entropy now contains a product of two zero-mode charges instead of a single one. The factor 4π in (<ref>) is naturally interpreted as volume of the round unit 2-sphere.§ ELABORATIONS In this section we discuss some elaborations. We address inner horizon entropy in section <ref>, mention Schwarzschild and extremal limits in section <ref>, and present the generalization to Kerr–Taub–NUT in section <ref>. We comment on some further interesting generalizations in the concluding section <ref>. §.§ Inner horizon entropy The “inner horizon entropy” defined and used in <cit.> is bilinear in the other pair of zero modes [the ones not appearing in the new entropy formula (<ref>)].S_inner = 2π K_0^+ K_0^- = 2π M^2 (1-√(1-a^2/M^2)) eq:k21 The result above suggests that the generators J_n^± are associated with the outer (black hole) horizon, while the generators K_n^± are associated with the inner (Cauchy) horizon.(Note that one can define generators T̃_(n,m) = K_n^+ K^-_m that also satisfy (<ref>), but with T replaced by T̃, which can be interpreted as generating supertranslations on the inner horizon.)This suggestion is confirmed by the expressions for outer and inner horizon radii (<ref>). As may have been anticipated, angular momentum (<ref>) is a quantity that involves information about both horizons. The same is true for the mass (<ref>). §.§ Schwarzschild and extremal Kerr For Schwarzschild the inner horizon charges vanish, K_0^±=0. Other than that all the results of section <ref> apply. By contrast, in the extremal limit the outer and inner horizon charges are identical, J_0^± = K_0^±, which physically makes sense as both horizons then coalesce to a single one. In that case the new entropy formula (<ref>) by virtue of the angular momentum formulas (<ref>) can be rewritten asS_EK = 4π √(J_0^+ K_0^+ J_0^- K_0^-) = 2π |J|eq:k34which coincides with the near horizon extremal Kerr result <cit.>. Note, however, that the boundary conditions of <cit.> on which our construction is based solely apply to non-extremal Kerr black holes, so one has to be careful with this limit. See <cit.> for more on extremal Kerr entropy in a near horizon context, including a proposal for its microstates. §.§ Kerr–Taub–NUT We consider now a generalization to Kerr–Taub–NUT <cit.> whose metric in Boyer–Lindquist coordinates is given bys^2 = - Δ/Σ (t - (a sin^2 θ- 2n cosθ)φ)^2 + Σ/Δ r^2 + Σ θ^2 + sin^2θ/Σ ( a t^2 -(r^2+ a^2 + n^2)φ)^2 eq:KTN with Kerr parameter a=J/M, nut charge n and the two functionsΔ= r^2 - 2 Mr +a^2 - n^2 and Σ= r^2 + ( n + a cosθ)^2 . eq:SDdef Rather than repeating in detail the analysis we did for Kerr we merely state the result for the zero mode charges.J_0^±= 1/2 (M + √(M^2-a^2 + n^2) ±i a) K_0^±=1/2 ( M - √(M^2-a^2 + n^2) ±i a) eq:k35 Besides mass M and angular momentum J the zero mode charges (<ref>) depend also on the nut charge n.Repeating the analysis of section <ref> yields expressions for the horizon radiir_+ = J_0^+ + J_0^- = M + √(M^2-a^2+n^2) r_- = K_0^+ + K_0^- = M - √(M^2-a^2+n^2)eq:k36 and the mass.M = J_0^+ + K_0^- = J_0^- + K_0^+eq:k39 The main change as compared to Kerr is that the left and right Witt zero mode charges L_0^±=J_0^± K_0^± do not add up to zero, L_0^+ + L_0^-≠ 0, but instead yield the square of the nut charge.-n^2/2= J_0^+ K_0^+ + J_0^- K_0^- eq:k41 The angular momentum J = a M is determined from the difference of the Witt zero mode charges. iJ= J_0^+ K_0^+ - J_0^- K_0^- eq:k40 The Bekenstein–Hawking entropy is again compatible with our main result (<ref>).S_Kerr-Taub-NUT=4πJ_0^+ J_0^- = 2π (M^2 +n^2 + M √(M^2-a^2+n^2)) eq:k37 For completeness we also state the corresponding result for inner horizon entropy.S_inner=4πK_0^+ K_0^- = 2π (M^2 +n^2 - M √(M^2-a^2+n^2)) eq:k38 §.§ Summary of relations between u(1) zero mode charges and black hole parameters Table <ref> summarizes various linear and bilinear relations between the 𝔲(1) zero mode charges J_0^±, K_0^± and black hole quantities. § OUTLOOK Our main result (<ref>) is a reformulation of the usual Kerr black hole entropy. As indicated, we view this as the first modest step in a more ambitious program, namely to provide near horizon boundary conditions, discuss associated symplectic and/or asymptotic symmetries, derive Cardyology and propose Hardyology for non-extremal Kerr. The endgame of this research avenue is to provide an explicit set of all microstates for semi-classical non-extremal Kerr black holes and to discuss consequences for black hole evaporation, (avoidance of) information loss and related semi-classical black hole puzzles.In four and five spacetime dimensions Cvetič and collaborators have found that the product of black hole and “inner horizon” entropies is some integer multiple of 4π^2, see <cit.> and references therein as well as papers by Ansorg and collaborators, e.g. <cit.>. Our results (<ref>) and (<ref>) actually suggest that already the individual entropies are some integer multiples of 2π, provided the zero mode charges J_0^± and K_0^± are quantized in the integers. Results for three-dimensional black holes suggest that this is the case <cit.>, but clearly it would be desirable to verify this in four dimensions.Another interesting research direction is to relate our results to the hidden conformal symmetry for Kerr black holes, see <cit.> and references therein. While we do not see direct evidence for conformal symmetry, our 𝔲(1) currents J_n^± and K_n^± can be Sugawara-combined to Virasoro generators, which then indirectly generate conformal symmetry, see <cit.> for a summary of the relevant algebraic statements.The new entropy formula for Kerr (<ref>) and some of the directions alluded to above should be generalizable to other black holes in four dimensions with cosmological constant, electromagnetic sources and acceleration (e.g. Carter–Kottler a.k.a. Kerr-de Sitter <cit.>, Kerr–Newman <cit.>, accelerated black holes <cit.>, Plebański–Demiański <cit.> or its cosmological generalization <cit.>, and black holes with toroidal or higher genus horizons <cit.>) as well as to black holes (and possibly other black objects such as black rings <cit.>) in more than four spacetime dimensions. Studying these generalization would also allow to verify if the simple entropy formula (<ref>) is again as universal as its 3-dimensional pendant (<ref>).Besides the four- and higher-dimensional applications mentioned above there is still some interesting work to be done in three spacetime dimensions. Namely, so far formula (<ref>) was verified solely for locally maximally symmetric metrics, such as Bañados–Teitelboim–Zanelli black holes <cit.> or flat space cosmologies <cit.>. However, the near horizon approximation (<ref>) should be valid for any horizon, even if the solution is not maximally symmetric. It would be interesting to verify whether our general conclusions still apply to such scenarios, for instance to warped black holes, see e.g. <cit.>. Further open issues in three spacetime dimensions are listed in the concluding section of <cit.>.Whether all these research avenues will turn out to be fruitful is unclear. However, in our opinion it is exciting that there is now a new approach available — based on near horizon considerations and soft Heisenberg hair — to address macro- and microscopic aspects of black hole entropy.§ ACKNOWLEDGMENTS DG thanks Soo-Jong Rey and the organizers of the 13^th International Conference on Gravitation, Astrophysics, and Cosmology and the 15^th Korea-Italy Relativistic Astrophysics Symposium for the invitation and the participants for enjoyable discussions. 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"authors": [
"Hernan Gonzalez",
"Daniel Grumiller",
"Wout Merbis",
"Raphaela Wutte"
],
"categories": [
"hep-th",
"gr-qc"
],
"primary_category": "hep-th",
"published": "20170927180000",
"title": "New entropy formula for Kerr black holes"
} |
Some New Results on Charged Compact Boson Stars [ December 30, 2023 =============================================== Given a triangle-free planar graph G and a 9-cycle C in G, we characterize situations where a 3-coloring of C does not extend to a proper 3-coloring of G.This extends previous results when C is a cycle of length at most 8. § INTRODUCTIONGiven a graph G, let V(G) and E(G) denote the vertex set and the edge set of G, respectively.We will also use |G| for the size of E(G).A proper k-coloring of a graph G is a function φ:V(G) →{1,2,…, k} such that φ(u)≠φ(v) for each edge uv∈ E(G). A graph G is k-colorable if there exists a proper k-coloring of G, and the minimum k where G is k-colorable is the chromatic number of G.Garey and Johnson <cit.> proved that deciding if a graph is k-colorable is NP-complete even when k=3.Moreover, deciding if a graph is 3-colorable is still NP-complete when restricted to planar graphs <cit.>. Therefore, even though planar graphs are 4-colorable by the celebrated Four Color Theorem <cit.>, finding sufficient conditions for a planar graph to be 3-colorable has been an active area of research.A landmark result in this area is Grötzsch's Theorem <cit.>, which is the following: Every triangle-free planar graph is 3-colorable.We direct the readers to a nice survey by Borodin <cit.> for more results and conjectures regarding 3-colorings of planar graphs. A graph G is k-critical if it is not (k-1)-colorable but every proper subgraph of G is (k-1)-colorable.Critical graphs are important since they are (in a certain sense) the minimal obstacles in reducing the chromatic number of a graph.Numerous coloring algorithms are based on detecting critical subgraphs. Despite its importance, there is no known characterization of k-critical graphs when k≥ 4.On the other hand, there has been some success regarding 4-critical planar graphs.Extending Theorem <ref>, the Grünbaum–Aksenov Theorem <cit.> states that a planar graph with at most three triangles is 3-colorable, and we know that there are infinitely many 4-critical planar graphs with four triangles.Borodin, Dvořák, Kostochka, Lidický, and Yancey <cit.> were able to characterize all 4-critical planar graphs with four triangles.Given a graph G and a proper subgraph C of G, we say G is C-critical for k-coloring if for every proper subgraph H of G where C⊆ H, there exists a proper k-coloring of C that extends to a proper k-coloring of H, but does not extend to a proper k-coloring of G. Roughly speaking, a C-critical graph for k-coloring is a minimal obstacle when trying to extend a proper k-coloring of C to a proper k-coloring of the entire graph.Note that (k+1)-critical graphs are exactly the C-critical graphs for k-coloring with C being the empty graph.In the proof of Theorem <ref>, Grötzsch actually proved that any proper coloring of a 4-cycle or a 5-cycle extends to a proper 3-coloring of a triangle-free planar graph.This implies that there are no triangle-free planar graphs that are C-critical for 3-coloring when C is a face of length 4 or 5.This sparked the interest of characterizing triangle-free planar graphs that are C-critical for 3-coloring when C is a face of longer length.Since we deal with 3-coloring triangle-free planar graphs in this paper, from now on, we will write “C-critical” instead of “C-critical for 3-coloring” for the sake of simplicity.The investigation was first done on planar graphs with girth 5. Walls <cit.> and Thomassen <cit.> independently characterized C-critical planar graphs with girth 5 when C is a face of length at most 11.The case when C is a 12-face was initiated in <cit.>, but a complete characterization was given by Dvořák and Kawarabayashi in <cit.>. Moreover, a recursive approach to identify all C-critical planar graphs with girth 5 when C is a face of any given length is given in <cit.>. Dvořák and Lidický <cit.> implemented the algorithm from <cit.> and used a computer to generate all C-critical graphs with girth 5 when C is a face of length at most 16. The generated graphs were used to reveal some structure of 4-critical graphs on surfaces without short contractible cycles. It would be computationally feasible to generate graphs with girth 5 even when C has length greater than 16.The situation for planar graphs with girth 4, which are triangle-free planar graphs, is more complicated since the list of C-critical graphs is not finite when C has size at least 6. We already mentioned that there are no C-critical triangle-free planar graphs when C is a face of length 4 or 5.An alternative proof of the case when C is a 5-face was given by Aksenov <cit.>. Gimbel and Thomassen <cit.> not only showed that there exists a C-critical triangle-free planar graph when C is a 6-face, but also characterized all of them.A k^--cycle, k^+-cycle is a cycle of length at most k, at least k, respectively.A cycle C in a graph G is separating if G-C has more connected components than G.Let G be a connected triangle-free plane graph with outer face bounded by a 6^--cycle C=c_1c_2⋯.The graph G is C-critical if and only if C is a 6-cycle, all internal faces of G have length exactly four and G contains no separating 4-cycles.Furthermore, if φ is a 3-coloring of C that does not extend to a 3-coloring of G, then φ(c_1)=φ(c_4), φ(c_2)=φ(c_5), and φ(c_3)=φ(c_6).Aksenov, Borodin, and Glebov <cit.> independently proved the case when C is a 6-face using the discharging method, and also characterized all C-critical triangle-free planar graphs when C is a 7-face in <cit.>. The case where C is a 7-face was used in <cit.>.Let G be a connected triangle-free plane graph with outer face bounded by a 7-cycle C=c_1⋯ c_7.The graph G is C-critical and ψ is a 3-coloring of C that does not extend to a 3-coloring of G if and only if G contains no separating 5^--cyclesand one of the following propositions is satisfied up to relabelling of vertices (see Figure <ref> for an illustration). (a) The graph G consists of C and the edge c_1c_5, and ψ(c_1)=ψ(c_5).(b) The graph G contains a vertex v adjacent to c_1 and c_4, the cycle c_1c_2c_3c_4v bounds a 5-face and every face drawn inside the 6-cycle vc_4c_5c_6c_7c_1 has length four; furthermore, ψ(c_4)=ψ(c_7) and ψ(c_5)=ψ(c_1).(c) The graph G contains a path c_1uvc_3 with u,v∉V(C), the cycle c_1c_2c_3vu bounds a 5-face and every face drawn inside the 8-cycle uvc_3c_4c_5c_6c_7c_1 has length four; furthermore, ψ(c_3)=ψ(c_6), ψ(c_2)=ψ(c_4)=ψ(c_7), and ψ(c_1)=ψ(c_5).Dvořák and Lidický <cit.> used a correspondence of nowhere-zero flows and colorings to give simpler proofs of the case when C is either a 6-face or a 7-face, and also characterized C-critical triangle-free planar graphs when C is an 8-face. For a plane graph G, let S(G) denote the set of multisets oflengths of internal faces of G with length at least 5. Let G be a connected triangle-free plane graph with outer face bounded by an 8-cycle C. The graph G is C-critical if and only if G contains no separating 5^--cycles, the interior of every non-facial 6-cycle contains only 4-faces, and one of the following propositions is satisfied (see Figure <ref> for an illustration).(a) S(G)=∅. (b) S(G)={6} and the 6-face of G intersects C in a path of length at least one. (c) S(G)={5,5} and each of the 5-faces of G intersects C in a path of length at least two. (d) S(G)={5,5} and the vertices of C and the 5-faces f_1 and f_2 of G can be labelledin clockwise order along their boundaries so that C=c_1c_2⋯ c_8, f_1=c_1v_1zv_2v_3, and f_2=zw_1c_5w_2w_3(where w_1 can be equal to v_1, v_1 can be equal to c_2, etc).Theorem <ref> has the following corollary that was not explicitly stated in <cit.>. Let G be a triangle-free plane graph and let v be a vertex of degree 4 in G. Then there exists a proper 3-coloring of G where all neighbors of v are colored the same.The corollary can be proven by splitting v into four vertices of degree 2 that are in one 8-face F and precoloring F by two colors, see Figure <ref>. In this paper, we push the project further and characterize all C-critical triangle-free planar graphs when C is a 9-face. Let G be a connected triangle-free plane graph with outer face bounded by a 9-cycle C.The graph G is C-critical for 3-coloring if and only if for every non-facial 8^--cycle of K the subgraph of G drawn in the closed disk bounded by K is K-critical and one of the following propositions is satisfied(see Figure <ref> for an illustration). * S(G)={5} and the 5-face of G intersects C in a path of length at least two.* S(G)={7}.* S(G)={5, 6} and the 5-face and 6-face of G intersects C in a path of length at least two and one, respectively.* S(G)={5, 6} and G is depicted as (d1) or (d2) in Figure <ref>.* S(G)={5, 5, 5} and G is depicted as (Bij) in Figure <ref> for all i,j.* G contains a chord.0.19The proof of Theorem <ref> involves enumerating all integer solutionsto several small sets of linear constraints. It would be possible to solve them by hand but we have decided to use computer programs to enumerate the solutions. Both computer programs and enumerations of the solutions are available online on arXiv and at .§ PRELIMINARIESOur proof of Theorem <ref> uses the same method as Dvořák and Lidický <cit.>. The main idea is to use the correspondence between colorings of a plane graph G and flows in the dual of G. In this paper, we give only a brief description of the correspondence and state Lemma <ref> from <cit.>, which is used throughout this paper. A more detailed and general description can be found in <cit.>.Let G^⋆ denote the dual of a 3-colorable plane graph G. Let φ be a proper 3-coloring of the vertices of G by colors {1,2,3}.For every edge uv of G, we orient the corresponding edge e in G^⋆ in the following way. Let e have endpoints f,h in G^⋆, where f,v,h is in the clockwise order from vertex u in the drawing of G. The edge e will be oriented from f to h if (φ(u),φ(v)) ∈{(1,2),(2,3),(3,1)}, and from h to f otherwise. See Figure <ref> for an example of the orientation.Since φ is a proper coloring, every edge of G^⋆ has an orientation. Tutte <cit.> showed thatthis orientation of G^⋆ defines a nowhere-zero ℤ_3-flow, which means that the in-degree and the out-degree of every vertex in G^⋆ differ by a multiple of three. Conversely, every nowhere-zero ℤ_3-flow inG^⋆ defines a proper 3-coloring of G up to the rotation of colors.Let h be the vertex in G^⋆ corresponding to the outer face of G. Edges oriented away from h are called source edges and the edges oriented towards h are called sink edges. The orientations of edges incident to h depend only on the coloring of C, where C is the cycle bounding the outer face of G. Denote bythe number of source edges and bythe number of sink edges. For a subgraph Z of G or a subset Z of E(G), we will use Z and Z to denote the number of source edges and sink edges in G^⋆ whose dual is in Z, respectively.Recall that only edges in C have source edges or sink edges in the dual. For a vertex f of G^⋆, let δ(f) denote the difference of the out-degree and in-degree of f. Possible values of δ(f) depend on the size of the face corresponding to f, denoted by |f|. Clearly |δ(f)| ≤ |f| and δ(f) has the same parity as |f|. Hence if |f| = 4, then δ(f) = 0.Similarly, if |f| ∈{5,7}, then δ(f) ∈{-3,3} and if |f| = 6 then δ(f) ∈{-6,0,6}.We call a function q assigning an integer to every internal face f of G a layout if q(f) ≤ |f|, q(f) is divisible by 3, and q(f) has the same parity as |f|. Notice that q(f) satisfies the same conditions as δ(f).Therefore it is sufficient to specify the q-values for faces of size at least 5, since q(f)=0 if f is a 4-face. A layout q is ψ-balanced if +m=, where m is the sum of the q-values over all internal faces of G.Our main tool is the following lemma from <cit.>. Let G be a connected triangle-free plane graph with outer face C bounded by a cycle and let ψ be a 3-coloring of C that does not extend to a 3-coloring of G. If q is a ψ-balanced layout in G, then there exists a subgraph K_0⊆ G such that either i) K_0 is a path with both ends in C and no internal vertex in C, and if P is a path in C joining the end vertices of K_0, n^s is the number of source edges of P, n^t is the number of the sink edges of P, and m is the sum of the q-values over all faces of G drawn in the open disk bounded by the cycle P+K_0, then | n^s+m-n^t| > | K_0|. In particular, | P| + | m | >| K_0|. Or,ii)K_0 is a cycle with at most one vertex in C, and if m is the sum of the q-values over all faces of G drawn in the open disk bounded by K_0, then | m | > | K_0 |.For a multiset of numbers F, let ℓ(F) denote the smallest integer ℓ such that there exists a triangle-free plane graph G with outer face bounded by an ℓ-cycle C, such that G is C-critical and S(G)=F. It is known from <cit.> that ℓ({i}) = i+2 and ℓ({5,6}) = 9. The next lemma from <cit.> describes interiors of cycles in critical graphs and will be used frequently in this paper.Let G be a plane graph with outer face C. Let K be a non-facial cycle in G,and let H be the subgraph of G drawn in the closed disk bounded by K. If G is C-critical for k-coloring, then H is K-critical for k-coloring.Next we include several definitions used throughout the rest of the paper. For the definitions, we assume that G is a graph with outer face bounded by a cycle C.An x, y-path is a path with endpoints x and y.Given a, b, c, d∈ V(C), let C(a, b; c, d) denote the a, b-subpath of C that does not contain vertices c and d as internal vertices.An x, y-path K is an (x, y; f)-cut if x, y are on C, no internal vertices of K are on C, and the face f is in the region bounded by K and the clockwise x, y-subpath of C. Let K_1 and K_2 be two distinct paths with endpoints on C that are internally disjoint from C. For i ∈{1,2} let P_i be a subpath of C with the same endpoints as K_i and label the endpoints of K_i by u_i and v_i, where u_i is the first vertex of P_i when traversing the cycle formed by K_i and P_i clockwise. The order of u_1,v_1,u_2,v_2 is an ordering of these vertices when traversing along C in the clockwise order. If x_1 ∈{u_1, v_1} and x_2 ∈{u_2,v_2} are the same vertex, we define the order of x_1 and x_2 in the following way. Let x_1=y_0,…,y_m be the longest common subpath of K_1 and K_2.We consider the neighbors N of y_m in the counterclockwise order ending with y_m-1 or a vertex of C if m=0.If a vertex of K_1 appears in N before every vertex of K_2, then x_1 is before x_2 in the ordering, otherwise x_2 is before x_1. Every order, or the pair K_1,K_2, is assigned a kind (t_1t_2), where t_iis the number of vertices from {u_3-i,v_3-i} that are in P_i. Hence there are only five possible kinds; namely (00), (02), (20), (22), and (11).If K_1 is the same path as K_2, we can pick any order that will give kind (00), (02),or (20). Suppose that the order is u_1,v_2,v_1,u_2, which gives kind (11). By planarity, there exists a vertex x such that x is an internal vertex of both K_1 and K_2. Denote byK_i^y a subpath of K_i with endpoints x and y for i ∈{1,2} and y ∈{u_i,v_i};see Figure <ref>. Let F_i be the set of 5^+-faces that are in the interior of the cycle bounded byP_i and K_i and in the exterior of the cycle bounded by P_3-i and K_3-i for i ∈{1,2}. The vertex x is a common point of K_1 and K_2 ifevery face in F_1 is in an interior face of the subgraph of G induced by P_1,K_1^v_1,K_2^v_2 and every face in F_2 is in an interior face of the subgraph of G induced by P_2,K_1^u_1,K_2^u_2.It is possible to show that there always exists a common point for the kind (11) if K_1 and K_2 are not too long.Let G be a triangle-free plane graph with outer face C where every 4-cycle bounds a face and let K_1 and K_2 be paths in G with endpointsu_1,v_1,u_2,v_2 in C. Let K_1 and K_2 be internally disjoint with C and the order of u_1,v_1,u_2, and v_2form (11). Then there exists a common pointfor K_1 and K_2 if eithermax{|K_1|,|K_2|}≤ 7 and min{|K_1|,|K_2|}≤ 6or |K_1|=|K_2|=7 and the endpoints of K_1 and K_2 are the same. We describe operations that eliminate candidates for common points. Eventually, we show that the situation is equivalent to the case where K_1 and K_2 share exactly one vertex and then it is easy to see it is the common point.By planarity and the kind (11), there must be at least one vertex of G that is internal vertex of both K_1 and K_2.When traversing K_2 from v_2 to u_2 we label the internal vertices of K_2 that are also vertices of K_1 by c_1,c_2,c_3,….These vertices are candidates to be common points.We order them by their distance from u_1 on the path K_1. Let P_1 be the clockwise path in C from u_1 to v_1.An edge of K_2 is inside if it is drawn inside of the open disk bounded by the cycle formed by P_1 and K_1 or if it is incident with v_2. An edge of K_2 is outside if it is drawn outside of the closed disk bounded by the cycle formed by P_1 and K_1 or if it is incident with u_2.Notice that if an edge of K_2 is neither inside nor outside, it is also an edge of K_1 and we call it shared. Now we do several modifications to G, K_1 and K_2 such that the 5^+-faces are not affected but some candidates for common vertices are eliminated. For some i, if c_i is adjacent to one shared edge and one not shared edge h, then we split c_i into two vertices c^1_i and c^2_i, creating a new 4-face containing c^1_i,c^2_i and the two other neighbors of c_i in K_1. We can replace c_i by c^1_i and c^2_i in K_1 and K_2 such thatc^2_i is inside or outside of the new cycle formed by P_1K_1 if h is inside or outside, respectively. By performing this operation, we decrease the number of vertices in the intersection of K_1 and K_2 and we can assume K_2 has no shared edges.If both edges of K_2 incident with c_i for some i are inside (or outside) we split c_i into two vertices c^1_i and c^2_i, creating a new 4-face containing c^1_i,c^2_i and the other neighbors of c_i in K_1. We can replace c_i by c^1_i and c^2_i in K_1 and K_2, respectively. We label the vertices such that c^2_i is in the interior (or exterior, respectively) of the cycle bounded by K_1 and P_1.By performing this operation, we decrease the number of vertices in the intersection of K_1 and K_2 and assume that c_i is incident to one inside edge and one outside edge for all i. If c_i and c_i+1 are consecutive in the order given by the distance from u_1 andthe subpaths of K_1 and K_2 with endpoints c_i and c_i+1 form a 4-cycle K (hence a 4-face), then we can reroute the paths such that the length of one of the paths is decreased or we create two vertices that are both incident with only inside or only outside edges. If one of the two paths forming K has length one, the other one has length three and replacing the longer one by an edge decreases the length of K_1 or K_2 (it also creates a new shared thatwe can eliminate).If both paths have length two, we swap them and now both c_i and c_i+1 are incident to two edges that are both inside or both outside and they can be eliminated.Notice that these operations do not increases the length of K_1 or K_2, do not create new vertices in the intersection of K_1 or K_2, do not affects locations or number of 5^+-faces of G with respect to regions formed by K_1C and K_2C. Hence a common point in the result would be a common point in the original configuration.With use of computer, we generate all possible patterns where none of the above operations can be applied. In all of the patters withmax{|K_1|,|K_2|}≤ 7 and min{|K_1|,|K_2|}≤ 6, there is only one vertex shared by K_1 and K_2, which is the common point.If |K_1| = |K_2| =7, there are eight patterns with more than one internal vertex in the intersection of K_1 and K_2. Four of them do not actually form (11) and none of the three operations was used on them which is a contradiction. The other four contain 4-cycles that do not bound a face which is also a contradiction. The program including the eight patterns is available with all the other programs used in this paper.§ PROOF OF THEOREM <REF>Let _k be the set of possible multisets of lengths of 5^+-faces in a connected plane graph of girth at least 4 where the length of the precolored face is k. The result of Dvořák, Král, and Thomas <cit.> implies among others that _6 = {∅}, _7 = {{5}}, _8 = {∅, {6},{5,5}}, and_9 = {{7},{5},{6,5},{5,5,5}}. By the previous paragraph, we have four cases to consider when C has length 9. The case of one 7-face was already resolved by Dvořák and Lidický <cit.>, and it is described in Theorem <ref>(b). We restate the result from <cit.> in the next subsection as Theorem <ref>. We resolve the remaining three cases in Lemmas <ref>, <ref>, <ref>, <ref>, <ref>, and <ref> in the following three subsections. In order to simplify the cases, we first solve the case when C has a chord.If G is C-critical and C has a chord, then Lemma <ref> implies that G can be obtained by identifying two edges of the outer faces of two different smaller critical graphs or cycles. Lemma <ref> shows that the converse is also true. Let G_i be either a cycle C_i or a triangle-free plane C_i-critical graph, where |C_i| ≥ 4 for i ∈{1,2}. Let G be the graph obtained by identifying e_1 ∈ E(C_1) and e_2 ∈ E(C_2) and let C be the longest cycle formed by E(C_1) ∪ E(C_2) after the identification.Then G is C-critical, where |C|=|C_1|+|C_2|-2. Let e∈ E(G)∖ E(C).Suppose first that e ∈ E(G_i)-e_i for some i ∈{1,2}. Since G_i is either a cycle or a C_i-critical triangle-free plane graph and it contains e that is not on the boundary, G_i is C_i-critical. Hence there exists a 3-coloring φ of C_i that extends to a proper 3-coloring of G_i-e but does not extend to a proper 3-coloring of G_i.Since G_3-i is triangle-free, there exists a proper 3-coloring ϱ of G_3-i by Grötzsch's Theorem <cit.>. By permuting colors we may assume that φ and ϱ agree on e_i and e_3-i. This gives a proper 3-coloring of C showing that G is C-critical with respect to e.The other case is when e is the result of the identification of e_1 and e_2. Let u,v be the vertices of e. Since G-e is a triangle-free planar graph, there exists a proper 3-coloring φ of G-e such that φ(u)=φ(v);this is a result of Aksenov et al. <cit.> that was simplified by Borodin et al. <cit.>. Let ϱ be the restriction of φ to C.Clearly, ϱ can be extended to a proper 3-coloring of G-e but not to a proper 3-coloring of G. Therefore, we can enumerate C-critical triangle-free plane graphs G where C has a chord and has length 9 by identifying edges from two smaller graphs with outer faces of lengths either 4 and 7 or 5 and 6.Since there are no C-critical graphs when |C|∈{4, 5},we just use a 4-cycle and a 5-cycle. The resulting graphs are depicted in Figure <ref> (a), (b), (c1),(c2), (f1), and (f2), where some of the vertices may be identified. §.§ One 7-face The case of one 7-face is solved by a more general result from <cit.>.The result works for graphs with an outer face of lengthk and one internal face of length k-2. Let r(k)=0 if k≡ 03, r(k)=2 if k≡ 13, and r(k)=1 if k≡ 23. Let G be a connected triangle-free plane graph with outer face bounded by a 7^+-cycle C of length k. Suppose that f is an internal face of G of length k-2 and that all other internal faces of G are 4-faces.The graph G is C-critical if and only if (a) f∩ C is a path of length at least r(k) (possibly empty if r(k)=0),(b) G contains no separating 4-cycles, and(c) for every (k-1)^--cycle K≠ f in G, the interior of K does not contain f.Furthermore, in a graph satisfying these conditions, a precoloring ψ of C extends to a 3-coloring of G if and only if E(C)∖ E(f) contains both a source edge and a sink edge with respect to ψ. In our case, we apply Theorem <ref> with k=9. Since r(9)=0, the 7-face does not have to share any edges with the outer face. The description is in Theorem <ref>(b) and it is depicted in Figure <ref>(b). §.§ One 5-face and one 6-faceLet G be a connected triangle-free plane graph with outer face bounded by a chordless 9-cycle C. Moreover, let G contain one 5-face f_5 and one 6-face f_6, all other internal faces are 4-faces, and all non-facial 8^--cycles K in G bound K-critical subgraphs. If ψ is a 3-coloring of C that does not extend to a 3-coloring of G, then ψ extends to G-e for every e ∈ E(G)∖ E(C). Let e∈ E(G)∖ E(C).We want to show that ψ extends to a proper 3-coloring of G-e. Suppose that ψ does not extend to a 3-coloring of G-e.Then there exists a C-critical subgraph H of G-e, such that the 3-colorings of C that extend to G-e are exactly the 3-colorings of C that extend to H. Since H is C-critical, its multiset of5^+-faces is one of {5}, {7}, {5,6}, {5,5,5}. Since all non-facial 8^--cycles K in G bound K-critical subgraphs, Lemma <ref> implies thatevery 5-face of H is a 5-face of G,every 7-face of H contains exactly one 5-face of G,and a 6-face of H contains no 5-faces in the interior. Hence, H contains one odd 5^+-face and one even 6^+-face or one odd 9^+-face, and the only option for the multiset of 5^+-faces of H is {5,6}. That would mean that G is the same graph as H, and this is a contradiction.Notice that Lemma <ref> implies that in order to prove C-criticality, it is enough to find one coloring that does not extend. In Figure <ref> we depict colorings that do not extend.Now we prove the other direction of the Theorem <ref>. We start by the following lemma that we prove separately for future reference and then continue with the main part Lemma <ref>.Let G be a connected triangle-free plane graph with outer face bounded by a chordless 9-cycle C. Moreover, let G contain one 5-face f_5 and one 6-face f_6 and all other internal faces are 4-faces. IfG is C-critical ψ is a 3-coloring of G with 9 source edges then ψ extends to a 3-coloring of G. Suppose for a contradiction that ψ does not extend to a 3-coloring of G. Hence there is just one ψ-balanced layout q with q(f_5)=-3 and q(f_6)=-6.Let K_0 be obtained from Lemma <ref>.If K_0 is a cycle, then Lemma <ref> implies 9 > |K_0|. Let m denote the sum of the q-values of the faces in the interior of K_0. By Lemma <ref>, | m | > k_0. If both f_5,f_6 are in the interior of K_0, then | m | = | q(f_5)+q(f_6) | = 9, contradicting the fact that | m | > k_0 since k_0≥ℓ({5,6})=9.If f_5 is in the interior of K_0, butf_6 is not, then | m | = 3, while ℓ({5})=5, a contradiction again. Similarly, we obtain a contradiction when f_6 is in the interior of K_0 but f_5 is not, since ℓ({6})=6 and | m |≤ 6. Therefore K_0 is always a path joining two distinct vertices of C. These endpoints of K_0 partition the edges of C into two paths X and Y intersecting at the endpoints of K_0. For Z ∈{X,Y}, recall that Z and Z denotes the number of source edges and sink edges, respectively, among the edges of Z in coloring ψ.The described structure is shown in Figure <ref>. Let R_X and R_Y be the subgraph of G induced by vertices in the closed interior of the cycle formed by K_0,X and K_0,Y respectively.Note that X+ Y=9 and X+ Y=0.If both f_5, f_6 belong to R_X, then Lemma <ref> implies 9 -n^s_X > k_0 and Lemma <ref> implies n^s_X+k_0 ≥ 9 since ℓ({5,6})=9, which is a contradiction. By symmetry, R_Y does not contain both f_5 and f_6.Without loss of generality, suppose f_6 belongs to R_X and f_5 belongs to R_Y. Lemma <ref> implies that X + k_0 ≥ 6, which gives k_0 ≥ 6 -X. Lemma <ref> implies | X - 6| > k_0.Combining the inequalities give | X - 6| >6 -X, which implies X > 6. Hence Y < 3. Analogously, we obtainY + k_0 ≥ 5 and | Y - 3| > k_0, whose combination gives 3 -Y> 5 -Y, which is a contradiction.Let G be a connected triangle-free plane graph with outer face bounded by a chordless 9-cycle C. Moreover, let G contain one 5-face f_5 and one 6-face f_6 and all other internal faces are 4-faces. IfG is C-critical,thenG is described by Theorem <ref>(c),(d), and is depicted in Figure <ref>(c1),(c2),(d1), and (d2). Since G is C-critical, from Lemma <ref> follows thatevery non-facial 8^--cycle K in G bounds a K-critical subgraph.Since G is C-critical, there exists a 3-coloring ψ of C that does not extend to a proper 3-coloring of G.By symmetry, we assume that C has more source edges than sink edges. Hence C has either 9 or 6 source edges. Lemma <ref> eliminates the case of 9 source edges hence C has 6 source edges. Let q be a ψ-balanced layout of G.Let K_0⊂ G be obtained by Lemma <ref> and let k_0 = | K_0|.First suppose that K_0 is a cycle. Let m denote the sum of the q-values of the faces in the interior of K_0. By Lemma <ref>, | m | > k_0. If both f_5,f_6 are in the interior of K_0, then | m | = | q(f_5)+q(f_6) | = 6, contradicting the fact that | m | > k_0 since k_0≥ℓ({5,6})=9.If f_5 is in the interior of K_0, butf_6 is not, then | m | = 3, while ℓ({5})=5, a contradiction again. Similarly, we obtain a contradiction when f_6 is in the interior of K_0 but f_5 is not, since ℓ({6})=6 and | m |≤ 6. Therefore K_0 is always a path joining two distinct vertices of C. These endpoints of K_0 partition the edges of C into two paths X and Y intersecting at the endpoints of K_0. For Z ∈{X,Y}, recall that Z and Z denotes the number of source edges and sink edges, respectively, among the edges of Z in coloring ψ.The described structure is shown in Figure <ref>. Let R_X and R_Y be the subgraph of G induced by vertices in the closed interior of the cycle formed by K_0,X and K_0,Y respectively.If q is a ψ-balanced layout either with q(f_5)=-3 and q(f_6)=0 or with q(f_5)=3 and q(f_6)=-6, then both R_X and R_Y contain exactly one of f_5 and f_6.By Lemma <ref>, C contains 6 source edges, hence X+ Y=6 and X+ Y=3.By symmetry, suppose for a contradiction that both f_5, f_6 belong to R_X. Notice that q(f_5)+q(f_6)=-3 in both layouts. By Lemma <ref>, X+ X+k_0≥ℓ({5,6})=9, and by Lemma <ref>, | X -3 -X| > k_0. If X -3 -X > k_0, then we obtain X -3 -X > k_0 ≥ 9 -X- X. This gives X > 6, which is a contradiction.If - X +3 + X > k_0, then we obtain - X +3 +X > k_0 ≥ 9 -X- X. This gives X > 3, which is a contradiction. Since C has 6 source edges, we have two different ψ-balanced layouts. Let q_1 and q_2 be the layout where q_1(f_5)=-3, q_1(f_6)=0, and q_2(f_5)=3, q_2(f_6)=-6, respectively. Let K and L be the subgraph of G obtained by Lemma <ref> applied to q_1 and q_2, respectively, and let k=|K|and l=|L|.Note that we already showed that each of K and L is a path joining pairs of distinct vertices of C; let K and L be a (v_1, v_2; f_5)-cut and (w_1, w_2; f_6)-cut, respectively.The paths K and L form a structure of kind (00), (11), (22), (20), or (02), see Figure <ref>, Figure <ref>, and Figure <ref> for illustration. We discuss these cases in separate claims.If K and L are of kind (00), then G is depicted in Figure <ref>(c1). Note that K, L are not necessarily disjoint. By symmetry, let X be C(w_1, w_2; v_1, v_2) such that the disk bounded by L and X contains f_6. Similarly, let Z be C(v_1, v_2; w_1, w_2) such that the disk bounded by K and Z contains f_5.Denote by Y the edges of C that are neither in X nor in Z. See Figure <ref> (00).By the assumption that C has no chord, k ≥ 2 and l ≥ 2.By Claim <ref>, we know X+ Y+ Z=6 and X+ Y+ Z=3.Lemma <ref> implies l+ X≥ℓ({6})=6.Moreover, by parity, l+ X must be even. Similarly, Lemma <ref> implies that k+ Z ≥ℓ({5}) = 5 and it is odd. Lemma <ref> applied to q_1 and q_2 implies | X + Y- X- Y| > k and |3+ Z+ Y - Z- Y| > l, respectively.Here is the summary of the constraints:| X + Y- X- Y|> k|3+ Z +Y- Z- Y|> l l+ X ≥6and evenk+ Z ≥5 and oddX+ Y+ Z =6X+ Y+ Z =3min{k,l} ≥ 2 All integer solutions to these constraints are in the following table:X X Y Y Z Z k l 0 1 5 0 1 2 2 50 1 6 0 0 2 3 51 1 4 0 1 2 2 41 1 5 0 0 2 3 42 1 3 0 1 2 2 32 1 4 0 0 2 3 33 1 2 0 1 2 2 23 1 3 0 0 2 3 2 From these eight solutions we obtain the graph depicted in Figure <ref>(c1), up to identification of vertices. If K and L are of kind (22), then G is depicted in Figure <ref>(c2). By symmetry, let X be C(v_1, v_2; w_1, w_2) such that the disk bounded by K and X contains f_6. Similarly, let Z be C(w_1, w_2; v_1, v_2) such that the disk bounded by L and Z contains f_5.Denote by Y the edges of C that are in neither X nor Z. See Figure <ref> (22).By the assumption that C has no chord, k ≥ 2 and l ≥ 2.By Claim <ref>, we know X+ Y+ Z=6 and X+ Y+ Z=3.Lemma <ref> implies k+ X≥ℓ({6})=6.Moreover, by parity, k+ X must be even. Similarly, Lemma <ref> implies l+ Z ≥ℓ({5}) = 5 and it is odd. Lemma <ref> applied to q_1 and q_2 implies | X - X| > k and | Z + 3- Z| > l, respectively.Here are the constraints:| X - X|> k| Z + 3- Z|> l k+ X≥6and evenl+ Z ≥5 and oddX+ Y+ Z =6X+ Y+ Z =3min{k,l} ≥ 2All integer solutions to these constraints are in the following table: X X Y Y Z Z k l 4 0 0 2 2 1 2 2 4 0 0 3 2 0 2 3From these two solutions we obtain the graph depicted in Figure <ref>(c2).If K and L are of kind (11), then G is depicted in Figure <ref>(d1) or (d2). Assume that K and L are of kind (11), so that the clockwise order of their endpoints on C is v_1,w_2,v_2,w_1.Let v_1,w_2,v_2,w_1 partition C into four paths X,Y,Z,W in the clockwise order such that X is an w_1, v_1-path.Moreover, the disk bounded by X,Y,L contains f_6 and the disk bounded by K,Y,Z contains f_5. See Figure <ref> for an illustration. First we show that |K| ≤ 6 and |L| ≤ 7. We obtain the following set of constraints by applyingLemma <ref> and Lemma <ref>.|X +W -X -W | > |K| |Z +W +3 -Z -W | > |L| X +Y + |L|≥ℓ({6})=6and evenIn all solutions, t |K| ≤ 6 and |L| ≤ 7. Hence Lemma <ref> applies and K and L have a common point v.Partition L into paths L_1 and L_2 such thatL_1 and L_2 is a v, w_2-path and a v, w_1-path, respectively.Do a similar partition of K into K_1 and K_2. Since v is a common point, f_6 and f_5 is contained in interior faces of subgraphs of G induced by X,K_1,L_2 and Z,L_1,K_2, respectively. Let k_i = |K_i| and l_i = |L_i| for i ∈{1,2}.Note that min{k_1,k_2,l_1,l_2}≥ 1 since v is an internal vertex.We obtain the following set of constraints by applyingLemma <ref> and Lemma <ref>.|X +W -X -W | > k_1+k_2|Z +W +3 -Z -W | > l_1 + l_2k_1+l_2+ X≥ℓ({6})=6and evenl_1+k_2+ Z≥ℓ({5})=5and oddl_2+k_2+ X+ Y+ Z≥ℓ({5,6})=9and oddX+ Y+ Z +W =6X+ Y+ Z +W =3Inequalities (<ref>) and (<ref>) come from Lemma <ref>. Inequalities (<ref>)–(<ref>) come from the fact that interiors of cycles are also critical graphs. This system of equations has 68 solutions. In all of them, X +X + k_1 + l_2=6 and Z +Z + k_2+l_1 = 5. Hence the region bounded by X,K_1,L_2 is a 6-face and the region bounded by Z,L_1,K_2 is a 5-face. In order to generate only general solutions, where faces share as little with C as possible, we add constraints X +X= 0 and Z +Z = 0. Then the system has only two solutions. X X Y Y Z Z W W k_1 k_2 l_1 l_2 0 0 0 3 0 0 6 01 3 2 50 0 0 3 0 0 6 02 2 3 4 From these solutions we obtain graphs depicted in Figure <ref>(d1) and (d2). We also checked that the 68 solutions can indeed be obtained from these two by identifying some vertices. The solutions were obtained by a computer program that is available on arXiv and at . The case where K and L are of kind (02) does not occur. Assume that K and L are ofkind (02), so that the clockwise order of their endpoints on C is v_1,v_2,w_2,w_1.Let Y and X be the clockwise v_1, v_2-subpath and w_2,w_1-subpath, respectively, of C. Let Z be the edges of C that are in neither X nor Y. See Figure <ref> (02) for an illustration. Observe that (by the structure of K and L) the subgraph of G formed by Z, K, and L contains in the internal faces both f_5 and f_6 and at least one additional 4-face. Hence k+l+|Z| ≥ 15. We obtain the following set of constraints by applyingLemma <ref> and Lemma <ref>.|X -X | > k |Y -6- Y| > l k +l + Z≥ 15 X+ Y+ Z =6X+ Y+ Z =3This set of equations has no solution.The case where K and L are of kind (20) does not occur. Assume that K and L are ofkind (20), so that the clockwise order of their endpoints on C is w_1,w_2,v_2,v_1.Let Y and X be the clockwise w_1, w_2-subpath and v_2,v_1-subpath, respectively, of C. Let Z be the edges of C that are in neither X nor Y. See Figure <ref> (20) for an illustration. Observe that (by the structure of K and L) the subgraph of G formed by Z, K, and L contains in the internal faces both f_5 and f_6 and at least one additional 4-face. Hence k+l+|Z| ≥ 15. We obtain the following set of constraints by applyingLemma <ref> and Lemma <ref>.|Y - 3 -Y | > k |X +3 - X| > l k +l + Z≥ 15 X+ Y+ Z =6X+ Y+ Z =3This set of equations has no solution.This finishes the proof of Lemma <ref>. §.§ One 5-faceLet G be a connected triangle-free plane graph with outer facebounded by a chordless 9-cycle C. Moreover, let G contain one 5-face f_5 that shares a path of length at least two with C, all other internal faces of G are 4-faces,and all non-facial 8^--cycles K in G bound a K-critical graph. Then G is C-critical. Let e ∈ E(G) ∖ E(C).We want to find a 3-coloring ψ of C that does not extend to a proper 3-coloring of G but does extends to a proper 3-coloring of G-e.Note that if e ∉E(f_5), thenG-e has a 5-face and a 6-face, and if e ∈ E(f_5), then G-e has a 7-face. If every coloring of C extends to G-e, then we can let ψ be a coloring with 9 source edges since ψ does not extend to G as there is no ψ-balanced layout for G. If not all colorings of C extend to G-e, then there is a C-critical subgraph H of G-e where the same set of precolorings of C extends to G-e as well as to H. The property that every 8^--cycle K either bounds a face or a K-critical subgraph gives that H contains either a 5-face and a 6-face or a 7-face.Case 1: H contains a 5-face and a 6-face. Let ψ be a 3-coloring of C containing 9 source edges; in other words, the colors of the vertices around C are 1,2,3,1,2,3,1,2,3. Then ψ extends to a 3-coloring of H by Claim <ref>. However, ψ does not extend to a 3-coloring of G since it is not possible to create a ψ-balanced layout for G. Case 2: H contains a 7-face f_7.By Theorem <ref>, if ψ is a 3-coloring of C containing 9 source edges, then ψ does not extend to a proper 3-coloring of H, and if ψ is a 3-coloring of C containing 6 source edges and 3 sink edges, then ψ extends to a proper 3-coloring of H if E(C)∖ E(f_7) contains botha sink edge and a source edge with respect toψ. Now it remains to observe that there exists a coloringψ of C such that E(f_5) ∩ E(C) contains two sink edges and the third sink edge is inE(C)∖ E(f_7).The other edges of C are source edges.Such a coloring does not extend to G but it does extend to G-e.Let G be a connected triangle-free plane graph with outer facebounded by a chordless 9-cycle. Moreover, let G contain one 5-face f_5 and all other internal faces of G are 4-faces. If G is C-critical, then G is described by Theorem <ref>(a) and is depicted in Figure <ref>(a). Let G be C-critical.By Lemma <ref>, every 8^--cycle K bounds a face or a K-critical subgraph in G.Let e ∈ E(G) ∖ E(C) such that G-e contains a 7-face f_7. Let ψ be a 3-coloring of C that extends to G-e but does not extend to G.By Theorem <ref>, if ψ is a 3-coloring of C containing 9 source edges, then ψ does not extend to a proper 3-coloring of G-e. Hence ψ is a 3-coloring of C containing 6 source edges and 3 sink edges. Let q be a ψ-balanced layout of G. The only possibility is q(f_5)=-3.Since ψ does not extend to G and q isψ-balanced layout of G, Lemma <ref> can be applied. Notice that Lemma <ref> cannot give that K_0 is a cycle since |m| ≤ 3 and there is no cycle of length at most 2. Hence K_0 is a path, and let k_0=|K_0|. Let the endpoints of K_0 partition C into two paths X and Y that are internally disjoint and have the same endpoints as K_0. Since ψ has six source edges, we obtain X+ Y=6 and X+ Y=3.By symmetry assume that f_5 is in the region bounded by Y and K_0. Lemma <ref> implies | Y -3 -Y| > k_0. Since Y contains f_5, k_0+ Y≥ℓ({5})=5 and odd.Because C has no chords, k_0 ≥ 2. We solve this system of constraints by a computer program.The solutions are in Table <ref>.Sketches of the solutions are in Figure <ref>.0.170.19 From the first three solutions we obtain that Y is part of a 5-face f_5 sharing at least two sink edges with C. This is the desired conclusion. The other three solutions give that Y,K_0 form a7-cycle sharing at least three sink edges with C. We need to rule out this case. Since the cycle formed by Y,K_0 does not bounda face in G, it must contain an edge e' in its interior. Since G is C-critical, there exists a proper 3-coloring ϱ of C that does not extend to G but does extend to G-e'.Notice that the solutions (d), (e), and (f)also describe all patterns of a 3-coloring of C that do not extend to G, in particular for ϱ. In all three cases, X contains only source edges and |X| > |K_0|. Since the cycle formed by X,K_0 contains only 4-faces in its interior, it is not possible to create a ρ-balanced layout in its interior. Hence ϱ does not extend to subgraph of G-e' bounded X,K_0 .Thereforeϱ does not extend G. This contradicts the C-criticality of G. Hence the cases(d), (e), and (f) do not correspond to C-critical graphs.This finishes the proof of Lemma <ref>.§.§ Three 5-facesLet G be a connected triangle-free plane graph with outer facebounded by a chordless 9-cycle C. Moreover, let G contain three 5-faces, all other internal faces are 4-faces, and all non-facial 8^--cycles K in G bound a K-critical graph. If there is a proper 3-coloring ψ of C that does not extend toa 3-coloring of G, then G is C-critical. Let e∈ E(G)∖ E(C).We want to show that ψ extends to a proper 3-coloring of G-e. Suppose that ψ does not extend to a 3-coloring of G-e.Then there exists a C-critical subgraph H of G-e, such that the 3-colorings of C that extend to G-e are exactly the 3-colorings of C that extend to H. Since H is C-critical, its multiset of5^+-faces is one of {5}, {7}, {5,6}, {5,5,5}. Since all non-facial 8^--cycles K in G bound K-critical subgraphs, Lemma <ref> implies thatevery 5-face of H is a 5-face of G,every 7-face of H contains exactly one 5-face of G,and a 6-face of H contains no 5-faces in the interior. Hence, H contains three odd faces, and the only option for the multiset of 5^+-faces of H is {5,5,5}. That would mean that G is the same graph as H, and this is a contradiction. Let G be a connected triangle-free plane graph with outer face bounded by a chordless 9-cycle C.Moreover, let G contain three 5-faces and let all other internal faces of G be 4-faces. If G is C-critical, then G is described by Theorem <ref>(e)and is depicted in Figure <ref>(Bij) for some i and j.Let G be a C-critical graph containing three 5-faces. Hence there is a proper 3-coloring ψ of C that does not extend to a proper 3-coloring of G.Without loss of generality, assume C has more source edges than sink edges in the coloring ψ. Either C contains 9 source edges and no sink edges or C contains 6 source edges and 3 sink edges. Given i∈{0, 1, 2, 3}, let ℓ_5(i)=ℓ(S) where S is a multiset of cardinality i containing only elements 5.Observe that ℓ_5(0)=4, ℓ_5(1)=5, ℓ_5(2)=8, and ℓ_5(3)=9.There are 6 source edges in C. Suppose for a contradiction that there are 9 source edges. Hence there is just one ψ-balanced layout q assigning -3 to every 5-face. Let K_0 and m be obtained from Lemma <ref>, which says |m|>|K_0|. Let k=|K_0|.Suppose K_0 is a cycle. When i of the 5-faces are in the interior of K_0, then 3i=|m|>k≥ℓ_5(i), which is a contradiction for all i∈{0, 1, 2, 3}. Therefore, K_0 is a path. Let C be partitioned into paths X and Y that both have the same endpoints as K_0.Note that X+ Y=9 and X+ Y=0, which implies X= Y=0.Since C is chordless, k ≥ 2. By symmetry assume that X,K_0 form a cycle that has i∈{0, 1} of the three 5-faces in its interior. Lemma <ref> implies that | X -3i| > k and Y + k ≥ℓ_5(3-i). This set of equations gives a contradiction for all i∈{0, 1}. Hence C contains 6 source edges and 3 sink edges.Let q be a ψ-balanced layout, and we know that the three 5-faces of G are assigned q-values 3, -3, -3. Notice there are three different ψ-balanced layouts. Let K_0 be obtained from Lemma <ref>.K_0 is a path with both endpoints in C.Suppose for a contradiction that K_0 is a cycle. Denote by m the sum of the q-values of the faces in the interior of K_0. Lemma <ref> implies that |m|>|K_0|. When i of the 5-faces are in the interior of K_0, then 3i≥ |m|>|K_0|≥ℓ_5(i), which is a contradiction for all i∈{0, 1, 2, 3}. Claim <ref> says that K_0 is a path. Let C be partitioned into paths X and Y that both have the same endpoints as K_0.Denote by R_X and R_Y the induced subgraph of G whose outer face is bounded by K_0, X and K_0, Y, respectively. Each R_X and R_Y contains at least one 5-face. Note that X+ Y=6 and X+ Y=3. Without loss of generality, R_X contains three 5-faces. Hence X+ X + k≥ 9, and Lemma <ref> gives | X -3 - X| > k. This set of constraints has no solution which is a contradiction.By Claim <ref> and by symmetry, we may assume that R_X contains exactly one 5-face f; we call f lonely with respect to K_0. If q(f) = -3, then we call this configuration type A and if q(f) = 3, then we call it type B; see Figure <ref>.Denote the three different ψ-balanced layouts by q_1, q_2, and q_3. For i ∈{1,2,3}, let K_i be K_0 obtained from Lemma <ref> when applied to q_i. By Claim <ref>, we can define f_i to be the lonely face for K_i. Notice that f_1, f_2, and f_3 are not necessarily pairwise distinct faces. Label the endpoints of K_i by u_i and v_i such that K_i is a (u_i,v_i,f_i)-cut. Define k,l,m to be the length of K_1, K_2, K_3, respectively.First we show that configurations of type A do not exist. Let q_1 be a configuration of type A and let q_2 be a layout where q_2(f_1)=3. Then q_2 is not a configuration of type A. Suppose for a contradiction that both q_1 and q_2 give a configuration of type A,so q_2(f_1)=3 and q_2(f_2)=-3. Sinceq_2(f_1)=3, and q_2(f_2)=-3, we have that f_1 and f_2 are distinct. Let f_0 be the third 5-face. By symmetry, paths K_1 and K_2 give one offour possible kinds (11), (00), (22),and (20). The kind (02) is symmetric with (20). For an illustration, see Figure <ref>.Suppose K_1 and K_2 are of kind (11).The situation is depicted in Figure <ref> (AA11). Let X,A,Y,Z be C(u_2, u_1; v_2, v_1),C(u_1,v_2;v_1,u_2),C(v_2, v_1; u_2, u_1),C(v_1, u_2; u_1, v_2) respectively. We obtain the following constrains that must be satisfied by using Lemma <ref> and Lemma <ref>.| X +Z -X - Z| > k_1+k_2 | Y +Z -X - Y| > l_1+l_2X+ A + l≥ 7and oddY+ A + k≥7and oddX+ Y + k + l≥ 10Inequalities (<ref>) and (<ref>) follow from Lemma <ref>. Inequalities (<ref>),(<ref>), and (<ref>) follow from Lemma <ref> and the structure of the (11) kind. Suppose K_1 and K_2 are of kind (00). The situation is depicted in Figure <ref> (AA00). Let X and Y be C(u_2, v_2; u_1, v_1) and C(u_1,v_1;u_2,v_2) respectively. Let Z be edges of C that are in neither X nor Y.As in the previous case we obtain the following set of constraints that must be satisfied.| X +Z -X - Z| > k| Y +Z -X - Y| > lY+k≥ 5and oddX+l≥ 5and oddInequalities (<ref>) and (<ref>) are obtained from Lemma <ref>. The other inequalities come from Lemma <ref>. Recall that we assumed that C has no chords, so we also include that min{k,l}≥ 2. The above set of constraints has no solution.Hence K_1 and K_2 cannot be of kind (00).The next case (22) is depicted in Figure <ref> (AA22). Let X and Y be C(v_1,u_1;v_1,u_2) and C(v_1, u_2; v_1, u_1), respectively. Let Z be edges of C that are in neither X nor Y.Using Lemmas <ref> and <ref> we obtain the following set of constraints that must be satisfied:| X +Z -X - Z| > k| Y +Z -X - Y| > lY+l≥ 8and evenX+k≥ 8and evenThis system has no solution. This finishes the case (22) of Claim <ref>.The last case (20) is depicted in Figure <ref> (AA20). Let X and Y be C(u_2,v_2;v_1,u_1) and C(v_1, u_1; u_2, v_2), respectively. Let Z be edges of C that are in neither X nor Y. Using Lemmas <ref> and <ref> we obtain the following set of constraints that must be satisfied:| Y -Y| > k| X -3-X| > lY+k≥ 8and evenX+l≥ 5and oddk +l+ Z+ Z ≥ 10The last equation was obtained from the fact that in the kind (20), the subgraph of G bounded by K_1, K_2, and Z contains at least two 5^+-faces. This system has no solutions. This finishes the proof of Claim <ref>.Let q_1 be a configuration of type A and let q_2 be a layout where q_2(f_1)=3. Then q_2 is not a configuration of type B. Suppose for a contradiction that q_1 gives a configuration of type A and q_2 gives a configuration of type B, whereq_2(f_1) = 3, hence f_1=f_2. We have four kinds depending on the order of the endpoints of K_1 and K_2. The cases are depicted in Figure <ref>. The kind (22) is not possible if f_1=f_2.Depending on the case,from by Lemma <ref> and Lemma <ref>.we obtain a set of constraints that must be satisfied.(AB11):| X +B -X - B| > k | X +A - 6 - X - A| > lY+ A +k≥ 7and oddX+ Y+k+l≥ 13and odd (AB20):|X-X|> k |Y + 3 -Y|> lY+l≥ 5and oddX+k≥ 8and even(AB02):| Y -3 - Y|> k| X - 6 -X|> l Y+k is ≥ 5and odd X+l is ≥ 8and even(AB00):| X -3 - X|> k| Y +3 -Y|> lX+k≥ 5and oddY+l≥ 5and oddA+ A+k+l≥ 13Inequality (<ref>) comes from the fact that the subgraph bounded by K_1, K_2 and A must contain all three 5-faces of G in its interior faces. In addition, we include that min{k_1,k_2,l_1,l_2}≥ 1 since v is not a vertex of C and min{k,l}≥ 2 since C has no chords.None of the four sets of constraints has any solution, which is a contradiction. By Claim <ref> and Claim <ref>, every layout gives a configuration of type B.Thus, we know that for each i ∈{1,2,3}, q_i(f_i) = 3, and f_1,f_2,f_3 are pairwise distinct.Let P_i be the subpath of C such that K_i and P_i bound a cycle that contains f_i. Let i,j ∈{1,2,3} and i ≠ j. Based on the order of u_i,v_i,u_j,v_j on C, and K_i and K_j we get four possible kinds (BB00), (BB11), (BB22), (BB20); see Figure <ref>. Note that (BB02) is symmetric to what would be (BB20). For all i,j∈{1,2,3} and i≠ j we get that K_i and K_j do not form (BB22). Suppose for a contradiction that K_i and K_j do form (BB22). See Figure <ref> (BB22) for a sketch of the situation. Let X and Y be C(v_j, u_j; v_i,u_i) and C(v_i, u_i; v_j,u_j), respectively. Let Z be the edges of C that are in neither X nor Y.Let t = 6-i-j. Since K_i and K_j form (BB22), the subgraph of G bounded by K_i ∪ Y contains faces that contain 5-faces f_j and f_t,and the subgraph of G bounded by K_j ∪ X contains faces that contain 5-faces f_i and f_t. This, Lemma <ref>, and Lemma <ref> give the following set of constraints. | Y -6 - Y|> k_i| X -6 - X|> k_jk_i +Y ≥ 8and even k_j +X ≥ 8and evenThis set of constraints has no solution.For all i,j∈{1,2,3} and i≠ j we get that K_i and K_j do not form (BB20) or there exist alternative paths that form (BB11) and no new (BB20) is created. Supposefor a contradiction that K_i and K_j form (BB20). See Figure <ref> (BB20) for a sketch of the situation. Let X and Y be C(u_j, v_j; v_i,u_i) and C(v_i, u_i; u_j,v_j), respectively. Let Z be the edges of C that are in neither X nor Y.First we will obtain a few potential solutions. The first four inequalities follow from Lemmas <ref> and <ref>. The inequality (<ref>) comes from the description of (BB20) wherethe subgraph of G bounded by K_i,K_j, and Z contains at least three interior faces where at least two are 5-faces. The inequality (<ref>) comes from (BB20) saying that X and K_j do not form the boundary of f_j.| Y + 3 -Y|> k_i| X + 3 - X|> k_jk_j +X ≥ 5and odd k_i +Y ≥ 8and even k_i+k_j+ Z≥ 14k_j+ X≥ 7This set of constraints has the following four solutions. X X Y Y Z Z k_i k_j 3 0 0 3 3 0 7 44 0 0 3 2 0 7 55 0 0 3 1 0 7 66 0 0 3 0 0 7 7Notice that in all the solutions k_i+k_j+ Z = 14.Hence the subgraph of G bounded by K_i,K_j, and Z has two 5-faces and one 4-face. We create a more detailed instance where we split Z into two paths C(v_j,v_i;u_j,u_j) that we keep calling Z and C(u_i,u_j;v_j,v_i) that we call W. Moreover, we partition K_i and K_j into three subpaths of lengths i_1,i_2,i_3 and j_1,j_2,j_3 respectively. See Figure <ref>.This leads to the following constraints, where the first six are the same as before. | Y + 3 -Y|> k_i| X + 3 - X|> k_jk_j +X ≥ 5and odd k_i +Y ≥ 8and even k_i+k_j+ Z≥ 14k_j+ X≥ 7 i_1+j_2+i_3+Y≥ 5and odd i_2+j_2 ≥ 5and odd i_1+j_1 +Z ≥ 5and oddThe system has 14 solutions.Create a path K_j' from K_j by dropping the piece corresponding to j_3 and replacing it by i_3 and potentially deleting repeated edges. The path K_j' is a path with endpoints in C and it makes f_j lonely. Moreover, all 14 solutions satisfy | X +W + 3 - X - W| > j_1+j_2+i_3.Hence K_j' can be used instead of K_j, and K_j' and K_i form configuration (BB11). Finally, we need to show that no new (BB20) or (BB02) is created by replacing K_j by K_j'. All 14 solutions satisfy i_3+j_3+ W+ W = 4, i_2+j_2=5, i_1+j_1+ Z + Z = 5, Y = 3, and Z ≤ 2. Hence the subgraph of G induced by W, Z, K_i, and K_j contains a 4-face and two 5-faces as internal faces and one of the 5-faces is sharing with C vertices v_j and v_i. Let K_a and K_b form (BB20) or (BB20) for some a,b ∈{1,2,3}. The previous paragraph implies that for any c ∈{a,b}one of the edges of K_cincident to v_c and u_c is incident to a 4-face and the other edge is incident to a 5-face f_a or f_b. Moreover, one of f_a and f_b is disjoint from C and the other one is sharing at most two edges with C.Let t = 6-i-j and K_t be in {K_1,K_2,K_3} with endpoints u_t and v_t. Suppose for contradiction that a new (BB20) or (BB02) is created by replacing K_j by K_j'. Since K_i is not changed, the new (BB20) or (BB02) is formed by K_j' and K_t. Hence K_j and K_t is neither (BB20) nor (BB02). The new(BB20) or (BB02) must satisfy the constraints from the previous paragraph. Since the edge of K_j' incident to v_j is incident to a 4-face and f_i, the edge e of K_j' incident to u_i must be incident to f_t. Notice that e is also incident to a 4-face h that is incident to W.Hence f_t must be on the opposite side of e than h. Letx ∈{u_t,v_t} be incident to an edge of K_t that is incident to f_t. Since Y = 3 and f_t is sharing at most two edges with C, we obtain that x ∈ Y and the order around C is u_iv_jx.Since K_j' and K_t form (BB20) or (BB02) and we know the order for x, the order of the endpoints of K_j' and K_t isu_iv_jv_tu_t. Hence K_j' and K_t form (BB02). Observe that the order of endpoints of K_j and K_t is u_jv_jv_tu_t. Hence K_jand K_t form (BB02), a contradiction.For all i,j∈{1,2,3} and i≠ j if K_i and K_j form (BB11) then they have a common point. In order to apply Lemma <ref> we need to verify thatmax{|K_i|,|K_j|}≤ 7 and if|K_i| = |K_j|=7 then K_i and K_j have common endpoints. Let K_i and K_j form (BB11), see Figure <ref> (BB11) for illustration. Let X, Z, Y, and W be C(u_i,u_j;v_i,v_j), C(u_j,v_i;v_j,u_i), C(v_i,v_j;u_i,u_j), and C(v_j,u_i;u_j,v_i), respectively.Denote |K_i| and |K_j| by k_i and k_j, respectively. Lemma <ref> and Lemma <ref> imply that the following constraints are satisfied. | X +Z + 3 -X -Z|> k_i| Y +Z + 3 -Y -Z|> k_j X +Z + k_i ≥ 7and oddY +Z + k_j ≥ 7and odd All solutions to these constraints satisfy that max{k_i,k_j}≤ 7. Moreover, if k_i=k_j=7 then X +Y = 0. HenceLemma <ref> applies and there is a common point.Now we know that we have only configurations (BB00) and (BB11) with common points.Denote the length of the path K_1, K_2, and K_3 by k, l, and m, respectively. We will use k_1, k_2, k_3 to denote the lengths of subpaths of k if some of the paths form (BB11); l_1, l_2, l_3, m_1, m_2, m_3 will be used similarly.See Figure <ref>.If there is a pair of layouts giving configuration (BB00), we distinguish the following cases:(B1) all pairs form (BB00).(B2) one pairforms (BB11).(B3) two pairs form (BB11)If all three pairs of layouts give (BB11), then we define v_K and v_L to be the common point of K_3 with K_1 and K_2, respectively. The vertex v_K is before v_L if v_K appears before v_L when traversing the cycle formed by K_3 and P_3 in the clockwise order and the starting point is on C. (B4) P_1, P_2, and P_3 have a common edge and v_K is before v_L or v_K = v_L. (B5) There is no common edge of P_1, P_2, and P_3 andv_K is before v_L or v_K=v_L. (B6) There is no common edge of P_1, P_2, and P_3, andv_L is before v_K and v_K ≠ v_L (B7) P_1, P_2, and P_3 have a common edge and v_L is before v_K and v_K ≠ v_L. See Figure <ref> for an illustration of the cases (B1)–(B7). Since one layout may contain several different configurations of type B, pickK_1, K_2, K_3 such that the number of (B11) pairs is minimized. Next we give constraints for each of the cases (B1)–(B7). Solutions to these constraints were obtained by simple computer programs. Critical graphs obtained from (Bi) are depicted in Figure <ref> as (Bij) for all i,j.Endpoints of K_1, K_2, and K_3 partition C into several internally disjoint paths. The paths have names in {X,Y,Z,W,A,D,E,F} with exception of W in (B1) and (B2), where W refers to a union of up to three and two paths, respectively. To simplify the write-up we refer the reader to Figure <ref> for the labelings of the paths. The configuration (B1) results in a critical graph where every 5-face shares at least two edges with the boundary.Moreover, in every non-extendable 3-coloring of the outer face, every 5-face contains two source edges.We refer the reader to Figure <ref> (B1) for the labelings of the paths.By Lemma <ref> we get the first three equations and by Lemma <ref> we get the remaining equations.| X + 3 - X|> k ,| Y + 3 - Y|> l ,| Z + 3 - Z|> m ,k+ X≥ 5and odd, l+ Y≥ 5and odd, m+ Z≥ 5and odd,In addition, we also include some constraints to break symmetry; for example X +X ≥ Y +Y ≥ Z +Z. All solutions to this system of equations are in Table <ref>.By inspecting the solutions from Table <ref>, we conclude that they satisfy the statement of the claim. For the remaining cases, we give the sets of constraints but we skip detailed justification since they all come from the description of the configurations, Lemma <ref>, Lemma <ref>, and the fact that C has no chords. We provide computer programs online for solving the sets of equations and to help with checking the solutions.The description of the configuration is the following, see Figure <ref>. If there is exactly one (B11) pair, then we get configuration (B2), where we assume it is pair q_1 and q_2.If there are two (B11) pairs, then assume that q_3 is in both pairs.There are two common points on K_3, where one is shared with K_1 and the other is shared with K_2. Depending on the order of these points we get either (B3) or (B3X). The last option is that all three pairs are (B11). By considering the order of the endpoints of K_1,K_2,K_3 and the order of the common points on K_3, we get (B4)–(B7).Configurations (B2)–(B7) result in critical graphs (B21)–(B52).Every graph in Figure <ref> represents several graphsthat can be obtained from the depicted graph by identifying edges and verticesand by filling every face of even size by a quadrangulation with no separating4-cycles. Moreover, the 5-faces in (B21) and (B22) that share two edges with Ccan be moved along C as long as they stay neighboring with a region with three sink edges. We slightly abuse notation and use k_i,l_i,m_i for subpaths of K_1,K_2,K_3 respectively as well as for lengths of these subpaths, where i ∈{1,2,3}. For a path in {X,Y,Z,W,A,D,E,F}, we use its lower case letter to denote its length.For each case we include constraints that all three layouts give configurations of type B using Lemma <ref> and Lemma <ref> analogously to Claim <ref>. In addition, we add the following set of constraints depending on the case: (B2):x + k_2 + l_1≥ 5and oddy + k_1 + l_2≥ 5and oddz+w+k_2+l_2≥ 7and odd if w > 0 x+d+y+k_2+l_2≥ 8and evenx+y+z+w+l_1+k_1≥ 9and odd(B3):e+x+k_1+k_2≥ 7f+y+l_1+l_2≥ 7 y+m_1+l_2≥ 5and oddz+k_1+m_2+l_1≥ 5and oddx+m_3+k_2≥ 5and oddy+z+f+k_1+m_2+l_2≥ 8and even(B3X):min{m_1,m_2,m_3,k_1,k_2,l_1,l_2} ≥ 1k_1+l_1+z ≥ 6ifl_2 = 1 thenx+m_3+k_2≥ 6y+m_1+m_2+l_2≥ 5and odd ifk_2 = 1 theny+m_1+l_2≥ 6x+k_2+m_2+m_3≥ 5and odd(B4):x+e+l_3+k_1+k_2 ≥ 5 and oddy+f+k_3+k_2+m_2+l_1≥ 8 and evenk_2+l_2+m_2 ≥ 5 and oddf+y+w+x+m_3+k_2+k_3≥ 9 and oddx+k_1+m_3≥ 5 and odde+x+k_1+m_2+l_2+l_3≥ 8 and eveny+l_1+m_1≥ 5 and oddy+w+x+e+l_3+l_2+m_1≥ 9 and oddf+y+w+x+e+l_3+k_3≥ 9 and odd (B5):min{k_2,l_2,m_2} ≥ 1or k_2=l_2=m_2 = 0 y+k_1+l_2+m_1≥ 5 and odd x+l_1+k_2+m_3≥ 5 and odd z+k_3+m_2+l_3≥ 5 and odd y+f+l_3+l_2+k_1 ≥ 7 and odd x+e+k_3+k_2+l_1 ≥ 7 and odd m_1+m_2+k_3+f+z≥ 7 and odd m_3+m_2+l_3+e+z≥ 7 and odd l_1+l_2+m_1+d+y ≥ 7 and odd (B6):m_2≥ 1y+k_1+m_1≥ 5 and odd x+l_1+m_3≥ 5 and odd z+k_3+l_3≥ 5 and odd(B7):k_1+l_1+m_1+m_3+x+y≥ 10e+x+w+y+f+l_2+k_2-5≥ 9We enumerated all solutions to all seven sets of constraints, and we checked that the resulting graphs are depicted in Figure <ref>. In order to eliminate mistakes in computer programs, we have two implementations by different authors and we checked that they give identical results.Sources for programs for cases (B2)–(B7) together with their outputs can be found on arXiv and at . The most general solution for each of the sets of equations is depicted in Figure <ref>. Notice that (B3X), (B6), and (B7)have no solutions. In (B7), inequality (<ref>) comes from a subgraph having two faces where each contains a 5-face in the interior and (<ref>) comes from Lemma <ref> and the -5 appears due to k_2 and l_2 enclosing f_3.Observe that (B34), (B35), and (B36) are special cases of(B41), (B42), and (B43), respectively. Hence we dropped (B34), (B35), and (B36) from Figure <ref>.One can think of (B41), (B42), and (B43) as being obtained from (B34), (B35), and (B36) by duplicating a subpath P of Cwhere all dual edges of P are oriented inside.Notice that by using this operation, (B44) could be obtained from (B21), also (B22) from (B11) and (B41) from (B22).We suspect that it is part of a more general description of C-critical graphs, where C is larger.We think the case (B4) is the most complicated case. We again used the trick to identify general solutions quickly by observing that regions bounding faces contain only the face and obtained seven solutions. We include sketches of the solutions generated by our program in Figure <ref>. Although there are seven solutions, they give only four distinct cases due to some vertex identifications.0.22This finishes the proof of Lemma <ref>.§ ACKNOWLEDGEMENTS We would like to thank Zdeněk Dvořák for fruitful discussions and we are very grateful to anonymous referee who spotted numerous mistakes in the paper and suggested simplification to the proofs. This work was supported by the European Regional Development Fund (ERDF), project NTIS - New Technologies for the Information Society,European Centre of Excellence, CZ.1.05/1.1.00/02.0090. The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2015R1C1A1A02036398). The second and third authors were supported by project P202/12/G061 of the Grant Agency of the Czech Republic. The last author was supported by NSF grants DMS-1266016 and DMS-1600390.A preliminary version of this paper without a complete proof was published in proceedings on IWOCA 2014 <cit.>. abbrv | http://arxiv.org/abs/1709.09428v1 | {
"authors": [
"Ilkyoo Choi",
"Jan Ekstein",
"Přemysl Holub",
"Bernard Lidický"
],
"categories": [
"math.CO"
],
"primary_category": "math.CO",
"published": "20170927100654",
"title": "3-coloring triangle-free planar graphs with a precolored 9-cycle"
} |
firstpage–lastpage 2010A new approach for short-spacing correction of radio interferometric data sets S. Faridani1Corresponding author: [email protected] ,F. Bigiel2,L. Flöer1,J. Kerp1 S. Stanimirović3Accepted ....... Received......; in original form ...... ================================================================================================================================= Blazars are classified into high, intermediate and low energy peaked sources based on the location of their synchrotron peak.This lies in infra-red/optical to ultra-violet bands for low and intermediate peaked blazars. The transition from synchrotronto inverse Compton emission falls in the X-ray bands for such sources. We present the spectral and timing analysis of 14low and intermediate energy peaked blazars observed with XMM–Newton spanning 31 epochs. Parametric fits to X-ray spectrahelps constrain the possible location of transition from the high energy end of the synchrotron to the low energy end ofthe inverse Compton emission. In seven sources in our sample, we infer such a transition and constrain the break energyin the range 0.6 − 10 keV. The Lomb-Scargle periodogram is used to estimate the power spectral density (PSD) shape. It iswell described by a power law in a majority of light curves, the index being flatter compared to general expectation fromAGN, ranging here between 0.01 and 1.12, possibly due to short observation durations resulting in an absence of long term trends.A toy model involving synchrotron self-Compton (SSC) and external Compton (EC; disk, broad line region, torus) mechanismsare used to estimate magnetic field strength ≤ 0.03 - 0.88 G in sources displaying the energy break and infer a prominentEC contribution. The timescale for variability being shorter than synchrotron cooling implies steeper PSDslopes which are inferred in these sources.galaxies: active - BL Lacertae objects: general - BL Lacertae objects: radiation mechanisms: non-thermal; relativistic processesindividual)§ INTRODUCTION Blazars constitute a class of Active Galactic Nuclei (AGN) characterized by their extreme properties including strong variability (flux andpolarization) and weak spectral lines (dominated by continuum), believed to be due to observer line of sight orientation based effects whichfrom AGN unification models (e.g., Urry & Padovani 1995) imply that the relativistic jet is directed towards the observer at small angles.Radiation from blazars span the whole electromagnetic spectrum from radio wavelengths upto γ-ray range. Their broadband spectral energy distribution (SED) is characterized by a double peaked structure. The low energy peak is mainly due to the synchrotronradiation from relativistic non-thermal electrons. The high energy peak can be due to the inverse Compton scattering of lower energy synchrotronphotons from the same electron population (synchrotron self Compton scenario; e.g. Kirk, Rieger & Mastichiadis 1998) or of external photons fromaccretion disc, broad line region or dusty torus in the leptonic scenarios (external Compton scenario; e.g. Sikora, Begelman & Rees 1994);while it can be due to synchrotron emission from protons or from secondary decay products of charged pions in the hadronic scenarios (e.g.Atoyan & Dermer 2003; Bottcher et al. 2013).Based on the location of the low energy or synchrotron peak, these sources are classified into high, intermediate and low energy peaked blazars(HBL, IBL and LBL, respectively; Padovani & Giommi 1995). Abdo et al. (2010) classified blazars based on the location of the synchrotronpeak frequency, ν_s. If ν_s ≤ 10^14 Hz (in the infrared), it is classified as a low spectral peak (LSP) source; if it is inoptical–ultra-violet range (10^14 ≤ ν_s ≤ 10^15 Hz), it is classified as an intermediate spectral peak (ISP) source; andif it lies in the X-ray regime (ν_s ≥ 10^15 Hz), it is classified as a high spectral peak (HSP) source. BL Lacertae (BL Lac) andFlat Spectrum Radio Quasars (FSRQs) together constitute the blazar class of sources. Low luminosity BL Lacs which are HSPs exhibit the synchrotronpeak in the UV-soft X-ray band and the inverse Compton peak between the GeV and the TeV gamma-ray band (Padovani & Giommi 1995). In mid luminositysources which are LSPs and ISPs, the synchrotron peak is in the near infrared band and the X-ray emission is due to either or both the synchrotronand inverse Compton components. In high luminosity FSRQ sources, the synchrotron peak is in the far infrared band and X-ray emission is ascribedto the inverse Compton component (e.g. Giommi et al. 1995; Fossati et al. 1998). The peak frequency of the synchrotron component is found toinversely correlate with the luminosity of the blazar and different kind of blazars can be classified based on their peak energy to form a“blazar sequence" (e.g. Fossati et al. 1998; Ghisellini et al. 1998; Giommi et al. 2012).The HSP blazars are brightest in X-ray bands and show strong flux variability over diverse time-scales ranging between minutes to years(e.g. Sembayet al. 1993; Brinkmann et al. 2005; Zhang et al. 2005, 2008; Gaur et al. 2010; Kapanadze et al. 2014 and references therein). The variabilitytime-scales can be used to constrain the emission region size (e.g. Tramacere et al. 2009; Mohan et al. 2016). The variability amplitude is foundto be correlated with energy as the hardest synchrotron radiation is produced by the most energetic electrons with the smallest cooling time-scales(e.g. Zhang et al. 2005; Gliozzi et al. 2006). Their X-ray spectra are generally characterized by a soft (γ > 2) convex shape or by continuoslydownward curved shape (e.g. Perlman et al. 2005; Zhang et al. 2008), and can originate from an energy dependent particle acceleration with suitablecooling timescales, and is accompanied by strong and rapid variability (e.g. Massaro et al. 2004). The X-ray emission from LSPs is believed to originate mainly from the inverse Compton scattering of seed photons by the low energy tail of the electronpopulation. In addition, it can include a contribution from the synchrotron emission from high energy particles. In ISP blazars, there is a clear turningpoint in the SED where synchrotron and inverse Compton components intersect e.g. S5 0716+714 ( Giommi et al. 1999; Tagliaferri et al.2003; Donato et al. 2005; Ferrero et al. 2006; Wierzcholska & Siejkowski 2015); ON 231 (Tagliaferri et al. 2000); BL Lacertae (Tanihata et al. 2000;Ravasio et al. 2002);AO 0235+164 (Raiteri et al. 2006); OQ 530 (Tagliaferri et al. 2003); 3C 66A (Donato et al. 2005; Wierzcholska & Wagner 2016);4C +21.35 (Wierzcholska & Wagner 2016), amongst others. The X-ray variability is also observed for ISP and LSP blazars on both long timescales (e.g. Donato et al. 2005;Ferrero et al. 2006; Raiteri et al. 2006; Wierzcholska & Siejkowski 2015). Recently, Gupta et al. (2016) studied a sample of LSP blazars in X-ray bandsand found the intra-day variability to be less pronounced as compared to that in HSP blazars, expected due to longer cooling time scales for the lowestenergy electrons responsible for the inverse Compton emission. In LSPs, short time scale (<hours) variability is prevalent only in the synchrotroncomponent, while inverse Comption emission appears to dominate over longer time scales (∼days). Unlike HSPs, their variability amplitudes are found tobe anti-correlated with the emission energy and no significant time lags are found between the hard and soft energy X-rays (Giommi et al. 1999; Ravasioet al. 2002; Ferrero et al. 2006). In addition, the X-ray band mostly lies at the transition from synchrotron to the inverse Compton component for LSPand ISP blazars and hence is key to disentangle the contribution of the two components to the broad band continuum. One expects that FSRQs would havea flatter spectra (index Γ < 2), while the LSPs and ISPs would have intermediate slopes (with index Γ=1.5–1.8), or even concave X-raycontinua (e.g. Donato et al. 2005; Massaro et al. 2008; Wierzcholska & Wagner 2016). In previous studies, it was inferred thatthe X-ray spectra of blazars is well described by a single power law or a broken power law (e.g. Perlmann et al. 1996; Urry et al. 1996). The log parabolamodel has also proven to describe the spectrum well with the power law index varying as log E (e.g. Massaro et al. 2004, 2008; Donato et al.2005; Tramacere et al. 2010)is not a constant but varies slowly with energy i.e. ∝ log E and hence the name log parabola (Massaro et al. 2004, 2008; Donato et al.2005; Tramacere et al. 2009).The model has often been invoked to fit the entire SED of blazars (e.g. Landau et al. 1986; Massaro et al. 2004; Chen et al. 2014) and such curvedspectra of blazars are known to arise due to log parabolic electron distributions (e.g. Tramacere et al. 2007,2009; Paggi et al. 2009).In the current study, we compile a sample of 14 LSP and ISP blazars observed with XMM-Newton spanning 31 observation epochs and study their spectraand timing information in the 0.6–10 keV energy range. Owing to a large effective area and small cadence, XMM-Newton offers good spectral resolutionand timing information for our study. We fit each spectra with parametric models to identify possible transitions from low to high energy component.We then study the timing properties of our sample including a measure of variability and the power spectral density shape to study any possible evolvingfeatures.Our motivations include the disentangling of low and high energy contribution from synchrotron and inverse Compton components respectively,the comparison of the inferred break energy across sources, and a comparison of our inferences with previous studies (e.g. Wierzcholska& Wagner 2016) which were similarly motivated through studies of mostly high peaked blazars and ISPs.The paper is structured as follows: in Section 2, we give a brief description of the sample selection and data reduction method;in Section 3, we present the results of the analysis which are then discussed and interpreted in Section 4.§SAMPLE SELECTION AND DATA ANALYSISThe sample of the blazars are selected from the catalogue of TeV sources (TeVCat[(TeVCat online catalogue provided by Scott Wakely& Deirdre Horan (http://tevcat.uchicago.edu/)]). For our sample, we selected all ISPs and LSPs from the XMM-Newton catalogue andconsidered all observations since its launch. The blazar sample and their observation log is provided in Table 1.The blazars in our sample are observed by the European Photon Imaging Camera (EPIC) on board the XMM-Newton satellite (Jansen et al. 2001). The EPIC is composed of three co-aligned X-ray telescopes which simultaneously observe a source by accumulating photons in the three CCD-based instruments: the twins MOS 1 and MOS 2 and the pn (Turner et al. 2001; Strüder et al. 2001). The EPIC instrument provides imaging and spectroscopy in the energy range from 0.2 to 15 keV with a good angular resolution (PSF = 6 arcsec FWHM) and a moderate spectral resolution (E/Δ E ≈ 20-50). We consider here only the EPIC-pn data as it is most sensitive andless affected by the photon pile-up effects. We used the XMM-Newton Science Analysis System (SAS) version 14.0.0 for the light curve extraction and spectral analysis. The observation summary or Observation Data File (ODF) and the calibration index file (CIF) are generated using updated calibration data files or Current Calibration Files (CCF) following “The XMM-Newton ABC Guide” (version 4.6, Snowden et al. 2013). XMM-Newton EPCHAIN pipeline is used to generate the event files. In order to identify intervals of flaring particle background, we extracted the high energy (10 keV < E < 12 keV) light curve for the full frame of the exposed CCD and found background flares in few light curves. The flaring portions are removed in affected light curves. Pile up effects are examined for each observation by using the SAS task EPATPLOT. We found that the observations are not affected by the pile-up effects. We read out source photons recorded in the entire 0.3 - 10 keV energy band, using a circle varying between 30–40 arcsec radius centered on the source. These radii have been chosen to sample most of the point spread function according to the observing mode. Background photons were read out from a circular region with an area comparable to the source region, located about 180 arcsec off the source on the same chip set. Redistribution matrices and ancillary response files were produced using the SAS tasks rmfgen and arfgen. The pn spectra were created by the SAS tool XMMSELECT and grouped to have at least 30 counts in each energy bin to ensure the validity of χ^2 statistics. The X-ray spectra are expected to be significantly affected by the instrumental uncertainties at energies below 0.5 keV, hence we consider only the 0.6 - 10 keV energy band for our studies.§.§ Timing Analysis§.§.§ Excess Variance and Variability Amplitude We calculate the excess variance (e.g. Edelson et al. 2002; Vaughan et al. 2003), which is an estimator of the intrinsic source variance over and above the underlying noise. The variance, after subtracting the excess contribution from the measurement errors isσ_NXS^2 = S^2 - σ_err^2,where σ_err^2 is the mean square error,σ_err^2 = 1/N∑_i=1^Nσ_ err, i^2.The normalized excess variance is given by σ_NXS^2=σ_XS^2/x̅^2 and the fractional root mean square (rms) variability amplitude (F_var; Edelson, Pike & Krolik 1990; Rodriguez-Pascual et al. 1997) is F_var = √(S^2 - σ_err^2/x̅^2).and the error on the fractional amplitude iserr(F_var) = 1/2 F_var err(σ_NXS^2) =√({√(1/2N)·σ_err^2/x̅^2F_var}^2 + {√(σ_err^2/N)·1/x̅}^2).We calculated F_var for all of our light curves and the results are presented in Table 2.§.§.§ Lomb-Scargle periodogram The Lomb-Scargle periodogram <cit.> is used to determine the power spectral density (PSD) generated by a physical process manifested through the observed light curve. It is an effective measure of variability in un-evenly sampled light curves including those with short data gaps such as those encountered here. The periodogram evaluated from a light curve x(t_k) spanning N points is determined by a least squares fit to a mean subtracted time series usingx(t_k) = C_1 sin 2 π f_j (t_k-τ)+C_2 cos 2 π f_j (t_k-τ)and is given by <cit.>P (f_j) = 1/2 σ^2[(∑^N_k=1 (x(t_k)-x̅) cos 2 π f_j (t_k-τ))^2/∑^N_k=1cos^2 2 π f_j (t_k-τ) . +. (∑^N_k=1 (x(t_k)-x̅) sin 2 π f_j (t_k-τ))^2/∑^N_k=1sin^2 2 π f_j (t_k-τ)] ,where τ is a time shift parameter and is given bytan(4 π f_j τ)=∑^N_k=1sin(4 π f_j t_k)/∑^N_k=1cos(4 π f_j t_k)where x̅ is the mean of the light curve and P (f_j) is evaluated at frequencies f_j = j/(t_N-t_1) where j = 1, 2, .., N/2 and (t_N-t_1) is the total duration of the observation. The periodogram is evaluated using the algorithm presented in <cit.> in order to achieve fast computational speeds.We performed a linear interpolation of the light curves to populate small (< 200 - 300 s; 2-3 points) data gaps and sample the lightcurves at regular intervals Δ t = 100 s. The LSP is the Fourier periodogram for an evenly sampled light curve which leads to a properestimation of the underlying PSD as noted in <cit.>, where it was inferred that the LSP of unevenly sampled light curvestended to be dominated by higher temporal frequencies thus yielding systematically flatter PSD slopes. We use two competing parametric models to infer the PSD shape, a power law modelI(f_j) = A f^μ_j+C, with amplitude A, slope μ, and a constant Poisson noise C, and a bending power law modelI(f_j) = A f^-1_j (1+(f_j/f_b)^-μ-1)^-1+C, with amplitude A, slope μ, bend frequency f_b, and a constant Poisson noise C. By maximizing the likelihood functionℒ (θ_k) = ∏^N/2_j = 11/I(f_j,θ_k) e^-P(f_j)/I(f_j,θ_k),we determine the parameters of the model θ_k. The Akaike informaion criteria (AIC) is then calculated for each competing model and the best fit model is that with higher likelihood of describing the data effectively. The best fit model parameters are employed in thesimulation of 1000 random realizations of light curves with similar statistical and spectral properties as the original light curve, including the same sampling pattern, using an algorithm similar to that prescribed in <cit.>. The periodogram of each of these simulated light curves is determined using at the same sampling frequencies as that used for the original light curve and at each frequency bin, we construct an empirical distribution function to determine the 3-σ model error at each ordinate. Details of the fitting procedure, model selection using the AIC and the light curve simulations procedure are presented in <cit.> and references therein. For a light curve populated by random Gaussian noise, its periodogram ordinates are expected to be χ^2_2 distributed <cit.>. The residuals of the best fit PSD are thus expected to be χ^2_2 distributed. From the cumulative distribution function of the χ^2_2 distribution, and after accounting for the number of frequencies sampled and the model errors inferred from the simulations described above, we set a significance threshold of 99 %, used to infer the presence of any statistically significant quasi-periodic oscillations in the data and to infer the statistical significance of each periodogram peak. §.§ Spectral Analysis We fit each spectra using following models:* Power law model defined by k E^Γ with fixed Galactic absorption. It is characterized by a normalization k and spectral index Γ. * Logarithmic parabola model defined by k E^(-α+β log(E/E_1) (e.g. Landau et al. 1986; Massaro et al. 2004; 2008) with fixed Galactic absorption. It is characterized by a normalization k, spectral index α, transition energy E_1 and curvature parameter β. We fix E_1 = 0.6 keV, the lowest probed energy without loss of generality as the particular choice does not drastically affect the results owing to its logarithmic dependance. * Broken power law model defined by k E^Γ_1 for E < E_ break and k E^Γ_2 otherwise, with fixed Galactic absorption. This model is also characterized by a normalization k, two spectral indices Γ_1 and Γ_2, and a break energy E_ break. This model is used only to fit those spectra when there is significant concave curvature (>99% significance) using the log parabolic model and is used to infer the break energy E_ break in the spectrum, which is the point at which synchrotron component turns over to inverse Compton component.Some FSRQs in our sample are at high redshift and can have an intervening absorption due to gravitational microlensing and damped Lyman-αsystems i.e. AO 0235+164, 3C 454.3 (e.g. Foschini et al. 2006, Raiteri et al. 2008). For these blazars, all models are fit after including anabsorber along the line of sight in addition to the Galactic absorption density. The spectra are fit using the XSPEC software package version 12.8.1. The XSPEC routine “cflux” is used to obtain unabsorbed flux and its error. The Galactic absorption is taken from the survey by Willingale et al. (2013) which includes both the atomic gas column density N_HI and the molecular hydrogen column density N_H_2. The N_HI is adopted from the Leiden Argenine Bonn Survey (Kalberla et al. 2005) which is obtained by merging two surveys: the Leiden/Dwingeloo Survey (Hartmann & Burton 1997) and the Instituto Argentino de Radioastronoma Survey (Arnal et al. 2000; Bajaja et al. 2005). The N_H_2 is estimated using the maps of dust infrared emission by Schlegel, Finkbeiner & Davis (1998) and the dust-gas ratio by Dame, Hartmann & Thaddeus (2001). § RESULTS AND INFERENCES§.§ Temporal Variability The light curves of the blazars are analyzed to infer their normalizedexcess variance and their PSD shape with the results being summarized inTable 1. The Lomb-Scargle periodogram analysis for some of the sources including the best fit model are plotted in Fig. 1. The variabilityamplitude shows a large spread, ranging between 0.80 ± 2.4 % (PKS 0521–365) and 52.12 ± 0.59 % (ON 231: 26-06-2002) with a median of5.76 ± 2.26 indicating a low to moderate overall variability with a few outliers. This is consistent with expectations for LSP and ISPsources, due to their X-ray emission mainly originating from the inverse Compton scattering of seed photons by the lower energy tail of theelectron population and thus indicating lower variability in the 0.6 - 10 keV band.From the parametric model fit to the periodogram, the power law model is inferred to be the best fit PSD shape in 29/31 light curves withthe power law index in the range -0.01 to -1.12 and many light curves being white noise dominated. This indicates that the inferredvariability here probes only the noise floor as opposed to intrinsic source based variability which often leads to indices in the range -1.5–2.5(e.g. Mohan et al. 2014, 2015, 2016, Agarwal et al. 2017; Goyal et al. 2017 and references therein). The bending power law is the best fit modelin the light curves of PKS 0235+164: 10-02-2002 and PKS 0528+134: 14-09-2009. However, as the bend frequency indicates that it is close to theedge of the observation duration, these results are similar to the power law PSD model. There are no inferred statistically significant quasi-periodicoscillations in any of the light curves. Thus, no conclusion can be drawn from the PSD slopes alone in relation to the typical distance ofvariability origin or the emission mechanism, necessitating the analysis of the spectra to clear these uncertainties. §.§ Spectral VariabilityThe spectral parameters of power law, log parabolic and broken power law model fits are presented in Table 3–4. The preferred models for eachspectra are shown in Fig 2–7. The goodness of fit of power law and log parabolic model are compared using the F-test (Bevington & Robinson 2003)and values are quoted in Table 3–4. More than half of the observation (18/31) are described well with the power law model and the remaining13/31 are well described by the log parabolic model. In 7 (PKS 0235+164, PKS 0426-380, PKS 0537-441, S5 0716+714, OJ 287, ON 231 and BL Lacertae) of the 14 blazars, a significant negative curvature is inferred. The spectra of such sources are also fit with a brokenpower law to constrain the E_ break.§.§ Notes on individual blazarsTXS 0106+612:This LSP was first discovered by Gregory & Taylor (1981) in a radio survey of the Galactic plane. This source is a TeV blazar (e.g. Tsujimoto et al. 2015). The X-ray spectrum of this blazar isbetter described with a power law with fixed N_Hrather than log parabolic model.Also, we fit the spectra with free N_H but did not get any improvement in theχ^2 values. We did not find any significant curvature in the studied energy range. 3C 66A: This source is a well known ISP (e.g. Aliu et al. 2009). In previous studies, significant negative curvature is foundin the SED using Swift/XRT observations (i.e. Wierzcholska & Wagner 2016) with break energy of ∼3.4 keV.We examined the spectra of this blazar using XMM-Newton pointing and found that it is well described with power law model.The spectral index (Γ=2.45) is well matched with the previous studies. However, we did not find significant spectral curvature for this source.Similar results are found by Donato et al. (2005) where they did not find any significant curvature using BeppoSAX observations and found the spectrumto be well fitted by the power law model with Γ=2.26. We also model our spectrum by making N_H as free parameter but χ^2 values did not improve significantly.PKS 0235+164: This source is a BL Lac object and a TeV γ-ray emitting blazar (e.g. Tsujimoto et al. 2015). Donato et al. (2005)studied theX-ray spectrum of this blazarusing BeppoSAX observations and found power law to be well fitted to the spectra. Raiteri et al. (2006) analyzed XMM-Newton and Chandra observations during the period 2000–2005 and found log-parabolic model to be superior than power law model. The source has an intervening system along the line of sight at z=0.524 which likely absorbs the soft X-ray spectrum (Raiteri et al. 2006). This intervening medium has been measured by ROSAT and ASCA obtaining a value of N_H=2.8 ×10^21 cm^-2 (Madejski et al. 1996) and ∼2.4–2.6 ×10^21 cm^-2 using XMM-Newton observations (Raiteri et al. 2005; 2006; Foschini et al. 2006).We analyzed four pointings of XMM-Newton satellite during the period 2002–2005. During the observation performed in 2002, the source wasin outburst state (Raiteri et al. 2005). The spectrum is found to be well fitted with log parabolic model with significant concave curvature with break energy at around 4 keV in 2002. The source is in faint state during the observation performed in the period 2004–2005 and wefound the spectra to be well described with a power law model with Γ=1.5–1.7. PKS 0426-380: The source is classified as a FSRQ (i.e. Ghisellini et al. 2011; Sbarrato et al. 2012). Tanaka et al. (2013) reported the discovery of the Very High Energy γ-ray emission from this blazar. We analyzed the XMM-Newton pointing of the blazarin 2012 and found it to be well described with a log-parabolic model which showed significant concave spectrum. The spectrum is alsofitted with the broken power law and the upturn break energy is found to be at around 2.1 keV. We fit the spectra by makingN_H as free parameter but did not get any significant improvement in χ^2 values. This the first occasion where the spectral flattening is found for this blazar.PKS 0521-365: The X-ray spectrum of this blazar is studied by Donato et al. (2005) and found to be well characterized by simple power law with Γ ∼1.7. Foschini et al. (2006) also fit the spectrum with broken power law and found break energy at ∼1.5 keV. In our studies, X-ray spectrum analyzed for one pointing of XMM-Newton is well fitted by the powerlaw when we fixed N_H. This spectrum shows significant improvement in the model when we kept N_H as free parameter. N_H is found to be 3.8 which is very close to the Galactic absorption column density [10^20 cm^-2] from Dickey & Lockman (1990). Since, this blazar is a FSRQ, we also fit these two observations using power law plus black body and power law plus bremsstrahlung butdid not find any improvement on the power law model. PKS 0537-441: This blazar is a FSRQ and a TeV source (Tsujimoto et al. 2015).The spectrum is well fitted by simple power law by Donato et al. (2005) with spectral index of 1.8. We observed three pointings of XMM-Newton in 2010 and found the X-ray spectrum of this blazar to be well described by log-parabolic model. There is significant spectral flattening with negative curvature. All the spectra are also fitted with the broken power law model withbreak energy at around 2.1–2.7 keV.S5 0716+714: This is a well known ISP (Giommi et al. 2008). The X-ray spectrum of this source is very well studied in literature by Tagliaferri et al. (2003); Ferrero et al. (2006); Donato et al. (2005); Wierzcholska & Wagner (2016) and has shown an upturn in the SED at the break energy varying between ∼ 1.5–2.73 keV. We studied one pointing of XMM-Newton in 2007 and found that the spectra is well described with a log parabolic model. The curvature is significantly negative. We fit the spectrumwith the broken power law to locate the break energy at ∼1.9. The spectral indices (Γ=2.63) are consistent with the previous studies. Fitting parameters are not significantly altered by making N_H as free parameter.OJ 287: The X-ray spectrum of this blazar is analyzed by Donato et al. (2005) in two occasions and found the spectra were well characterized by simple power law with Γ=1.6–1.9. Massaro et al. (2003) analyzed the X-ray spectra of this source during its optical bright state in 2001 and found the spectrum to be described by power law model. Seta et al. (2009) analyzed Suzaku observations of the blazar in April and November 2007, in quiescent and flaring states,respectively. They found the simple power law to describe the spectra with photon indices Γ=1.65 and 1.50 in quiescent and flaring states,respectively. In more recent studies Siejkowski & Wierzcholska (2017) found flat spectrum of OJ 287, which can be interpreted as an indication ofthe spectral upturn located in the X-ray regime. We analyzed five pointings of XMM-Newton and found one observation in April 2005 and April 2008 to be well fitted by simple power law.Other three pointings in November 2005, November 2006 and October 2011 have shown significant negative spectral curvature withbreak energy varying between 1.5–3.0 keV. We did not find any significant improvement by keeping N_H as free parameter in the models.S4 0954+65: This source is a TeV blazar. Tanaka et al. (2016) analyzed Swift/XRT observations of this source in optical flaring state and found softening of the X-ray spectrum, with a photon index of Γ ∼1.7 (compared to the earlier outburst with Γ ∼1.38) possibly indicating a modest contribution of synchrotron emission by the highest-energyelectrons superposed on the inverse Compton component. We analyzed the X-ray spectra of this blazar on two occasions in 2007 andfound them to be well described with simple power law model with photon index Γ ∼1.96. Free N_H in the fitting modelsdid not alter the results. ON 231: This blazar has shown significant concave curvature in literature (i.e. Donato et al. 2005; Tagliaferri et al. 2000; Wierzcholska & Wagner 2016). Wierzcholska & Wagner (2016) fitted the Swift/XRT observation of this source using the broken power law and found a break at 2.01 keV. Donato et al. (2005) found the break energy to be at 3.09_-1.092 ^+0.557.We analyzed the spectra of this source on four occasions and found significant curvature in one of the observations performed in 2002. We found a break energy of 4.33_-1.87 ^+0.51 keV which is inconsistent with previous studies within errorbars. We did not find any significant improvement by keeping N_H as free parameter in the models. 3C 279: This blazar is very well studied FSRQ and is classified as an LSP (Ackermann et al. 2011). In previous studies by Donato et al.(2005), its spectra were fitted by power law and broken power lawand break energy is found at low energy (∼ 0.5–0.7) keV. Wierzcholska & Wagner (2016) studied the source and found it to be well described by the log parabola model with positive curvature. We analyzed the spectra of 3C 279 in two occasions in 2009 and 2011 using XMM-Newton. In both the pointings, we did not find any significantcurvature and F-test gave reasonable fit using power law model. The photon indices are varying between 1.77–1.80 which arein accordance with previous studies.PKS 1334-127: This FSRQ is a γ-ray loud blazar studied by Foschini et al. (2006) using XMM-Newton observationsand found it to be well fitted with power law model with Γ=1.8 and Galactic absorption equal to 6.7±0.9 in 0.4–10 keV energy band.We also analyzed this spectrum in the (0.6–10) keV and found it to be well described by the power law model with α=1.8. We did not find any significant improvement by keeping N_H as free parameter in our fitting parameters. BL Lac: The X-ray spectra of this source is very well studied with log-parabolic as well as with power law model in previous studies.Tanihata et al. (2000) found that a soft steep component dominates below 1 keV and hard component dominates otherwise. Ravasio et al. (2002)observed BL Lac in two observations and found significant negative curvature above 5-6 keV in July 1999. However, December 1999observation was well fitted by a simple power law. Donato et al. (2005) also found significant concave curvature after the breakenergy of ∼1.71 keV. Massaro et al. (2008) also preferred log-parabolic model for the BeppoSAX observations of this source.Raiteri et al. (2009) analyzed the XMM-Newton observations of this source during the period 2007–2008 in its low state and found significantconcave curvature in all of the observations. Wierzcholska & Wagner (2016) also found the signature of negative curvature in the BL Lac spectrum and found the break energy at ∼1.1 keV. We analyzed three pointings of XMM-Newton and found that the spectra shows an upturn at the break energy of around ∼1.25–1.55 keV.Fitting parameters are not significantly alteredby making N_H as free parameter. 3C 454.3: This source is studied by Donato et al. (2005) and found that it is well described by simple power law with Γ=1.34. XMM-Newton observations are analyzed by Raiteri et al. (2007); 2008 in its post outburst and outburst states, respectively. We analysed fourobservations of XMM-Newton and found that all are well described by the power law model. The spectral indices varies between 1.5–1.7 which is in accordsnce with the previous findings.§.§ Blazars with concave spectrum In 7 (PKS 0235+164, PKS 0426-380, PKS 0537-441, S5 0716+714, OJ 287, ON 231 and BL Lacertae) of 14 blazars in our sample, we infer a significantnegative curvature suggesting that both synchrotron and inverse Compton components are located in the X-ray regime. The spectra of these sourcesare also fit by a broken power law model to infer the E_ break, with two energy bands for each source. The soft energy band is in the energyrange 0.6 keV–E_ break, representing the synchrotron component and the hard energy band is in the range E_break–10 keV, representingthe inverse Compton component. Though the power law PSD slopes based on the 0.6–10 keV band integrated light curves for these sources are dominated bywhite noise, if the underlying physical mechanism as inferred from the spectral analysis shows a demarcation between the synchrotron and inverse Comptoncomponents, it is expected that the variability in the above defined soft and hard energy bands must show clear differences. The F_ var. is thencalculated for both segments of each of the light curves for the sources displaying a spectral break to help distinguish their origin. Below, wedetermine the spectral and temporal variability of the soft and the hard components separately for all those sources and the figures for these blazarsare provided in Fig 10–12. AO 0235+164: The light curve of this source has shown pronounced variability in one of the occasion in 10.02.2002 withF_var of 11.76% and is shown in fig 10. The hardness ratio versus counts/sec is also showed in lower panel. We did not find any correlation betweenthese two. To separate the synchrotron and inverse Compton components, we divided the lightcurve in two bands i.e. soft (0.6–4.03 keV) and hard (4.03–10 keV) energy bands. The fractional variability amplitude for the soft and hard energy bands are 11.94±0.56% and 8.88±5.45%, respectively. It can be seen that the synchrotron components is highly variable and contributing mostly to the total emission. Hard IC component appears to be almost stable. PKS 0426-380: The source has not shown any variability during the observation. A weak positive correlation is found between HR andcounts/sec with (Pearson correlation coefficient, r=0.20, p=0.007). We separated the soft(0.6-2.1 keV) and hard (2.1-10 keV) energy components and found that the hard energy band representing the IC component showed higher fractionalvariability amplitude with F_var=21.94±6.25 as compared to stable synchrotron component (1.82±15.15).PKS 0537-441: The source has not shown significant variability in X-ray bands and are shown in fig 10 along with their respective HR versus counts/sec. We found weak positive correlation between HR versus counts/sec for both the pointings i.e. 27.02.2010 (r=0.239,p=4.49×10^-5) and 04.03.2010 (r=0.222, p=6.6×10^-4). It indicates that both spectra are showing bluer-when-brighter trend. We divided the light curve of 27.02.2010 into (0.6–2.28) and (2.28–10 keV),respectively. The hard energy band representing the IC component is highly variable (F_var=11.47±3.77) in this case.Synchrotron component is almost constant. Also, in the other observation performed on 04.03.2010, we divided the light curveinto soft (0.6–2.16 keV) and hard (2.16–10 keV) energy bands. The synchrotron component has shown more pronounced variability withF_var=7.10±1.52 as compared to stable IC component. For the above two observations, E_break is consistent within the errorbars.S5 0716+714: This blazar has shown significant variability in the X-ray band with F_var=17.94%. We found a negative correlation between HR versus counts/sec (r=-0.397, p <2.2×10^-16)which indicates redder-when-brighter trend. We divided the light curve in soft (0.6–1.9 keV) and hard (1.9–10 keV) energy bands.For this sources, synchrotroncomponent reveals more pronounced variability with F_var=38.03±0.39 as compared to lesser variable IC component (28.40±1.10). OJ 287: This blazar has shown significant variability in one occasion of 17.11.2006 with F_var=7.98±0.96. We did not find any correlation between HR versus counts/sec during two pointings of 03.11.2005 and 15.10.2011. However, positive correlation of r=0.256, p=4.078×10^-8 is found during the observation dated 12.11.2006. We divided the light curve into soft (0.6–3.08 keV) and hard (3.08–10 keV) and found that IC component (17.84±3.90) has higher variability amplitude as compared to the synchroton component (7.16±1.23). Similarly, during the pointing on 15.10.2011, E_break shifts to 1.66 keV and IC component shows more pronounced variability of 5.47±1.48 as compared to the flatter synchrotron component (2.88±1.42). However, the case is very different on 03.11.2005 whereE_break is consistent with theE_break of pointing on 15.10.2011, but synchrotron component is highly variable (7.20±1.98) as compared to almost constant IC component 3.17±5.77.ON 231: This ISP has shown strong variability during the pointing on 26.06.2002. The HR versus counts/sec is also shown in fig 8 and wefound strong negative correlation ( r=-0.414,p=3.739×10^-11) which represents redder-when-brighter trend.In the spectral fitting, we found a break at 4.1 keV and divided the spectra into soft (0.6–4.3) keV and hard (4.3–10) keV energy bands, respectively. It has been found that the hard energy component has more variability amplitude (F_var=65.72±3.43) as compared to the synchrotron component (52.86±3.61).BL Lacertae: This blazar has not shown variability in all of the three pointings it was observed with XMM-Newton. Significant correlation is not found between HR versus counts/sec for two pointings of 05.12.2007 and 08.01.2008. However, significant negative correlation (r=-0.468, p=1.9×10^-11) is found for the pointing dated on 10.07.2007. We divided the light curve observed on 10.07.2007 into soft (0.6-1.26 keV) and hard (1.26–10 keV) energy bands and found that synchrotroncomponent is highly variable with F_var=8.92±0.99 than the constant IC component (F_var=1.87±2.13). We divided the pointing on 05.12.2007 at E_break=1.56 and we found that both, synchrotron as well as IC components are stable with F_var=2.09±2.49 and 2.69±2.61, respectively. Third observation is divided at E_break=1.43 and found the IC component to be more variable (F_var=4.19±1.29) as compared to flat synchrotron component (F_var=1.18±4.79). § DISCUSSION AND CONCLUSIONS We study a sample of 14 blazars (5 ISPs and 9 LSPs) in the 0.6 – 10 keV X-ray energy band spanning 31 observation epochs.The analyses includes a timing study using the Lomb-Scargle periodogram to infer the power spectral density shape and possiblequasi-periodic oscillations in their light curves and the fitting of the 0.6–10 keV spectrum with parametric models depending onsource properties. We did not find any characteristic timescale in any light curve. The PSD shape is also well fitted with a powerlaw with no temporal breaks or quasi periodic components. This is expected for this class of blazars as their synchrotron componentpeaks in optical/UV and hence are highly variable in these bands. X-ray emission generally comes from less variable and flatterinverse Compton component (e.g. Gupta et al. 2016). The inferred fractional variability amplitude F_var=52% for one pointing ofON 231 which is the highest in the present sample. In 20/31 of the pointings, we found small amplitude flickering (F_var<5%) in the X-ray light curves. In six of the observations, we found significant (>3 σ) variability amplitude in the range varying from 5–52 %. This includes one pointing of PKS 0235+164, S5 0716+714, two poinings of S4 0954+65 and two pointings of ON 231. Spectral analysis is performed to check whether power law or log parabolic model well describes the LBL and IBL spectra. In 18/31 epochs, the spectra are well described by the power law model. The remaining 13/31 are well described by the log parabolic model. The spectral indices Γ for our sample of blazars lie in the range 1.2–2.7. The results are consistent with the previous studies where for LSPs and ISPs, flatter X-ray spectral slopes varying between 1.5–2 are found (e.g. Donato et al. 2005; Massaro et al. 2008; Wierzcholska & Wagner 2016). In seven of these blazars (PKS 0235+164, PKS 0426-380, PKS 0537-441, S5 0716+714, OJ 287, ON 231 and BL Lacertae), we found significantconcave or negative curvature. The spectral fits with their residuals are shown in figure 2–9. It is clear from the figure that thereare discrepancies around 1–3 keV with significant spectrum flattening. As the synchrotron peak lies in the optical/IR for LSPs and ISPs,it is expected that the transition from the synchrotron emission to inverse Compton occurs in the X-ray bands. Hence, the concave curvaturecould be interpreted as the beginning of a spectral upturn from the steep component which is the high energy tail of the synchrotroncomponent to more flatter low-energy side of the inverse Compton component. The inferred break energy for these sources are consistentwithin error bars and do not vary significantly with flux changes. Similar spectral upturns are found in previous studies also i.e. 3C 66A (Donato et al. 2005, Wierzcholska & Wagner 2016); S5 0716+714 (Giommi et al. 1999; Tagliaferri et al.2003; Donato et al. 2005; Ferrero et al. 2006; Wierzcholska & Siejkowski 2015); ON 231 (Tagliaferri et al. 2000); BL Lacertae (Tanihata et al. 2000; Ravasio et al. 2002);AO 0235+164 (Raiteri et al. 2006); OQ 530 (Tagliaferri et al. 2003); 4C +21.35 (Wierzcholska & Wagner 2016). For the well known HBL Mrk 421, Kataoka & Stawarz (2016) found that the X-ray observations from NuStar Satellite is dominated by the highest-energy tail of the synchrotron continuum till 20 keV and the variable excess hard X-ray emission is related to inverse Compton emission in its very low state.All these sources are TeV blazars and have shown variability in γ-bands (Abdo et al. 2010).Differences in the fractional variability amplitude for the soft and hard components is additionally used to demarcate the emission mechanisms.Four sources do not show significant variability based on their 0.6 – 10 keV band integrated light curves (marked in Table 5). Though, on five of the twelve occasions, a more pronounced variability is inferred in the soft energy band i.e the high energy tail of the synchrotron component. In the remaining seven occasions (including five blazars PKS 0426-380, PKS 0537-441, OJ 287, ON 231 and BL Lacertae), the variability is more pronounced in the hard energy band i.e. inverse Compton component. The study of Wiercholska & Wagner (2016) also found more pronounced variability of inverse Compton component for the blazar BL Lacertae. For the ISP S5 0716+714, this component appears to show pronounced variability with F_var=28.40% and the synchrotron component also highly variable with F_var=38.03%.It must be noted however that as the spectral slopes after E_ break in some cases (PKS 0537-441: 2010.02.27, OJ 287: 2006.11.17 andON231: 2002.06.26) are still soft, in addition to a possible onset of transition from synchrotron to inverse Compton emission, this may also bedue to a hardening of synchrotron emission from the highest energy electrons which is expected in physical scenarios including a dominant inverseCompton cooling in the Klein-Nishina regime <cit.> or a flattening intrinsic to the mechanisms causing the electronacceleration. Such scenarios can also produce an increased F_ var. We infer a negative spectral curvature in seven sources (Table 5), four of which are IBLs (BL Lacs) and the remaining three, LSPs (both BL Lacs and FSRQ). Their high energy emission (X-ray–γ ray) in the leptonic scenario is produced by inverse-Compton scattering of lower energy seed photons. The seed photons can originate from the synchrotron emitting electrons (synchrotron self-Compton, SSC process) or external sources (external Compton, EC process), including the accretion disk, broad line region clouds, dust torus or the cosmic microwave background (e.g. Sikora, Begelman & Rees 1994; Dermer, Sturner & Schlickeiser 1997). From a systematic study of blazar broadband spectra, Ghisellini et al. (1998) suggest that along the sequence HSP–LSP–FSRQ, there exists an increasing contribution to the energy density from external radiation fields in turn leading to an increasing incidence of Compton cooling.The high energy spectrum of FSRQs thus contains a prominent contribution from EC processes (e.g. Mukherjee et al. 1999; Hartman et al. 2001) while that of HSPs can be sufficiently fit with SSC models (Pian et al. 1998; Bottcher et al. 2002). The ISPs and LSP BL Lacs being intermediate between these classes then require to account for a non-neglible contribution from EC processes to reproduce the high energy spectrum (Madejski et al. 1999; Bottcher & Bloom 2000). Owing to this, we incorporate all these contributory sources in the following model to estimate physical properties of the emission region.The SSC and EC emission processes may be disentangled by appealing to clear differences in the variability pattern between the hard and soft X-ray bands. The hard band variability can be similar to or less than the soft band in the SSC process and can be larger than the soft band in the EC process if it contains imprints of the rapid microvariability (hour timescales) from the accretion flow (e.g. inner accretion disk, corona and the disk-jet transition region).However, in the strong cooling regime, changes in the seed photons may not necessarily cause rapid changes in the EC emission. Hence, no strict conclusionmay be drawn from the current small sample of sources indicating different F_ var. For both the SSC and EC scenarios, the size of the region along the jet participating in the scattering isΔ r ≤c δτ/1+z,where c is the speed of light, δ is the Doppler factor, τ is the variability timescale and z is the redshift. For the range ofz = 0.07 - 1.11 corresponding to BL Lac and PKS 0426-380 respectively (Table 1), using a δ∼ 6 obtained from studies of relativistic beamingin BL Lac objects <cit.>, and τ∼ 1 day (as a majority of the light curves do not show clear trendsand are consistent with Poisson noise), Δ r ≤ (0.002 - 0.005) pc. As this small region is along the beamed jet, we can estimate the distanceto this region r from the central black hole by assuming a conically shaped jet with half opening angle θ_0 ∼ 1/Γ where Γ isthe bulk Lorentz factor and r ∼ΓΔ r. For Γ∼ 10 <cit.>, r ≤ (0.02 - 0.05) pc ∼ (249 - 492)r_S where r_S = G M_∙/c^2 is the Schwarzschild radius for a black hole of mass M_∙, taken to be ∼ 10^9 M_⊙. In canonicalmodels of the AGN jet structure <cit.>, the region at ∼ 10^2 - 10^3 r_S hosts helical magnetic fields whichcan lead to the appearance of quasi-periodic flux and polarization variability for favourable viewing angles due to beamed emission<cit.>.In the EC scenarios relating to strong variability, we can roughly estimate the relevant scattering energies involved. For a thin accretion diskemitting black body radiation (e.g. Shakura & Sunyaev 1973), the temperature structure in scaled units isk T ∼ 54 (ṁ/m_9 r̃^3(1-(1/r̃)^1/2))^1/4 eVwhere m_9 = M_∙/(10^9 M_⊙), r̃ = r/r_S and ṁ = Ṁ/Ṁ_ Edd. is the accretion rate Ṁ scaled bythe Eddington accretion rate Ṁ_ Edd. = L_ Edd./(η c^2) with L_ Edd. = 1.3 × 10^47 m_9 and η = 0.007.For r̃∼ 3 - 50 and ṁ = 0.1 - 0.3 <cit.>, k T ∼ (1.7 - 15.5) eV,corresponding to optical/ultra-violet emission. For a corona (composed of electrons) in virial equilibrium,k T ∼m_e c^2/2 r̃ = 0.511/2 r̃ MeVwhere m_e = 0.511 MeV/c^2 is the electron rest mass. For r̃∼ 3 - 50, the accretion energy per electron k T ∼ (5.1 - 85.2) keV. If the lower energy optical/ultra-violet disk based photons scatter off sufficiently dense regions of the corona, this can result in upscattering of soft X-ray photons to higher energy further downstream in the jet by relativistic electrons.The relative strength of the emission processes can be tested and physical parameters of the pc-scale jet roughly estimated using a toy model involving synchrotron and inverse Compton emission. The synchrotron and Compton power losses L_S and L_C (effectively the luminosity) during the emission and scattering from a relativistic electronrespectively are in the ratioL_S/L_C = U_B/U_ ph.,where U_B = B^2/(8 π) and U_ ph. are the magnetic field and radiation energy densities respectively. We can infer the magnetic field strength asB = (8 π U_ ph.L_S/L_C)^1/2,once we estimate luminosities L_S and L_C and the energy density in the radiation field. It is assumed that the electrons emitting synchrotrontraverse a region in the jet with a strong magnetic field such as that argued above (in the region ∼ 10^2 - 10^3 r_S). Further, if we assumethat the majority of scattered Compton energy density is from these synchrotron emitting electrons being Doppler beamed along the observerline of sight U_ ph.,S and an EC process composed of energy densities from the accretion disk, broad line region clouds surrounding the disk and the dust torus <cit.>,U_ ph. = U_ ph.,S+ U_ ph.,D+ U_ ph.,BLR+ U_ ph.,T= 1/4 π c{L_S/δ^4 Δ r^2+Γ^2 (L_D/r^2_D+L_ BLR/r^2_ BLR+L_T/r^2_T)},where L_ D,BLR,T and r_ D,BLR,T are the accretion disk, broad line region and torus luminosities and radii respectively. The luminosity L_S is scaled by the Doppler factor δ and the luminosities L_ D,BLR,T by the Lorentz factor for an energy density as measured in the jet co-moving frame <cit.>. For a radiatively efficient accretion disk,L_D =ηṀ c^2 = (9.1 × 10^44 erg/s) ṁ m_9,and r_D is the radius of the accretion disk participating in the contribution to the EC scattered seed photons. An approximation for r_D can bethe distance at which the disk transitions from being gravitationally bound to the central supermassive black hole to being self gravitational(King 2016) in which caser_D = (6.46 × 10^16 cm) ṁ^-8/27 m^1/27_9,which is typically 0.01 - 0.1 pc.If we assume that all radiation emitted from the disk participates in ionizing the BLR clouds and results in a re-processed emission from the BLR, the emergent luminosity L_ BLR is similar to L_D owing to ionization timescale being much lesser compared to the scattering timescale (mostly dominated by the geometry of the BLR). Then, L_ BLR = η_ BLR L_D where η_ BLR∼ 0.1 is a disk covering factor attributable to the BLR and represents the fraction of disk radiation it re-processes <cit.> andr_ BLR = (10^17 cm) (L_D/10^45)^0.5 = (9.54 × 10^16 cm) ṁ^0.5 m^0.5_9.The dust torus emission is expected to be thermal and dominant in the infra-red wavelengths. If we assume that L_T = η_T L_D where η_T∼ 0.5 is a disk covering factor attributable to the torus <cit.> and a torus temperature T_T below the dust sublimation temperature, the torus radius can be scaled in terms of the disk luminosity as <cit.> r_T= (3.5 × 10^18 cm) (L_D/10^45)^0.5(T_T/10^3 K)^-2.6= (3.5 × 10^18 cm) ṁ^0.5 m^0.5_9 T^-2.6_T,3, where T_T,3 = T_T/(10^3K).For the sources indicating a spectral upturn, we can approximate L_C ≥ L_X = 4 π D^2_L f_X where f_X are the X-ray fluxes obtained from the spectral fits and listed in Table 3. We take L_S based on the observed synchrotron peak flux densities (Fan et al. 2016 and references therein) for PKS 0235+164, PKS 0537-441, S5 0716+714, OJ 287, ON 231 and BL Lac and L_S = 4 π D^2_L f_R based on a 1.4 GHz flux density of 4 × 10^-4 Jy (Tingay et al. 2003) for PKS 0426-380. Input parameters include m_9 = 1, ṁ = 0.1 - 0.3, δ = 6, η = 0.007, T_T,3 = 0.2 (based on a typical range of torus temperatures between 138 - 300 K; <cit.>) and the resulting estimates of B are presented in Table 6. In these estimates, we assume a flat cold dark matter dominated cosmology with matter energy density Ω_m = 0.308 and Hubble constant H_0 = 67.8km s^-1 Mpc^-1 (Planck Collaboration et al., 2016) to calculate the luminosity distance D_L (Wright, 2006). The contribution of the dust torus to the estimated U_ ph. is negligible in these cases (typically at 0.0001 - 0.001 %) but is included in the calculation for completeness and for cases where larger T_T which can result in a non-negligible contribution. The B values estimated here are strictly upper limits owing to L_C ≥ L_X and are in the range 0.03 - 0.88 G consistent with estimates from the core shift method <cit.> for some of these sources (S5 0716+714 and BL Lac with B = 0.22 ± 0.36 G and 0.02 ± 0.06 G respectively), eventhough the region being probed here is ≤ 0.05 pc compared to the pc-scale jet in the core shift method, thus indicating a similar magnetic field energy density from sub-pc–pc scale possibly implying that magnetic field structuring develops in the innermost jet and is retained upto the pc-scale jet, consistent with the canonical perspective.The method can thus be applied in a self consistent manner, accounting for all sources contributing to the energy density including the broad line region clouds, the torus and the microwave background, and is an independent check on estimates from core shift measurements which assume equipartition between the magnetic and particle kinetic energy densities. The above simplistic calculation is able to capture a general agreement with other methods. For a more rigourous approach, it can be extended to include the distribution function of the particles composing the jet (which can include electron-positron pairs; leptons and protons), the bulk properties of the jet <cit.>, treatment of the γ-ray regime and a careful assessment of the covering factors and various geometries of the sources of external seed photons <cit.>, and the relative motion between the emitting source and the observer which can introduce systematic variability <cit.>.The energy density in the magnetic field encountered by an electron composing the jet is U_B = |B|^2/(8 π) with a median U_B = 0.01 erg cm^-3 and thecorresponding synchrotron cooling time is t_ cool = 3 m_e c/4 σ_T β^2 γ U_B = (0.91 days) γ̃^-1 B^-2_1, where m_e = 9.11 × 10^-28 g is the electron mass, σ_T = 6.65 × 10^-25 cm^2 is the Thomson cross section for electronscattering, β∼ 1 is the speed of the relativistic electrons, γ̃ = γ/10^3 is the Lorentz factor γ of theinjected electrons scaled by ∼ 10^3 which can typically be reached, and B_1 = B/(1G). For the inferred range of B, the mediant_ cool∼ 29 days. The ratio between the Compton energy density from various sources contributing to EC (U_ ext.,here the disk, BLR and torus), and the total scattered photon energy density (including SSC) is U_ ext./U_ ph. =0.74–0.99 indicating that the EC seed photons prominently contribute to the observed luminosity and variability.In this scenario, the jet can undergo strong radiative cooling due to the Compton drag caused resulting in maximum limits forthe bulk Lorentz factor and is useful in constraining the jet composition <cit.>. If the relativisticelectrons within a spherical cloud of radius Δ r are variable over a median timescale t_ var = δτ/(1+z) = 4.6 days(jet co-moving frame), t_ var < t_ cool in general for these sources. The synchrotron electrons then participate in multipleepochs of inverse Compton-scattering before cooling down and the resultantvariability will tend towards red noise behaviour in their power spectral density shapes (slope μ = -1 to -2). As the energy ofscattered photons from a given epoch depends on the available energy from scattering events in previous epochs, thus exhibiting long termtrends in the light curve within and across epochs. Here, the weighted mean μ = - 0.77 ± 0.26 indicating that in addition to rednoise behaviour as expected from the above discussed process, random processes in the jet contribute white noise (slope μ = 0)variability thus causing flattening of the slopes. Such jet based processes operational at ∼ day type timescales can includeturbulence and shocks <cit.> or magnetic re-connection events<cit.> amongst others.In the earlier studies, Giommi et al. (1999) and Tagliaferri et al. (2003) found lack of significant variability in the higher energybands i.e. 3 - 5 keV and hence concluded a scenario of a variable synchrotron component and a stable IC component, on hour like timescales. But, Ferroro et al. (2006) found that even at higher energies i.e. above ∼ 3 keV, where the IC component is expected todominate, the fractional variability amplitude was significantly high. They argued in a favour of a variable IC component on shorttimescales. For our sample of blazars, it might be the case, supported by the above analysis of IC emission sources. The hardness ratios are found to be anti-correlated with respect to the total count rate for S5 0716+714, ON 231 and BL Lacertae.It represents the redder-when-brighter trend which is different for the usual bluer-when-brighter trend for HSPs. The redder-when-brightertrend could be explained by the relative contribution of softer synchrotron component, getting higher during the enhanced emissionwith respect to the more stable IC component. It implies an overall steeping of the spectra, however the actual slope becomes flatter. § ACKNOWLEDGEMENTS We thank the referee for thoughtful and constructive comments which have improved the context of our work. This research is based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directlyfunded by ESA member states and NASA. We thank Lucasz Stawarz for critical comments and discussions which have helped improvethe manuscript. H.G. is sponsored by the Chinese Academy of Sciences (CAS) Visiting Fellowship for Researchers from DevelopingCountries; CAS Presidents International Fellowship Initiative (PIFI) (grant No. 2014FFJB0005); supported by the National NaturalScience Foundation of China (NSFC) Research Fund for International Young Scientists (grant No. 11450110398, 11650110434) andsupported by a Special Financial Grant from the China Postdoctoral Science Foundation (grant No. 2016T90393). P.M. is supported bythe CAS-PIFI (grant no. 2016PM024) post-doctoral fellowship and the NSFC Research Fund for International Young Scientists(grant no. 11650110438). A.W. acknowledges support by the Foundation for Polish Science (FNP).MF.G. is supported by the National Science Foundation of China (grants 11473054 and U1531245). [Abdo et al.2010]2010ApJ...716...30A Abdo A. 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"authors": [
"Haritma Gaur",
"P. Mohan",
"Alicja Wierzcholska",
"Minfeng Gu"
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"published": "20170927054133",
"title": "Signature of Inverse Compton emission from blazars"
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Convergence of utility indifference prices to the superreplication price in a multiple-priors frameworkRomain Blanchard, E.mail :[email protected]. Laurence Carassus, E.mail : [email protected] Léonard de Vinci Pôle Universitaire, Research Center, 92 916 Paris La Défense, France and LMR, UMR 9008, Université Reims Champagne-Ardenne.===================================================================================================================================================================================================================================================================== This paper formulates autility indifference pricing model for investors trading in a discrete time financial market under non-dominated model uncertainty.Investor preferences are described bypossibly random utility functions defined on the positive axis. We prove thatwhen the investors's absolute risk-aversion tends to infinity, the multiple-priors utility indifference prices of acontingent claim converge to itsmultiple-priors superreplication price. We alsorevisitthe notion of certainty equivalent for multiple-priorsand establish its relation with risk aversion.Key words: utility indifference price; superreplication price; absolute risk aversion, Knightian uncertainty; multiple-priors;non-dominated model AMS 2000 subject classification: Primary 91B70, 91B16, 91G20 ; Secondary91G10, 91B30, 28B20JELclassification: C61, D81, G11, G13§ INTRODUCTIONIn this paper, weexamine different definitions of prices for a contingent claim and their relation in the context of uncertainty. Risk and uncertainty are at the heart of economic lifeand modeling the way an agent will react to themis a central thematic ofeconomic research (see for instance <cit.>). Knightian uncertainty (see <cit.>) means that the agent is notcertain about the choice of a given priormodeling the outcome of asituation. It is a kind of“unknown unknown in opposition to the risk where the agent is confident about her prior and only faces the randomness of the outcome, which is somehow the “known unknown.Issues related to uncertainty arise in various concrete situations in social sciences and economics, such as policy-making.Theyalso affectmany aspects of modern finance such asmodel risk whenpricing and risk-managingcomplex derivatives productsorcapital requirement quantification when looking at regulation for banksand others financial entities.As illustrated through the Ellsberg Paradox (see <cit.>), when facing uncertainty an agentdisplays uncertainty aversion:She tends to prefer a situation where the uncertainty is reduced.This is the pendant of the risk aversion when the agent faces only risk.It is well known that if one wants to represent the preferences of the agent in this context, theaxiomatic of the von Neumann and Morgenstern expected utility criterium (see <cit.>) is not verified. The Savage's extension (see <cit.>), where subjective probability measures depending on each agent are introduced, does not solve this issue. Thus, in this paper, we follow the pioneering approach introduced by<cit.>where undersuitable axiomatic on the investor preferences,the form of the utility functional isa worst case expected utility: inf_P ∈𝒬 E_P U( X), where 𝒬 is the set of allpossible probability measures representingthe agent'sbeliefs on the market model. Somehow the larger𝒬,the less confident the agent is in a specific model and the more she wishes to take into account as many scenarii as possible.For example, the set 𝒬 may be constructed starting from a givenunderlying model whereall the parameters arenot available but some of them may beinferred or estimatedfrom observable prices.The agentmight also want to add her own belief or view on the “correct"valueof these parameters. Such a worst case expected utility representation can also be used for robustness considerations when𝒬 is a set of modelsresulting fromsmall perturbations of an initial reference model. This isrelated for instance to the work of<cit.> where a term corresponding to therelative entropy given a certain reference probability measureis added to the utility functional.The frameworkof<cit.>was extended by <cit.> who introduced a penalty termto the utility functional. Finally, <cit.> representthe preferences by a more general functionalinf_P ∈𝒬 g(E_P U(X),P) where g is a so-called uncertainty index reflecting the decision-maker's attitudes towards uncertainty.An important feature is to allow the set of probability measures 𝒬to be non-dominated. This means that no probability measuredetermines the set ofevents that can happen or not. The relevance of this idea is illustrated by the concreteexample of an underlying market model with volatility uncertainty, see <cit.>, <cit.> and<cit.>. For a simple binomial model where the up and down multiples belong to intervals, the set of priors is non-dominated as soon as for one scenario, one of the interval is not trivial (see <cit.>). However considering non-dominated models increases significantly the mathematical difficulties assome of the classical tools of probability theory such as conditional expectation or essential supremum are ill-suited to this framework (since they are defined with respect to a given probability measure). These type ofissues havecontributedto the development of innovative mathematical tools such as quasi-sure stochastic analysis, non-linear expectations, G-Brownian motions. On these topics,we refer amongothers to<cit.>,<cit.> or <cit.>.In quantitative finance, the No-Arbitrage (NA) notion iscentral to many problems and crucial when it comes to pricing questions. It asserts that starting fromzero wealth it is not possible to reach a positive one (i.e. non negative almost surely and strictly positive with a strictly positive probability). The characterisation of this condition or of the No Free Lunch condition is called the Fundamental Theorem of Asset Pricing (FTAP in short) and makes the link betweenthese notionsand the existence ofequivalent risk-neutral probability measures (also called martingale measures or pricing measures) which are equivalent probability measures that turn the (discounted) asset price process into a martingale. This was initially formalised in <cit.>, <cit.> and <cit.> while<cit.>obtained the FTAPin a general discrete-time setting under the NA condition. The literature on the subject is hugeand we refer to <cit.> for a general overview.The martingale measure is usedto price contingent claims. However, inincomplete marketsi.e. when not allcontingent claims can be perfectly replicated by dynamic trading, the risk-neutral probability measure is not unique and this leadsto different possible evaluations for a given claim. The superreplication price is the minimum amountneeded for an agent selling a claim in order to superreplicateit by trading in the market. This is the hedging price with no risk and to the best of our knowledge it was first introduced in <cit.> in the context of transaction costs. In complete markets the superreplication costis just the cash flow expectation computed under the unique martingale measure. But in incomplete markets,the superreplication costis equal to the supremum of those expectations computed under the different risk-neutral probability measures. This isthe so called dual formulation of the superreplication price orSuperhedging Theorem (see for instance <cit.> or <cit.>).Naturally, all these concepts have seen a renewed interest in the context of uncertainty, seeamongothers <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.> and <cit.>. One may wonder if the superreplication price is not too high to be used practically infinancial markets and if it should only be seenas anupper-bound for pricing issues. On one hand, in <cit.> it is proved that when the support of the conditional law of the risky asset is bounded, the superreplication price of some convex option is equal to the replication price in a binomial model (see <cit.>) whose parameters are the law support boundaries. So, if this support is not too large, the superreplication price can be of practical use. This type of result has been generalized in the robust setting by <cit.>: The mutilple-priors superreplication price corresponds to theuni-prior superreplication price for an extreme prior in 𝒬. On the other hand,the supperreplication price is sometimes too onerous:For examplethesuperreplication price of a call option may be equal to the underlying initial pricein a stochastic volatility model (see <cit.>). Thus it may be interestingto consider another concept of pricing,especially since the superreplication price does not take into account the preferences of the agents.The reservation price (or utility indifference price)is an alternativeapproach to pricing contingent claims. In the context of quantitative financeit was first introduced in <cit.> in the presenceof transaction costs. This is the minimum amount of money to be paid to an agent selling a contingent claimsuch that, added to her initial capital,herutility when selling and hedging it by trading dynamically in the market is greater than or equalto the one she would get without selling the product. Importantly, this notion of price allows to take into account the preferences of the agent andallows for somerisk-seeking behavior while the superreplication price corresponds to a totally risk averse agent. Hence, the reservation price should provide a cheaper alternative to the suppereplication price.But can it be used in practice? Considerthe case of illiquidity and basisrisk:Options are sometimes written onilliquid underlying assets where a liquid marketexists in some closely related asset (as for example in commodity markets or for real options). To find an appropriate price and the best hedging strategy using only tradable assets, a widely used approach is the reservation price(see <cit.> and the reference therein). In the case of the exponential utility functions the price can be computed in a reasonably explicit form using convex duality. Moreover <cit.> shows that the optimal strategy based on exponential utility maximization gives a superior hedging performance than a naive Black-Scholes strategy that assumes that the traded asset is a good proxy for the non-traded asset. However the reservation price for exponential utility functions is wealth independent, which is quite unrealistic since agents with different endowments will not have the same attitude towards risk. This is a good argument for considering other utility functions. In these cases, the reservation price is difficult to compute but it is still possible to obtain power series expansions (see <cit.>).In this paper, we fill a gap in the literature introducing the reservation pricein the multiple-priors set-up and studying its links with the (multiple-priors) superreplication price. Our convergence result asserts that even in a multiple-priors set-up when the absolute risk aversion increases the preferences of the agent is less and less relevant for pricing issues: The preference based prices of the agent converge to the preference risk free one. Proving this, we extend an important literaturestarting with <cit.>for exponential utility functions and aBrownian model. Then,<cit.> extended the result to a general semimartingale setting while a nonexponential case was treated in <cit.>, but with severe restrictions on the utility functions. The case of general utility functions was considered in <cit.> and <cit.> in discrete-time market models and in <cit.> for continuous time ones.We have chosen towork in discrete time and to considerutility functions defined onthe half real line rather than the whole real line. We believe that this is actuallyrelevant in practice as it corresponds tosituations where the agent is totally averse to bankruptcy. Wealso consider random utility functions to allow for state dependent absolute risk aversion orrandom reference point. To the best of our knowledge Theorems <ref>, <ref>(for non random utility functions) and <ref> (for random utility functions) are the first general asymptotic results in the multiple-priors framework. We treat the case of general concave utility functions in a regular enough market (see Assumption <ref> and Theorem <ref>) andthe case of general markets for sequence of functions which are possibly non concave butbounded from above uniformly in n (see Theorem <ref>). Note that simultaneously <cit.> obtains someconvergence result for yet anotherutility based price and agents with constant absolute risk aversion.Even if the paper follows a long line of research, one could questionthe theoretical (and practical) value of our asymptotic result. First, it proves that the superreplication price is a kind of universal price even when taking into account the preferences of the agent in the case of high absolute risk aversion. Indeed, empirical evidence related to the risk premium puzzle (see <cit.>) has shown that therisk aversion of an agent can be very high. Thus, in theses cases the superreplication price is a good approximation, even in the multiple-priors case. As already mentioned,in some quite general cases (if the set 𝒬 is not too wide) it can be computed as in the uni-prior case andused in practice. Outside these cases (very high risk aversion and “reasonable superreplication price), taking into account the preferences of the agent is a good way to obtain a lower price (see the basis risk example and also Section <ref>). We also revisit in a static context the notion of certainty equivalent introduced in <cit.>. We extend itin the presence of multiple-priors and give some conditions for existence and uniqueness of certainty equivalent (see Proposition <ref>). Weestablish that the absolute risk aversionallows the ranking of the multiple-priorscertainty equivalent despitethe presence of uncertainty aversion (see Proposition <ref>). This part isrelated to <cit.>wherean alternative notion of (static) indifferenceprices are introduced for non-random utility functionsunder the representation of <cit.>.Finally we present a detailed example where all the concepts and results of the paper are illustrated.We have chosen towork under the discrete-time framework introduced in <cit.>. We outline briefly in Section<ref> some of the interesting features of this framework, in particular with respect to time-consistency. To solve our problem, we use some arguments of <cit.> that are adapted to our multiple-priorsframework together with <cit.>. We also usesomeelements of quasi-sure stochastic analysis as developed in <cit.> and <cit.>. The article is structured as follows:Section <ref> presentsthe framework and the definitionsneeded in the rest of the paper. Section <ref>presents the main theorem on the convergence of the utility indifference prices to the superreplication price for non-random utility function. This section alsorevisits the link between certainty equivalent and absolute risk aversion in our set-up. Section <ref> proposes a detailed example illustrating our results. The proofsare reported in Section <ref>. Finally,in Section <ref> theconvergence result is extended to random utility functions. § THE MODEL This section presents our multiple-priorsframework.§.§ Framework overview We fixa time horizon T∈ℕ andintroducea sequence (Ω_t)_1 ≤ t ≤ Tof Polish spaces. Each Ø_t+1 contains all possible scenarii between time t and t+1. For some 1 ≤ t ≤ T, let Ω^t:=Ø_1×…×Ø_t(with the convention that Ω^0 is reduced to a singleton). We denote byℬ(Ø^t)its Borel sigma-algebra and by 𝔓(Ø^t) the set of all probability measures on (Ø^t,ℬ (Ø^t)).An element of Ω^t will be denoted by ø^t=(ω_1,…, ω_t) for (ø_1,…,ø_t) ∈Ω_1×…×Ω_t. We also introduce the universal sigma-algebra ℬ_c(Ø^t) which is the intersection of all possible completions of ℬ(Ø^t). A function f: Ø^t→ Y (where Y is an other Polish space) is universally-measurable(resp. Borel-measurable) if for all B ∈ℬ(Y) (the Borel sigma-algebra on Y), f^-1(B) ∈ℬ_c(Ø^t) (resp. f^-1(B) ∈ℬ(Ø^t)).Similarly we will speak of universally-adapted or universally-predictable (resp. Borel-adapted or Borel-predictable) processes. §.§.§ Uncertainty modelisationUncertainty is modeled as in<cit.> (see also<cit.>). We refer to these papers for a thorough technical presentation of the framework.For each 0≤ t≤ T-1 we consider some random set𝒬_t+1 : Ω^t ↠𝔓(Ø_t+1), where 𝒬_t+1(ø^t) can be understood asthe set of all possible models, from the agent perspective, for the t+1-th period if the scenario ø^t occurs until time t. Those setsare the blocksfrom which the set 𝒬^t of all priorson Ø^t is built.𝒬^t:={ Q_1⊗ q_2⊗…⊗ q_t, Q_1∈𝒬_1, q_s+1(·,ø^s) ∈𝒬_t+1(ø^s), ∀ø^s∈Ø^s, s ∈{1,…, t-1}},where for all 1≤ s ≤ T-1, q_s+1(·,ø^s) is a universally-measurable stochastic kernel on Ø_s+1 given ø^s ∈Ø^s (see <cit.>), and wherethe notation Q_s:=Q_1⊗ q_2⊗…⊗ q_s stands for the probability measureresulting from the composition using Fubini's Theorem: For all A ∈ℬ_c(Ø^s)Q_s(A)=∫_Ω_1⋯∫_Ω_s 1_A(ω_1,⋯, ω_s) q_s(dω_s,(ω_s-1,⋯ø_1)) ⋯ q_2(dø_2,ø_1) Q_1(dω_1).The set 𝒬^T governs the market until time T anddetermines which events are relevant or not for the agents. Note that tomake this construction mathematically rigorous, a measurable selection theorem is applied in order to pick up some universally-measurable q_s+1(·,ø^s) ∈𝒬_t+1(ø^s). To do that we rely on the following technical assumption which is now classical in the recent literature on multiple-priors models.For all 0≤ t≤ T-1,𝒬_t+1 is a non-empty and convex valued random set such that(𝒬_t+1):={(ω^t,P) ∈Ω^t×𝔓(Ω_t+1),P ∈𝒬_t+1(ω^t)}is an analytic set. [Recall that an analytic set is the continuous image of some Polish space, see<cit.>, and also <cit.>for more details on analytic sets.] Apart from Assumption <ref>, no specific assumption on the set of priors is made:𝒬^Tis neither assumed to bedominated by a given reference probability measure nor to be weakly compact. This setting allows for various general definitions of thesets 𝒬^T. We propose below some examples andrefer to <cit.> and <cit.> for other examples.Our model includes the (non-dominated) case where the physical measure is not known a priori but is rather a result of collecting data and estimation, so that some kind of “neighborhood” is added to the estimator P^*=P^*_1⊗ p^*_2⊗⋯⊗ p^*_T of the physical measure𝒬_t+1(ω^t)={P ∈𝔓(Ø_t+1),(P,p^*_t(ø^t)) ≤_t(ø^t) },where dist isa distance between distributions (a popular choice is the Wassersteindistance). Then if _t, p_t^* and dist are Borel-measurable, (𝒬_t+1) isan analytic set (see <cit.>). Another interesting non-dominated caseis when the increments of the price process are bounded. Working on the canonical space of a one-dimensional stock Ω= ℝ^T, S_t(ø^t)=ø_t,the set of possible priors is given by𝒬_t+1(ω^t) ={P ∈𝔓((ℝ)),(P) ⊂ [ø_t d_t+1,ø_tu_t+1]},where (P) is the support of the measure P. ThenAssumption <ref> is satisfied(see <cit.>). §.§.§ Time-consistency and related commentsThe fact that our sets of probability measures are uniquely determined by the set of one-step-ahead probability measures is related to the notion of time-consistency on which wefocusbrieflynow.Roughly speaking, time-consistency means that a decision taken tomorrow will satisfy today's objective. Recall that this issue appears already in a uni-prior setting, in the study of dynamic risk measures for instance, andislinkedto the law of iterated conditional expectations and the dynamic programming principle.We refer to the surveys <cit.> and <cit.>fordetailed overviews. When introducing multiple-priorsone has to be even more careful with time-consistency. In <cit.>a simple exampleillustrates what can happenifone is not cautiouson the structure of the initial set of priors: One cannot hope to find an optimal solution using thedynamic programming principle.To deal with this, one has to assume that the set of priors is stable under pasting whichroughly meansthatdifferent priorscan be mixedtogether (see<cit.>). It is clear that the set of priors 𝒬^T is stable under pasting. Indeed, given(<ref>), if Q^1,Q^2 ∈𝒬^T with Q^1=Q^1_1⊗ q^1_2⊗⋯⊗ q^1_T, Q^2=Q^2_1⊗ q^2_2⊗⋯⊗ q^2_T, then R:=Q^1_1⊗ q^1_2⋯⊗ q^1_t-1⊗ q^2_t⊗⋯⊗ q^2_T∈𝒬^T for all 2 ≤ t ≤ T-1. In a sense, the set 𝒬^T is large enough (unlike in the exampleconsidered in<cit.>). In <cit.> the equivalent notion of rectangularity isintroduced (see also<cit.> for more details and a graphical interpretation).§.§.§ The traded assets and the trading strategiesLet S:={S_t, 0≤ t≤ T} be a universally-adapted d-dimensionalprocess where for 0≤ t≤ T, S_t=(S^i_t)_1 ≤ i ≤ d represents theprice of d risky securities in the financial market in consideration. To solve measurability issuesthe following assumption already present in <cit.> is made. The price process S is Borel-adapted.Trading strategies are given by universally-adapted d-dimensionalprocesses ϕ:={ϕ_t, 1 ≤ t ≤ T} where for all 1 ≤ t ≤ T, ϕ_t=(ϕ^i_t)_1 ≤ i ≤ d represents the investor's holdings ineach of the d assets at time t. The set of trading strategies is denoted by Φ.The trading is assumed to be self-financed andthe riskless asset's price constant equal to 1. The value at time t of a portfolio ϕ starting from initial capital x∈ℝ is then given byV^x,ϕ_t=x+∑_s=1^tϕ_s Δ S_s.§.§ Multiple-priors no-arbitrage conditionAs already eluded to in the introduction, the issue of no-arbitrage in the context of uncertainty has seen a renewed interest. In this paper we follow the definition introduced by <cit.>.The NA(𝒬^T) condition holds true ifV_T^0,ϕ≥ 0𝒬^T for some ϕ∈Φ implies that V_T^0,ϕ= 0𝒬^TA set N ⊂ X is a 𝒬^T-polar setif for all P ∈𝒬^T, there exists some A_P∈ℬ(Ø^T) such that P(A_P)=0 and N ⊂ A_P. A property holds true 𝒬^T-quasi-surely (q.s.), if it is true outside a 𝒬^T-polar set. Finallya set is of 𝒬^T-full measureif its complement is a 𝒬^T-polar set.We outline briefly some of the interesting features of this definition. First it isanatural and intuitive extension of the classical uni-prior no-arbitrage condition. This argumentis strengthened by the FTAP generalisation proved by <cit.>. Under Assumptions <ref> and <ref>, the NA(𝒬^T) is equivalent to the following: For all Q ∈𝒬^T, there exists some P ∈ℛ^T such that Q ≪ P whereℛ^T:={P ∈𝔓(Ø^T), ∃Q^'∈𝒬^T, P ≪ Q^' }.The classical notion of equivalent martingale measures is replaced by the fact that for all priors Q ∈𝒬^T, there exists a martingale measure P such that Q is absolutely continuous with respect to P and one can find an other priorQ' ∈𝒬^T such that P is absolutely continuous with respect to Q'. The extension in the same multiple-priorssetting ofthe Superhedging Theorem and subsequent results onworst-case expected utility maximisation (see for example <cit.>)is an other convincing element.We present now an alternative characterisation of the NA(𝒬^T) condition which was proved in <cit.>. Assume that the NA(𝒬^T) condition andAssumptions <ref>, <ref> hold true. Then for all 0≤ t≤ T-1, there exists some 𝒬^t-full measure set Ω^t_NA∈ℬ_c(Ø^t) such that for all ω^t∈Ω^t_NA, there exists α_t(ω^t)>0 such that for allh ∈D^t+1(ω^t)there exists P_h∈𝒬_t+1(ø^t) satisfyingP_h(h/|h|Δ S_t+1(ω^t,.)<-α_t(ω^t))> α_t(ω^t),where D^t+1(ω^t) is the affine hull of the multiple-priorsconditional support of the price increments D^t+1(ø^t) := (⋂{ A ⊂ℝ^d, , P_t+1(Δ S_t+1(ø^t,.) ∈ A)=1,∀ P_t+1∈𝒬_t+1(ø^t) }). Note that D^t+1(ø^t) is a vector space for all ω^t∈Ω^t_NA.In the case where there is only one risky asset and one period,the interpretation of (<ref>) is straightforward. It simply meansthat there exists a prior (i.e. some probability measure P^+) for which the price of the risky asset increases enough and an other one (P^-) for which the price decreases, i.e. P^±( ∓Δ S(·)<-α)> α where α>0. The numberserves as a measure of the gain/loss and of their size. In the general casethere is always a prior in which an agent is exposed to a potential loss when buying or selling some quantity of risky assets. Note that in <cit.>, the equivalence between Assumption <ref> and condition (<ref>) is established.We present now the measurability assumption needed for our convergence results (see Theorems <ref> and <ref> below) when the sequence of utility functions is not bounded from above (uniformly in n). First, weintroduce the following spaces that will be used throughout the paper 𝒲^0_t:={ X: Ω^t→ℝ∪{±∞} },𝒲^r_t:={ X ∈𝒲^0_t,sup_P ∈𝒬^tE_P |X|^r <∞}𝒲^∞_t:= { X∈𝒲^0_t,∃ M≥ 0, |X| ≤ M 𝒬^t},recall(<ref>) for the definition of𝒬^t. We also consider𝒲^0,bo_T the set of contingent claims bounded from below𝒬^T-q.s. i.e.G ∈𝒲^0,bo_T if and only if G ∈𝒲^0_Tandthere exists some constant b ≥ 0 such that G ≥ -b 𝒬^T-q.s.Note that a superscript ^+ will be added for non-negative elements (it will be also used for denoting positive parts). We have that Δ S_t, 1/α_t∈𝒲^r_t for all 1 ≤ t ≤ T and 0 < r <∞.Inlight of Proposition <ref>, the condition 1/α_t∈𝒲^r_t is akind of strong form of no-arbitrage. Note that if α_tis not constant, then even in the uni-prior casethe utility maximisation problem may be ill posed (see Example 3.3 in <cit.>). Hence our integrability assumption on 1/α_t looks reasonable.Some concrete examples where _t is computed and Assumption <ref> is verified are presented in <cit.>. Assumption <ref> could be weakened to the existence of the 𝒲_t^N-th moment for N large enough but this would lead to complicated book-keeping with no essential gain in generality, which we prefer to avoid. Note finally that as in <cit.> one can prove that for all r ∈ [1,∞],𝒲^r_t is aBanach space (up to the usual quotient identifying two random variables that are 𝒬^t-q.s. equal) for the norm ||·||_r,t where||X||_r,t:=(sup_P ∈𝒬^tE_P |X|^r)^1/r||X||_∞,t:=inf{ M ≥ 0, X(·) ≤ M𝒬^t}.We will omit the index t when t=T. § MAIN RESULTS This section contains our main results as well asthe definitionsof the superreplicationand of the(seller) utilityindifference prices.§.§ Multiple-priors superreplication priceThe multiple-priorssuperreplication price (a seller price) is the minimum initial amount thatan agent will ask fordelivering some contingent claim G ∈𝒲_T^0 so that she is fully hedged at T when trading in the market. The set of strategies which dominateG𝒬^T-q.s. starting from a given wealth x ∈ℝ is defined by𝒜(G,x):={ϕ∈Φ,V_T^x,ϕ≥ G𝒬^T}. Let G ∈𝒲_T^0. The multiple-priorssuperreplicationprice of G is defined byπ(G):=inf{z ∈ℝ,𝒜(G,z) ≠∅}and π(G)=+∞ if 𝒜(G,z) = ∅ for all z ∈ℝ.Note that π(G)=+∞ if and only if 𝒜(G,z) = ∅ for all z ∈ℝ. The corresponding buyer price is the multiple-priorssubreplicationprice of G defined byπ^sub(G):=sup{z ∈ℝ, 𝒜(-G,-z) ≠∅} and π^sub(G)=-∞ if 𝒜(-G,-z) = ∅ for all z ∈ℝ. It is clear that for G ∈𝒲_T^0, π^sub(G)=-π(-G).We recall now for the convenience of the reader<cit.>.Assume that Assumptions<ref> and <ref> hold true and let G ∈𝒲_T^0 befixed. Then π(G)>-∞ and 𝒜(G,π(G)) ≠∅.To apply <cit.> Assumption <ref>is not needed and one may also usea weaker form ofAssumption <ref>.As usual if the market is complete (i.e. ifany bounded contingent claim is replicable), the superreplication price is equal to the replication price. Indeed, ifG is replicable,i.e. if there exists some x_G and some ϕ_G∈Φsuch that G=V_T^x_G,ϕ_G 𝒬^T-q.s., then π(G)=x_G=π (V_T^x_G,ϕ_G). Moreover, under some measurability assumption on G, the Superreplication Theorem holds: π(G)=sup_P ∈ℛ^T E_P G, see <cit.> and (<ref>) for the definition of ℛ^T.We now turn to some pricing rules which take into account the preferences of the agents. §.§ Utility function and utility indifference priceIn Section <ref> we focus on deterministic utility functions defined on the half-real line. An extension to random utility functions is proposed in Section <ref>. In the rest of the section, we consider utility functionU:ℝ→ℝ∪{- ∞} such that U(x)=-∞ if x<0 and such that there exists some x_0 ∈ (0,∞) verifyingU(x_0)>-∞. Considering such functions corresponds to the concrete situation where the investor is infinitely averse to bankruptcy. Up to a translation one may consider investor with limited credit line. The restriction of U on (0,∞)is strictly increasing, twice continuously differentiable and we set U(0):=lim_x→ 0^+ U(x).For any function U satisfying Assumption <ref>,the absolute riskaversion is defined for all x ∈(0,+∞) byr(x) :=-U^”(x)/U^'(x). We now turn to pricing issues. First we define some particular sets of strategies for a contingent claim G ∈𝒲^0_T and some initial wealth x ∈ℝ (recall (<ref>))Φ(U,G,x):={ϕ∈Φ,E_P U^+(V_T^x,ϕ(·) -G(·))<+∞, ∀ P ∈𝒬^T} 𝒜(U,G,x) := Φ(U,G,x) ∩𝒜(G,x).If ϕ∈Φ(U,G,x), the integrals E_P U (V_T^x,ϕ(·)-G(·)) are well-defined for all P ∈𝒬^T and belong to [-∞,∞). However, even for x≥π(G), 𝒜(U,G,x) might beempty. Indeed, fromTheorem <ref> there exists some ϕ∈𝒜(G,x), but ϕ might not belong to Φ(U,G,x).In order to fix these integrability issues,the following proposition establishes that 𝒜(U,G,x)=𝒜(U,x) under suitable assumptions.Suppose that Assumptions <ref>, <ref> and <ref> hold true.Suppose that either U is bounded from above or that U is a concave function satisfying Assumption <ref> and that Assumption <ref>holds true. Fix some contingent claimG ∈𝒲_T^0,bo and some x ∈ℝ.Then, 𝒜(U,G,x)=𝒜(G,x).If 𝒜(G,x) ≠∅, there exists some constant M_x∈ [0,∞) such that -∞≤ E_P U ( V_T^x,ϕ(·)-G(·)) ≤ M_x,for all P ∈𝒬^T and ϕ∈𝒜( G,x).Note that (<ref>) and Theorem <ref> imply that 𝒜(G,x) ≠∅ if and only if x ≥π(G). If 𝒜(G,x)= ∅ then𝒜(U,G,x)=∅.Suppose now that 𝒜(G,x) ≠∅ (which is equivalent to x ≥π(G)). Assume first that U is a concave function satisfyingAssumption <ref> and thatAssumption <ref>holds true. The proof relies on Lemma <ref>.The monotonicity, concavity and differentiabilityof U imply thatfor all y ∈ℝ and x_0>0U(y) ≤ U(max(y,x_0)) ≤ U(x_0)+ max(y-x_0,0) U'(x_0). ThusU^+(y) ≤ U^+(x_0)+ |y| U'(x_0).For any ϕ∈𝒜(G,x) andP ∈𝒬^T, using the monotonicity of U, the existence of some constant b ≥ 0 such that G ≥ -b 𝒬^T-q.s. and Lemma <ref>, we get thatE_P U^+ (V_T^x,ϕ(·)-G(·))≤E_P U^+ (V_T^x+b,ϕ(·)) ≤U^+(1) +sup_P ∈𝒬^T E_P(|V_T^x+b,ϕ(·)|) U'(1)≤U^+(1) + (|x|+b)U'(1) sup_P ∈𝒬^TE_PM_T(·):=M_x<∞since M_T∈𝒲^1_T and both U(1) and U'(1) are finite valued.Assume now that U is bounded from above. Then the last boundness result in (<ref>) is still valid choosing for M_x the upper bound of U^+.We now introduce the quantityu(G,x) which represents,the maximum worst-case expected utility starting from initial capital x ∈ℝ anddelivering G ∈𝒲^0_Tat Tu(G,x):= sup_ϕ∈𝒜(U,G,x)inf_P ∈𝒬^T E_P U (V_T^x,ϕ(·)-G(·))whereu(G,x)=-∞ if 𝒜(U,G,x)=∅. Note that the expectations inu(G,x) are well-defined (by definition of 𝒜(U,G,x)) and without further assumptionsu(G,x) ∈ [-∞,∞]. The aim ofthis paper is not to find an optimal solution for (<ref>).As already mentioned, without further assumption 𝒜(U,G,x) might be empty and in this case u(G,x)=-∞: The agent will never sell the contingent claims and (<ref>) is not really an interesting problem. If x ≥π(G), under the assumptions of Proposition <ref>,we can rely onTheorem <ref>to find strategies in𝒜(U,G,x) since any super-replication strategy (which exists)automatically belongs to 𝒜(U,G,x). In this case,we have that u(G,x) ≤ M_x<∞, see (<ref>). Note that if 𝒜(U,G,x) is not empty, findingan optimalsolution for (<ref>) requiresto study how the terminal constraint V_T^x,ϕ -G ≥ 0 𝒬^T-q.s. together with the integrability condition on E_P U^+ (V_T^x,ϕ(·)-G(·))are “propagated" to the previous periods through dynamic programming. This is a non-trivial problem. This was done for G=0 in <cit.> andcould be done in thegeneral case along the lines of <cit.>, see also <cit.>.The problem has been solvedfor bounded from above utility function in<cit.>. We are now in a position to definethe(seller) multiple-priorsutility indifferenceprice or reservation price, which generalizes in the presence of uncertainty the conceptintroduced by <cit.>. It represents the minimum amount of money to be paid to an agent selling a contingent claim G so that added to her initial capital,her multiple-priors utility when selling G and hedging it by trading dynamically in the market is greater than or equal tothe one she would get without selling this product. Let G ∈𝒲^0_T and x ∈ℝ.The (seller) multiple-priorsutility indifferencepriceisdefined byp(G,x):= inf{z ∈ℝ,u(G,x+z) ≥ u(0,x)}where p(G,x)= +∞ ifu(G,x+z) < u(0,x) for all z ∈ℝ.In the previous definition x represents the initial wealth of the agent. It could be the accumulated trading gain and hence much higher than the price of a given contingent claim. However,nothing prevents x from being smaller than the superreplication price, see Section <ref>. The utility indifference pricein (<ref>) is static. Neverthelesswe believe it is important to allow for a dynamic evolution of the price process S and of the uncertainty on the priors between the initial dateand the maturity. Moreover, it could be proved that p(G,x) coincides at t=0 with the dynamic version of the utility indifference price. One may also introduce the(buyer) multiple-priorsutility indifferenceprice p^B(G,x):= sup{z ∈ℝ,u(-G,x-z) ≥ u(0,x)}where p^B(G,x)= -∞ if u(-G,x-z) < u(0,x) for all z ∈ℝ. If G ∈𝒲^∞_T then under the assumptions of Proposition <ref>, 𝒜(-G,z)=𝒜(U,-G,z) for all z ∈ℝ and p^B(G,x)=-p(-G,x).The next proposition shows that under suitable assumptions whatever the preferences of the agent are, the reservation priceis lower than the superreplication price.The superreplication price is, in the sense that we will define below, the price corresponding to an infinite absolute risk averse agent. Assume that Assumptions <ref>, <ref> and <ref> hold true. Suppose that either U is a non decreasing and bounded from above function or that U is a concave function verifying Assumption <ref> and that Assumption <ref>holds true. Fix some contingent claimG ∈𝒲_T^0,bo and some x ∈ℝ.Let x ∈ℝ. Thenp(G,x)≤π(G).Assume thatu(0,x)>-∞. Then p(G,x)≥π(G)-x and lim_x → 0 p(G,x)=π(G).Fix some x∈ℝ. First Theorem <ref> gives some ϕ_G∈𝒜(G,π(G)). As U is non-decreasing, we get that u(0,x) = sup_ϕ∈𝒜(U,0,x)inf_P ∈𝒬^T E_P U( V_T^x,ϕ(·))≤sup_ϕ∈𝒜(U,0,x)inf_P ∈𝒬^T E_P U(V_T^x+π(G), ϕ+ ϕ_G(·) - G(·))≤sup_ϕ∈𝒜(U,G,x+ π(G))inf_P ∈𝒬^T E_P U(V_T^x+π(G), ϕ(·) - G(·))=u(G,x+π(G)),where Proposition <ref> has been used for the second inequality : If ϕ∈𝒜(U,0,x) = 𝒜(0,x),then ϕ+ϕ_G∈𝒜(G,π(G)+x)= 𝒜(U,G,π(G)+x). So p(G,x) ≤π(G) follows from (<ref>).Assume now thatu(0,x)>-∞.If x+z< π(G), thenu(G,x+z) =-∞< u(0,x), which implies that p(G,x)≥π(G)-x andlim_x → 0 p(G,x)=π(G) follows.§.§ Asymptotic result Intuitively speaking an agent who is totally risk averse will use the superreplication price: Whatever the possible outcomes (where possible outcomesare defined bya set of probability measures), she does not want to incur any loss (see (<ref>)). Weprove now thatthe utility indifference price goes to the superreplication price when the absolute risk aversion goes to infinity. We willsee alsoin Proposition <ref> that the absolute risk aversion remains a good tool to rank a proper notion of certainty equivalent in the presence of multiple-priors. The result of <cit.> remains true: Increasing absolute risk-aversion implies decreasingcertainty equivalent.We consider some contingent claim G ∈𝒲^0,bo_T and a sequence of non-randomfunctions(U_n)_n ≥ 1 where for all n ≥ 1, U_n:ℝ→ℝ∪{- ∞}is such that U_n(x)=-∞ if x<0 and such that there exists some x_n ∈ (0,∞) verifying U_n(x_n)>-∞. For all n≥ 1, we denote byu_n(G,x) the value function in (<ref>) for U_n, p_n(G,x) the indifference price for U_n (see (<ref>)) and r_n the absolute risk aversion for U_n (see (<ref>)) when U_n satisfies Assumption <ref>.Assume that Assumptions <ref>, <ref>, <ref>and <ref> hold true.Let (U_n)_n≥ 1 be a sequenceofconcaveutility functions satisfying Assumption <ref> such thatlim_n → +∞ r_n(x)=+∞ for all x>0. Then lim_n →+ ∞ p_n(G,x)=π(G) for all x>0 and G ∈𝒲^0,bo_T. Assume that Assumptions <ref>, <ref> and <ref>hold true.Let G ∈𝒲^0,bo_T and x_0 > 0. Thefollowing two assertions hold true. 1. Let (U_n)_n≥ 1 be a sequenceof concave functions satisfying Assumption <ref>. Assume that the sequence (U_n/U_n(x_0)-U_n(x_0)/U_n(x_0))_n≥ 1 is bounded from above uniformly in n for n big enough and thatlim_n → +∞ r_n(x)=+∞ for all 0<x<x_0. Then lim_n →+ ∞ p_n(G,x_0)=π(G). 2. Let (U_n)_n≥ 1 be a sequenceof non decreasing functions which are bounded from above uniformly in n for n>N for some N≥ 1. Assume that inf_n≥ NU_n(x_0)>-∞and thatlim_n → +∞ U_n(x)=-∞ for all 0<x<x_0. Thenlim_n →+ ∞ p_n(G,x)=π(G). We comment on Theorem <ref>. For the first item, note that if for some N>1, the (U_n)_n≥ Nare bounded from above uniformly in n, inf_n≥ N U_n(x_0) >-∞ and inf_n≥ N U_n(x_0) >0 then the sequence(U_n/U_n(x_0)-U_n(x_0)/U_n(x_0))_n≥ N is bounded from above uniformly in n. This is not a necessary condition. Let U_n(x)=-e^-r_nxfor x≥ 0 and U_n(x)=-∞ otherwise where lim_n → +∞ r_n=+∞.Then for all N ≥ 1 and all x_0>0, inf_n≥ N U_n(x_0) =0. Nevertheless, for all x_0>0, U_n (x)/U_n(x_0)-U_n(x_0)/U_n(x_0)=-1/r_ne^-r_n(x-x_0) + 1/r_n is bounded from above uniformly in n for n big enough. Finally, if inf_n≥ N U_n(x_0) >0, a straightforward adaptation of <cit.> shows thatlim_n → +∞ U_n(x)=-∞ for all 0<x<x_0. Butif this last condition holds true, theconvergence result does not even needthe functions to be either concave or differentiable, see item 2. The proof of Theorems <ref> and <ref> are reported in Section <ref>.Under the same assumptions,Theorem <ref> implies that ifG ∈𝒲^∞_T then lim_n → +∞ p^B_n(G,x)=π^sub(G).We give some intuition on Theorem <ref>:For a utility function that has a sort of infinite absolute riskaversion, we show that the utility indifference price is equal to the superreplication price. Let G ∈𝒲^0,bo_T and assume for sake of simplicity that π(G) ≥ 0. Fix some x ≥π(G) and introduce the following utility function U_∞: ℝ→ℝ∪{-∞} where U_∞(y):= -∞ 1_(-∞,x)(y). The absolute riskaversion of U_∞ is not defined, but U_n(y) := - e^-n(y-x) for y ≥ 0 and U_n(y):=-∞ for y<0 verifies Assumption <ref> andfor y ≥ 0 fixed with y ≠ x, lim_n → +∞ U_n(y)=U_∞(y). Since the absolute riskaversion of the utility functions U_n goes to +∞, one may say that U_∞ has an infinite absolute riskaversion. We now show that the superreplication price of G is equal to its utility indifference price evaluated with the functionU_∞. First, it is easy to see that u_∞(0,x)=0.Since for all ϕ∈Φ, y ∈ℝ, U_∞^+(V_T^y,ϕ(·)-G(·))=0, we have that Φ(U_∞,G,y)=Φ(U_∞,0,y)=Φ and 𝒜(G,y)=𝒜(U_∞,G,y).Theorem <ref> implies that 𝒜(U_∞,G,y) is not empty for all y ≥π(G). Now fix some z< π(G) and ϕ∈𝒜(U_∞,G,x+z). There exists some P ∈𝒬^T such that P(V_T^z,ϕ(·) -G(·)<0)>0 or equivalently P(V_T^x+z,ϕ(·) -G(·)<x)>0 which implies that E_P U_∞(V_T^x+z,ϕ(·)-G(·)) =-∞.Hence for all ϕ∈𝒜(U_∞,G,x+z), inf_P ∈𝒬^T E_P U_∞(V_T^x+z,ϕ(·)-G(·)) =-∞ andu_∞(G,x+z)=-∞<u_∞(0,x) follows.The definitionof p(G,x) implies that p(G,x) ≥ z and letting z go to π(G), p(G,x) ≥π(G). Proposition <ref> shows that the equality holds true. The dominated case is of special interest. Assume that there exists some P^* ∈𝒬^T such that for all P ∈𝒬^T, P is absolutely continuous with respect to P^*. Then a set is of 𝒬^T-full measure if and only if it is of P^*-full measure. So NA(𝒬^T) is equivalent to NA(P^*) i.e.V_T^0,ϕ≥ 0 P^* for ϕ∈Φ implies that V_T^0,ϕ= 0P^* Moreover π(G)=inf{ϕ∈Φ,V_T^x,ϕ≥ G P^*}:=π^P^*(G).Theorem <ref> rephrases as follows: For G ∈𝒲^0,bo_T, if Assumptions <ref>, <ref>, <ref> and NA(P^*)hold true andif lim_n → +∞ r_n(x)=+∞ for all x>0 then lim_n →+ ∞ p_n(G,x)=π^P^*(G) for all x>0. At the limit we are back to a uni-prior setup. The same holds true for Theorem <ref>changing Assumption <ref> by NA(P^*).§.§ Absolute risk aversion and certainty equivalent In the uni-prior case, we know from <cit.> that theabsolute risk-aversion allows to rank the certainty equivalent:An increasing absolute risk-aversionimplies a decreasing certainty equivalent. In this section we will see how this property extend to the multiple-priors case.Recall the uni-prior case where 𝒬^T={P} and the preferences are represented by autility function U. Fora givenasset whose payoff at maturity isG, the certainty equivalent e(G,P) is the amount of cash that will make the agent indifferent (in the sense of the expected utility evaluation) between receiving the cashand the assetGE_P U(e(G,P))=U(e(G,P))=E_P U(G(·)).The risk premium λ(G,P):= E_P G(·)-e(G,P) is the amountthe agent is willing to lose in order to be indifferent (in the sense of the expected utility evaluation) between the sure quantity E_P G(·)-λ(G,P) and the random variable G.The following proposition gives the definition of the certainty equivalent in a multiple-priorsframework and provides conditions for existence and uniqueness. It also establishesthat under suitable assumptions λ(G,P) ≥ 0. The risk premiumis thus a measure ofthe risk-aversion of the agent: The higher the risk premium, the more risk-averse the agent is. LetU bea concave utility function satisfying Assumption <ref> and let 𝒲^+_T(U):= {G ∈𝒲^0,+_T, G(·) <+∞ 𝒬^T E_P U^+(G(·)) <∞, ∀ P∈𝒬^T, sup_P ∈𝒬^TE_P U^-(G(·))<∞} Assume that G ∈𝒲^+_T(U).Then, there exists unique e(G,P) and e(G) in [0,∞) such thatE_P U(e(G,P))=U(e(G,P))=E_P U(G(·)),∀ P ∈𝒬^T inf_P ∈𝒬^T E_P U(e(G))=U(e(G)) = inf_P ∈𝒬^T E_P U(G(·)).Moreover, e(G,P) ≤ E_P G(·) for all P ∈𝒬^T ande(G)=inf_P ∈𝒬^T e(G,P) ≤inf_P ∈𝒬^T E_P G(·).Furthermore the multiple-priorsrisk premium defined byλ(G):= sup_P ∈𝒬^Tλ(G,P) satisfies 0 ≤λ(G) ≤sup_P ∈𝒬^TE_PG(·) -e(G). Note that (<ref>) is true if we assume only that E_P U^-(G(·))<∞ for all P ∈𝒬^T. See Section <ref>. Finally, we consider two investors A and B whose respectiveutility functions U_A and U_B satisfyAssumption <ref> and are concave. Recall that in the uni-prior case with𝒬^T={P} investor A has greater absolute risk-aversion than investor B (i.e. r_A(x) ≥ r_B(x)for all x>0) if and only if investor A is globally more risk averse than investor B, in the sense that the certainty equivalent of every contingent claim is smaller for A than for B (i.e. e_A(G,P) ≤ e_B(G,P) for any G ∈𝒲_T^0,+) see <cit.>. We propose the followinggeneralization of this result in the multiple-priorsframework. Let U_A, U_B be concaveutility functions satisfying Assumption <ref>. Let 𝒲^+_T(U_A,B):=𝒲^+_T(U_A) ∩𝒲^+_T(U_B) (see (<ref>)). 1. If for all x > 0, r_A(x) ≥ r_B(x) then e_A(G) ≤e_B(G) for all G∈𝒲^+_T(U_A,B).2.If forall G ∈𝒲^+_T(U_A,B),e_A(G) < e_B(G) then r_A(x) ≥ r_B(x) for all x > 0. See Section <ref>.Proposition <ref> shows that the absolute risk aversionallows the ranking of the multiple-priorscertainty equivalent despitethe presence of uncertainty (and thus uncertainty aversion). The reason for this is relatedto the multiple-priorsrepresentation we have chosen and the fact that the two agents have the same set of priors.We briefly make the link with the monetaryrisk measures introduced in <cit.>, see also <cit.>. Let 𝒳⊂𝒲^0_T bea linear space of random variables (containing the constant random variables). A monetary risk measure is a mapping ρ: G ∈𝒳↦ρ(G) ∈ℝ∪{±∞} that verifies the monotonicity and the cash invariance properties (see<cit.>).The measureis said to be a normalizedif ρ(0)=0. One may measure the risk of a position usingρ_x:G ∈𝒳↦ p(-G,x) for somex ≥ 0 fixed (see for example <cit.>) orconsider ρ :G ∈𝒳↦π(-G). IndeedunderAssumptions<ref> and <ref>, ρ is anormalizedconvex monetary risk measureon𝒲_T^0. If U satisfies Assumption <ref> and under the assumptions of Proposition <ref>, ρ_x is a monetary risk measure on𝒲_T^0,bo. If one also assumes that u(0,x)>-∞, then ρ_x is aconvex monetary riskmeasureon{G ∈𝒲_T^0,bo, u(-G,z)<∞, ∀ z ∈ℝ}. If furthermore u(0,x-δ)<u(0,x) for all δ>0, then ρ_x is normalized.The proof is classical (see for instance <cit.>) and therefore omitted. § EXAMPLE To illustrate the previous results we provide the following one period non-dominated example.LetΩ=[-1,∞). We assume that the risky asset is given by S_0=1 andS_1(ø)=1+ø.Here Φ=ℝ and Assumption <ref> is satisfied. We will compute the utility indifference price and the superreplication price ofthe contingent claim G(ø)=1_[1,∞)(ø). Fix somep_1,p_-1∈ (0,1) such that 0<p_1+p_-1<1. For c ≥ 1, let P_c be the probability measure on Ω be defined byP_c(·)=p_1δ_c(·)+p_-1δ_-1(·)+ (1-p_1-p_-1) δ_0(·) where δ_x(B):=1 ⇔ x ∈ B for all x∈Ω and B∈(Ω).We assume furthermore that p_1 < p_-1: The agents think that the risky asset is more likely to go down than up. The set of priors is given by𝒬:={P_c, c ≥ 1}.Using <cit.>, T:c ∈ℝ↦ P_c(·) ∈𝒫(Ø) is a homeomorphism, so {P_c, c ≥ 1}=T([1,∞)) is a Borel set of𝒫(Ø). As in<cit.>, we find that 𝒬 is a Borel set.So (𝒬)=Ø×𝒬 is also a Borel set anda fortiori an analytic set in Ø×𝒫(Ø). Assumption <ref>is verified.It isclear that NA(P) holds true for any P ∈𝒬. SoNA(𝒬) also holds true.Moreover the set 𝒬 is not dominated. Indeed for a dominating measure P, P({c})>0 for all c ≥ 1 which is impossible.Finally, we takea sequence of utility functionsdefined by U_n(x)=-e^-r_nxfor x≥ 0 and U_n(x)=-∞ otherwise where lim_n → +∞ r_n=+∞. The assumptions of Theorem <ref> item 1 are satisfied for all x_0>0, see Remark <ref>. Note that as lim_n → +∞ r_n=+∞ we will assumethat r_n>(p_-1/p_1).First, we focus on the admissibility condition. Forx ≥ 0,we get that 𝒜(0,x) ={h ∈ℝ,-x/c≤ h≤ x,∀c ≥ 1}={h ∈ℝ,0 ≤ h ≤ x} 𝒜(G,x) ={h ∈ℝ, 1-x/c≤ h ≤ x,∀c≥ 1}= {h ∈ℝ, max(0,1-x) ≤ h ≤ x }.It is easy to seethat π(G)=1/2.Note that both 𝒜(0,x) and 𝒜(G,x) are included in the half-real line: The agent cannot sell the risky asset in (<ref>). This implies thatthe worst case prior in 𝒬corresponds toP_1 (recall that c≥ 1).We evaluate first u_n(0,x) for x ≥ 0. u_n(0,x) =sup_0 ≤ h ≤ x -e^-r_nx(p_1 e^-r_n h + 1-p_1-p_-1+ p_-1 e^-r_n(-h)) =-e^-r_nx. Indeed, the minimum of h ↦ p_1 e^-r_n h + 1-p_1-p_-1+ p_-1 e^r_nh is reached for 1/2r_n(p_1/p_-1)<0sincep_1 < p_-1.Ify<1/2=π(G), then 𝒜(G,y) ≠∅ and u_n(G,y)=-∞. Let y ≥1/2. u_n(G,y) =sup_max(0,1-y) ≤ h≤ y -e^-r_ny(p_1 e^-r_n (h-1) + 1-p_1-p_-1+ p_-1 e^-r_n(-h)).Remark that the minimum of h ↦ p_1 e^-r_n (h-1) + 1-p_1-p_-1+ p_-1 e^r_nh is reached for h^*_n:=1/2- 1/2r_n(p_-1/p_1). As we have assumed that p_-1>p_1, r_n>(p_-1/p_1) for all n, 0<h^*_n<1/2 and we obtain thatu_n(G,y)= -e^-r_ny( 2e^r_n/2√(p_1p_-1)+(1-p_1-p_-1) )-e^-r_ny( p_1e^r_ny+p_-1e^r_n(1-y)+(1-p_1-p_-1)).Now starting from a wealth x≥ 0, we want to find the smallest z ∈ℝ such thatu_n(G,x+z) =u_n(0,x).Fix x >0. As lim_n → +∞ u_n(0,x)=0 there is some N_x such that for n ≥ N_x,u_n(0,x) ≥ -p_1/2.If 1/2≤ x+z < 1/2+1/2r_n(p_-1/p_1),for n ≥ N_x,(<ref>) showsthatu_n(G,x+z)≤ -p_1< u_n(0,x).Thus for n ≥ N_x, we need to assume thatx+z ≥1/2+ 1/2r_n(p_-1/p_1). Then (<ref>) implies thatp_n(G,x)= 1/2+ 1/r_n( 2√(p_1p_-1)+e^-r_n/2(1-p^_1-p^_-1)) ≤π(G)as soon as x+ 1/r_n( 2 p_1+e^-r_n/2√(p_1/p_-1)(1-p^_1-p^_-1)) ≥ 0. In this case p_n(G,x) is well-defined for x<π(G). For example, ifp_-1= 2/3, p_1=1/4 and r_n=n, this last inequality is always satisfied for x=0.1 as soon as n≥ 7 and thenp_7(G,0.1)=0,471or for x=0.4 as soon as n≥ 1 and then p_1(G,0.4)=0,417.Note that the convergence rate, even in the presence of uncertainty, is only driven by the risk aversion r_n (see (<ref>)).The optimal strategy for u_n(G,y)=u^P_1_n(G,y) (recall that P_1 is the worst-case scenario) is given, for n large enough, by 1/2-1/2r_n(p_-1/p_1) which is different from the superreplication strategy h=1/2 but converges to it with the risk aversion coefficient. Proving such result in a general setting i.e. finding a kind of accumulation point of the optimalstrategies's sequence which is a superhedgingprice, is left for further research. § PROOFS OF THE RESULTS Weborrow some ideas from <cit.> adapted to the multiple-priors set-up.§.§ Proof of Theorems<ref> and <ref>The proof is based on two ingredients: Some closure property (see Lemma <ref>) and some boundness property (see Lemma <ref> which has already been used in Proposition <ref> to fix some integrability issues).We firstintroduce, the set of terminal wealth including the possibility of throwing away money starting from capital x ∈ℝ𝒞_x^T:={V_T^x,ϕ, ϕ∈Φ} - 𝒲^0,+_T.In the sequel we will write X ∈𝒞_x^T if there exists some ϕ∈Φ and Z ∈𝒲^0,+_T such that X=V_T^x,ϕ - Z 𝒬^T-q.s. Under the assumptions of Lemma <ref> the set 𝒞_x^T has a classical closure property (in the 𝒬^T quasi-sure sense, see <cit.>). Note that the same comment as inRemark <ref> applies. Assume that Assumptions <ref> and<ref> hold true. Fix some z ∈ℝ and let B ∈𝒲_T^0 such that B∉𝒞^T_z.Then there exists some ε>0 such thatinf_ϕ∈Φsup_P ∈𝒬^T P(V_T^z,ϕ < B -ε)> ε.Assume that (<ref>) does not hold true.Then, for all n ≥ 1, there exists some ϕ_n∈Φ such thatP(V_n < B -1/n)≤1/n for all P ∈𝒬^T, where V_n:=V_T^z,ϕ_n. Set K_n:= (V_n - (B- 1/n) ) 1_{V_n≥ B -1/n}∈𝒲_T^0,+. Then V_n - K_n∈𝒞^T_z. Moreover, for allP ∈𝒬^T, P( |V_n -K_n -B|> 1/n) = P(V_n < B -1/n) ≤1/n. Thus lim_n → +∞sup_P ∈𝒬^TP( |V_n -K_n -B|> 1/n) =0.If we prove that there exists a subsequence (n_k)_k ≥ 1 such that (V_n_k - K_n_k)_k ≥ 1 converges to B 𝒬^T-q.s. (i.e. on a 𝒬^T-full measure set), <cit.> implies thatB ∈𝒞^T_z. This contradiction will achieve the proof. So fix η>0 and consider thesub-sequence (V_n_k - K_n_k)_k ≥ 1 such thatsup_P ∈𝒬^T P(A_k) ≤1/2^k A_k:={ |V_n_k - K_n_k-B| > 1/n_k}. As ∑_k ≥ 1sup_P ∈𝒬^T P(A_k) <∞, Borel-Cantelli's Lemma for capacity (see <cit.>)implies that sup_P ∈𝒬^T P(lim sup_k A_k)=0. HenceØ^T\lim sup_k A_k is a 𝒬^T-full measure set on which |V_n_k(·) - K_n_k(·)-B(·)| ≤η holds true for k big enough. Under suitable assumptions, the next propositionestablishes thatwhatever the strategy is, the wealth at time T starting from capital x is uniformly bounded.Assume that Assumptions <ref>, <ref>, <ref> and <ref>hold true.Then, for all x∈ℝ, ϕ∈𝒜(0,x) and0 ≤ t ≤ T,|V_t^x,ϕ(·)| ≤|x| M_t(·)𝒬^t where M_0:=1 and M_t(ø^t):=∏_s=1^t(1+ |Δ S_s(ø^s)|/α_s-1(ø^s-1)). Moreover, M_t and V_t^x,ϕ belong to 𝒲^r_tfor all 0 ≤ t ≤ T and0 < r<∞.Weusesimilar argumentsas in the proof of <cit.>.Let x ≤ 0 and ϕ∈𝒜(0,x). Then V_T^0,ϕ≥ 0 𝒬^T and byNA(𝒬^T) and <cit.>,V_t^0,ϕ≥ 0 𝒬^t and V_t^0,ϕ = 0 𝒬^t. So (<ref>) holds trivially true.So fixx > 0 and ϕ∈𝒜(0,x).For all 1 ≤ t ≤ Tand ø^t-1∈Ø_NA^t-1 (recall Proposition <ref> for the definition of Ø_NA^t-1), we denote by ϕ_t^⊥(ø^t-1) theorthogonal projection of ϕ_t(ø^t-1) on the vector space D^t(ø^t-1) (see again Proposition <ref>). We have for all ø^t-1∈Ø^t-1_NA that ϕ_t(ø^t-1) Δ S_t(ø^t-1,·)=ϕ^⊥_t(ø^t-1) Δ S_t(ø^t-1,·)𝒬_t(ø^t-1)see <cit.>. AsV_T^x,ϕ≥ 0 𝒬^T-q.s. and as Assumptions <ref>, <ref> and <ref> hold true, <cit.>applies together with <cit.> andℋ^t-1:={ø^t-1∈Ø^t-1, V_t-1^x,ϕ(ø^t-1) +ϕ_t(ø^t-1) Δ S_t(ø^t-1,·) ≥ 0𝒬_t(ø^t-1) }∈_c(Ø^t-1) is a 𝒬^t-1-full measure set. Fix now some 1 ≤ t ≤ T, ø^t-1∈ℋ^t-1∩Ø^t-1_NA. We prove that|ϕ^⊥_t(ø^t-1)| ≤|V_t-1^x,ϕ(ø^t-1)|/α_t-1(ø^t-1).If ϕ^⊥_t(ø^t-1)=0 there is nothing to prove and one may assumethat ϕ^⊥_t(ø^t-1) ≠0. First, using (<ref>) and ø^t-1∈ℋ^t-1∩Ø^t-1_NA, we get that V_t-1^x,ϕ(ø^t-1)+ ϕ^⊥_t(ø^t-1)Δ S_t(ø^t-1,·) ≥ 0𝒬_t(ø^t-1) .Now, we proceed by contradiction and assume that (<ref>) does not hold true. LetB:= {ϕ^⊥_t(ø^t-1)Δ S_t(ø^t-1,·) < -α_t-1(ø^t-1) |ϕ^⊥_t(ø^t-1)|}.From Proposition <ref>, there exists some P_ϕ∈𝒬_t(ø^t-1) such that P_ϕ(B) > α_t-1(ø^t-1)>0. But,for all ø_t∈ BV_t-1^x,ϕ(ø^t-1)+ϕ^⊥_t(ø^t-1) Δ S_t(ø^t-1,ø_t) < |V_t-1^x,ϕ(ø^t-1)| - α_t-1(ø^t-1) |ϕ^⊥_t(ø^t-1)| <0,a contradiction with (<ref>) and therefore (<ref>) holds true.Now, we prove by induction that (<ref>) holds true for all 0 ≤ t ≤ T. For t=0 this is trivial. Assumethat for some t ≥ 1, there exists some 𝒬^t-1-full measure setØ^t-1∈ℬ_c(Ø^t-1) on which(<ref>) is true at stage t-1. Let Ø^t_EQ:={(ø^t-1,ø_t) ∈Ø^t-1×Ø_t, ϕ_t^⊥(ø^t-1) Δ S_t(ø^t-1,ø_t)=ϕ_t(ø^t-1) Δ S_t(ø^t-1,ø_t)} .It is clear that Ø^t_EQ∈ℬ_c(Ø^t). For some P=P_t-1⊗ p_t∈𝒬^t,(<ref>) and Fubini's Theorem (see<cit.>) imply that P(Ø^t_EQ)=1 (recall that Ø^t-1_NA is of 𝒬^t-1 full measure). Set Ø^t-1:=Ø^t-1∩ℋ^t-1∩Ø^t-1_NA and Ø^t= Ω^t_EQ∩(Ø^t-1×Ø_t). It is clear that Ø^t∈_c(Ω^t) and is a 𝒬^t-full measure set. For allø^t=(ø^t-1,ø_t) ∈Ø^t|V_t^x,ϕ(ø^t-1,ø_t)|= |V_t-1^x,ϕ(ø^t-1)+ ϕ^⊥_t(ø^t-1) Δ S_t(ø^t-1,ø_t)| ≤ |V_t-1^x,ϕ(ø^t-1)| (1 + |Δ S_t(ø^t-1,ø_t)|/α_t-1(ø^t-1)) ≤x M_t-1(ø^t-1)(1 + |Δ S_t(ø^t-1,ø_t)|/α_t-1(ø^t-1)) and(<ref>) is proved.For all 0 ≤ r <∞ and 1 ≤ s ≤ T,Δ S_s,1/α_s∈𝒲^r_s(see Assumption<ref>), soboth M_t and V_t^x,ϕ belong to 𝒲^r_tfor all 1 ≤ t ≤ T.We are now in position to prove Theorem <ref> and <ref>. Let G ∈𝒲^0,bo_T and b ≥ 0 such that G ≥-b 𝒬^T-q.s. and fix somex_0 >0.We prove first Theorem <ref> and the first item of Theorem <ref>. As in <cit.>, we may replace U_n by Û_n:=α_n U_n + β_n forsome α_n >0, β_n∈ℝ. If U_n isconcave, strictly increasing or twice continuously differentiable, Û_n will display the same property. Thus under the assumptions of Theorem <ref>, the Û_n are concave and satisfy Assumption <ref>.Moreover, the absolute risk aversion and the utility indifference price for U_n and Û_n (which will be denoted by p̂_n(G,x_0))are the same (see (<ref>),(<ref>) and(<ref>)).Thusit is enough to show thatlim_n → +∞p̂_n(G,x_0)=π(G). AsU'_n(x_0) ∈ (0,∞) (recall that U_n is concave and strictly increasing), we chooseα_n=1/U'_n(x_0) and β_n=-U_n(x_0)/U'_n(x_0)which leads toÛ_n(x_0)=0 andÛ'_n(x_0)=1 for all n ≥ 1. Note that under the assumptions of Theorem <ref> item 1,the Û_n are concave and there exist some N>1 and k>0 such that Û_n(x) ≤ k for all n ≥ N and x ∈ℝ.We denote by û_n(G,x) the value function for Û_n. The choice ofα_n and β_n implies that 0 ∈ A(Û_n,0,x_0) and that û_n(0,x_0) ≥inf_P ∈𝒬^T E_PÛ_n(x_0) = 0.We treat first the caseπ(G)=+∞. By definitionfor all z ∈ℝ, n ≥ 1, ∅=𝒜(G,z)=𝒜(Û_n,G,z) and û_n(G,x_0+z)=-∞ (see(<ref>)).So, (<ref>) and (<ref>) show that p̂_n(G,x_0)=+∞ for all n ≥ 1. The claim is proved.Assume now that π(G)<∞.Proposition <ref>implies that p̂_n(G,x_0) ≤π(G)<∞ and lim_n → +∞p̂_n(G,x_0)=π(G) will hold if lim inf_np̂_n(G,x_0) ≥π(G). Assume that this is not the case. Hence there is subsequence (n_k)_k ≥ 1 and some η >0 such that p̂_n_k(G,x_0) ≤π(G)-η for all k ≥ 1. Since x_0>0, we may and will assume that η < x_0.By definition of p̂_n_k(G,x_0) we have thatû_n_k(G,x_0 + π(G)-η) ≥û_n_k(0,x_0).If lim_k → +∞û_n_k(G,x_0 + π(G)-η) = -∞ is proved,lim inf_k → +∞û_n_k(0,x_0)=-∞ follows andcontradicts (<ref>). So, it remains to provethat lim_k → +∞û_n_k(G,y) = -∞ with y:=x_0 + π(G)-η∈ (π(G),x_0 + π(G)).First we show thatx_0+ G ∉𝒞^T_y.Indeed if this is not the case,there exists some X ∈𝒲_T^0,+ and ϕ∈Φ such that x_0+G=V_T^y,ϕ-X 𝒬^T-q.s. Hence G ≤ V_T^y-x_0,ϕ 𝒬^T-q.s. and y-x_0 ≥π(G) follows: A contradiction. ApplyingLemma <ref>, we get some ε>0 such that inf_ϕ∈Φsup_P ∈𝒬^T P (A_ϕ) > ε, where A_ϕ:={V_T^y,ϕ < x_0+ G-ε}. Note that we can always assume that x_0 > ε.Hence for all ϕ∈Φ, there exists some P_ε,ϕ∈𝒬^T such that P_ε,ϕ (A_ϕ) > ε.Proposition <ref> and Theorem <ref> imply thatfor all n≥ 1,𝒜(Û_n,G,y)=𝒜(G,y) ≠∅ since y > π(G). Choose some ϕ∈𝒜(G,y). We first postulatethe assumptions of Theorem <ref>. Using the monotonicity of Û_n, recalling G(·) ≥-b 𝒬^T-q.s.,(<ref>) and Lemma <ref>, we get for all n ≥ 1 thatE_P_ε,ϕ 1_Ω^T\A_ϕÛ_n(V_T^y,ϕ(·)- G(·))≤E_P_ε,ϕÛ^+_n(V_T^y+b,ϕ(·))≤Û_n^+(x_0) +sup_P ∈𝒬^T E_P(|V_T^y+b,ϕ(·)|) Û_n'(x_0)≤ (|y|+b)sup_P ∈𝒬^T E_P(M_T(·))≤ (x_0+ |π(G)|+b)||M_T||_1=:K<∞.Under the assumptions of Theorem <ref> item 1, the last boundnessfrom above property in (<ref>) is still valid forK=k andn≥ N.Now, as Û_n(x_0-ε) ≤Û_n(x_0)=0 we get thatE_P_ε,ϕ 1_A_ϕÛ_n(V_T^y,ϕ(·)-G(·))≤Û_n(x_0-ε) E_P_ε,ϕ 1_A_ϕ≤εÛ_n(x_0-ε).So(<ref>) and (<ref>) imply thatfor all n ≥ N inf_P ∈𝒬^T E_PÛ_n(V_T^y,ϕ(·)-G(·)) ≤ E_P_ε,ϕÛ_n(V_T^y,ϕ(·)-G(·)) ≤K +εÛ_n(x_0-ε).As this is true for all ϕ∈𝒜(G,y)=𝒜(Û_n,G,y),û_n(y,G) ≤K +εÛ_n(x_0-ε) followsfor all n≥ N. Finally, <cit.> (which uses the concavity of Û_n)implies thatlim_n → +∞Û_n(x_0-ε)=-∞ and thus lim_n → +∞û_n(G,y)=-∞ as claimed.We now prove the second item of Theorem <ref>. The proof is similar to the one ofthe first item of Theorem <ref> and we only enlighten the main changes.First, we do not modify the functions U_n. Let k_1 >0 be such that U_n(x) ≤ k_1 for all n ≥ N and x ∈ℝ.As 0 ∈ A(U_n,0,x_0) for all n ≥ Nu_n(0,x_0) ≥U_n(x_0)>inf_n≥ NU_n(x_0)>-∞,which is the pendant of (<ref>). If π(G)=+∞ the same arguments as above apply. Assume that π(G)<+∞: Proposition <ref> still applies and p̂_n(G,x_0) ≤π(G).Fix ϕ∈𝒜(G,y) and let y, A_ϕ and P_ε,ϕ be as before: We prove directly that lim_n → +∞ u_n(G,y)=-∞.Fix some J>0 and C_J := 1/(J+k_1).As lim_n→ + ∞U_n(x_0 -)=-∞, there existsN_J≥ N such that for all n ≥ N_J, U_n(x_0 -)≤ -C_J. E_P_ε,ϕ U_n(V_T^y,ϕ(·)-G(·))≤E_P_ε,ϕ 1_Ω^T\A_ϕ U_n(V_T^y,ϕ(·)- G(·)) + E_P_ε,ϕ 1_A_ϕU_n(V_T^y,ϕ(·)-G(·)) ≤ k_1 +U_n(x_0 -) P_ε,ϕ(A_ϕ) ≤ k_1 + U_n(x_0 -)ε≤ -J.ThusasN_J does not depend on ϕ,we obtain that for alln ≥ N_J, u_n(y,G) ≤ -J.§.§ Proofs of Propositions <ref> and <ref> Fix some P ∈𝒬^T and G ∈𝒲_T^+(U). As E_P U^-(G(·))<+∞, G ∈𝒲_T^0,+ and U is non-decreasing, E_P U(G(·)) -U(0) ≥ 0 (note that U(0) may be equal to -∞).AsP(G(·)<∞)=1 and U isstrictly increasing U(G(·))< lim_y → +∞ U(y) P-a.s. Together withE_P U^+(G(·))<+∞, one concludes that E_P U(G(·))-U(y)<0 for y large enough and the intermediate value theorem implies that (<ref>)holds true.NowE_P U(G(·)) ≥ U(0) for all P ∈𝒬^T implies that inf_P ∈𝒬^TE_P U(G(·)) -U(0) ≥ 0 (recall that sup_P ∈𝒬^TE_P U^-(G(·))<∞). Moreover for some P ∈𝒬^T as inf_P ∈𝒬^TE_P U(G(·))-U(y) ≤ E_P U(G(·))-U(y)<0 for y large enough, the intermediate value theorem implies(<ref>). Now for any Q ∈𝒬^T, (<ref>)implies thatU(inf_P ∈𝒬^T e(G,P)) ≤U(e(G,Q))= E_Q U(G(·)).This is true for any Q ∈𝒬^T, so(<ref>)implies thatU(inf_P ∈𝒬^T e(G,P)) ≤inf_P ∈𝒬^T E_P U(G(·)) =U(e(G))andinf_P ∈𝒬^T e(G,P) ≤e(G) follows from strict monotonicity of U.Now for any Q ∈𝒬^T, (<ref>), (<ref>) and Jensen's inequality imply thatU(e(G))= inf_P ∈𝒬^T E_P U(G(·)) ≤ E_Q U( G(·))=U(e(G,Q))≤ U (E_Q G(·)).Thus, by strict monotonicity of U, e(G) ≤ e(G,Q) ≤ E_Q G(·) and since this is true for all Q ∈𝒬^T, we find that e(G) ≤inf_P ∈𝒬^Te(G,P) ≤inf_P ∈𝒬^T E_P G(·).We adapt theproofof <cit.> to the multiple-priors framework.1. We first show that if for all x > 0, r_A(x) ≥ r_B(x), thene_A(G,P) ≤e_B(G,P)for all G∈𝒲^+_T(U_A,B) and P ∈𝒬^T.This will imply that e_A(G) ≤e_B(G) using Proposition <ref>.Fix some G∈𝒲^+_T(U_A,B)andP ∈𝒬^T. Let D:=U_B((0,∞)) ⊂ (-∞,∞) and define F: D →ℝby F(y)= U_A(U^-1_B(y)). Then on DF'(·)=U'_A(U_B^-1(·))/U^'_B(U_B^-1(·)) F”(·)=U'_A(U^-1_B(·))/(U'_B(U^-1_B(·)))^2( r_B(U_B^-1(·))-r_A(U_B^-1(·))).As U^-1_B(·)>0 on D,F isincreasing and concave on D andU_A(x)=F (U_B(x)) for all x >0.Now let:̣=U_B(0) ∈ [-∞,∞) bethe lower bound ofD. We distinguish between two cases. First if >̣-∞, weextend F by continuity in $̣, settingF()̣= U_A(U^-1_B()̣)=U_A(0) ∈[-∞,∞). It is clear thatF()̣ ≤F(y)for ally ∈[,̣+∞), thatFis concave on[,̣+∞)and thatU_A(x)=F (U_B(x))holds also true for allx ≥0. Now, using (<ref>) andJensen's inequality, we get thatU_A(e_A(G,P))=E_P U_A (G(·))= E_P F(U_B(G(·))) ≤ F( E_P(U_B(G(·))))=F(U_B(e_B(G,P)))=U_A(e_B(G,P)).SinceU_Ais strictly increasing, we obtain thate_A(G,P) ≤e_B(G,P)as claimed.Now we treat the case where=̣-∞. FirstP(G>0)=1. Indeed ifP(G=0)>0,E_P U^-_B (G(·))= E_P U^-_B(G(·))1_{G>0}(·)+ U^-_B(0) P(G=0)=+∞, a case that we have excluded. ThusP(G>0)=1. Moreovere_A(G,P)ande_B(G,P)are positive. ElseU_A(e_A(G,P))=-∞whileE_P U_A(G(·)) ≥-E_P U^-_A(G(·))>-∞. Thus the previous arguments apply andwe also obtaine_A(G,P) ≤e_B(G,P).2.Assume thate_A(G) < e_B(G)for allG ∈𝒲_T^+(U_A,B)and there exists somex_0>0such thatr_A(x_0) < r_B(x_0). By continuity, there existsα>0, such thatr_A(x) < r_B(x)on(x_0-α,x_0+α). We can chooseαsuch thatx_0-α>0. LetI:=(U_B(x_0-α), U_B(x_0+α)) ⊂D, thenFis strictly convex onI(see (<ref>)). FixG ∈𝒲^+_T(U_A,B)and setG:= x_0-α+ 2 αG/G+1 ∈𝒲^+_T(U_A,B). It is clear thatG(·) ∈(x_0-α,x_0+α).As in (<ref>), using Jensen inequality, the fact thatFis (strictly) convex onIwe get that for anyP∈𝒬^TU_A(e_A(G,P)) =E_P F(U_B(G(·))) ≥ F(E_P (U_B(G(·)))=U_A(e_B(G,P)).This implies thate_A(G,P) ≥e_B(G,P)for allP ∈𝒬^T, thuse_A(G)≥e_B(G): A contradiction. Note that ifPis such that one can find someGwhich is not constant then theinequality in (<ref>) is strict and one gets thate_A(G,P)>e_B(G,P).§ EXTENSION TORANDOM UTILITY FUNCTIONSRandom utility functions capture very general situations where the preferences of the agent depend not only on her wealth but also on the path. At the starting date,the agent might not knowexactly how her utility function will depend on her wealth. Moreover the shape of her utility function vary with the contextand can be updated as information is released. For instance, she could become more risk averse if the market exhibits a tendency to move lower and could take more risk in the opposite situation. Such behaviors are oftenobserved in financial markets.An example of state-dependent utility function is the forward investment performance process introduced in <cit.>, see also <cit.> for an extension to the multiple-priors framework.A random utility functionU:Ω^T × (0,∞) →ℝ∪{- ∞}satisfies the followingconditions i) for every x>0, U (·,x): Ω^T→ℝ is universally-measurable, ii)for all ω^T∈Ω^T,U (ω^T,·): (0,∞)→ℝ isnon decreasing on (0,∞) and such U (ω^T,x_ø)>-∞ for some x_ø>0. We extend U by (right) continuity in 0 and setU(·,x)=-∞ if x<0.The next exampleexhibits random utility functionssuch that in additionU_n (ω^T,·)isconcave, strictly increasing and twice continuously differentiable on(0,∞).Assume that the agent analyzes her gain or loss with respect to a (random) reference point B rather than with respect to zero as suggested for instance by<cit.>. Let U be a non-random concave function satisfying Assumption<ref> and B ∈𝒲_T^∞,+ and set for all ø^T ∈Ø^T, x ≥ 0, U(ø^T,x)=U(x+||B||_∞-B(ø^T)) and U(ø^T,x)=-∞ for x<0.The second exampleproposes to consider random absolute risk aversion. The idea is to use classical utility functions but with random coefficients. For example, we may consider U(ø^T,x)= x^β_1(ø^T) or U(ø^T,x)= -e^-β_2(ø^T)x for x ≥ 0 (and U(·,x)=-∞ for x<0) where β_1,β_2∈𝒲_T^0 and 0<β_1(·) <1, β_2(·) >0𝒬^T-q.s.We can imagine various situations for β_2(which can be easily adapted for β_1):The law ofβ_2 under P can be uniformly distributed on[β^P_min,β^P_max] for all P ∈𝒬^T (with β^P_max≥β^P_min >0), alternatively it could follow a Poisson lawof parameter ł_P>0 for all P ∈𝒬^T. It could also be a function of some market parameters to model situations where the agent updates her utility function depending on market conditions. Let(U_n)_n ≥1be a sequence of utility functions satisfying Definition <ref>.The following definition is(<ref>)adaptedfor general random utility functions. u_n(G,x) := sup_ϕ∈𝒜(U_n,G,x)inf_P ∈𝒬^T E_P U_n(·, V_T^x,ϕ(·)-G(·))The generalization of Theorem <ref>for random utility functions will be stated for some fixedx_0>0and requiressome further assumptions. The first one replaces the convergence of the absolute risk aversion to infinity:U_n(·,x)goes to-∞with respect toinf_P ∈𝒬^T Pfor all0 < x <x_0. This is explainedin Lemma <ref>, where we also give alternative conditions to (<ref>). It also requires some uniform boundness from below assumption in x_0.We have thatsup_n ||U^-_n(·,x_0)||_1<∞ and that for all 0 < x <x_0 and M > 0,lim_n → +∞inf_P ∈𝒬^T P(U_n(·,x) ≤ -M)=1.The second assumption allows to fix integrability issues for unbounded from above utility functions and states that theU_nare sufficiently measurable and regular. For all ω^T∈Ω^T,U (ω^T,·) is concaveandtwice continuously differentiable on (0,∞). There exist some x_1≥ x_0 and some q>1 such thatsup_n ||U^+_n(·,x_1)||_1<∞sup_n ||U'_n(·,x_1)||_q<∞.Assume that Assumptions <ref>, <ref> and <ref> holds true. Let (U_n)_n ≥ 1 be a sequence ofrandom utility functions satisfying Definition <ref> and let G ∈𝒲^0,bo_T.Assume that Assumption <ref> holds true for some x_0>0. Suppose that either there exist some N>1 and B∈𝒲_T^1,+ such thatU_n(·,x) ≤ B(·)𝒬^T-q.s. for all x>0 and n ≥ N or thatAssumptions <ref>and <ref> hold true.Thenlim_n → +∞ p_n(G,x_0)=π(G).We give a concrete example for Theorem <ref>. For all n ≥ 1, let R_nbe a random variableuniformly distributed in [b_n,B_n] for all P ∈𝒬^Twith b_n>0, lim_n → +∞ b_n=+∞.Setfor all ω^T∈Ω^TU_n(ø^T,x)=-e^-R_n(ø^T)(x-1) for x ≥ 0 and U_n(ø^T,x)=-∞ for x<0. We choose x_0=1. Thenfor all M>0 and 0<x<1, U_n(·,x)≤ -M if and only if R_n (·)≥ln M/1-x. As lim_n → +∞ b_n=+∞, Assumption <ref>is verified.Weoutline briefly how the proof of Theorem <ref> (and Theorem <ref>) is modified.The structure of the proof is similar to the one of item 2 of Theorem <ref>, in particular we do not modify the functions U_n. Under the assumptions of Theorem <ref>, Proposition <ref> (and thus Proposition <ref>) is still valid replacing (<ref>) by (<ref>) below.Assume first that Assumptions <ref>and <ref> hold true. Using (<ref>) for U_n(ø^T,·) (recall x_1>0), we get thatfor all x>0, ø^T∈Ø^T, for allP ∈𝒬^T E_P U_n^+(.,V_T^x,ϕ(.))≤ sup_P ∈𝒬^T E_P U_n^+(·,x_1) + sup_P ∈𝒬^T E_P(|V_T^x,ϕ(·)|U_n'(·,x_1)) ≤||U_n^+(·,x_1)||_1+ |x| ||M_T(·) U_n'(·,x_1)||_1≤ ||U_n^+(·,x_1)||_1+ |x| ||M_T(·)||_p||U_n'(·,x_1)||_q ≤ sup_n ||U_n^+(·,x_1)||_1+|x| ||M_T(·)||_psup_n||U_n'(·,x_1)||_q=:K(x_1, x)<∞,whereLemma <ref>, M_T∈𝒲^p_T(where p verifies 1/p+1/q=1),Assumption <ref> and <cit.> have been used.Now, if Assumptions <ref>and <ref> do not hold true but there exists some B∈𝒲_T^1,+ such thatU_n(·,x) ≤ B(·) for all x>0 and n ≥ N, the last boundness property in (<ref>) is still valid forK(x_1, x)=||B(·)||_1. Remark now that 0 ∈ A(U_n,0,x_0) for all n ≥ N (recall that x_0>0 and (<ref>)). Now Assumption <ref> implies that for all n ≥ N u_n(0,x_0) ≥inf_P ∈𝒬^T E_P U_n(·,x_0) ≥ -sup_n ||U^-_n(·,x_0)||_1 >-∞.Let G ∈𝒲^bo_Tand b ≥ 0 such that G ≥ -b𝒬^T-q.s. Let ϕ∈𝒜(G,y) and let y, A_ϕ and P_ε,ϕ be as in the proof of Theorems <ref> and <ref>. Instead of(<ref>) (recall (<ref>)), we use that E_P_ε,ϕ 1_Ω^T\A_ϕ U_n(·,V_T^y,ϕ(·)- G(·))≤K(x_1,x_0+ |π(G)|+b).The arguments to obtain (<ref>) are more involved. First, fix some J>0 and setC_J := 2/(J+K(x_1,x_0)+ K(x_1,x_0+ |π(G)|+b))B_J,n:={U_n(·,x_0-ε) ≤ -C_J}.Weapply Assumption <ref>(recall that x_0 >) and obtain some N_J≥N (which does not depend on ϕ) such that for all n ≥ N_J, P_ε,ϕ(B_J,n) ≥inf_P ∈𝒬^T P(B_J,n) > 1-ε/2. Then, for all n ≥ N_J, P_ε,ϕ(B_J,n∩ A_ϕ) >ε/2 (recall that P_ε,ϕ (A_ϕ) > ε) andwe get thatE_P_ε,ϕ 1_A_ϕU_n(·,V_T^y,ϕ(·)-G(·))≤E_P_ε,ϕ 1_A_ϕ∩ B_J,n U_n (·,x_0-ε ) + E_P_ε,ϕ 1_A_ϕ\B_J,nU_n (·,x_0 )≤-ε C_J/2 + K(x_1,x_0)= -J - K(x_1,x_0+ |π(G)|+b),using (<ref>) and the definition of C_J.Combining the previous equationwith (<ref>),we obtain thatfor all n ≥ N_Jinf_P ∈𝒬^T E_PU_n(·,V_T^y,ϕ(·)-G(·)) ≤ E_P_ε,ϕU_n(·,V_T^y,ϕ(·)-G(·)) ≤ -J.AsN_J does not depend on ϕ,we obtain that for alln ≥ N_J, u_n(y,G) ≤ -J. Sincethis istrue for all J ≥ 0, lim_n → +∞ u_n(G,y)=-∞ and the proof is complete.We now make the link between (<ref>) and the convergence of the absolute risk aversion. From now we take a sequence ofutility functionssatisfying Assumption <ref> and such that for all ω^T∈Ω^T,U_n (ω^T,·) is concaveandtwice continuously differentiable on (0,∞). The generalisation of the absolute risk aversion (see(<ref>)) isr_n(ø^T,x):=-U_n^”(ø^T,x)/U_n^'(ø^T,x).Assume thatsup_n U_n(·,x_0)_1 <+ ∞and that there exists some N≥ 1, astrictly positive random variable łandsome deterministic functions(ρ_n)_n ≥ 1 such that for alln≥ N, U'_n(·,x_0) ≥ł(·), r_n(·,x) ≥ρ_n(x) and lim_n → +∞ρ_n(x)=+∞ for all x ∈ (0,x_0].Then(<ref>) holds true.Supposeforall >0 such that x_0> and all C≥ 0, we have thatlim_n → +∞inf_P ∈𝒬^T P( {∫_x_0-/2^x_0U_n”(·,v)dv <-C/})=1.First we prove that (<ref>)holds true. Fix some >0 such that x_0> and M > 0. For all ø^T∈Ø^TU_n(ø^T,x_0-) = U_n(ø^T,x_0)- ∫_x_0-^x_0 U'_n(ø^T,u)du. Using that U'_n(ø^T,·) is non-negative and non increasing,we obtain that U_n(ø^T,x_0-) + /2U'_n(ø^T,x_0-/2)≤ U_n(ø^T,x_0-) + ∫_x_0-^x_0-/2U'_n(ø^T,v)dv ≤ U_n(ø^T,x_0).NowU'_n(ø^T,x_0-/2) =U'_n(ø^T,x_0)-∫_x_0-/2^x_0 U”_n(ø^T,v)dv ≥ -∫_x_0-/2^x_0 U”_n(ø^T,v)dvand all togetherU_n(ø^T,x_0-)≤ |U_n(ø^T,x_0)|+ /2∫_x_0-/2^x_0 U”_n(ø^T,v)dv.We fix some η>0 andshowthat there exists some N_η>0 such thatfor all ninf_P ∈𝒬^T P(|U_n(·,x_0)| ≤ N_η)>1-η/2.As sup_n U_n(·,x_0)_1 <+ ∞,<cit.> implies that for all k ≥ 1sup_P ∈𝒬^T P(|U_n(·,x_0)|>k ) ≤1/ksup_P ∈𝒬^TE_P(|U_n(·,x_0)|)≤1/ksup_n U_n(·,x_0)_1. Thus there exists N_η>0 such that sup_P ∈𝒬^T P(|U_n(·,x_0)|>N_η)<η/2 for all n.From (<ref>) withC=2(N_η+M), there exists N=N(η,M,) such that for all n ≥ N,inf_P ∈𝒬^T P( U_n(·,x_0-)≤ -M) ≥inf_P ∈𝒬^TP({ |U_n(·,x_0)| ≤ N_η}∩{∫_x_0-/2^x_0 U”_n(·,v)dv <-2(N_η+M)/}) ≥inf_P ∈𝒬^TP({ |U_n(·,x_0)| ≤ N_η}) + inf_P ∈𝒬^T P({∫_x_0-/2^x_0 U”_n(·,v)dv <-2(N_η+M)/})-1 >1-η.Thus, (<ref>)is proved for all x=x_0- >0. We are left with the proof of(<ref>). Going back to the assumption of the lemma,there exists some N ≥ 1and astrictly positive random variable ł such that U'_n(·,x_0) ≥ł(·) for alln ≥ N. Sowe get that ∫_x_0-/2^x_0U_n”(·,v)dv =-∫_x_0-/2^x_0 U_n'(·,v)r_n(·,v) dv ≤ -ł(·)∫_x_0-/2^x_0 r_n(·,v)dv.Thus to prove that(<ref>) holds true, it is enough to show thatlim_n → +∞inf_P ∈𝒬^T P(ł(·) ∫_x_0 -/2^x_0 r_n(·,v)dv > C/)=1.As for all x ∈ (0,x_0]lim_nρ_n(x)=+∞, we get thatlim_n → +∞∫_x_0-/2^x_0ρ_n(v)dv=+∞ by Fatou's Lemma.Now (<ref>) holds true asr_n(·,x) ≥ρ_n(x), forn ≥ N and x ∈ (0,x_0].1. We indicatewhy in our proof we cannot use directly the assumption that lim_n → +∞ r_n(·,x_0)=+∞ instead of Assumption <ref>. Indeed lim_n → +∞ r_n(ø^T,x)=+∞ for all x ∈ (0,x_0], ø^T∈Ø^T together with Fatou's Lemma imply that for all ø^T ∈Ω^T, there exists N_ø^T such that for all k ≥ N_ø^T, ł(ø^T) ∫_x_0 -/2^x_0 r_n(ø^T,v)dv > C/, which means that Ω^T=∪_n ∩_k ≥ n{ł(·)∫_x_0 -/2^x_0 r_k(·,v)dv>C/} and using <cit.> this implies thatlim_n → +∞sup_P ∈𝒬^T P(ł(·) ∫_x_0 -/2^x_0 r_n(·,v)dv > C/)=1.But this does not imply that (<ref>) holds true, hence we cannot conclude as in the proof of Lemma <ref>that (<ref>) holds true. 2. In the course of the proof of Lemma <ref>we saw that if sup_n U_n(·,x_0)_1 <+ ∞, (<ref>)is satisfied under other sets of assumptions.*Of course (<ref>) implies that (<ref>) holds true.*Ifthere exists some N ≥ 1 and astrictly positive random variable ł such that U'_n(·,x_0) ≥ł(·) for all n ≥ N, then (<ref>) also implies (<ref>).*If furthermoreU”_n(ø^T,·) is non decreasing for all n ≥ N and ø^T ∈Ω^T, thenlim_n → +∞inf_P ∈𝒬^T P( {ł(·) r_n(·,x_0)>2C/^2})=1.implies that(<ref>) holds true.Indeed for the last assertion, sincefor all n ≥ N and ø^T ∈Ω^T, U”_n(ø^T,·) is non decreasing and U'_n(ø^T,x_0) ≥ł(ø^T), we get that ∫_x_0-/2^x_0U_n”(·,v)dv ≤/2U_n”(·,x_0)=-/2U_n'(·,x_0)r_n(·,x_0) ≤ -/2ł(·)r_n(·,x_0).Thus (<ref>) implies (<ref>) and (<ref>) holds true. Note that power utility functions or exponential utility functions (with random coefficients, see Example <ref> for the precise conditions) are examples where U”_n(ø^T,·) is non decreasing for all n and ø^T ∈Ω^T.We revisit briefly the notion of certainty equivalent (see Proposition <ref>)but for random utility functions in both the uni and multiple-priors framework. Let G ∈𝒲_T^0 such that0 ≤ G(·) <+∞𝒬^T-q.s. and assume that U isa utility function verifying Definition <ref>and such that for all ω^T∈Ω^T,U (ω^T,·) is concaveandtwice continuously differentiable on (0,∞). Moreover suppose thatsup_P ∈𝒬^T E_P U^-(·,y)<+∞ for all y>0, E_P U^+(·,1)<+∞and E_P| U(·,G(·))|<+∞ for all P ∈𝒬^T. Then*For all P ∈𝒬^T, there exists a unique constant e(G, P) ∈ [0,+∞) such thatE_P U(·,e(G,P)) =E_P U(·,G(·)).*If furthermoreG ∈𝒲_T^∞,+, sup_P ∈𝒬^T E_P U^-(·,G(·))<∞ andinf_P ∈𝒬^TE_P U'(·,z)>0 for all z>0,then therealso existsa unique e(G) ∈ [0,||G||_∞) such thatinf_P ∈𝒬^T E_PU(·,e(G))=inf_P ∈𝒬^T E_P U(·,G(·))and in this case, we have thate(G) ≥inf_P ∈𝒬^T e(G,P). We call e(G) the multiple-priorscertainty equivalent of G.As for Proposition <ref>, the (omitted) proof relies on a careful application of the intermediate value theorem. | http://arxiv.org/abs/1709.09465v2 | {
"authors": [
"Romain Blanchard",
"Laurence Carassus"
],
"categories": [
"q-fin.MF"
],
"primary_category": "q-fin.MF",
"published": "20170927120623",
"title": "Convergence of utility indifference prices to the superreplication price in a multiple-priors framework"
} |
headings 18SubNumber1648Fast Convolutional Sparse Coding in the Dual DomainL. Affara, B. Ghanem, P. Wonka King Abdullah University of Science and Technology (KAUST), Saudi [email protected], [email protected], [email protected] Fast Convolutional Sparse Coding in the Dual Domain Lama Affara, Bernard Ghanem, Peter Wonka December 30, 2023 =================================================== Convolutional sparse coding (CSC) is an important building block of many computer vision applications ranging from image and video compression to deep learning. We present two contributions to the state of the art in CSC. First, we significantly speed up the computation by proposing a new optimization framework that tackles the problem in the dual domain. Second, we extend the original formulation to higher dimensions in order to process a wider range of inputs, such as RGB images and videos. Our results show up to 20 times speedup compared to current state-of-the-art CSC solvers. § INTRODUCTION Human vision is characterized by the response of neurons to stimuli within their receptive fields, which is usually modeled mathematically by the convolution operator. Correspondingly for computer vision, coding the image based on a convolutional model has shown its benefits through the development and application of deep Convolutional Neural Networks. Such a model constitutes a strategy for unsupervised feature learning, and more specifically to patch-based feature learning also known as dictionary learning. Convolutional Sparse Coding (CSC) is a special type of sparse dictionary learning algorithms. It uses the convolution operator in its image representation model rather than generic linear combinations. This results in diverse translation-invariant patches and maintains the latent structures of the underlying signal. CSC has recently been applied in a wide range of computer vision problems such as image and video processing <cit.>, structure from motion <cit.>, computational imaging <cit.>, tracking <cit.>, as well as the design of deep learning architectures <cit.>. Finding an efficient solution to the CSC problem however is a challenging task due to its high computational complexity and the non-convexity of its objective function. Seminal advances <cit.> in CSC have shown computational speed-up by solving the problem efficiently in the Fourier domain where the convolution operator is transformed to element-wise multiplication. As such, the optimization is modeled as a biconvex problem formed by two convex subproblems, the coding subproblem and the learning subproblem, that are iteratively solved in a fixed point manner. Despite the performance boost attained by solving the CSC optimization problem in the Fourier domain, the problem is still computationally expensive due to the dominating cost of solving large linear systems. More recent work <cit.> makes use of the block-diagonal structure of the matrices involved and solves the linear systems in a parallel fashion, thus leveraging hardware acceleration.Inspired by recent work on circulant sparse trackers <cit.>, we model the CSC problem in the dual domain. The dual formulation casts the coding subproblem into anAlternating Direction Method of Multipliers (ADMM) framework that involves solving a linear system with a lower number of parameters than previous work. This allows our algorithm to achieve not only faster convergence towards a feasible solution, but also a lower computational cost. The solution for the learning subproblem in the dual domain is achieved by applying coordinate ascent over the Lagrange multipliers and the dual parameters. Our extensive experiments show that the dual framework achieves significant speedup over the state of the art while converging to comparable objective values. Moreover, recent work on higher order tensor formulations for CSC (TCSC) handles the problem with an arbitrary order tensor of data which allows learning more elaborate dictionaries such as colored dictionaries. This allows a richer image representation and greatly benefits the applicability of CSC in other application domains such as color video reconstruction. Our dual formulation provides faster performance compared to TCSC by eliminating the need for solving a large number of linear systems involved in the coding subproblem which dominates the cost for solving the problem.Contributions. We present two main contributions. (1) We formulate the CSC problem in the dual domain and show that this formulation leads to faster convergence and thus lower computation time. (2) We extend our dual formulation to higher dimensions and gain up to 20 times speedup compared to TCSC.§ RELATED WORKAs mentioned earlier, CSC has many applications and quite a few methods have been proposed to solve the non-convex CSC optimization. In the following, we mainly review the works that focus on the computational complexity and efficiency aspects of the problem.The seminal work of <cit.> proposes Deconvolutional Networks, a learning framework based on convolutional decomposition of images under a sparsity constraint. Unlike previous work in sparse image decomposition <cit.> that builds hierarchical representations of an image on a patch level, Deconvolutional Networks perform a sparse decomposition over entire images. This strategy significantly reduces the redundancy among filters compared with those obtained by the patch-based approaches. Kavukcuoglu <cit.> propose a convolutional extension to the coordinate descent sparse coding algorithm <cit.> to represent images using convolutional dictionaries for object recognition tasks. Following this path, Yang <cit.> propose a supervised dictionary learning approach to improve the efficiency of sparse coding.To efficiently solve the complex optimization problems in CSC, mostexisting solvers attempt to transform the problem into the frequency domain. Šorel and Šroubek <cit.> propose a non-iterative method for computing the inversion of the convolutional operator in the Fourier domain using the matrix inversion lemma. Bristow <cit.> propose a quad-decomposition of the original objective into convex subproblems andexploit the ADMM approach to solve the convolution subproblems in the Fourier domain. In their follow-up work <cit.>, a number of optimization methods for solving convolution problems and their applications are discussed. In the work of <cit.>, the authors further exploit the separability of convolution across bands in the frequency domain. Their gain in efficiency arises fromcomputing a partial vector (instead of a full vector). To further improve efficiency, Heide <cit.> transform the original constrained problem into an unconstrained problem by encoding the constraints in the objective using some indicator functions. The new objective function is then further split into a set of convex functions that are easier to optimize separately. They also devise a more flexible solution by adding a diagonal matrix to the objective function to handle the boundary artifacts resulting from transforming the problem into the Fourier domain.Various CSC methods have also been proposed for different applications. Zhang <cit.> propose an efficient sparse coding method for sparse tracking. They also solve the problem in the Fourier domain, in which the ℓ_1 optimization is obtained by solving its dual problem and thus achieving more efficient computation. Unlike traditional sparse coding based image super resolution methods that divide the input image into overlapping patches, Gu <cit.> propose to decompose the image by filtering. Their method is capable of reconstructing local image structures. Similar to <cit.>, the authors also solve the subproblems in the Fourier domain. The stochastic average and ADMM algorithms <cit.> are used for a memory efficient solution. Recent work <cit.> has also reformulated the CSC problem by extending its applicability to higher dimensions <cit.> and to large scale data <cit.>.In this work, we attempt to provide a more efficient solution to CSC and higher order CSC by tackling the optimization problem in its dual form.§ CSC FORMULATION AND OPTIMIZATIONIn this section, we present the mathematical formulation of the CSC problem and show our approach to solving its subproblems in their dual form. There are multiple slightly different, but similar formulations for the CSC problem. Heide <cit.> introduced a special case for boundary handling, but we use the more general formulation that is used by most authors. Thus, unlike <cit.>, we assume circular boundary conditions in our derivation of the problem. Brisow <cit.> verified that this assumption has a negligible effect for small support filters, which is generally the case in dictionary learning where the learned patches are of a small size relative to the size of the image. In addition, they show that the Fourier transform can be replaced by the Discrete Cosine Transform when the boundary effects are problematic. §.§ CSC ModelThe CSC problem is generally expressed in the form_𝐝,𝐳1/2𝐱-∑_k=1^K 𝐝_k * 𝐳_k _2^2 + β∑_k=1^K 𝐳_k_1 subject to 𝐝_k_2^2 ≤ 1 ∀ k ∈{1,...,K}where 𝐝_k ∈ℝ^M are the vectorized 2D patches representing K dictionary elements, and 𝐳_k ∈ℝ^D are the vectorized sparse maps corresponding to each of the dictionary elements (see Figure <ref>). The data term represents the image 𝐱∈ℝ^D modelled by the sum of convolutions of the dictionary elements with their correspondingsparse maps, and β controls the tradeoff between the sparsity of the feature maps and reconstruction error. The inequality constraint on the dictionary elements assumes Laplacian distributed coefficients, which ensures solving the problem at a proper scale for all elements since a larger value of 𝐝_k would scale down the value of the corresponding 𝐳_k. The above equation shows the objective function for a single image, and it can be easily extended to multiple images, where K corresponding sparse maps are inferred for each image and all the images share the same K dictionary elements. §.§.§ CSC SubproblemsThe objective in Eq. <ref> is not jointly convex. However, using a fixed point approach (iteratively solving for one variable while keeping the other fixed) leads to two convex subproblems, which we refer to as the coding subproblem and the dictionary learning subproblem. For ease of notation, we represent the convolution operations by multiplication of Toeplitz matrices with the corresponding variables.Coding Subproblem. We infer the sparse maps for a fixed set of dictionary elements as shown in Eq. <ref>._𝐳1/2𝐱-𝐃𝐳_2^2 + β𝐳_1Here, 𝐃=[𝐃_1…𝐃_K] is of size D × DK and is a concatenation of the convolution matrices of the dictionary elements, and 𝐳=[𝐳_1^T…𝐳_K^T]^T is a concatenation of the vectorized sparse maps. Learning Subproblem. We learn the dictionary elements for a fixed set of sparse feature maps as shown in Eq. <ref>._𝐝 1/2𝐱-𝐙𝐒^T𝐝_2^2 subject to 𝐝_k_2^2 ≤ 1 ∀ k ∈{1,...,K}Similar to above, 𝐙=[𝐙_1…𝐙_K] is of size D × DK and is a concatenation of the sparse convolution matrices, 𝐝=[𝐝_1^T…𝐝_K^T]^T is a concatenation of the dictionary elements, and 𝐒 projects the filter onto its spatial support. The above two subproblems can be optimized iteratively using ADMM <cit.>, whereeach ADMM iteration requires solving a large linear system of size DK for each of the two variables 𝐳 and 𝐝. Moreover, when applied to multiple images, solving the linear systems for the coding subproblem can be done separably, but should be done jointly for the learning subproblem, since all images share the same dictionary elements (see Section <ref> for more details on complexity analysis).§.§ CSC Dual OptimizationIn this section, we show our approach to solving the CSC subproblems in the dual domain. Formulating the problems in the dual domain reduces the number of parameters involved in the linear systems from DK to D, which leads to faster convergence towards a feasible solution and thus better computational performance. Since the two subproblems are convex, the duality gap is zero and solving the dual problem is equivalent to solving the primal form. In addition, similar to <cit.>, we also solve the convolutions efficiently in the Fourier domain as described below. §.§.§ Coding SubproblemTo find the dual problem of Eq. <ref>, we first introduce a dummy variable 𝐫 with equality constraints to yield the following formulationmin_𝐳,𝐫 1/2𝐫_2^2 + β𝐳_1 subject to 𝐫=𝐃𝐳-𝐱The Lagrangian of this problem would be:ℒ ( 𝐳,𝐫,λ )=1/2𝐫_2^2 + β𝐳_1+λ^T(𝐃𝐳-𝐱-𝐫)which results in the following dual optimization withdual variable λ:max_λ[ min_𝐫1/2𝐫_2^2 -λ^T𝐱 -λ^T𝐫 + min_𝐳β𝐳_1+λ^T𝐃𝐳]Solving the minimizations over 𝐫 and 𝐳 and using the definition of the conjugate function to the l_1 norm, we get the dual problem of Eq. <ref> as:min_λ 1/2λ^Tλ+λ^T𝐱 subject to 𝐃^Tλ_∞≤β Coding Dual Optimization. Now, we show how to solve the optimization problem in Eq. <ref> using ADMM. ADMM generally solves convex optimization problems by breaking the original problem into easier subproblems that are solved iteratively. To apply ADMM here, we introduce an additional variable θ=𝐃^Tλ,which allows us to write the problem in the general ADMM form as shown in Eq. <ref>. Since the dual solution to a dual problem is the primal solution for convex problems, the Lagrange multiplier involved in the ADMM update step is the sparse map vector 𝐳 in Eq. <ref>.min_λ,θ h(λ) + g(θ)s.t. θ=𝐃^Tλ whereh(λ) =1/2λ^Tλ+λ^T𝐱 and g(θ)=ind_C(θ)Here, ind_C(.) is the indicator function defined on the convex set of constraints C={θ | θ_∞≤β}. Deriving the augmented Lagrangian of the problem and solving for the ADMM update steps <cit.> yields the following iterative solutions to the dual problem with i representing the iteration number.λ^i+1 =( 𝐃𝐃^T + 1/ρ𝐈 )^-1 ( 𝐃θ^i + 1/ρ𝐃𝐳^i -1/ρ𝐱 ) θ^i+1 =Π_C ( 𝐃^Tλ^i+1-1/ρ𝐳^i) 𝐳^i+1 = 𝐳^i + ρ(θ^i+1-𝐃^Tλ^i+1)The parameter ρ∈ℝ+denotes the step size for the ADMM iterations, and Π represents the projection operator onto the set C. The linear systems shown above do not require expensive matrix inversion or multiplication as they are transformed to elementwise divisions and multiplications when solved in the Fourier domain. This is possible because ignoring the boundary effects leads to a circulant structure for the convolution matrices, and thus they can be expressed by their base sample as follows:𝐃_𝐤=ℱdiag(𝐝̂_̂𝐤̂)ℱ^Hwhere 𝐝̂ denotes the Discrete Fourier Transform (DFT) of 𝐝, ℱ is the DFT matrix independent of 𝐝, and X^H is the Hermitian transpose.In our formulation, the λ-update step requires solving a linear system of size D. Heide <cit.> however solve the problem in the primal domain in which the 𝐝-update step involves solving a much larger system of size KD. Clearly, our solution in the dual domain willlead to faster ADMM convergence. Figure <ref>-left shows the coding subproblem convergence of our approach and that of Heide Our dual formulation leads to convergence within less iterations compared to the primal domain. In addition, our approach achieves a lower objective in general at any feasible number of iterations.§.§.§ Learning SubproblemTo find a solution for Eq. <ref>, we minimize the Lagrangian of the problem (see Eq. <ref>) assuming optimal values for the Lagrange multipliers μ_k. This results in the optimization problem shown in Eq. <ref>.ℒ(𝐝,μ)=1/2𝐱-𝐙𝐒^T𝐝_2^2+∑_k=1^Kμ_k(𝐝_k_2^2-1)_𝐝1/2𝐱-𝐙𝐒^T𝐝_2^2+∑_k=1^Kμ_k^*𝐝_k_2^2To find the dual problem of Eq. <ref>, we follow a similar approach to the inference subproblem by introducing a dummy variable 𝐫 with equality constraints such that 𝐫=𝐙𝐒^T𝐝-𝐱. Deriving the Lagrangian of the problem and minimizing over the primal variables yields the dual problem shown in <ref>.min_γ1/2γ^Tγ+γ^T𝐱+∑_k=1^K1/4μ_k^*𝐒_k𝐙_k^Tγ_2^2 Learning Dual Optimization. The optimization problem in Eq. <ref> has a closed form solutionas shown in Eq. <ref>.γ^*=-(𝐈+∑_k=1^K1/2μ_k^*𝐙_k𝐒^T_k𝐒_k𝐙_k^T)^-1𝐱Given the optimal value for the dual variable γ, we can compute the optimal value for the primal variable 𝐝 as follows:𝐝_k^*=-1/2μ_k^*𝐒_k𝐙_k^Tγ^*∀ k={1,...,K} To find the optimal values for the Lagrange multipliers μ_k, we need to assure that the KKT conditions are satisfied. At optimal (μ_k^*,𝐝_k^*), the solution to the primal problem in Eq. <ref> and its Lagrangian are equal. Thus, we end up with the below iterative update step for μ_k. μ_k^i+1=μ_k^i 𝐝_k^i_2The learning subproblem is then solved iteratively by alternating between updating 𝐝_𝐤 as per Eqs. <ref>,<ref> and updating μ_k as per Eq. <ref> until convergence is achieved.We use conjugate gradient to solve the system involved in the γ-update step by applying the heavy convolution matrix multiplications in the Fourier domain. The computation cost for solving this system decreases with ADMM iterations, since we employ a warm start where weinitializeγ with the solutionfrom the previous iteration. Figure <ref>-right shows the decreasing computation time of the learning subproblem of our approach.For more details on the derivations of the coding and learning subproblems as well as the solutions to the equations in the Fourier domain, you may refer to the supplementary material.§.§.§ Coordinate DescentNow that we derived a solution to the two subproblems, we can use coordinate descent to solve the joint objective in Eq. <ref> by alternating between the solutions for 𝐳 and 𝐝. The full algorithm for the CSC problem is shown in Alg. <ref>.The coordinate descent algorithm above guarantees a monotonically decreasing joint objective. We keep iterating until convergence is reached, when the change in the objective value, or the solution for the optimization variables 𝐝 and 𝐳 reaches a user-defined threshold τ=10^-3. For solving the coding and learning subproblems, we also run the algorithms until convergence.§.§ Higher Order Tensor CSCHigher order tensor CSC <cit.> allows convolutional sparse coding over higher dimensional input such as a set of 3D input images as well as 4D videos. Similar to before, given the input data, we seek to reconstruct using K patches convolved with K sparse maps. In this formulation, each of the patches is of the same order of dimensionality as the input data with the possibility of a smaller spatial support. In addition, TCSC allows high dimensional correlation among features in the data. In this sense, unlike traditional CSC in which a separate sparse code is learned for separate features, the sparse maps in TCSC are shared along one of the dimensions such as the color channels for images/videos. The reader may be referred to the paper by Bibi <cit.> for more details of the derivations for TCSC. Below we give our approach to solving the TCSC coding and learning subproblems in the dual domain.The dictionary elements are represented by a tensor 𝒟 ∈ R^J× K× n_1 × ... × n_d where J represents the correlated input dimension (usually referring to data features/channels), K is the number of elements, and n_1,...,n_d are the uncorrelated dimensions (d=2 representing the spatial dimensions for images, and d=3 representing the spatial and time dimensions for videos). In our dual formulation, we perform circulant tensor unfolding <cit.> resulting in a block circulant dictionary matrix 𝐃 of size JD× KD where D= n_1 × ... × n_d.Thus, each of the K convolution matrices is now of size DJ× D where 𝐃_k=[𝐃_1,k^T…𝐃_J,k^T]^T.The solution to the coding dual problem is shown in Eq. <ref> where the inverse in the λ-update step involves now a block diagonal matrix. Thus, the inversion can be done efficiently by parallelization over the D blocks while making use of the Woodburry inversion formula <cit.>.The solution to the dictionary learning subproblem is straightforward, since the solution is separable along the dimension J. We solve for γ_j^* and 𝐝_k,j^* as previously shown in Eqs. <ref> and <ref> for all 𝐱_j where j=1… J.§ RESULTS In this section, we give an overview of the implementation details and the parameters selected for our dual CSC solver. We also show the complexity analysis and convergence of our approach compared to <cit.>, the current state-of-the-art CSC solver. Finally, we show results on 4D TCSC using color input images as well as 5D TCSC using colored videos. §.§ Implementation Details We implemented the algorithm in MATLAB using the Parallel Computing Toolbox and we ran the experiments on an Intel 3.1GHz processor machine. We used the code provided by <cit.> in the comparisons for regular CSC and by <cit.> in the comparisons to TCSC. We evaluate our approach on the fruit and city datasets formed of 10 images each, the house dataset containing 100 images, and basketball video from the OTB50 dataset selecting 10 frames similar to <cit.>.We apply contrast normalization to the images prior to learning the dictionaries for both gray scale and color images; thus, the figures show normalized patches. We show results by varying the sparsity coefficient β, the number of dictionary elements K, andthe number of images N. In our optimization, we choose a constant value of ρ=0.1 for the ADMM step size. We also initialize 𝐳 and θ with zeros for the first iteration of the learning subproblem, and 𝐝 with random values in the coding subproblem. Our results compare with Heide <cit.> for regular CSC, as it is the fastest among the published methods discussed in the related work section, and with Bibi <cit.> for TCSC as it is the only method that deals with higher order CSC. §.§ Complexity AnalysisIn this section, we analyze the per-iteration complexity of our approach compared to <cit.> and <cit.> with respect to each of the subproblems as shown in Table <ref>. In the equations below, D corresponds to the product of the order of the uncorrelated dimensions (e.g. number of pixels for images), J is the number of channels in the correlatedinputdimension, and Q is the number of conjugate gradient iterations within the learning subproblem. For regular CSC on high dimensional data, we assume that the problem is solved separately for each of the J channels.Coding Subproblem. In the coding subproblem, the complexity of our approach is similar to that of <cit.> for regular 2D CSC (J=1). The computational complexity is dominated by solving the linear system by elementwise product and division operations in the Fourier domain. Althoughthe two approaches are computationally similar here, it is important to notethat the number of variables involved in solving the systems is much less in our approach. In practice, we observe that this also leads to faster convergence for the subproblem as shown in Figure <ref>. For higher order dimensional CSC (J>1), our formulation is linear in the number of filters compared to a cubic cost for TCSC. In TCSC, the Sherman Morrison formula no longer applies and the computation is dominated by solving D linear systems of size K× K.Learning Subproblem. Here, our approach solves the problem iteratively using conjugate gradient to solve the linear system. Thus, the computational cost lies in solving elementwise products and divisions, with the additional cost of applying the Fourier transforms to the variables. On the other hand, Heide <cit.> and Bibi <cit.> need to solve the subproblem by applying ADMM.We observe that the performance of our dictionary learning approach improves by increasingthe number of conjugate gradient iterations involved in solving the linear system. This number decreases after each inner iteration and its cost becomes negligible within 4-6 iterations due to the warm-start initialization of γ at each step as shown in Figure <ref>. On the other hand, Heide <cit.> and Bibi <cit.> incurthe additional cost of solving the linear systems. Thus, our method has better scalability compared to the primal methods in which the linear systems solving step dominates as the number of images and filters increase (see section <ref>). §.§ CSC Convergence In this section, we analyse the convergence properties of our approach to regular convolutional sparse coding. In Figure <ref>, we plot the convergence of our method compared to the state of the art <cit.> on the city dataset for fixed β = 0.5 and K = 49. We also show the progression of the learnt filters in correspondence with the curves. As shown in the figure, the two methods converge to the same solution. Figure <ref> plots how the objective value decreases with time for each of the two methods using the same parameters as above. This shows that our method converges significantly faster than <cit.>. We also plot in Figure <ref> the objective value as a function of the sparsity coefficient β. The plot shows how increasing the sparsity coefficient results in an increase in the objective value, and more importantly, it verifies that our method converges to an objective value similar to that of <cit.> even though we reach a solution faster as shown in Figure <ref>. Figure <ref> also shows how the dictionary elements vary withβ. §.§ Scalability In this section, we analyze the scalability of the CSC problem with increasing number of filters and images. We compare our approach to Heide et. al. <cit.> in terms of the overall convergence time and average time per iteration for reaching the same final objective value. To ensure that the two problems achieve similar overall objective values, we make sure that each of the methods runs until convergence for both the coding and learning convex subproblems with the same initial point. Figure <ref> shows that the computation time of each of the methods increases when the number of filters and images increases. It also shows a significant speedup (about 2.5x) for our approach over that of <cit.>. §.§ TCSC Results In this section, we show results for TCSC on color images and videos. We compare the performance of our dual formulation for TCSC with that of Bibi <cit.>. Figures <ref> and <ref> show iteration time results on color images from the fruit and house datasets respectively with varying the number of filters and number of images. Correspondingly, we show the speedup acheived by our dual formulation for images and videos in Figure <ref>. As shown, our dual formulation achieves up to 20 times speed-up compared to the primal solution. We can observe higher speedups for smaller number of filters and we can also observe that the speedup is approximately constant as the number of images increases. This is inline with our complexity analysis in which we verified that Sherman Morrison formula is no longer applicable in the primal domain for TCSC, while parallelization is still applicable in the dual.We also show in Figure <ref> the iteration time as the number of filters varies on colored video. Since the video is considered as a 5D tensor, 3D Fourier transforms are applied resulting in a lower speedup but still maintaining a lower computation time compared to <cit.>. § CONCLUSION AND FUTURE WORKWe proposed our approach for solving the convolutional sparse coding problem by posing and solving each of its underlying convex subproblems in the dual domain. This results in a lower computational complexity than previous work. We can also easily extend our proposed solver to CSC problems ofhigher dimensional data. We demonstrated that tackling CSC in the dual domain results in up to 20 times speedup compared to the current state of the art. In future work, we would like to experiment with additional regularizers for CSC. We could make use of the structure of the input signal and map the regularizer over to the sparse maps to reflect this structure. For example, for images with a repetitive pattern, a nuclear norm can be added as a regularizer, which is equivalent to making the sparse maps low rank.splncs | http://arxiv.org/abs/1709.09479v2 | {
"authors": [
"Lama Affara",
"Bernard Ghanem",
"Peter Wonka"
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"categories": [
"cs.CV"
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"primary_category": "cs.CV",
"published": "20170927125500",
"title": "Fast Convolutional Sparse Coding in the Dual Domain"
} |
Joshua Erde was supported by the Alexander von Humboldt Foundation. University of Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany [email protected] Lehner was supported by the Austrian Science Fund (FWF) Grant no. J 3850-N32 University of Warwick, Mathematics Institute, Zeeman Building, Coventry CV4 7AL, United Kingdom [email protected] of Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany [email protected] [2010]05C45, 05C63, 20K99 We prove that any one-ended, locally finite Cayley graph with non-torsion generators admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the n-dimensional grid ℤ^n admits a decomposition into n edge-disjoint Hamiltonian double-rays for all n ∈. Hamilton decompositions of one-ended Cayley graphs Max Pitz December 30, 2023 ==================================================§ INTRODUCTION A Hamiltonian cycle of a finite graph is a cycle which includes every vertex of the graph. A finite graph G=(V,E) is said to have a Hamilton decomposition if its edge set can be partitioned into disjoint sets E=E_1 ∪̇E_2 ∪̇⋯∪̇E_r such that each E_i is a Hamiltonian cycle in G. The starting point for the theory of Hamilton decompositions is an old result by Walecki from 1890 according to which every finite complete graph of odd order has a Hamilton decomposition (see <cit.> for a description of his construction). Since then, this result has been extended in various different ways, and we refer the reader to the survey of Alspach, Bermond and Sotteau <cit.> for more information. Hamiltonicity problems have also been considered for infinite graphs, see for example the survey by Gallian and Witte <cit.>. While it is sometimes not obvious which objects should be considered the correct generalisations of a Hamiltonian cycle in the setting of infinite graphs, for one-ended graphs the undisputed solution is to consider double-rays, i.e. infinite, connected, 2-regular subgraphs. Thus, for us a Hamiltonian double-ray is then a double-ray which includes every vertex of the graph, and we say that an infinite graph G=(V,E) has a Hamilton decomposition if we can partition its edge set into edge-disjoint Hamiltonian double-rays. In this paper we will consider infinite variants of two long-standing conjectures on the existence of Hamilton decompositions for finite graphs. The first conjecture concerns Cayley graphs: Given a finitely generated abelian group (Γ,+) and a finite generating set S of Γ, the Cayley graph G(Γ,S) is the multi-graph with vertex set Γ and edge multi-set{(x,x+g):x ∈Γ, g ∈ S }.If Γ is an abelian group and S generates G, then the simplification of G(Γ,S) has a Hamilton decomposition, provided that it is 2k-regular for some k. Note that if S ∩ -S = ∅, then G(Γ,S) is automatically a 2|S|-regular simple graph. If G(Γ,S) is finite and 2-regular, then the conjecture is trivially true. Bermond, Favaron and Maheo <cit.> showed that the conjecture holds in the case k=2. Liu <cit.> proved certain cases of the conjecture for finite 6-regular Cayley graphs, and his result was further extended by Westlund <cit.>.Our main theorem in this paper is the following affirmative result towards the corresponding infinite analogue of Conjecture <ref>: Let Γ be an infinite, finitely generated abelian group, and let S be a generating set such that every element of S has infinite order. If the Cayley graph G=G(Γ,S) is one-ended, then it has a Hamilton decomposition. We remark that under the assumption that elements of S are non-torsion, the simplification of G(Γ,S) is always isomorphic to a Cayley graph G(Γ,S') with S' ⊆ S and S' ∩ -S' = ∅, and so our theorem implies the corresponding version of Conjecture <ref> for non-torsion generators, in particular for Cayley graphs of ℤ ^n with arbitrary generators.In the case when G=G(Γ,S) is two-ended, there are additional technical difficulties when trying to construct a decomposition into Hamiltonian double-rays. In particular, since each Hamiltonian double-ray must meet every edge cut an odd number of times, there can be parity reasons why no decomposition exists. One particular two-ended case, namely where Γ≅ℤ, has been considered by Bryant, Herke, Maenhaut and Webb <cit.>, who showed that when G(ℤ,S) is 4-regular, then G has a Hamilton decomposition unless there is an odd cut separating the two ends.The second conjecture about Hamiltonicity that we consider concerns Cartesian products of graphs: Given two graphs G and H the Cartesian product (or product) G □ H is the graph with vertex set V(G) × V(H) in which two vertices (g,h) and (g',h') are adjacent if and only if either * g = g' and h is adjacent to h' in H, or* h = h' and g is adjacent to g' in G.Kotzig <cit.> showed that the Cartesian product of two cycles has a Hamilton decomposition, and conjectured that this should be true for the product of three cycles. Bermond extended this conjecture to the following: If G_1 and G_2 are finite graphs which both have Hamilton decompositions, then so does G_1□ G_2. Alspach and Godsil <cit.> showed that the product of any finite number of cycles has a Hamilton decomposition, and Stong <cit.> proved certain cases of Conjecture <ref> under additional assumptions on the number of Hamilton cycles in the decomposition of G_1 and G_2 respectively.Applying techniques we developed to prove Theorem <ref>, we show as our second main result of this paper that Conjecture <ref> holds for countably infinite graphs.theoremmain If G and H are countable graphs which both have Hamilton decompositions, then so does their product G □ H. The paper is structured as follows: In Section <ref> we mention some group theoretic results and definitions we will need. In Section <ref> we state our main lemma, the Covering Lemma, and show that it implies Theorem <ref>. The proof of the Covering Lemma will be the content of Section <ref>. In Section <ref> we apply our techniques to prove Theorem <ref>. Finally, in Section <ref> we listopen problems and possible directions for further work.§ NOTATION AND PRELIMINARIES If G=(V,E) is a graph, and A,B ⊆ V, we denote by E(A,B) the set of edges between A and B, i.e. E(A,B) = (x,y) ∈ E:x ∈ A, y ∈ B. For A ⊆ V or F ⊆ E we write G[A] and G[F] for the subgraph of G induced by A and F respectively. For A,B ⊆Γ subsets of an abelian group Γ we write -A := -a:a ∈ A and A + B := a+b:a ∈ A, b ∈ B⊆Γ. If Δ is a subgroup of Γ, and A ⊂Γ a subset, then A^Δ = a + Δ:a ∈ A denotes the family of corresponding cosets. If g ∈Γ we say that the order of g is the smallest k ∈ such that k· g = 0. If such a k exists, then g is a torsion element. Otherwise, we say the order of g is infinite and g is a non-torsion element. For k ∈ we write [k] = 1,2,…, k.The following terminology will be used throughout.Given a graph G, an edge-colouring cE(G) → [s] and a colour i ∈ [s], the i-subgraph is the subgraph of G induced by the edge set c^-1(i), and the i-components are the components of the i-subgraph.Let Γ be an infinite abelian group, S = g_1,g_2, … , g_s a finite generating set for Γ such that every g_i ∈ S has infinite order, and let G be the Cayley graph G(Γ,S). * The standard colouring of G is the edge colouring c_std E(G) → [s] such that c_std((x,x + g_i)) = i for each x ∈Γ, g_i ∈ S. * Given a subset X ⊆ V(G) we say that a colouring c is standard on X if c agrees with c_std on G[X]. Similarly if F ⊂ E(G) we say that c is standard on F if c agrees with c_std on F.* A colouring cE(G) → [s] is almost-standard if the following are satisfied: * there is a finite subset F ⊆ E(G) such that c is standard on E(G) ∖ F;* for each i ∈ [s] the i-subgraph is spanning, and each i-component is a double-ray.Let Γ and S be as above. Given x∈Γ and g_i ≠ g_j ∈ S, we call xg_ig_j := (x,x+g_i),(x,x+g_j),(x+g_i ,x + g_i + g_j),(x+g_j,x + g_i + g_j)an (i,j)-square with base point x, and xg_i := (x+n g_i,x+(n+1)g_i): n ∈an i-double-ray with base point x. Moreover, given a colouring cE(G(Γ,S)) → [s] we call xg_ig_j and xg_i an (i,j)-standard square and i-standard double-ray if c is standard onxg_ig_j and xg_i respectively. Since Γ is an abelian group, every xg_ig_j is a 4-cycle in G(Γ,S) (provided g_i ≠ -g_j), and since S contains no torsion elements of Γ, xg_k really is a double-ray in the Cayley graph G(Γ,S).Let Γ be a finitely generated abelian group. By the Classification Theorem for finitely generated abelian groups (see e.g. <cit.>), there are integers n,q_1,…,q_r such that Γ≅ℤ^n ⊕⊕_i=1^r ℤ_q_i, where ℤ_q is the additive group of the integers modulo q. In particular, for each Γ there is an integer n and a finite abelian group Γ_fin such that Γ≅ℤ^n ⊕Γ_fin.The following structural theorem for the ends of finitely generated abelian groups is well-known: For a finitely generated group Γ≅ℤ^n ⊕Γ_fin, the following are equivalent: * n ≥ 2,* there exists a finite generating set S such that G(Γ,S) is one-ended, and* for all finite generating sets S, the Cayley graph G(Γ,S) is one-ended. See e.g. <cit.> for the fact the number of ends of G(Γ,S) is independent of the choice of the generating set S, and <cit.> for the equivalence with the first item. A group Γ satisfying one of the conditions from Theorem <ref> is called one-ended. Let Γ be an abelian group, S={g_1,…, g_s} be a finite generating set such that the Cayley graph G(Γ,S) is one-ended. Then, for every g_i ∈ S of infinite order, there is some g_j ∈ S such that g_i,g_j≅ (^2,+). Suppose not. It follows that in Γ / ⟨ g_i ⟩ every element has finite order, and since it is also finitely generated, it is some finite group Γ_f such that Γ≅⊕Γ_f. Thus, by Theorem <ref>, G is not one-ended, a contradiction. § THE COVERING LEMMA AND A HIGH-LEVEL PROOF OF THEOREM <REF> Every Cayley graph G(Γ,S) comes with a natural edge colouring c_std, where we colour an edge (x,x+g_i) with x ∈Γ and g_i ∈ S according to the index i of the corresponding generating element g_i. If every element of S has infinite order, then every i-subgraph of G(Γ,S) consists of a spanning collection of edge-disjoint double-rays, seeDefinitions <ref> and <ref>. So, it is perhaps a natural strategy to try to build a Hamiltonian decomposition by combining each of these monochromatic collections of double-rays into a single monochromatic spanning double-ray. Rather than trying to do this directly, we shall do it in a series of steps: given any colour i ∈ [s] = |S| and any finite set X ⊂ V(G), we will show that one can change the standard colouring at finitely many edges so that there is one particular double-ray in the colour i which covers X. Moreover, we can ensure that the resulting colouring maintains enough of the structure of the standard colouring that we can repeat this process inductively: it should remain almost standard, i.e. all monochromatic components are still double-rays, see Definition <ref>. By taking a sequence of sets X_1 ⊆ X_2 ⊆⋯ exhausting the vertex set of G, and varying which colour i we consider, we ensure that in the limit, each colour class consists of a single spanning double-ray, giving us the desired Hamilton decomposition.In this section, we formulate our key lemma, namely the Covering Lemma <ref>, which allows us to do each of these steps. We will then show how Theorem <ref> follows from the Covering Lemma. The proof of the Covering Lemma is given in Section <ref>.Let Γ be an infinite, one-ended abelian group, S = g_1,g_2, … , g_s a finite generating set such that every g_i ∈ S has infinite order, and G=G(Γ,S) the corresponding Cayley graph.Then for every almost-standard colouring c of G, every colour i and every finite subset X ⊆ V(G), there exists an almost-standard colouring ĉ of G such that * ĉ=c on E(G[X]), and* some i-component in ĉ covers X. Fix an enumeration V(G)=v_n:n ∈. Let X_0 = D'_0 = {v_0} and c_0 = c_std. For each n ≥ 1 we will recursively construct almost standard colourings c_nE(G) → [s], finite subsets X_n ⊂ V(G), (n mod s)-components D_n of c_n and finite paths D'_n ⊆ D_n such that for every n ∈ * X_n-1∪v_n⊆ X_n,* V(D'_n-1) ⊆ X_n,* X_n ⊆ V(D'_n),* D'_n properly extends the path D'_n - s (the `previous' path of colour n mod s) in both endpoints of D'_n - s, and* c_n agrees with c_n-1 on E(G[X_n]). Suppose inductively for some n ∈ that c_n, X_n, D_n and D'_n have already been defined. Choose some X_n +1⊇ X_n ∪v_n large enough such that (1) and (2) are satisfied. Applying Lemma <ref> with input c_n and X_n+1 provides us with a colouring c_n+1 such that (5) is satisfied and some (n+1 mod s)-component D_n+1 covers X_n+1. Since c_n+1 is almost standard, D_n+1 is a double-ray. Furthermore, since c_n+1 agrees with c_n on E(G[X_n+1]), by the inductive hypothesis it agrees with c_k on E(G[X_k+1]) for each k ≤ n.Therefore, since D'_n+1-s⊂ X_n-s+2 is a path of colour (n+1 mod s) in c_n+1-s, it follows that D'_n+1-s⊂ D_n+1 and so we can extend D'_n+1-s to a sufficiently long finite path D'_n+1⊂ D_n+1 such that (3) and (4) are satisfied at stage n+1.Once the construction is complete, we define T_1, …, T_s⊂ G by T_i = ⋃_n ≡ isD'_nand claim that they form a decomposition of G into edge-disjoint Hamiltonian double-rays. Indeed, by (4), each T_i is a double-ray. That they are edge-disjoint can be seen as follows: Suppose for a contradiction that e ∈ E(T_i) ∩ E(T_j). Choose n(i) and n(j) minimal such that e ∈ E(D'_n(i)) ⊂ E(T_i) and e ∈ E(D'_n(j)) ⊂ E(T_j). We may assume that n(i) < n(j), and so e ∈ E(G[X_n(i)+1]) by (2). Furthermore, by (5) it follows that c_n(j) agrees with c_n(i) on E(G[X_n(i)+1]). However by construction c_n(j)(e) = j ≠ i = c_n(i)(e) contradicting the previous line.Finally, to see that each T_i is spanning, consider some v_n ∈ V(G). By (1), v_n ∈ X_n. Pick n' ≥ n with n' ≡ is. Then by (3), D'_n'⊂ T_i covers X_n' which in turn contains v_n, as v_n ∈ X_n⊆ X_n' by (1).§ PROOF OF THE COVERING LEMMA§.§ Blanket assumption.Throughout this section, let us now fix* a one-ended infinite abelian group Γ with finite generating set S = g_1, …, g_s such that every element of S has infinite order,* an almost-standard colouring c of the Cayley graph G=G(Γ,S),* a finite subset X ⊆Γsuch that c is standard on V(G) ∖ X,* a colour i, say i = 1, and corresponding generator g_1 ∈ S, for which we want to show Lemma <ref>, and finally* a second generator in S, say g_2, such that Δ:=g_1,g_2≅ (^2,+), see Corollary <ref>. §.§ Overview of proofWe want to show Lemma <ref> for the Cayley graph G, colouring c, generator g_1 and finite set X. The cosets of g_1,g_2 in Γ cover V(G), and in the standard colouring the edges of colour 1 and 2 form a grid on g_1,g_2. So, since c is almost-standard, on each of these cosets the edges of colour 1 and 2 will look like a grid, apart from on some finite set.Our aim is to use the structure in these grids to change the colouring c to one satisfying the conclusions of Lemma <ref>. It will be more convenient to work with large finite grids, which we require, for technical reasons, to have an even number of rows. This is the reason for the slight asymmetry in the definition below. Let g_i,g_j ∈Γ. For N,M ∈ we writeg_ig_jNM := n g_i + m g_j:n,m ∈, -N ≤ n ≤ N, -M < m ≤ M⊆g_i,g_j⊆Γ.The structure of our proof can be summarised as follows. First, in Section <ref>, we will show that there is some N_0 and some `nice' finite set of P of representatives of cosets of g_1,g_2 such that P + g_1g_2N_0N_0 covers X. We will then, in Section <ref> pick sufficiently large numbers N_0 < N_1 < N_2 < N_3 and consider the grids P + g_1g_2N_3N_1. Using the structure of the grids we will make local changes to the colouring inside P + (g_1g_2N_3N_1∖g_1g_2N_0N_0) to construct our new colouring ĉ. This new colouring ĉ will then agree with c on the subgraph induced by P + g_1g_2N_0N_0⊇ X, and be standard on V(G) ∖(P + g_1g_2N_3N_1), and hence, as long as we ensure all the colour components are double-rays, almost-standard.These local changes will happen in three steps. First, in Step <ref>, we will make local changes inside x_ℓ + (g_1g_2N_3N_1∖g_1g_2N_2N_1) for each x_ℓ∈ P, in order to make every i-component meeting P + g_1g_2N_2N_1 a finite cycle. Next, in Step <ref>, we will make local changes inside x_ℓ + (g_1g_2N_2N_1∖g_1g_2N_1N_1) for each x_ℓ∈ P, in order to combine the cycles meeting this translate of the grid into a single cycle. Finally, in Step <ref>, we will make local changes inside P + (g_1g_2N_1N_1∖g_1g_2N_0N_0), in order to join the cycles for different x_ℓ into a single cycle covering P + g_1g_2N_0N_0. We then make one final local change to turn this finite cycle into a double-ray.§.§ Identifying the relevant cosets There exist N_0 ∈ and a finite set P = x_0, …, x_t⊂Γ such that * P^Δ = x_0 + Δ, …, x_t + Δ is a path in G(Γ / Δ, S ∖g_1,g_2^Δ), and* X ⊆ P + g_1g_2N_0N_0.Since X is finite, there is a finite set Y = y_1, …, y_k⊂Γ such that the cosets in Y^Δ=y_1 + Δ, …, y_k + Δ are all distinct and cover X. Moreover, since every (y_ℓ + Δ ) ∩ X is finite, there exists N_0 ∈ such that (y_ℓ + g_1,g_2) ∩ X = (y_ℓ + g_1g_2N_0N_0 ) ∩ Xfor all 1 ≤ℓ≤ k. Then X ⊆ Y + g_1g_2N_0N_0.Next, by a result of Nash-Williams <cit.>, every Cayley graph of a countably infinite abelian group has a Hamilton double-ray, and it is a folklore result (see <cit.>) that every Cayley graph of a finite abelian group has a Hamilton cycle. So in particular, the Cayley graph of (Γ / Δ, S ∖g_1,g_2^Δ), has a Hamilton cycle / double-ray, say H. Let P ⊇ Y be a finite set of representatives of the cosets of Δ which lie on the convex hull of Y^Δ on H. It is clear that P is as required. * For the rest of this section let us fix N_0 ∈ℕ and P = x_0, …, x_t⊂Γ to be as given by Lemma <ref>.§.§ Picking sufficiently large gridsIn order to choose our grids large enough to be able to make all the necessary changes to our colouring, we will first need the following lemma, which guarantees that we can find, for each k ≠ 1,2 and x ∈Γ, many distinct standard k-double-rays which go between the cosets x + Δ and (x + g_k) + Δ.For any g_k ∈ S ∖g_1,g_2 and any pair of distinct cosets x+Δ and (g_k+x)+Δ, there are infinitely many distinct standard k-double-rays R for the colouring c with E(R) ∩ E(x+Δ,(g_k+x)+Δ) ≠∅.It clearly suffices to prove the assertion for c = c_std. We claim that eitherR_1=x+mg_1g_k:m ∈orR_2=x+mg_2g_k:m ∈is such a collection of disjoint standard k-double-rays.Suppose that R_1 is not a collection of disjoint double-rays. Then there are m ≠ m' ∈ and n,n' ∈ such that m g_1 +ng_k = m' g_1 +n'g_k.Since g_k has infinite order, it follows that n ≠ n', too, and so we can conclude that there are ℓ, ℓ' ∈∖0 such that ℓ g_1 = ℓ' g_k. Similarly, if R_2 was not a collection of disjoint double-rays, then we can find q, q' ∈∖0 such that q g_2 = q' g_k. However, it now follows that q'ℓ g_1 = q'(ℓ' g_k) = ℓ' (q'g_k) = ℓ' q g_2,contradicting the fact that ⟨ g_1, g_2 ⟩≅ (^2,+). This establishes the claim.Finally, observe that if say R_1 is a disjoint collection, then for every R_m = x+mg_1g_k∈R_1 we have (x + m g_1,x + m g_1 + g_k) ∈ E(R_m) ∩ E(x+Δ,(g_k+x)+Δ) as desired. We are now ready to define our numbers N_0 < N_1 < N_2 < N_3. Recall that N_0 and P=x_0, …, x_t are given by Lemma <ref>. For each ℓ∈ [t], let g_n(ℓ) be some generator in S ∖g_1,g_2 that induces the edge between x_ℓ-1 + Δ and x_ℓ+ Δ on the path P^Δ. Note that n(ℓ) ∈ [s] ∖1,2 for all ℓ. By Lemma <ref>, we may find t^2 many disjoint standard double-raysR = R^k_ℓ:1 ≤ k,ℓ≤ tsuch that for every ℓ, the double-rays in R^k_ℓ = y^k_ℓg_n(ℓ):k ∈ [t] are standard n(ℓ)-double-rays containing an edgee^k_ℓ=(y^k_ℓ,y^k_ℓ + g_n(ℓ)) ∈ E(R^k_ℓ) ∩ E(x_ℓ-1+Δ, x_ℓ + Δ)so that all T^k_ℓ = y^k_ℓg_ig_n(ℓ) are (1,n(ℓ))-standard squares for c which have empty intersection with x_ℓ-1,x_ℓ + g_1g_2N_0N_0. Furthermore we may assume that these standard squares are all edge-disjoint. Then * let N_1 > N_0 be sufficiently large such that the subgraph induced by P + g_1g_2N_1-3N_1-3 contains all standard squares T^k_ℓ mentioned above. * Let N_2 be arbitrary with N_2 ≥ 5N_1.* Let N_3 be arbitrary with N_3 ≥ N_2 + 2N_1.§.§ The cap-off stepOur main tool for locally modifying our colouring is the following notion of `colour switchings', which is also used in <cit.>. Given an edge colouring cE(G(Γ,S)) → [s] and an (i,j)-standard square xg_ig_j, a colour switching on xg_ig_j changes the colouring c to the colouring c' such that * c'=c on E ∖xg_ig_j,* c'((x,x+g_i)) = c'((x + g_j,x+ g_i + g_j)) = j,* c'((x,x+g_j)) = c'((x + g_i,x+ g_i + g_j)) = i. It would be convenient if colour switchings maintained the property that a colouring is almost-standard. Indeed, if c is standard on E(G) ∖ F then c' is standard on E(G) ∖ (F ∪xg_ig_j). Also, it is a simple check that if the i and j-subgraphs of G for c are 2-regular and spanning, then the same is true for c'. However, some i or j-components may change from double-rays to finite cycles, and vice versa. [Cap-off step]There is a colouring c' obtained from c by colour switchings of finitely many (1,2)-standard squares such that * c'=c on E(G[X]);* every 1-component in c' meeting P + g_1g_2N_2N_1 is a finite cycle intersecting both P + (g_1g_2N_3N_1∖g_1g_2N_2N_1) and P + g_1g_2N_1N_1;* every other 1-component, and all other components of all other colour classes of c' are double-rays;* c' is standard outside of P + g_1g_2N_3N_1 and inside of P + (g_1g_2N_2N_1∖g_1g_2N_0N_0);* for each x_ℓ∈ P, the sets of verticesx_l + n g_1 + mg_2N_1 ≤ |n| ≤ N_2 , m ∈{ N_1, N_1 -1}are each contained in a single 1-component of c'.For ℓ∈ [t] and q ∈ [N_1] letR^ℓ_q =v^ℓ_qg_1g_2 and L^ℓ_q =w^ℓ_qg_1g_2 be the (1,2)-squares with base point v^ℓ_q = x_ℓ + (N_3 + 1 -2q) · g_1 + (N_1 + 1 - 2q) · g_2 and w^ℓ_q = x_ℓ - (N_3 + 2 -2q) · g_1 + (N_1 + 1 - 2q) · g_2 respectively. The square L^ℓ_q is the mirror image of R^ℓ_q with respect to the y-axis of the grid x_ℓ + g_1,g_2, however the base points are not mirror images, accounting for the slight asymmetry in the definitions.Since N_3 ≥ N_2 + 2N_1, it follows that R^ℓ_q ∪ L^ℓ_q ⊆ E(x_ℓ + (g_1g_2N_3N_1∖g_1g_2N_2N_1))for all q ∈ [N_1], and so by assumption on c, all R^ℓ_q and L^ℓ_q are indeed standard (1,2)-squares. We perform colour switchings on R^ℓ_q and L^ℓ_q for all ℓ∈ [t] and q ∈ [N_1], and call the resulting edge colouring c'. It is clear that c'=c on E(G[X]) and that c' is standard outside of P + g_1g_2N_3N_1 and inside of P + (g_1g_2N_2N_1∖g_1g_2N_0N_0).Let C ⊂ G denote the region consisting of all vertices that lie in x_ℓ + (g_1g_2N_3N_1 for some ℓ between a pair L^ℓ_q and R^ℓ_q for some q, i.e.C = ⋃_ℓ=1^t⋃_q=1^N_1⋃_m=1^2 x_ℓ + n g_1 +(N_1 + m- 2q)g_2:|n| ≤ N_3 + 1 -2q.Then P + g_1g_2N_2N_1⊆ C. By construction, there are no edges of colour 1 in c' leaving C, that is, E(C,V(G) ∖ C) ∩ c'^-1(1) = ∅. In particular, since the 1-subgraph of G under c' remains 2-regular and spanning, as remarked above, all 1-components under c' inside C are finite cycles, whose union covers C. Also, since each 1-component of c is a double-ray, it must leave the finite set P + g_1g_2N_3N_1 and hence meets some R_q^ℓ or L_q^ℓ. Therefore, by construction each 1-component of c' inside C meets some R_q^ℓ or L_q^ℓ and so, since c' is standard outside of P + g_1g_2N_0N_0 except at the squares R_q^ℓ or L_q^ℓ, each such 1-component meets both P + (g_1g_2N_3N_1∖g_1g_2N_2N_1) and P + g_1g_2N_1N_1.Moreover, all other colour components remain double-rays. This is clear for all k-components of G if k ≠ 1,2 (as the colours switchings of (1,2)-standard squares did not affect these other colours). However, it is also clear for the 1-coloured double-rays outside of C and also for all 2-coloured components, as we chose our standard squares R_q^ℓ and L_q^ℓ `staggered', so as not to create any finite monochromatic cycles, see Figure <ref> (recall that every x_ℓ + Δ is isomorphic to the grid). Finally, since N_1 > N_0, the edge set{(x_ℓ+ n g_1 + N_1 g_2, x_ℓ + (n+1) g_1 + N_1 g_2) -N_3 ≤ |n| < N_3-1}∪(v^ℓ_1,v^ℓ_1 + g_2),((w^ℓ_1+g_1,w^ℓ_1 + g_1 + g_2))∪(x_ℓ + n g_1 + (N_1-1) g_2, x_ℓ + (n+1) g_1 + (N_1-1) g_2):-N_3 ≤ n < -N_1∪(x_ℓ + n g_1 + (N_1-1) g_2, x_ℓ + (n+1) g_1 + (N_1-1) g_2):N_1 ≤ n < N_3meets only R^ℓ_1 and L^ℓ_1 and therefore is easily seen to be part of the same 1-component of c'. In Figure <ref>, these edges correspond to the red line at the top, and the two lines below it on either side of x_ℓ + g_1g_2N_1N_1. §.§ Combining cycles inside each coset of ΔIn the previous step we chose the (1,2)-standard squares at which we performed colour switchings in a staggered manner in the grids x_l + g_1g_2N_3N_1, so that we could guarantee that all the 2-components were still double-rays afterwards. In later steps we will no longer be able to be as explicit about which standard squares we perform colour switchings at, and so we will require the following definitions to be able to say when it is `safe' to perform a colour switching at a standard square. Suppose R =(v_i,v_i+1):i ∈ℤ is a double-ray and e_1 = (v_j_1,v_j_2) and e_2 = (v_k_1,v_k_2) are edges with j_1 < j_2 and k_1 < k_2. We say that e_1 and e_2 cross on R if either j_1 < k_1 < j_2 < k_2 or k_1<j_1<k_2<j_2. For an edge-colouring cE(G(Γ,S)) → [s], suppose that xg_ig_k is an (i,k)-standard square with g_i ≠ g_k, and further that the two k-coloured edges (x,x+g_k) and (x+g_i,x+g_i+g_k) of xg_ig_k lie on the same standard k-double-ray R = xg_k. Then the two i-coloured edges of xg_ig_k cross on R.Write e_1 = (x,x+g_i) and e_2 = (x+g_k,x+g_k+g_i) for the two i-coloured edges of xg_ig_k. The assumption that (x,x+g_k) and (x+g_i,x+g_i+g_k) both lie on xg_k implies that g_i = rg_k for some r ∈∖-1,0,1. If r > 1, we have x < x + g_k < x+g_i < x+g_k+g_i (where < denotes the natural linear order on the vertex set of the double-ray), and if r < -1, we have x + g_i < x + g_k + g_i < x < x+g_k, and so the edges e_1 and e_2 indeed cross on R.Given an edge colouring cE(G(Γ,S)) → [s] we say an (i,k)-standard square xg_ig_k is safe if g_i ≠ -g_k and either * the k-components for c meeting T are distinct double-rays, or* there is a unique k-component for c meeting T, which is a double-ray on which (x,x+g_i) and (x+g_k,x+g_i+g_k)cross.The following lemma tells us, amongst other things, that if we perform a colour switching at a safe (1,k)-standard square then the k-components in the resulting colouring meeting that square will still be double-rays.Let cE(G(Γ,S)) → [s] be an edge colouring, T = xg_ig_k be an (i,k)-standard square with g_i ≠ -g_k, and c' be the colouring obtained by performing a colour switching on T. Suppose that the i and k-components for c meeting T are all 2-regular, and that there are two distinct i-components C_1 and C_2 meeting T, at least one of which is a finite cycle. Then the following statements are true: * There is a single i-component for c' meeting T which covers V(C_1) ∪ V(C_2);* If the k-components for c meeting T are distinct double-rays then the k-components for c' meeting T are distinct double-rays;* If there is a unique k-component for c meeting T, which is a double-ray on which (x,x+g_i) and (x+g_k,x+g_i+g_k) cross, then there is unique k-component for c' meeting T, which is a double-ray. Let us write e_i=(x,x+g_i), e_k=(x,x+g_k), e'_i=(x+g_k,x + g_i + g_k) and e'_k=(x+g_i,x + g_i + g_k), so that xg_ig_j = e_i,e_k,e'_i,e'_k.For the first item, let the i-components for c be e_i ∈ C_1 and e'_i∈ C_2, where without loss of generality C_2 is a finite cycle. Then C_2 - e'_i is a finite path, and C_1 - e_i has at most 2 components, one containing x and one containing x + g_i. Hence, the i-component for c' meeting T, (C_1 ∪ C_2) - e_i,e'_i +e_k,e'_k, is connected and covers V(C_1) ∪ V(C_2).For the second item, let the k-components for c be e_k ∈ D_1 and e'_k∈ D_2. Then D_1 - e_k has two components, a ray starting at x and a ray starting at x+g_k. Similarly, D_2 - e'_k has two components, a ray starting at x+g_i and a ray starting at x + g_i + g_k. Hence, the k-components for c' meeting T, which are the components of (D_1 ∪ D_2) -e_k,e'_k +e_i,e'_i, are distinct double-rays.Finally, if there is a single k-component D for c meeting T such that D is a double-ray, then D - e_k,e'_k consist of three components. Since e_i and e'_i cross on D there are two cases as to what these components are. Either the components consist of two rays, starting at x and x + g_i + g_k and a finite path from x+g_k to x+g_i, or the components consist of two rays, starting at x+g_i and x+g_k, and a finite path from x+g_i+g_k to x. In either case, the k-component for c' meeting T, namely D -e_k,e'_k +e_i,e'_i, is a double-ray. Lemma <ref> is also useful as the first item allows us to use (1,k) colour switchings to combine two 1-components into a single 1-component which covers the same vertex set.[Combining cycles step]We can change c' from Step <ref> via colour switchings of finitely many (1,2)-standard squares to a colouring c” satisfying * c”=c'=c on E(G[X]);* every 1-component in c” meeting P + g_1g_2N_2N_1 is a finite cycle intersecting both P + (g_1g_2N_3N_1∖g_1g_2N_2N_1) and P + g_1g_2N_1N_1;* every other 1-component, and all other components of all other colour classes of c” are double-rays;* every 1-component in c” meeting some x_k + (g_1g_2N_2N_1∖g_1g_2N_0N_0) covers x_k + (g_1g_2N_2N_1∖g_1g_2N_0N_0);* c” is standard outside of P + g_1g_2N_3N_1 and inside of P + (g_1g_2N_1N_1∖g_1g_2N_0N_0).Our plan will be to go through the `grids' x_k + g_1g_2N_2N_1 in order, from k=0 to t, and use colour switchings to combine all the 1-components which meet x_k + (g_1g_2N_2N_1∖g_1g_2N_0N_0) into a single 1-component. We note that, since c' is not standard on X, it may be the case that these 1-components also meet x_k' + g_1g_2N_2N_1 for k' ≠ k.We claim inductively that there exists a sequence of colourings c' = c_0,c_1, …, c_t = c” such that for each 0 ≤ℓ≤ t:* c_ℓ=c'=c on E(G[X]);* every 1-component in c_ℓ meeting P + g_1g_2N_2N_1 is a finite cycle intersecting both P + (g_1g_2N_3N_1∖g_1g_2N_2N_1) and P + g_1g_2N_1N_1;* for every k ≤ℓ, every 1-component in c_ℓ meeting x_k + (g_1g_2N_2N_1∖g_1g_2N_0N_0) covers x_k + (g_1g_2N_2N_1∖g_1g_2N_0N_0);* for every k > ℓ, c_ℓ=c' on x_k +g_1g_2N_2N_1* every other 1-component, and all other components of all other colour classes of c_ℓ are double-rays;* c_ℓ is standard outside of P + g_1g_2N_3N_1 and inside of P + (g_1g_2N_1N_1∖g_1g_2N_0N_0). In Step <ref> we constructed c_0=c' such that this holds. Suppose that 0 < ℓ≤ t, and that we have already constructed c_k for k < ℓ. For q∈ [4N_1 - 2] we define T_q = v_qg_1g_2 to be the (1,2)-square with base point v_q =x_ℓ + (N_2 +2 - 2q)g_1 + (N_1 - q)g_2 if q ≤ 2N_1 - 1, and x_ℓ - (N_2 + 3- 2q')g_1 + (N_1 - q')g_2 if q'= q-(2N_1-1) ≥ 1.With these definitions, T_2N_1-1+q is the mirror image of T_q for all q ∈ [2N_1-1] along the y-axis. Moreover, since N_2 ≥ 5N_1, each T_q is contained within x_k + (g_1g_2N_2N_1∖g_1g_2N_1N_1).We will combine the 1-components in c_ℓ-1 which meet x_ℓ + (g_1g_2N_2N_1∖g_1g_2N_0N_0) into a single component by performing colour switchings at some of the (1,2)-squares T_q. Let us show first that most of the induction hypotheses are maintained regardless of the subset of the T_q we make switchings at. We note that, since c_ℓ-1 is standard inside of x_ℓ + (g_1g_1N_2N_1∖g_1g_2N_0N_0) and outside of P + g_1g_2N_3N_1, and g_1 ≠ -g_2, each T_q is a safe (1,2)-standard square for c_ℓ-1. Furthermore, by construction, even if we perform colour switchings at any subset of the T_q, the remaining squares remain standard and safe.Hence, by Lemma <ref> and the induction assumption, after performing colour switchings at any subset of the standard squares T_q all 2-components of the resulting colouring will be double-rays. Secondly, these colour switching will not change the colouring outside of P + g_1g_2N_2N_1 and inside of P + g_1g_2N_1N_1, or in any x_k + g_1g_2N_2N_1 with k ≠ℓ. In particular, every 1-component not meeting P + g_1g_2N_2N_1 will still be a double-ray. Finally, again by Lemma <ref>, every 1-component of the resulting colouring meeting P + g_1g_2N_2N_1 will be a finite cycle which covers the vertex set of some union of 1-components in c_ℓ-1, and hence will intersect both P + (g_1g_2N_3N_1∖g_1g_2N_2N_1) and P + g_1g_2N_1N_1.Let us write e_q = (v_q,v_q + g_1) for each q ∈ [4N_1-2]. Since c_ℓ-1 = c' on x_ℓ +g_1g_2N_2N_1, and by Step <ref> c' is standard on x_ℓ + (g_1g_2N_2N_1∖g_1g_2N_0N_0), each 1-component of c_ℓ-1 that meets x_ℓ + (g_1g_2N_2N_1∖g_1g_2N_0N_0) contains at least one e_q. Also, e_1 and e_2N_1 belong to the same 1-component by the last claim in Step <ref>. Let us write C for the collection of such cycles, and consider the mapαC→1, …, 4N_1-1, C ↦minq:e_q ∈ E(C),which maps each cycle to the first e_q that it contains. Since C is a disjoint collection of cycles, the map α is injective. Now let c_ℓ be the colouring obtained from c_ℓ-1 by switching all standard squares inT=T_q:q ∈ran(α)∖{T_1}. We claim that c_ℓ satisfies our induction hypothesis for ℓ. By the previous comments it will be sufficient to show Every 1-component in c_ℓ meeting x_ℓ + (g_1g_2N_2N_1∖g_1g_2N_0N_0) covers x_ℓ + (g_1g_2N_2N_1∖g_1g_2N_0N_0). To see this, we index C = C_1, …, C_r such that u< v implies α(C_u) < α(C_v), and consider the sequence of colourings c^z: z∈ [r] where c^1 = c_ℓ and each c^z is obtained from c^z-1 by switching the standard square T_α(C_z).Let us show by induction that for every z ∈ [r] there is an 1-component of c^z which covers ⋃_y ≤ z C_y. For z=1 the claim is clearly true. So, suppose z > 1. Since α(C_z) is minimal in {α(C_y)y ≥ z } it follows that e_q ∈⋃_y < z C_y for every q < α(C_z). Note that, since c_ℓ-1 = c' on x_ℓ + g_1g_2N_2N_1, it follows from the final claim in the Cap-off step that C_1 contains both e_1 and e_2N_1, and so α(C_z) ≠ 2N_1.Consider the standard square T_α(C_z). Since c_ℓ-1 = c' on x_ℓ + g_1g_2N_2N_1, by construction the edge `opposite' to e_α(C_z) in T_α(C_z), that is, e_α(C_z) + g_j, is in the same 1-component in c_ℓ-1 as e_α(C_z) - 1, and hence is contained in ⋃_y < z C_y.Therefore, by Lemma <ref>, after performing an (1,2)-colour switching at T_α(C_z), the 1-component in c^z contains ⋃_y ≤ z C_y.Hence, there is an 1-component of c_ℓ = c^r which covers ⋃_y ≤ r C_y, and so there is a unique 1-component of c_ℓ meeting x_ℓ + (g_1g_2N_2N_1∖g_1g_2N_0N_0) which covers it, establishing the claim.§.§ Combining cycles across different cosets of Δ In the third and final step we join the finite cycles covering each x_ℓ + (g_1g_2N_1N_1∖g_1g_2N_0N_0) into a single finite cycle, and then make one final switch to absorb this cycle into a double-ray. The resulting colouring will then satisfy the conditions of Lemma <ref>. [Combining cosets step]We can change c” from the previous lemma to an almost-standard colouring ĉ such that* ĉ=c”=c'=c on E(G[X]);* Some component in colour 1 covers P + g_1g_2N_1N_1. Recall that P = x_0, …, x_t is such that P^Δ = x_0 + Δ, …, x_t + Δ is a finite, graph-theoretic path in the Cayley graph of the quotient Γ / Δ with generating set S ∖{g_1,g_2}. Moreover, recall from Section <ref> that N_1 > N_0 was chosen so that for the initial colouring c there were t^2 many disjoint standard double-raysR = R^k_ℓ:1 ≤ k,ℓ≤ tsuch that for every ℓ, the double-rays in R^k_ℓ = y^k_ℓg_n(ℓ):k ∈ [t] are standard n(ℓ)-double-rays containing an edgee^k_ℓ=(y^k_ℓ,y^k_ℓ + g_n(ℓ)) ∈ E(R^k_ℓ) ∩ E(x_ℓ-1+Δ, x_ℓ + Δ)so that all T^k_ℓ = y^k_ℓg_1g_n(ℓ) are edge-disjoint (1,n(ℓ))-standard squares for the colouring c contained in the subgraph induced by P + g_1g_2N_1-3N_1-3 which have empty intersection with x_ℓ-1,x_ℓ + g_1g_2N_0N_0. However, since we only altered the (1,2)-subgraphs of G in Step <ref> and <ref>, it is clear that all these standard double-rays and standard squares for c remain standard also for the colourings c' and in particular c”.We claim that there exists a function k[t] → [t] ∪ such that iteratively switching T^k(ℓ)_ℓ (or not doing anything at all if k(ℓ) =) results in a sequence of colourings c” = c_0,c_1, …, c_t such that for each 0 ≤ℓ≤ t, *a single finite 1-component in c_ℓ covers x_0, …, x_ℓ + (g_1g_2N_1N_1∖g_1g_2N_0N_0),*for every k, every 1-component in c_ℓ meeting x_k + (g_1g_2N_1N_1∖g_1g_2N_0N_0) is a finite cycle covering x_k + (g_1g_2N_1N_1∖g_1g_2N_0N_0), and*every other 1-component, and all other components of all other colour classes in c_ℓ are double-rays. In Step <ref> we constructed a colouring c_0=c” for which properties (1)–(3) are satisfied. Now suppose that ℓ≥ 1, and that the colouring c_ℓ-1 obtained by switching the standard squares T^k(ℓ')_ℓ':ℓ' ∈ [ℓ-1] satisfies (1)–(3). By construction, each such standard square T^k(ℓ')_ℓ' is incident with the ray R^k(ℓ')_ℓ' and potentially one further n(ℓ')-component. But since we had reserved more that ℓ-1 different rays R^1_ℓ, …, R^t_ℓ, it follows that some ray R^k(ℓ)_ℓ remains a standard n(ℓ)-coloured component for c_ℓ-1. Both edges (y^k(ℓ)_ℓ,y^k(ℓ)_ℓ +g_i) and (y^k(ℓ)_ℓ+g_n(ℓ),y^k(ℓ)_ℓ+g_n(ℓ)+g_i) of T^k(ℓ)_ℓ are contained inx_ℓ-1,x_ℓ + (g_1g_2N_1N_1∖g_1g_2N_0N_0), and hence are, by assumption (<ref>), covered by finite 1-cycles in c_ℓ-1. If both edges lie in the same finite 1-cycle, there is nothing to do (and we redefine k(ℓ) :=, and let c_ℓ= c_ℓ-1). However, if they lie on different finite cycles, we perform a colour switching on the standard square T^k(ℓ)_ℓ and claim that the resulting c_ℓ is as required. ByLemma <ref>, the two finite 1-components merge into a single finite cycle, and so (<ref>) and (<ref>) are certainly satisfied for c_ℓ. To see (<ref>), we need to verify that T^k(ℓ)_ℓ is, when we perform the switching, safe. However, T^k(ℓ)_ℓ was chosen so that the edge (y^k(ℓ)_ℓ,y^k(ℓ)_ℓ + g_n(ℓ))∈ T^k(ℓ)_ℓ lies on a standard double-ray R=R^k(ℓ)_ℓ of c_ℓ-1. Also, by the inductive assumption (<ref>), the second n(ℓ)-coloured edge (y^k(ℓ)_ℓ +g_i,y^k(ℓ)_ℓ+g_i+g_n(ℓ)) ∈ T^k(ℓ)_ℓ lies on an n(l)-coloured double-ray R' in c_ℓ-1. If R and R' are distinct, then T^k(ℓ)_ℓ is safe, and if R=R' then, since R is a standard n(ℓ)-double-ray, Lemma <ref> implies that T^k(ℓ)_ℓ is safe. Hence c_ℓ satisfies (<ref>). This completes the induction step.Thus, by (<ref>) and (<ref>), we obtain an edge-colouring c_t for G such that a single finite 1-component covers P + (g_1g_2N_1N_1∖g_1g_2N_0N_0), and all other 1-components and all other components of other colour classes in c_t are double-rays. Furthermore, since every 1-component which meets P+g_1g_2N_0N_0 must meet P + (g_1g_2N_1N_1∖g_1g_2N_0N_0), it follows that the 1-component in fact covers P+g_1g_2N_0N_0. Moreover, since T^k(ℓ)_ℓ⊂ P + g_1g_2N_1-3N_1-3 for all ℓ∈ [t], it follows that c_t is standard on x_0 + g_1g_2N_1∞∖g_1g_2N_1-3N_1-3, and that it is standard outside of P + g_1g_2N_3N_1. Hence, the square xg_1g_2 with base point x = x_0 + (N_1-2)g_1 + N_1 g_2 is a standard (1,2)-square such that* the edge (x,x+g_1) lies on the finite 1-cycle of c_t,* the edge (x+g_2,x+g_2+g_1) lies on standard 1-double-ray x+g_2g_1 (lying completely outside of P + g_1g_2N_3N_1) of c_t, and* the edges (x,x+g_2) and (x+g_1,x+g_2+g_1) lie on distinct standard 2-double-rays xg_2 and x+g_1g_2⊆ x_0 + g_1g_2N_1∞∖g_1g_2N_1-3N_1-3.Therefore, we may perform a colour switching on xg_1g_2, which results, by Lemma <ref>, in an almost standard colouring of G such that a single 1-component covers P + g_1g_2N_1N_1, and hence X.§ HAMILTONIAN DECOMPOSITIONS OF PRODUCTSThe techniques from the previous section can also be applied to give us the following general result about Hamiltonian decompositions of products of graphs. * Suppose that R_ii ∈ I and S_jj ∈ J form decompositions of G and H into edge-disjoint Hamiltonian double-rays, where I,J may be finite or countably infinite. Note that, for each i ∈ I, j ∈ J, R_i □ S_j is a spanning subgraph of G □ H, and is isomorphic to the Cayley graph of (^2,+) with the standard generating set.Let π_GG □ H → G and π_HG □ H → H the projection maps from G □ H onto the respective coordinates. As our standard colouring for G □ H we take the mapcE(G □ H) → I ∪̇J, e ↦ i ife ∈π_G^-1(E(R_i)), j ife ∈π_H^-1(E(S_j)).Then each R_i □ S_j is 2-coloured (with colours i and j), and this colouring agrees with the standard colouring of C_^2 = G((^2,+), {(1,0),(0,1)}) from Section <ref>.We may suppose that V(G) == V(H). Fix a surjection f → I ∪ J such that every colour appears infinitely often.By starting with c_0 = c and applying Lemma <ref> recursively inside the spanning subgraphs R_f(k)□ S_1, if f(k) ∈ I, or inside R_1 □ S_f(k), for f(k) ∈ J, we find a sequence of edge-colourings c_kG □ H → I ∪ J and natural numbers M_k ≤ N_k < M_k+1 such that * c_k+1 agrees with c_k on the subgraph of G □ H induced by [0,M_k+1]^2,* there is a finite path D_k of colour f(k) in c_k covering [0,N_k]^2, and* M_k+1 is large enough such that D_k ⊂ [0,M_k+1]^2.To be precise, suppose we already have a finite path D_k of colour f(k) in c_k covering [0,N_k]^2, and at stage k+1 we have say f(k+1) ∈ I, and so we are considering R_f(k+1)□ S_1 ≅ C_^2. We choose * M_k+1 > N_k large enough such that D_k ⊂ [0,M_k+1]^2 ⊂ G□ H, and* N_k+1 > M_k+1 large enough such that Q_1=[0,N_k+1]^2 ⊂ G□ H contains all edges where c_k differs from the standard colouring c.Next, consider an isomorphism h R_f(k+1)□ S_1 ≅ C_^2. Pick a `square' Q_2 ⊂ R_f(k+1)□ S_1 with Q_1 ⊂ Q_2, i.e. a set Q_2 such that h restricted to Q_2 is an isomorphism to the subgraph of C_^2 induced by[-Ñ_k+1,Ñ_k+1]^2 ⊆^2 for some Ñ_k+1∈, and then apply Lemma <ref> to R_f(k+1)□ S_1 and Q_2 to obtain a finite path D_k+1 of colour f(k+1) in c_k+1 covering Q_2.It follows that the double-rays T_ii∈ I∪T_jj∈ J with T_ℓ = ⋃_k ∈ f^-1(ℓ)D_k give the desired decomposition of G □ H. § OPEN PROBLEMSAs mentioned in Section <ref>, the finitely generated abelian groups can be classified as the groups ℤ^n ⊕⊕_i=1^r ℤ_q_i, where n,r,q_1,…,q_r ∈. Theorem <ref> shows that Alspach's conjecture holds for every such group with n ≥ 2, as long as each generator has infinite order. The question however remains as to what can be said about Cayley graphs G(Γ,S) when S contains elements of finite order. Let Γ be an infinite, finitely-generated, one-ended abelian group and S be a generating set for Γ which contains elements of finite order. Show that G(Γ,S) has a Hamilton decomposition. Alspach's conjecture has also been shown to hold when n=1, r=0, and the generating set S has size 2, by Bryant, Herke, Maenhaut and Webb <cit.>. In a paper in preparation <cit.>, the first two authors consider the general case when n=1 and the underlying Cayley graph is 4-regular. Since the Cayley graph is 2-ended, it can happen for parity reasons that no Hamilton decomposition exists. However,this is the only obstruction, and in all other cases the Cayley graphs have a Hamilton decomposition. Together with the result of Bermond, Favaron and Maheo <cit.> for finite abelian groups, and the case Γ≅ (^2,+) of Theorem <ref>, this fully characterises the 4-regular connected Cayley graphs of finite abelian groups which have Hamilton decompositions. A natural next step would be to consider the case of 6-regular Cayley graphs.Let Γ be a finitely generated abelian group and let S be a generating set of Γ such that C(Γ,S) is 6-regular. Characterise the pairs (Γ,S) such that G(Γ,S) has a decomposition into spanning double-rays. plain | http://arxiv.org/abs/1709.09463v1 | {
"authors": [
"Joshua Erde",
"Florian Lehner",
"Max Pitz"
],
"categories": [
"math.CO",
"05C45, 05C63, 20K99"
],
"primary_category": "math.CO",
"published": "20170927120158",
"title": "Hamilton decompositions of one-ended Cayley graphs"
} |
Millisecond Pulsars, their Evolution and Applications R. N. ManchesterCSIRO Astronomy and Space Science, PO Box 76, Epping NSW 1710, [email protected] ============================================================================================================================= Millisecond pulsars (MSPs) are short-period pulsars that are distinguished from “normal” pulsars, not only by their short period, but also by their very small spin-down rates and high probability of being in a binary system. These properties are consistent with MSPs having a different evolutionary history to normal pulsars, viz., neutron-star formation in an evolving binary system and spin-up due to accretion from the binary companion. Their very stable periods make MSPs nearly ideal probes of a wide variety of astrophysical phenomena. For example, they have been used to detect planets around pulsars, to test the accuracy of gravitational theories, to set limits on the low-frequency gravitational-wave background in the Universe, and to establish pulsar-based timescales that rival the best atomic-clock timescales in long-term stability. MSPs also provide a window into stellar and binary evolution, often suggesting exotic pathways to the observed systems. The X-ray accretion-powered MSPs, and especially those that transition between an accreting X-ray MSP and a non-accreting radio MSP, give important insight into the physics of accretion on to highly magnetised neutron stars.pulsars: general—stars: evolution—gravitation § INTRODUCTIONThe first pulsars discovered <cit.> had pulse periods between 0.25 s and 1.3 s. Up until 1982, most of the 300 or so known pulsars had similar periods, with the notable exceptions of the Crab pulsar <cit.>, the Vela pulsar <cit.>, and the Hulse-Taylor binary pulsar <cit.>. These had periods of 33 ms, 89 ms and 59 ms respectively. The Crab and Vela pulsars had rapid slow-down rates showing that were very young and almost certainly associated with their respective supernova remnants. Discovery and timing of these and other pulsars led to the conclusion that pulsars are rotating neutron stars, born in supernova explosions with periods 10 – 20 ms and gradually slowing down to periods of order 1 s over millions of years <cit.>.This cosy consensus was somewhat shaken in 1982 by <cit.> announcing the discovery of the first “millisecond pulsar” (MSP), PSR B1937+21, with the amazingly short period of just 1.558 ms. This pulsar was found in September 1982 at Arecibo Observatory in a very high time-resolution search of the enigmatic steep-spectrum compact source 4C21.53W. This source also showed strong interstellar scintillation and strong linear polarisation. All of these properties suggested an underlying pulsar, but previous searches with lower time resolution (including one by the author) had failed to reveal any periodicity.Within days of the announcement of the discovery of PSR B1937+21, <cit.> and <cit.> proposed that the MSP resulted from the “recycling” of an old, slowly rotating and probably dead neutron star through accretion from a low-mass companion. The mass transferred from the companion also carries angular momentum from the orbit to the neutron star, spinning it up and reactivating the pulsar emission process. This idea built on earlier work by <cit.> and <cit.> in which the relatively short periodand large age of the original binary pulsar, PSR B1913+16, <cit.> and maybe some other binary pulsars <cit.>, were explained by invoking such an accretion process.[For the purposes of this article, we define an MSP to be a pulsar with period less than 100 ms and period derivative less than 10^-17. The somewhat generous period limit allows recycled pulsars such as PSR B1913+16, to be included and the period-derivative limit excludes young pulsars such as the Crab and Vela pulsars.] The next two MSPs to be discovered, PSR B1953+29 <cit.> and PSR B1855+09 <cit.>, were both members of a binary system, consistent with the recycling idea. At the time, only five of the more than 400 “normal” (non-millisecond) pulsars known were binary, compared to three of the four known MSPs. At first glance, the absence of a binary companion for PSR B1937+21 was surprising, but it was quickly recognised, even in the Backer et al. discovery paper and by <cit.>, that this could be explained by disruption of the binary by asymmetyric mass loss in an accretion-induced collapse of the likely white-dwarf remnant of the companion star. With the later discovery of the “black widow” pulsar, PSR B1957+20 <cit.>, it was realised that complete ablation of the companion star was another viable mechanism for formation of solitary MSPs. The first MSP in a globlular cluster, PSR B1821-24A in M28, was discovered in 1987 by <cit.>. This set off an avalanche of discoveries of MSPs in globular clusters, with 21 MSPs being discovered in globular clusters by 1993, with eight in both M15 <cit.> and 47 Tucanae <cit.>. There are now 145 globular-cluster pulsars known, all but a handful of them MSPs. Clearly globular clusters are efficient factories for the production of MSPs, see <cit.>.Another important development in MSP research was the discovery that MSPs are relatively strong emitters of pulsed γ-rays. Although predicted by Srinivasan in 1990 <cit.>, the first observational evidence was the tentative detection by <cit.> of pulsed γ-ray emission from PSR J0218+4232, a known binary radio MSP with a period of 2.3 ms, in EGRET data, later confirmed as one of eight MSPs detected with the Fermi Gamma-ray Space Telescope <cit.>. About 25 previously known radio MSPs have now been detected as γ-ray pulsars by folding the Fermi data at the known radio period <cit.>.It was soon recognised that γ-ray detected pulsars had rather unusual γ-ray properties compared to other classes of γ-ray sources, for example, they are steady emitters over long intervals and have characteristic power-law spectra with an exponential cutoff at a few GeV <cit.>. Radio searches of previously unidentified Fermi sources with these properties have been extraordinarily successful in uncovering MSPs, with about 50 so far identified, e.g., <cit.>. In about half of these, γ-ray pulsations have subsequently been detected by folding the γ-ray data with the precise period ephemeris from the radio observations. One particularly interesting aspect of these MSP discoveries is that many are in short-period binary orbits (P_b1 day) with low-mass companions (M_c0.3 M_⊙) and exhibit radio eclipses due to gas ablated from the companion, forming black widow or redback systems[The names “black widow” and “redback” were coined by<cit.> and<cit.>, respectivly, after the rather ungracious female spiders that have a tendency to consume their much smaller male companion after mating. The pulsar analogy is that, in these close binary systems, ablation of the companion star by the pulsar wind may destroy it, with no thanks for the fact that earlier accretion from the companion star gave the pulsar its rapid spin and energetic wind.]In parallel with these developments, the wide-field radio searches for pulsars continued, discovering many MSPs. Particularly successful were the Parkes Swinburne mid-latitude survey (14 MSPs, see <cit.>), the Parkes Multibeam Survey (28 MSPs, <cit.>), the Parkes “HTRU” surveys (28 MSPs, <cit.>), the Arecibo “PALFA” survey (21 MSPs, <cit.>) and the Green Bank low-frequency surveys (15 MSPs, <cit.>). Analysis or re-analysis of many of these surveys is continuing and more discoveries can be expected.These various searches have revealed a total of 255 MSPs, roughly 10% of the known pulsar population. Of these, more than 180 are members of binary systems, with orbital periods ranging from 1.5 hours (PSR J1311-3430) to nearly 700 days (PSR J0407+1607)[Other longer-period binary systems are known, but in these cases the pulsar is probably not recycled.]. By comparison, only 24 or about 1% of the normal pulsar population, are binary. Figure <ref> illustrates the distinct properties of MSPs compared to normal pulsars, viz., much shorter period, very small slow-down rate and predominance of binary membership. If one assumes period slow-down due to emission of magnetic-dipole radiation (electromagnetic waves with a frequency equal to the pulsar spin frequency) or acceleration of pulsar winds in a dipole magnetic field, the characteristic age τ_c and surface-dipole field strength B_s can be estimated. τ_c is a reasonable estimator of the true age of normal pulsars, but only an upper limit on the true age of MSPs since it assumes that the pulsar was born with infinite spin frequency with regular magnetic-dipole slow-down after that. MSPs have a much more complicated spin history, see <cit.>.One of the main reasons that MSPs are so important is that their spin periods are extraordinarily stable. This enables their use as “celestial clocks” in a wide variety of applications. Studies of pulsar “timing noise”, e.g., <cit.>, show that MSP periods are typically more than three orders of magnitude more stable than those of normal pulsars, and for the best cases, e.g., PSR J1909-3744, see, e.g., <cit.>, have a stability rivalling that of the best atomic clocks. This great period stability may be related to the very weak external magnetic fields of MSPs (Figure <ref>).Despite having been proposed as early as 1969 <cit.>, the issue of magnetic field decay in pulsars is not yet resolved, with recent population studies of normal pulsars, e.g.,<cit.> and the discovery of low-field but young pulsars, e.g., PSR J1852+0040 which is associated with the supernova remnant Kesteven 79 but has a dipole surface field of only 3× 10^10 G <cit.>, suggesting that decay of normal pulsar magnetic fields is not required. If this remains true over the ∼ 10^9 yr timescale for recycling, then the low fields of MSPs must be a by-product of the recycling process, for example, through burial of the field by accreted material <cit.>. On the other hand, if fields do decay on timescales of 10^9 yr or less, then the problem becomes accounting for the low but finite field strengths of MSPs. An innovative solution to this problem in which field decay is related to the variable spin-down rate across the whole history of the present-day MSP was presented by <cit.>.In <ref> we discuss the use of MSPs as probes of binary motion, including the detection of planetary companions and the investigations of relativistic perturbations leading to tests of gravitational theories. The search for nanoHertz gravitational waves using pulsar timing arrays (PTAs) is described in <ref> and the use of PTA data sets to establish a pulsar-based timescale is dicussed in <ref>. The different classes of binary and millisecond pulsars and their formation from X-ray binary systems through the recycling process are discussed in <ref>. In the concluding section (<ref>) we highlight the rich research fields opened up by the discovery of binary and millisecond pulsars and the important contributions of Srinivasan to many aspects of this work.§ MSPS AS PROBES OF BINARY MOTION §.§ Planets around pulsarsThe first detection of a planet around a star other than the Sun was made by <cit.> who discovered two planets orbiting PSR B1257+12, an MSP with a pulse period of 6.2 ms. The planets have orbital periods of about 66 and 98 days, are in circular orbits of radii 0.36 and 0.47 AU and have masses of 3.4/sin i and 2.8/sin i Earth masses, respectively, where i is the (unknown) orbital inclination. In 1994,<cit.> announced the discovery of a third planet in the system with a mass close to that of the Moon and an orbital period of approximately 25 days. This remains (by a wide margin) the least massive planet known for any star, and its detection is an excellent demonstration of the power of pulsar timing. Figure <ref> shows the timing signatures of the three planets. The 1994 paper also announced the detection of predicted small perturbations in the orbital periods of the two larger planets. This observation unequivocally confirmed that the timing modulations observed in this pulsar are caused by orbiting planetary bodies.PSR B1620-26, located in the globular cluster M4, has a pulse period of about 11 ms and a binary companion of mass about 0.3 M_⊙ in a 191-day, almost circular, orbit <cit.>. Continued timing observations showed evidence for additional perturbations to the pulsar period that could result from the presence of a third body, possibly of planetary mass <cit.>. Analysis of an 11-year timing dataset by <cit.> showed that the results were consistent with a Jupiter-mass planet with an orbital period of the order of 100 years. <cit.> used Hubble Space Telescope observations to determine a mass for the wide-dwarf companion which in turn fixed the orbital inclination and constrained the outer planetary companion to have an orbital radius of about 23 AU and mass about 2.5 Jupiter masses. The third pulsar known to have a planetary-mass companion is PSR J1719-1438, an MSP with a period of 5.7 ms <cit.>. This system is somewhat different to those described above in that it is more akin to the binary systems (often known as “black-widow” systems) which have very low-mass companions, e.g., PSR J0636+5129 <cit.>, but more extreme. In most black widow systems, the radio emission is periodically eclipsed and they are believed to be systems in which the companion is a stellar core being ablated by the pulsar wind. They can have companion masses as low as 0.007 M_⊙ (about 7 Jupiter masses). PSR J1719-1438 does not show eclipses and appears to be an ex-black-widow system in which the companion narrowly survived the wind-blasting with a mass approximately equal to that of Jupiter. <cit.> make a case for the companion having a very high density, greater than 23 g cm^-3, probably composed mostly of carbon, leading to its moniker “the diamond planet”.Although all MSPs are believed to have passed through an evolutionary phase where they had an accretion disk, it is rare for this accretion disk to spawn a planetary system. The precise timing of MSPs limits the number of planetary systems to those described above, only about 1% of the population, although the relatively large intrinsic timing noise of PSR B1937+21 can be intepreted as resulting from the perturbations due to an asteroid belt surrounding the pulsar <cit.>.§.§ Tests of gravitational theoriesThe discovery of the first binary pulsar, PSR B1913+16, by <cit.> marked a turning point in pulsar science. With its short orbital period (7.75 h) and high orbital eccentricity (0.617), it was immediately clear that precise timing of this pulsar would allow detection of relativistic perturbations in the orbital parameters. As mentioned above (<ref>), the short period (59 ms) and large characteristic age (10^8 yr) of PSR B1913+16 indicated that this pulsar was recycled. What turned out to be unusual was that the Keplerian orbital parameters showed that the companion was massive (minimum mass about 0.86 M_⊙) and probably another neutron star. The two largest of the so-called “post-Keplerian” parameters[See <cit.> for a description of the post-Keplerian parameterisation.] periastron precession (ω̇) and relativitic time dilation, usually described by the parameter γ, were detected at close to their predicted values within a few years <cit.>. In Einstein's general theory of relativity (GR), these two parameters depend on just the masses of the two stars plus the Keplerian parameters (which were well known). Consequently, the observation of these two parameters allowed the masses to be derived. Both were close to 1.4 M_⊙, confirming that PSR B1913+16 was a member of a double-neutron-star system. Given these two masses, other relativistic parameters could be predicted including, most importantly, orbital decay due to emission of gravitational waves from the system. This too was observed by <cit.>, fully consistent with the GR prediction.Continued observations of this system have refined these parameters, with the latest results (Figure <ref>) showing that the orbital decay term is in agreement with the GR prediction (after compensation for differential acceleration of the PSR B1913+16 system and the solar system in the Galactic gravitational field) to better than 0.2%. These observations also allowed detection of the relativistic Shapiro delay parameters, r and s, which depend on the orbit inclination and companion mass, for the first time in this system. The measured parameters are consistent with the GR predictions although not very constraining since the orbital inclination is close to 45^∘. The discovery of the Double Pulsar system, PSR J0737-3039A/B, at Parkes <cit.> made possible even more stringent tests of GR. Its orbital period is only 2.4 h and the predicted periastron advance, 16^∘.9 yr^-1, is more than four times that of PSR B1913+16. Also, its orbital plane is almost exactly edge-on to us, making the Shapiro delay large and easily measurable. Finally, the companion star was observed as a pulsar (PSR J0737-3039B) with a long period (2.8 s) but a much younger age than the A pulsar. Not only did this still unique discovery allow a direct measurement of the mass ratio of the two stars, it was fully consistent with the idea that the A pulsar was (partially) recycled prior to the explosion of the companion star that formed the B pulsar.Continued timing observations using the Parkes, Green Bank and Jodrell Bank telescopes have resulted in the measurement of five post-Keplerian parameters, several to unprecedented levels of precision: ω̇ to 0.004%, γ to 0.6%, the Shapiro delay terms r and s to 5% and 0.03% respectively and the orbital period decay Ṗ_b to 1.4% <cit.>. Also, the mass ratio R was measured to 0.1%. The s ≡sin i measurement implies an orbital inclination angle of 88^∘.7 ± 0^∘.7. This is sufficiently close to edge-on that the radiation from the A pulsar is eclipsed by the magnetosphere of the B pulsar for just 30 s per orbit. Remarkably, high time-resolution observations made with the Green Bank Telescope showed that the eclipse is modulated at the rotation period of star B <cit.>. Modelling of this eclipse pattern by Lyutikov & Thompson <cit.> allowed determination of the system geometry, including showing that the rotation axis of B was inclined to the orbit normal by about 60^∘. Even more remarkably, observations of the eclipse pattern over a four-year data span gave a measurement of a sixth post-Keplerian paramter, the rate of geodetic precession, (4^∘.8± 0^∘.7) yr^-1, consistent with the GR prediction <cit.>.As shown in Figure <ref>, all of these measurements can be plotted on the so-called “mass-mass” diagram, a plot of companion mass versus pulsar mass. Within the framework of GR, each post-Keplerian measurement defines a zone on this plot within which the two masses must lie. The Newtonian mass function and mass ratio also define allowed regions. If GR is an accurate theory of gravity, there will be a region on this diagram consistent with all constraints, defining the masses of the two stars. Although it is difficult to see, even on the inset, there is such a region on this plot. These updated results verify that GR accurately descibes the motion of the stars in the strong gravitational fields of this binary system with a precision of better than 0.02% <cit.>.Although GR has been incredibly successful as a theory of relativistic gravity, passing every test so far with flying colours, it is by no means the only possible theory of gravity. Departures from GR and the equivalence principles that it is based on can be quantified in a theory-independent way using the “Parameterised post-Newtonian” (PPN) parameters, see <cit.>. Pulsars provide a variety of tests that limit various combinations of these parameters, see <cit.>. Here, we describe just one such test: the effect of “self-gravitation” on the acceleration of objects in an external gravitational field, a test of the Strong Equivalence Principle (SEP). This test was first applied to solar-system dynamics by <cit.>, looking for a “polarisation” of the Moon's orbit in the direction of the solar gravitational field. The test depends on the different gravitational self-energy of the two binary components and so can be tested using binary pulsars with low-mass companions and very low eccentricity with the Galactic gravitational field as the polarising agent. There is a large sample of such systems known and <cit.> analysed these to put a limit on the PPN parameter Δ, which measures the ratio of the gravitational and inertial masses, effectively of the neutron star, of 4.3× 10^-3.In 2014, <cit.> announced the discovery of the fascinating stellar triple system containing the pulsar J0337+1715, an MSP with pulse period of 2.73 ms. The pulsar is in a relatively tight orbit, orbital period 1.63 days, with a white dwarf of mass about 0.197 M_⊙. On a nearly co-planar orbit about this inner system, there is a second white dwarf of mass about 0.41 M_⊙ and orbital period about 327 days. The inner white dwarf has been optically identified, leading to an estimate of the distance to the system, about 1300 pc. Analysis of the complex interactions between the three stars, illustrated in Figure <ref>, enabled determination of the component masses (the neutron star has a mass close to 1.44 M_⊙) and the precise orbital inclinations for the two systems which are equal to within 0^∘.01 and close to 39^∘.2.An interesting aspect of this system is that it will provide a much more sensitive test of the SEP than the analysis of wide asymmetric binaries described above, through a potential induced eccentricity of the inner orbit in the gravitational field of the outer white dwarf <cit.>. The gravitational field of the outer star at the inner orbit is about six orders of magnitude greater than the Galactic gravitational field in the local neighbourhood, leading to the high expected sensitivity of the test. However, observations over several orbits of the outer white dwarf will be required to separate any SEP-violation effect from the intrinsic eccentricity of the inner orbit, about 7× 10^-4. § THE SEARCH FOR GRAVITATIONAL WAVESDirect detection of the gravitational waves (GWs) predicted by Einstein in 1916 <cit.> has been a major scientific goal over many decades. With the remarkable discovery of a GW burst from two coalescing 30-M_⊙ black holes by the LIGO-Virgo consortium in September, 2015 <cit.>, this goal was achieved. Laser-interferometer systems such as LIGO are sensitive to GWs with frequencies in the range 10 – 1000 Hz, the frequencies expected from coalescing stellar-mass objects. The observed periods of pulsars will also be perturbed by GWs passing through the Galaxy. But because data spans of many years are required to reach the highest precision, pulsar detectors are sensitive to much lower frequencies, in the nanoHertz range. Likely astrophysical sources of such waves are very different - most probably super-massive black-hole binary (SMBHB) systems in the cores of distant galaxies. Studies of such waves are therefore complementary to investigations using laser-interferometer signals.Pulsar GW-detection efforts depend on the great stability of MSP periods. However, even the most stable pulsars can in principle have intrinsic period irregularities, so observations of an ensemble of MSPs, called a Pulsar Timing Array (PTA), is needed to detect GWs. The detection method is based on searching for correlated signals among the pulsars of a PTA which have the quadrupolar spatial signature expected for GWs <cit.>. Other correlated signals, such as those produced by irregularities in the reference time standard, can also be detected and distinguished from GWs by the different spatial response pattern <cit.>.Up to now, there has been no positive detection of nanoHertz GWs by a PTA. However, limits on the strength of such a signal are beginning to place interesting constraints on the source population and properties. Although correlated signals among the pulsars of a PTA must be observed to claim a detection, a limit on the strength of nanoHertz GWs in the Galaxy can be obtained by placing limits on the low-frequency signals in the modulation spectra of just the few best pulsars in a PTA. The best such limit so far comes from analysis of data from the Parkes Pulsar Timing Array (PPTA) <cit.>. <cit.> used 10 cm (3 GHz) observations of four of the most stable PPTA pulsars to set a limit on the characteristic strain amplitude h_c < 3× 10^-15 at a GW frequency of 0.2 cycles per year (6.3 nHz) of a power-law GW background (assumed spectral index -2/3) in the Galaxy. This corresponds to an energy density of GWs at this frequency that is a fraction 2.3× 10^-10 of the closure energy density of the Universe. As Figure <ref> shows, the new limit rules out a number of models for SMBH evolution in galaxies with high confidence.In their paper, <cit.> identified enviromental effects affecting the late evolution of SMBHBs as the most likely reason for the current non-detection. If the SMBHB system loses energy to surrounding stars or gas in its late evolution, it will pass through this evolutionary phase more quickly than if GW emission were the sole energy-loss process. This means that less energy will be emitted in the form of GWs, thereby lowering the observed GW signal, particularly at the lower observed frequencies (i.e., periods of decades) <cit.>. However, the evidently low amplitude of nanoHertz GW in the Galaxy can have other explanations. For example, the number density and/or merger rate of SMBHs in the early Universe may be less than assumed, see, e.g., <cit.>, or eccentricities of merging SMBHB may be relatively large <cit.>. Both of these would have the effect of reducing the GW amplitude at the low end of the PTA band where the sensitivity is greatest.Clearly, to achieve a detection of nanoHertz GWs and to begin to explore their properties, increased PTA sensitivities are required. As <cit.> pointed out, the most effective way to increase the senstivity of a PTA is to increase the number of pulsars observed with high timing precision. A start on this is being made by combining the data sets of the three existing PTAs to form the International Pulsar Timing Array (IPTA) <cit.>. Observations with future telescopes such as FAST <cit.> and the SKA <cit.>, combined with existing data sets,will almost certainly result in a detection and open up an era of nanoHertz-GW astronomy and astrophysics.§ PULSAR-BASED TIMESCALESAs mentioned in the previous section, PTA data sets can also be usedto investigate irregularities in the reference atomic timescales and therefore to establish a pulsar-based timescale. International reference timescales are currently based on a large number of atomic frequency standards distributed across the world at many different time and frequency standard laboratories, see <cit.>. These measurements are collated at the Bureau International des Poids et Mesures (BIPM)[www.bipm.org] in Paris to produce the timescale TT(TAI) which is a continuous timescale with a unit that is kept as close as possible to the SI second by reference to a few primary caesium standards. Although atomic frequency standards are improving all the time and have reached incredible stabilities, of the order of a part in 10^18 averaged over an hour or so for some optical lattice clocks, see <cit.>, the long-term stability, over years and decades, of these clocks is unknown. MSPs are highly stable clocks over intervals of years and hence are well suited as an alternate reference. Irregularities in the reference timescale would result in inverse irregularities in the apparent period of all pulsars in a PTA. This “common-mode” signal is relatively easy to identify and separate from other perturbations in the pulsar periods <cit.>. Figure <ref> shows the common-mode signal derived from a reanalysis of the 20 PPTA pulsars used by <cit.> with TT(TAI) as a reference timescale. Because intrinsic pulsar periods and slow-down rates are unknown and must be solved for as part of the analysis, the pulsar timescale is insensitive to linear and quadratic variations in the reference timescale. TT(TAI) is “real-time” and also contains known corrections to its rate. The BIPM regularly reanalyses the atomic clock data to derive an improved timescale TT(BIPMxx) where xx signifies the year of reanalysis – this is believed to be the most accurate long-term timescale available to us. In Figure <ref>, the line shows the quadratic-subtracted difference between TT(BIPM11) and TT(TAI). Within the uncertainties, the pulsar timescale accurately follows the known differences between TT(BIPM11) and TT(TAI), demonstrating both that TT(BIPM11) is a more uniform timescale than TT(TAI) and that pulsar timescales can have comparable precision to the best international atomic timescales over long time intervals. While current realisations of pulsar timescales are not quite at the same level of stability as the best atomic timescales, they are nevertheless valuable as an independent check on the long-term stability of the atomic timescales. Firstly, they are completely independent of terrestrial timescales and the terrestrial environment. Secondly, they are based on entirely different physics, rotation of a massive object, compared to the quantum-based atomic timescales. Thirdly, the vast majority of MSPs will continue spinning in a predictable way for billions of years, whereas the lifetime of an atomic frequency standard is typically of the order of a decade. In view of these points, the continued development and improvement of pulsar timescales is certainly desirable and will happen as PTAs improve. It is a nice thought that, in some sense, pulsar timescales return time-keeping to its astronomical roots.§ BINARY AND STELLAR EVOLUTIONAs mentioned in the Introduction, the issue of how binary and millisecond pulsars evolved to their present state has been a topic of great interest right from the discovery of the first binary pulsar. In the intervening 40 years or so the topic has become even more fascinating with the discovery of MSPs in globular clusters, triple systems, pulsars in orbit with very low-mass companions and eclipsing systems. This diversity is illustrated in Figure <ref> which shows the median companion mass (computed from the binary mass function with assumed orbit inclination i=60^∘ and pulsar mass 1.35 M_⊙) of pulsar binary systems as functions of the pulsar period P and the orbital period P_b.The most striking aspect of these plots is that systems with very low-mass companions (<0.08 M_⊙) are concentrated at both short pulsar periods (mostly P<5 ms) and short orbital periods (P_b<1 day). Most of these are black-widow systems, i.e., a non-degenerate or partially degenerate stellar core that is being irradiated and ablated by the pulsar wind. The ablated gas often results in eclipses of the pulsar radio emission when the companion is close to inferior conjunction, i.e., between the pulsar and us. PSR B1957+20, discovered in 1988 <cit.>, is the prototype for these systems. It has a very low-mass companion (median mass 0.024 M_⊙) in a 9.17 h circular orbit around an MSP with period 1.607 ms. The pulsar is eclipsed for about 50 min each orbit with additional dispersive delays before and after the eclipse as the pulsar enters and leaves the gaseous wind from the companion. Initially, most of these PSR B1957+20-like systems were associated with globular clusters, but in recent years, largely thanks to the great success of pulsar searches in the direction of unidentified Fermi γ-ray sources <cit.>, the population is roughly evenly divided between globular clusters and the Galactic field (Figure <ref>). Another population of binary systems with very short orbital periods (P_b<1 day) but somewhat larger companion masses, mostly between 0.1 and 0.3 M_⊙, can be identified on the m_c – P_b plot. These are the other type of “spider” pulsar known as redbacks <cit.>. The companion is a non-degenerate star that fills its Roche lobe and is losing mass, partly as a result of heating by the pulsar wind, resulting in extended and variable eclipses of the pulsar radio signal – see <cit.> for a discussion of the formation processes of black-widow and redback eclipsing binaries. The prototype redback (with hindsight) was PSR B1744-24A, the first pulsar discovered in the globular cluster Terzan 5 <cit.>, which has the very short orbital period of 1.81 h, a median companion mass of 0.10 M_⊙ and is eclipsed for intervals that vary from orbit to orbit from about 30% of the orbital period to all of it. The two next similar systems discovered (PSRs J0024-7204W <cit.> and J1740-5340A <cit.> also lie in globular clusters (47 Tucanae and NGC 6397 respectively), suggesting that exchange interactions in clusters is an important if not the only formation route for these systems (see <cit.> for a discussion of stellar interactions in globular clusters).However, the discovery of thebinary pulsar PSR J1023+0038 by <cit.> changed this picture. This system has a 1.69-ms pulsar in a 4.75-h circular orbit, with a companion of mass about 0.2 M_⊙, and shows deep and variable eclipses covering about 30% of the period at frequencies about 1.4 GHz and more at lower frequencies. These properties indicate a non-degenerate companion and that the system is a member of the redback group. With Galactic coordinates of l=243^∘.5, b=45^∘.8 and a distance of about 1.4 kpc, the PSR J1023+0038 system lies far from any globular cluster and, hence, an origin in a cluster is very unlikely. This showed that such systems could result from evolution of a binary system without the need to invoke exchange interactions; possible evolutionary paths are discussed by <cit.>. Since then, the Fermi-related searches <cit.> have uncovered many more redback systems in the Galactic field, such that they now out-number the globular-cluster redbacks (see right panel of Figure <ref>), reinforcing this conclusion.Compared to PSR B1913+16, the next two binary pulsars discovered, PSR B0820+02 <cit.> and PSR 0655+64 <cit.> had very different properties. PSR B0820+08 has a relatively long pulsar period, 0.865 s, a very long orbital period, about 3.3 years, a median companion mass of 0.22 M_⊙ and no sign of eclipses. PSR B0655+64 has a pulse period of 0.196 s and is in a circular orbit with a period of just over one day, giving a median companion mass of 0.79 M_⊙. Again, there is no sign of any eclipse. These properties suggest that the companions in these two systems are compact degenerate stars and this was later confirmed by optical identifications of white dwarf companions <cit.>. These were the first of many discoveries of pulsars with white dwarf companions, some believed (or known) to be helium white dwarfs with masses in the range 0.1 – 0.3 M_⊙, and others with more massive carbon-oxygen (CO) or oxygen-neon-magnesium (ONeMg) white dwarf companions. Between them these white-dwarf systems account for about 60% of the known binary pulsars, with about three-quarters of them having He white dwarf companions. As Figure <ref> illustrates, many of them have highly recycled pulsars (pulse period of a few milliseconds or less) and most have relatively long orbital periods. The lower-mass He white dwarf systems are believed to have evolved from low-mass X-ray binary (LMXB) systems whereas the more massive systems with a CO white dwarf are probably formed in intermediate-mass X-ray binary (IMXB) systems - see <cit.> for a detailed discussion of these evolutionary processes.Another important group of pulsars is those having neutron star companions. These double-neutron-star systems have intermediate pulsar periods, most between 20 ms and 100 ms, indicating a short accretion and spin-up phase. This is consistent with the faster evolution of the more massive stars needed to form neutron stars by collapse of the stellar core at the end-point of their evolution. Their orbital periods range between 0.1 day (for the Double Pulsar) and several tens of days and they have relatively high eccentricities (Figure <ref>). Only one, B2127+11C, lies in a globular cluster (M15).Finally, we have the binary systems with massive main-sequence companions. The prototype is PSR B1259-63, a 47.7 ms pulsar in a 3.4 year highly eccentric orbit (e ∼ 0.808) around a 20 M_⊙ Be star LS 2883 <cit.>. The pulsar is eclipsed in the radio for about 100 days around periastron and emits high-energy (X-ray and γ-ray) unpulsed emission <cit.> as it passes through the circumstellar disk of the Be star. Only a handful of similar systems are known and they are shown in the upper-right region of both panels in Figure <ref>.The idea that binary (and single) MSPs get their short periods and low magnetic fields by a recycling process occuring in X-ray binary systems, was first discussed by <cit.> and <cit.> in the context of PSR B1913+16, and then extended to account for the properties of the first MSP, PSR B1937+21, by<cit.> and <cit.>. Since that time, the field has become incredibly richer with the discovery of the diverse types of binary and triple systems described in the preceding sections. This diversity can be accommodated within the recycling model by invoking different initial conditions (component masses, orbital periods etc.) and different environments, e.g., globular clusters or Galactic field, for the progenitor binary systems - see <cit.> and <cit.> for extensive reviews of the evolution of compact X-ray binary systems.Many observations have supported the recycling hypothesis. For example, the properties of the second-formed B pulsar in the Double Pulsar system are completely in accord with expectations from the recycling model <cit.>. But much more direct evidence has recently been found with the discovery of the “transitional” systems, PSR J1023+0038 <cit.>, PSR J1824-2452I (M28I) <cit.> and PSR J1227-4853 <cit.>. In the first case, a relatively bright MSP with an orbital period of 0.198 days and wide eclipses, was discovered in a wide-area 350-MHz survey at Green Bank. Within the uncertainties, the MSP is coincident with a previously known solar-type star that showed brightness variations with a 0.198-day periodicity <cit.>, thereby clinching the identification. This combination of properties identified the system as a redback, with the solar-type star being the accretion donor in the earlier X-ray accretion phase. What makes this system especially interesting (besides not being in a globular cluster) is that optical spectra taken in 2000 – 2001 showed a very different blue and double-lined spectrum characteristic of an accretion disk <cit.>. This suggests that the system is bistable, oscillating between an accretion IMXB phase and a MSP redback phase. This was confirmed by observations of a sudden quenching of the pulsar radio emission and a coincident return to the optical and X-ray behaviour expected for an IMXB in 2013 <cit.>. Also, a coincident sudden increase in the γ-ray flux from the source was reported by <cit.>.In 2013, <cit.> also reported clear evidence for the connection between recycled MSPs and X-ray binary systems. In April, 2013, XMM Newton observations revealed coherent pulsations with a period of 3.931 ms in an X-ray transient source detected a few weeks earlier by the X-ray satellite INTEGRAL, marking this as an accretion-powered MSP similar to SAX J1808.4-3658 <cit.>. It was quickly realised that this is the same source as the binary radio pulsar J1824-2452I in M28. Archival observations from 2008 showed that the radio pulsar was detected just two months before a Chandra X-ray detection, showing that the change of state was rapid. This was confirmed by X-ray and radio observations in mid-2013 showing that the system swapped between X-ray and radio-pulsar states on timescales as short as a few days. The PSR J1227-4853 system appears to be similar to that of PSR J1023+0038. Both are members of systems in which there was a rapid transition between an accreting binary X-ray source and a radio MSP. The binary X-ray source, XSS J12270-4859, was observed to transition to a quiet state with none of the optical or X-ray signatures of an accretion disk in late 2012 <cit.>. A radio search at the X-ray position using the Giant Metrewave Radio Telescope (GMRT) in India and subsequent timing observations using the GMRT and the Parkes 64-m radio telescope revealed a binary eclipsing 1.686 ms pulsar with the same orbital period as the X-ray source, 6.91 days <cit.>. The binary parameters are consistent with a companion mass in the range 0.17 – 0.46 M⊙, suggesting that the system is a member of the redback group.These observations of binary systems which transition between X-ray accretion-powered systems to rotation-powered radio MSPs provide the best evidence yet that MSPs are indeed old and slowly rotating neutron stars that have been spun up, or recycled, by accretion in an X-ray binary system. Observations of binary MSPs, including those detected in X-ray accreting and burst systems, also provide important constraints on the structure and the equation of state for neutron stars and other hybrid or quark stars (for recent reviews see <cit.> and <cit.>). The most important constraints come from the large pulsar masses implied by either Shapiro delay measurements, as for PSR J1614-2230 <cit.> and PSR J1946+3417 <cit.>, or by optical identifcation and mass measurement of the companion star as for PSR J0348+0432 <cit.> and PSR J1012+0507 <cit.>. These four systems have estimated pulsar masses of 1.928± 0.017 M_⊙, 1.828± 0.022 M_⊙, 2.01± 0.04 M_⊙ and 1.83± 0.11 M_⊙, respectively, all above 1.8 M_⊙. These high masses are inconsistent with equations of state that are “soft” at high densities <cit.>. On the other hand, many “hard” equations of state are ruled out with X-ray measurements that constrain the stellar radius to values around 10 – 12 km <cit.>.Another important aspect of MSP properties that impacts on neutron-star physics is the maximum observed spin rate. Figure <ref> shows the observed spin frequency distribution of the fastest MSPs according to source type. Despite the discovery of several eclipsing MSPs with pulse frequencies above 500 Hz in radio searches of Fermi sources in the past few years, the spin rate record (716 Hz) is still held by PSR J1748-2446ad in the globular cluster Terzan 5, found more than a decade ago <cit.>. Searches for short-period (sub-millisecond) MSPs at both radio or X-ray frequencies have not been instrumentally limited since that time or even before. However, there is likely to be a selection against short-period pulsars in lower-frequency radio searches because of absorption and scattering in the circumstellar plasma of eclipsing binary radio pulsars – PSR J1748-2446ad was found in a 2-GHz search using the Green Bank Telescope. It is clear that there is a physical mechanism limiting the maximum spin frequency of neutron stars in accreting binary systems to something around 700 Hz despite most neutron-star equations of state allowing maximum spin frequencies of more than 1 kHz <cit.>. Gravitational “r-mode” instabilities were suggested by <cit.> as a mechanism, but these appear to be too efficient and must be suppressed in some way to allow the higher observed spin frequencies <cit.>.It is possible that spin rates are naturally limited by a balance between spin-up and spin-down torques <cit.>.§ CONCLUSIONThe discovery of the first binary pulsar in 1975 and the first MSP in 1982 opened the door to a huge diversity of research fields, ranging from tests of theories of relativistic gravitation to understanding the evolutionary pathways that could lead to such objects. In this brief review, I have just skimmed the surface of a few of these research fields, omitting mention of many altogether. I hope though that this has been enough to illustrate the enormous power and potential of the study of millisecond pulsars and pulsar binary systems. In this volume celebrating the contributions of Srinivasan to astrophysics, I have highlighted his contributions to the topics of MSP and binary evolution. With colleagues including Bhattacharya, Radhakrishnan and van den Heuvel, he has made ground-breaking contributions to the field right from the early days following the discovery of the first binary pulsar. Many of these contributions have proved prescient and are part of the present-day understanding of these topics. 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"authors": [
"R. N. Manchester"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170927102248",
"title": "Millisecond Pulsars, their Evolution and Applications"
} |
OT11011gobble Distributed Information Bottleneck Method for Discrete and Gaussian SourcesIñaki Estella Aguerri ^† Abdellatif Zaidi ^†^^† Mathematics and Algorithmic Sciences Lab. France Research Center, Huawei Technologies, Boulogne-Billancourt, 92100, France^ Université Paris-Est, Champs-sur-Marne, 77454, France{[email protected], [email protected]}December 30, 2023 ==============================================================================================================================================================================================================================================================================================================We study the problem of distributed information bottleneck, in which multiple encoders separately compress theirobservations in a manner such that, collectively, the compressed signals preserve as much information as possible about another signal. The model generalizes Tishby's centralizedinformation bottleneck method to the setting of multiple distributed encoders. We establish single-letter characterizations of the information-rate region of this problem for both i) a class of discrete memoryless sources and ii) memoryless vector Gaussian sources. Furthermore, assuming a sum constraint on rate or complexity, for both models we develop Blahut-Arimoto type iterative algorithms that allow to compute optimal information-rate trade-offs, by iterating over a set of self-consistent equations. § INTRODUCTION The information bottleneck (IB) method was introduced by Tishby <cit.> as an information-theoretic principle for extracting the relevant information that some signal Y ∈𝒴 provides about another one, X ∈𝒳, that is of interest.The approach has found remarkable applications in supervised and unsupervised learning problems such as classification, clustering and prediction, wherein one is interested in extracting the relevant features, i.e., X, of the available data Y <cit.>. Perhaps key to the analysis, and development, of the IB method is its elegant connection with information-theoretic rate-distortion problems. Recent works show that this connection turns out to be useful also for a better understanding ofdeep neural networks <cit.>.Other connections, that are more intriguing, exist also with seemingly unrelated problems such as hypothesis testing <cit.> or systems with privacy constraints <cit.>. Motivated by applications of the IB method to settings in which the relevant features about X are to be extracted from separately encoded signals, we study the model shown in Figure <ref>. Here, X is the signal to be predicted and (Y_1,,Y_K) are correlated signals that could each be relevant to extract one or more features of X. The features could be distinct or redundant. We make the assumption that the signals (Y_1,,Y_K) areindependent given X. This assumption holds in many practical scenarios. For example,the reader may think of (Y_1,...,Y_K) as being the results of K clinical tests that are performed independently at different clinics and are used to diagnose a disease X. A third party (decoder or detector) has to decide without access to the original data. In general, at every encoder k there is a tension among the complexity of the encoding, measured by the minimum description length or rate R_k at which the observation is compressed, and the information that the produced description, say U_k, provides about the signal X. The relevance of (U_1,,U_K) is measured in terms of the information that the descriptions collectively preserve about X; and is captured by Shannon's mutual information I(U_1,,U_K;X). Thus, the performance of the entire system can be evaluated in terms of the tradeoff between the vector (R_1,,R_K) of minimum description lengths and the mutual information I(U_1,,U_K;X). In this paper, we study the aforementioned tradeoff among relevant information and complexity for the model shown in Figure <ref>. First, we establish a single-letter characterization of the information-rate region of this model for discrete memoryless sources. In doing so, we exploit its connection with the distributed Chief Executive Officer (CEO) source coding problem under logarithmic-loss distortion measure studied in <cit.>. Next, we extend this result to memoryless vector Gaussian sources. Here, we prove that Gaussian test channels are optimal, thereby generalizing a similar result of <cit.> and <cit.> for the case of a single encoder IB setup. In a second part of this paper, assuming a sum constraint on rate or complexity,we develop Blahut-Arimoto <cit.> type iterative algorithms that allow to compute optimal tradeoffs between information and rate, for both discrete and vector Gaussian models. We do so through a variational formulation that allows the determination of the set of self-consistent equations satisfied by the stationary solutions. In theGaussian case, the algorithm reduces to an appropriate updating of the parameters ofnoisy linear projections. Here as well, our algorithms can be seen as generalizations of those developed for the single-encoder IB method, for discrete sources in <cit.> and for Gaussian sources in <cit.>; as well as a generalization of the Blahut-Arimoto algorithm proposed in <cit.> for the CEO source coding problem for K=2 and discrete sources, toK≥ 2 encoders and for both discrete and Gaussian sources. Notation: Upper case lettersdenote random variables, e.g., X;lower case letters denote realizations of random variables, e.g., x; and calligraphic letters denote sets, e.g., 𝒳. The cardinality of a set is denoted by | X|. For a random variable X with probability mass function (pmf) P_X, we use P_X(x)=p(x), x∈ X for short. Boldface upper case letters denote vectors or matrices, e.g., X, where contextmakes the distinction clear. For an integer n∈ℕ, we denote the set [1,n]:={1,2,…, n}.We denote by D_KL(P,Q) the Kullback-Leibler divergence between the pmfs P and Q. For a set of integers K⊆N,X_ K denotes the setX_ K={X_k:k ∈ K }.We denote the covariance of a zero-mean vector 𝐗 by Σ_𝐱:=E[𝐗𝐗^H];Σ_𝐱,𝐲 is the cross-correlationΣ_𝐱,𝐲:= E[𝐗𝐘^H], and the conditional correlationof 𝐗 given 𝐘 as Σ_𝐱|𝐲:= Σ_𝐱-Σ_𝐱,𝐲Σ_𝐲^-1Σ_𝐲,𝐱.§ SYSTEM MODEL Consider the discrete memoryless D-IB model shownin Figure <ref>. Let {X_i, Y_1,i,…, Y_K,i}_i=1^n = (X^n,Y_1^n,…, Y_K^n) be a sequence of n independent, identically distributed (i.i.d.) random variables with finite alphabets 𝒳,𝒴_k, k∈𝒦:= {1,…, K} and joint pmf P_X,Y_1,…,Y_K. Throughout this paper, we make the assumption that the observations at the encoders are independent conditionally on X, i.e., Y_k,i X_i Y_𝒦/k,ifork ∈ K andi ∈ [1,n].Encoder k∈𝒦 maps the observed sequence Y_k^n to an index J_k:= ϕ_k(Y_k^n), whereϕ_k: 𝒴_k^n→ℳ_k is a given map and ℳ_k:= [1,M_k^(n)]. The index J_k is sent error-free to the decoder.The decoder collects all indices J_𝒦:= (J_1,…,J_K) and then estimates the source X^n as X̂^n=g^(n)(J_ J), where g^(n): ℳ_1×⋯×ℳ_L→𝒳̂^n is some decoder map and 𝒳̂^n is the reconstruction alphabet of the source. The quality of the reconstruction is measured in terms of the n-letter relevant information between the unobserved source X^n and its reconstruction at the decoder X̂^n, given byΔ^(n) := 1/n I(X^n;g^(n)(ϕ_1^(n)(Y_1^n),…,ϕ_K^(n)(Y_K^n) )). A tuple (Δ,R_1,…, R_K) is said to be achievable for the D-IB model if there exists a blocklength n, encoder maps ϕ^(n)_k for k∈𝒦, and a decoder map g^(n), such thatR_k ≥1/nlog M_k^(n),k∈ K, andΔ≤1/nI(X^n;X̂^n).where X̂^n=g^(n)(ϕ_1^(n)(Y_1^n),…,ϕ_K^(n)(Y_K^n) ).The information-rate region ℛ_IB is given by the closure of all achievable rates tuples (Δ,R_1,…, R_K).We are interested in characterizing theregion ℛ_IB. Due to space limitations, some results are only outlined or provided without proof. We refer to <cit.> for a detailed version. § INFORMATION-RATE REGION CHARACTERIZATION In this section wecharacterize the information-rate region ℛ_IB for a discrete memoryless D-IB model.It is well known that the IB problem is essentially a source-coding problem where the distortion measure is of logarithmic losstype <cit.>. Likewise, the D-IB model of Figure <ref> is essentially a K-encoder CEO source coding problem under logarithmic loss (log-loss) distortion measure. The log-loss distortion between sequences is defined asd_LL(x^n,x̂^n):= -1/nlog(1/x̂^n(x^n)),where x̂^n = s(x^n|j_ K) and s is a pmf on X^n.The rate-distortion region of the K-encoder CEO source coding problem under log-loss, with K≥ 2 which we denote hereafter as RD_CEO, has been established recently in <cit.> for the case in which the Markov chain (<ref>) holds. We first state the following proposition, the proof of which is easy and omitted for brevity.A tuple (Δ,R_1,…, R_K)∈ R_IB if and only if (H(X)-Δ,R_1,…, R_K)∈RD_CEO. Proposition <ref> implies that <cit.> can be applied to characterize the information-rate region ℛ_IB as given next.In the case in which the Markov chain (<ref>) holds, the rate-information region R_IB of the D-IB model is given by the set of all tuples (Δ, R_1,…, R_K) which satisfy for𝒮⊆𝒦Δ≤∑_k∈𝒮 [R_k-I(Y_k;U_k|X,Q)]+ I(X;U_𝒮^c|Q),for some joint pmf p(q) p(x)∏_k=1^K p(y_k|x)∏_k=1^Kp(u_k|y_k,q).§.§ Memoryless Vector Gaussian D-IB Consider now the following memoryless vector Gaussian D-IB problem. In this model, the source vector 𝐗∈C^N is Gaussian and has zero mean andcovariance matrix Σ_ x, i.e., 𝐗∼CN( 0, Σ_ x).Encoder k, k∈ K, observes a noisy observation Y_k∈C^M_k, that is given by𝐘_k = 𝐇_k𝐗+𝐍_k,where 𝐇_k∈C^M_k× N is the channel connecting the source to encoder k, and 𝐍_k∈C^M_k, k∈ K, is the noise vector at encoder k, assumed to be Gaussian, with zero-mean and covariance matrix Σ_ n_k, and independent from all other noises and the source vector X.The studied Gaussian model satisfies the Markov chain (<ref>); and thus,the result of Theorem <ref>, which can be extended to continuous sources using standard techniques, characterizes the information-rate region of this model. The following theorem characterizesR_IB for the vector Gaussian model, shows that the optimal test channels P_U_k|Y_k, k∈ K, are Gaussian and that there is not need for time-sharing, i.e., Q=∅.If ( X, Y_1,…,Y_K) are jointly Gaussian as in (<ref>), the information-rate region R is given by the set of alltuples (Δ, R_1,…,R_L) satisfying that for all 𝒮⊆𝒦Δ≤∑_k∈𝒮[R_k+log|𝐈-𝐁_k|] + log|∑_k∈𝒮^c𝐇̅_k^H𝐁_k𝐇̅_k+𝐈| ,for some 0≼𝐁_k≼𝐈 and where 𝐇̅_k = Σ_ n_k^-1/2𝐇_kΣ_ x^1/2. In addition, the information-rate tuples in R_IB^G are achievable with Q= ∅ and p^*( u_k| y_k,q) = CN( y_k, Σ_ n_k^1/2( B_k- I)Σ_ n_k^1/2). An outline of the proof is given in Appendix <ref>. § COMPUTATION OF THE INFORMATION RATE REGION UNDER SUM-RATE CONSTRAINT In this section, we describe an iterative Blahut-Arimoto (BA)-type algorithm to compute the pmfs P_U_k|Y_k, k∈ K,that maximize information Δunder sum-rate constraint, i.e., R_sum := ∑_k=1^KR_k, for tuples (Δ,R_1,…, R_K) in R_IB. From Theorem <ref> we have:R_sum:= convex-hull{(Δ,R_sum):Δ≤Δ_sum( R_sum )},where we define the information-rate functionΔ_sum(R) :=max_ Pmin{I(X; U_ K),R - ∑_k=1^KI(Y_k;U_k|X)},and where the optimization is over the set of K conditional pmfs P_U_k|Y_k, k∈ K, which, for short, we define asP := {P_U_1|Y_1,…, P_U_K|Y_K}. Next propositionprovides acharacterization of the pairs (Δ,R_sum)∈ R_sum in terms of a parameter s≥ 0.Each tuple (Δ,R_sum) on the information rate curveΔ = Δ_sum(R_sum), can be obtained for somes ≥ 0, as (Δ_s, R_s), parametrically defined by (1+s)Δ_s = (1+sK)H(X) +s R_s - min_ PF_s( P),R_s = I(Y_ K;U_ K^*) + ∑_k=1^K [I(Y_k;U_k^*) - I(X;U_k^*)],where P^* are the pmfs yielding the minimum in (<ref>) and F_s( P) :=H(X|U_ K) + s ∑_k=1^K [I(Y_k;U_k) + H(X|U_k)].The proof of Proposition <ref> follows along the lines of <cit.> and is omitted for brevity.Note that the rate expression in Theorem <ref> is different to that in <cit.>. Suppose that P^* yields the minimum in (<ref>). Then, (1+s)Δ_s =(1+sK)H(X)+sR_s-F_s( P^*) = (1+sK) H(X) + sR_ s-(H(X|U_ K^*) + s(R_s- I(X;U^*_ K)+KH(X)) )= (1+s) I(X;U_ K^*) ≤ (1+s)Δ_sum(R_s),where (<ref>) follows since for P^*, we have from (<ref>) that ∑_k=1^K [I(Y_k;U_k^*) + H(X|U_k^*)] = R_s- I(X;U^*_ K)+KH(X). Conversely, if P^* is the solution to the maximization defining Δ_sum(R)=Δ_s, then Δ_s≤ I(X;U_K^*) and Δ_s≤ R_s -∑_k=1^K I(Y_k;U^*_k|X) and we have, for any s, thatΔ _sum(R) =Δ_s ≤ Δ_s -(Δ_s- I(X;U_K^*)) -s(Δ_s- R_s +∑_k=1^K I(Y_k;U^*_k|X)) =H(X) - sΔ_s+ sR - (H(X)+ sR_s -(1+s)Δ_s)= Δ_s + s(R -R_s) ,where (<ref>) follows from (<ref>) and since ∑_k=1^KI(Y_k;U_k|X)= -KH(X)+∑_k=1^KI(Y_k;U_k)+H(X|U_k),due to the Markov chain U_k - Y_k - X - (Y_ K∖ k , U_ K∖ k ).Given s, and hence (Δ_s, R_s), choosing R = R_s yields Δ_sum(R_s) ≤Δ_ s.Together with (<ref>), this completes the proof. FromProposition <ref>, the information-rate function can be computed by solving (<ref>) and evaluating (<ref>) for all s≥ 0. Inspired by the standard Blahut-Arimoto (BA) method<cit.>, and following similar steps as for the BA-type algorithm proposed in <cit.> for the CEO problem with K=2 encoders, we show that problem (<ref>) can be solved with an alternate optimization procedure, with respect to P and some appropriate auxiliary pmfs Q_X|U_k, k∈ K and Q_X|U_1,…, U_K, denoted for short asQ := {Q_X|U_1,…, Q_X|U_K,Q_X|U_1,…, U_K}.To this end, we define thefunction F̅_s(·) and write (<ref>) as a minimization over the pmfs P and pmfs Q, where F̅_s( P,Q) :=s ∑_k=1^K I(Y_k;U_k)- s ∑_k=1^K E_X,U_k[logq(X|U_k)]- E_X,U_K[log q(X|U_1,…, U_K)]. We haveF^*:=min_ PF_s( P) = min_ Pmin_ QF̅_s( P, Q). Algorithm <ref> describes the steps tosuccessively minimize F̅_s( P, Q) by optimizing a convex problem over P and over Q at each iteration.The proof of Lemma <ref> and the steps of the proposed algorithm are justified with the following lemmas, whose proofs are along the lines of Lemma 1, Lemma 2, Lemma 3 in <cit.>, and are omitted due to space limitations. F̅_s( P,Q) is convex in P and convex in Q.F̅_s( P,Q) is convex in P and convex in Q.Follows from the log-sum inequality <cit.>.For fixed pmfs P, F̅_s( P,Q) ≥ F_s( P) for all pmfs Q, andthere exists a unique Q that achieves the minimum min_ QF̅_s( P,Q) = F_s( P), given byQ^*_X|U_k = P_X|U_k, k∈ K, Q^*_X|U_1,…,U_k = P_X|U_1,…, U_K,where P_X|U_k and P_X|U_1,…, U_K are computed from P.For fixed Q, there exists aP that achieves the minimum min_ PF̅_s( P,Q), where P_U_k|Y_k is given byp^*(u_k|y_k) = p(u_k)exp(-ψ_s(u_k,y_k))/∑_y_k∈ Y_kexp(-ψ_s(u_k,y_k)),for u_k∈ U_k and y_k∈ Y_k, k∈ K,and where we define ψ_s(u_k,y_k):= D_KL(P_X|y_k||Q_X|u_k)+1/sE_U_ K∖ k|y_k[D_KL(P_X|U_ K∖ k,y_k||Q_X|U_ K∖ k,u_k))].It follows from the log-sum inequality <cit.> that F̅_s( P,Q) is a convex function of P andof Q. To minimizeF̅_s( P,Q)w.r.t. P, for given Q, we add the Lagrange multipliers λ_y_k≥ 0for each constraint ∑_u_k∈ U_kp(u_k|y_k) = 1 with y_k∈ Y_k.For each s, λ_y_k≥ 0 and p(u_k|y_k) can be explicitly found by solving the KKT conditions. Algorithm <ref> essentially falls in the Successive Upper-Bound Minimization (SUM) framework <cit.>in which F̅_s( P,Q) acts as a globally tight upper bound on F_s( P). Algorithm <ref> provides a sequence P^(t) for each iteration t, whichconverges to a stationary point of the optimization problem (<ref>). Every limit point of the sequence P^(t) generated by Algorithm <ref> converges to a stationary point of (<ref>). Let Q^*( P):= min_ QF̅_s( P,Q). From Lemma <ref>,F̅_s( P,Q^*( P'))≥F̅_s( P,Q^*( P)) = F_s( P) for P'≠ P.It follows that F_s( P) and F̅_s( P,Q^*( P')) satisfy <cit.>and thus F̅_s( P,Q^*( P'))satisfies A1-A4 in <cit.>. Convergence to a stationary point of (<ref>) follows from <cit.>. The resulting set of self consistent equations(<ref>), (<ref>) and (<ref>) satisfied by any stationary point of the D-IB problem, remind that of the original IB problem <cit.>.Note the additional divergence term in(<ref>) for encoder k averaged over the descriptions at the other K∖ k encoders.§ COMPUTATION OF THE INFORMATION RATE REGION FOR THE VECTOR GAUSSIAN D-IB Computing the maximum information under sum-rate constraint from Theorem <ref> is a convex optimization problem on B_k, which can be efficiently solved with generic tools.Alternatively, next we extend Algorithm <ref> for Gaussiansources.For finite alphabet sources the updates ofQ^(t+1) and P^(t+1) in Algorithm <ref> are simple, but become unfeasible for continuous alphabet sources. We leverage on the optimality of Gaussian descriptions, shown inTheorem <ref>, to restrict the optimization of P to Gaussian distributions, which are easily represented by a finite set of parameters, namely its mean and covariance. We show that if P^(t) are Gaussian pmfs, then P^(t+1) are alsoGaussian pmfs, which can be computed with an efficient update algorithm of its representing parameters. In particular,if at time t, the k-th pmf P_ U_k| Y_k^(t) is given byU_k^t =A_k^t Y_k + Z_k^t,where Z_k^t∼CN( 0,Σ_ z_k^t); we show that for P^(t+1)updated as in (<ref>),P_ U_k| Y_k^(t+1) corresponds to U_k^t+1 =A_k^t+1 Y_k+ Z_k^t+1, where Z_k^t+1∼CN( 0, Σ_ z_k^t+1)andA_k^t+1, Σ_ z_k^t+1 are updated asΣ_ z_k^t+1 = ((1+1/s)Σ_ u_k^t| x^-1- 1/sΣ_ u_k^t| u_ K∖ k^t^-1)^-1,A_k^t+1 = Σ_ z_k^t+1^-1((1+1/s)Σ_ u_k^t| x^-1A_k^t( I - Σ_ y_k| xΣ_ y_k^-1)..-1/sΣ_ u_k^t| u_ K∖ k^t^-1A_k^t( I - Σ_ y_k| u_ K∖ k^tΣ_ y_k^-1)).The detailed update procedure is given in Algorithm <ref>.Algorithm <ref> generalizes the iterative algorithmfor single encoder Gaussian D-IB in <cit.>to the Gaussian D-IBwith K encoders and sum-rate constraint. Similarly to the solution in <cit.>, the optimal description at each encoder is given by a noisy linear projection of the observation, whose dimensionality is determined by the parameter s and the second order moments between the observed data and the source of interest, as well as a term depending on theobserved data with respect to the descriptions at the other encoders. §.§ Derivation of Algorithm <ref> In this section, we derive the update rules in Algorithm <ref> and show that the Gaussian distribution is invariant to the update rules in Algorithm <ref>, in line with Theorem <ref>.First, we recall that if ( X_1, X_2) are jointly Gaussian, thenP_ X_2| X_1 =x_1 = CN(μ_ x_2| x_1,Σ_ x_2| x_1),where μ_ x_2| x_1:=K_ x_2| x_1 x_1, with K_ x_2| x_1:=Σ_ x_2, x_1Σ_ x_1^-1 .Then, for Q^(t+1) computed as in (<ref>) and (<ref>) from P^(t), which is a set of Gaussian distributions,we haveQ^(t+1)_ X|u_k = CN(μ_ x| u_k^t , Σ_ x| u_k^t), Q^(t+1)_ X|u_ K = CN(μ_ x| u_ K^t, Σ_ x| u_ K^t).Next, we look at the updateP^(t+1) as in (<ref>) from given Q^(t+1). First, we have that p( u_k^t)is the marginal of U_k^t, given by U_k^t∼CN( 0,Σ_ u_k^t )where Σ_ u_k^t =A_k^tΣ_ y_k A_k^t,H + Σ_ z_k^t. Then, to compute ψ_s( u_k^t, y_k), first, we note that E _ U_ K∖ k|y_k [D_KL(P_ X| U_ K∖ k,y_k||Q_X| U_ K∖ k,u_k)]= D_KL(P_ X, U_ K∖ k|y_k||Q_ X,U_ K∖ k|u_k)-D_KL(P_ U_ K∖ k|y_k||Q_ U_ K∖ k|u_k),and that for two generic multivariate Gaussian distributions P_1∼CN(μ_1,Σ_1) andP_2∼CN(μ_2,Σ_2) in C^N,D_KL(P_1,P_2) =(μ_1-μ_2)^HΣ_2^-1(μ_1-μ_2)+log |Σ_2Σ_1^-1| - N +tr{Σ_2^-1Σ_1}. Applying (<ref>) and (<ref>) in (<ref>) and noting that all involved distributions are Gaussian, it follows that ψ_s( u_k^t, y_k) is a quadratic form. Then, since p( u_k^t) is Gaussian, the product log (p( u_k^t)exp(-ψ_s( u_k^t, y_k))) is also a quadratic form, and identifying constant, first and second order terms, we can writelog p^(t+1)( u_k| y_k) =Z( y_k)+ ( u_k-μ_ u_k^t+1|y_k)^HΣ_ z_k^t+1^-1· ( u_k-μ_ u_k^t+1| y_k),where Z( y_k) is a normalization term independent of u_k, and Σ_ z_k^t+1^-1 = Σ_ u_k^t^-1 +K_ x| u_k^t^H Σ_ x|u_k^-1 K_ x| u_k^t +1/s K_ x u_ K∖ k^t| u_k^t^H Σ_ x u_ K∖ k^t|u_k^-1 K_ x u_ K∖ k^t| u_k^t - 1/s K_ u_ K∖ k^t| u_k^t^H Σ_ u_ K∖ k^t|u_k^-1 K_ u_ K∖ k^t| u_k^t,μ_ u_k^t+1= Σ_ z_k^t+1( K_ x| u_k^t^HΣ_ x| u_k^t^-1μ_ x| y_k.+1/s K_ x, u_ K∖ k^t| u_k^tΣ_ x, u_ K∖ k^t| u_k^t^-1μ_ x, u_ K∖ k^t| y_k.-1/s K_ u_ K∖ k^t| u_k^tΣ_ u_ K∖ k^t| u_k^t^-1μ_ u_ K∖ k^t| y_k).This shows that p^(t+1)( u_k| y_k) is a Gaussian distribution and that U_k^t+1 is distributed as U_k^t+1∼CN(μ_ u_k^t+1,Σ_ z_k^t+1). Next, we simplify (<ref>) and (<ref>) to obtain the update rules (<ref>) and (<ref>). From the matrix inversion lemma, similarly to <cit.>, for ( X_1, X_2) jointly Gaussianwe have Σ_ x_2| x_1^-1 = Σ_ x_2^-1 +K_ x_1| x_2^HΣ_ x_1| x_2^-1 K_ x_1| x_2.Applying (<ref>), in (<ref>) we haveΣ_ z_k^t+1^-1 =Σ_ u_k^t| x^-1 +1/sΣ_ u_k^t| xu_ K∖ k^t^-1 - 1/sΣ_ u_k^t| u_ K∖ k^t^-1,=(1+1/s)Σ_ u_k^t| x^-1- 1/sΣ_ u_k^t| u_ K∖ k^t^-1,where(<ref>) is due to the Markov chain U_k X U_ K∖ k.Then, also from the matrix inversion lemma,we have for jointly Gaussian ( X_1, X_2), Σ_ x_2| x_1^-1Σ_ x_1, x_2Σ_ x_1^-1 =Σ_ x_2^-1Σ_ x_1, x_2Σ_ x_1| x_2^-1.Applying (<ref>) in (<ref>), for the first term, we haveK_ x| u_k^t^HΣ_ x| u_k^t^-1μ_ x| y_k= Σ_ u_k^t| x^-1Σ_ x, u_k^tΣ_ x^-1μ_ x| y_k = Σ_ u_k^t| x^-1A_k^tΣ_ y_k, xΣ_ x^-1Σ_ x, y_kΣ_ y_k^-1 y_k= Σ_ u_k^t| x^-1A_k^t( I - Σ_ y_k| xΣ_ y_k^-1)y_k,where Σ_ x, u_k^t=A_k^tΣ_ y_k, x; and (<ref>) is due to the definition of Σ_ y_k | x. Similarly, for the second term, we haveK_ x u_ K∖ k^t| u_k^t Σ_ x u_ K∖ k^t| u_k^t^-1μ_ x, u_ K∖ k^t| y_k= Σ_ u_k^t| x u_ K∖ k^t^-1A_k^t( I - Σ_ y_k| x u_ K∖ k^tΣ_ y_k^-1)y_k,= Σ_ u_k^t| x^-1A_k^t( I - Σ_ y_k| xΣ_ y_k^-1)y_k,where we use Σ_ u_k^t, x u_ K∖ k^t=A_k^tΣ_ y_k, x u_ K∖ k^t; and (<ref>) is due to the Markov chain U_k X U_ K∖ k. For the third term,K_ u_ K∖ k^t| u_k^t Σ_ u_ K∖ k^t| u_k^t^-1μ_ u_ K∖ k^t| y_k= Σ_ u_k^t| u_ K∖ k^t^-1A_k^t( I - Σ_ y_k| u_ K∖ k^tΣ_ y_k^-1)y_k. Equation (<ref>) follows by noting that μ_ u_k^t+1 =A_k^t+1 y_k, and that from (<ref>) A_k^t+1 is given as in (<ref>). Finally, we note that due to (<ref>), Σ_ u_k| x and Σ_ u_k| u_ K∖ k^t are given as in (<ref>) and (<ref>), where Σ_ y_k| x=Σ_ n_k and Σ_ y_k| u_ K∖ k^t can be computed from its definition.§ NUMERICAL RESULTSIn this section, we consider the numerical evaluation of Algorithm <ref>, and compare the resulting relevant information to two upper bounds on the performance for the D-IB: i) the information-rate pairs achievable under centralized IB encoding,i.e., if (Y_1,…, Y_K) are encoded jointly at a rate equal to the total rate R_sum= R_1+⋯+R_K, characterized in <cit.>; ii) the information-rate pairs achievable under centralized IB encoding when R_sum→∞, i.e., Δ = I(X;Y_1,…, Y_K).Figure <ref> shows the resulting (Δ,R_sum) tuples for a Gaussian vector model with K=2 encoders, source dimension N=4, and observations dimension M_1 = M_2 = 2 for different values of s calculated as in Proposition <ref> using Algorithm <ref>, and its upper convex envelope. As it can be seen, the distributed IB encoding of sources performs close to the Tishby's centralized IB method, particularly for low R_sum values. Note the discontinuity in the curve caused by a dimensionality change in the projections at the encoders. § PROOF OUTLINE OF THEOREM <REF> Let (𝐗,𝐔) be two complex random vectors. The conditional Fischer information is defined as𝐉(𝐗|𝐔) := E[∇log p(𝐗|𝐔)∇log p(𝐗|𝐔)^H],and the MMSE is given by mmse( X| U) := E[( X-E[ X| U])( X-E[ X| U])^H]. Then <cit.> log|(π e) 𝐉^-1(𝐗|𝐔)|≤ h(𝐗|𝐔)≤log|(π e) mmse( X| U)|. We outer bound the information-rate region in Theorem <ref> for ( X, Y_ K) as in (<ref>).For q∈Q and fixed ∏_k=1^Kp(𝐮_k|𝐲_k,q), choose 𝐁_k,q, k∈ K satisfying 0≼𝐁_k,q≼Σ_ n_k^-1 such that mmse(𝐘_k|𝐗, 𝐔_k,q,q) = Σ_ n_k-Σ_ n_k𝐁_k,qΣ_ n_k.Such 𝐁_k,q always exists since 0≼mmse(𝐘_k|𝐗,𝐔_k,q,q)≼Σ_ n_k^-1, for all q∈ Q, and k∈ K. We have from (<ref>),I(𝐘_k;𝐔_k|𝐗,q) ≥log|Σ_ n_k| -log|mmse(𝐘_k|𝐗,𝐔_k,q,q) |= - log| I-Σ_ n_k^1/2𝐁_k,qΣ_ n_k^1/2|,where the inequality is due to (<ref>), and (<ref>) is due to (<ref>). Let 𝐁̅_k:= ∑_q∈𝒬p(q)𝐁_k,q. Then, we have from (<ref>)I(𝐘_k;𝐔_k|𝐗,Q)≥ -∑_q∈𝒬p(q) log| I-Σ_ n_k^1/2𝐁_k,qΣ_ n_k^1/2|≥-log| I-Σ_ n_k^1/2𝐁̅_kΣ_ n_k^1/2|,where(<ref>) follows from the concavity of the log-det function and Jensen's inequality.On the other hand, we haveI(𝐗;𝐔_S^c,q|q) ≤log|Σ_ x |-log|𝐉^-1(𝐗|𝐔_S^c,q,q)|, = log| ∑_k∈𝒮^cΣ_ x^1/2𝐇_k^H𝐁_k,q𝐇_kΣ_ x^1/2+𝐈|,where (<ref>) is due to (<ref>); and (<ref>) is due to to the following equality, which can be proved using the connection between the MMSE matrix (<ref>) and the Fisher information along the lines of<cit.> (We refer to <cit.> for details):𝐉(𝐗|𝐔_S^c,q,q) = ∑_k∈𝒮^c𝐇_k^H𝐁_k,q𝐇_k+Σ_ x^-1.Similarly to (<ref>), from (<ref>) and Jensen's Inequality we haveI(𝐗;𝐔_S^c|Q) ≤log| ∑_k∈𝒮^cΣ_ x^1/2𝐇_k^H𝐁̅_k𝐇_kΣ_ x^1/2+𝐈|.Substituting (<ref>) and (<ref>) in (<ref>) and letting 𝐁_k := Σ_ n_k^-1/2𝐁̅_kΣ_ n_k^-1/2 gives the desired outer bound. The proof is completed by noting that the outer bound is achieved with Q= ∅ and p^*( u_k| y_k,q) = CN( y_k, Σ_ n_k^1/2( B_k- I)Σ_ n_k^1/2).ieeetran | http://arxiv.org/abs/1709.09082v3 | {
"authors": [
"Inaki Estella Aguerri",
"Abdellatif Zaidi"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170926151356",
"title": "Distributed Information Bottleneck Method for Discrete and Gaussian Sources"
} |
Zhuang and othersTreatment Effect Modification with Baseline Sub-samplingEvaluation of Treatment Effect Modification by Biomarkers Measured Pre- and Post-randomization in the Presence of Non-monotone Missingness YINGYING ZHUANG^∗, YING HUANG, PETER B. GILBERT Department of Biostatistics, University of Washington, Seattle, WA, [email protected] December 30, 2023 ================================================================================================================================================To whom correspondence should be addressed. In vaccine studies, investigators are often interested in studying effect modifiers of clinical treatment efficacy by biomarker-based principal strata, which is useful for selecting biomarker study endpoints for evaluating treatments in new trials, exploring biological mechanisms of clinical treatment efficacy, and studying mediators of clinical treatment efficacy. However, in trials where participants may enter the study with prior exposure therefore with variable baseline biomarker values, clinical treatment efficacy may depend jointly on a biomarker measured at baseline and measured at a fixed time after vaccination. Therefore, it is of interest to conduct a bivariate effect modification analysis by biomarker-based principal strata and baseline biomarker values. Previous methods allow this assessment if participants who have the biomarker measured at the the fixed time point post randomization would also have the biomarker measured at baseline. However, additional complications in study design could happen in practice. For example, in the Dengue correlates study, baseline biomarker values were only available from a fraction of participants who have biomarkers measured post-randomization. How to conduct the bivariate effect modification analysis in these studies remains an open research question. In this article, we propose an estimated likelihood method to utilize the sub-sampled baseline biomarker in the effect modification analysis and illustrate our method with datasets from two dengue phase 3 vaccine efficacy trials.§ INTRODUCTION A common problem of interest within a randomized clinical trial is the evaluation of an inexpensive intermediate study endpoint, typically a biomarker, as an effect modifier of the clinical treatment efficacy which can accelerate research to apply and develop effective treatments against the clinical outcome. Motivated by randomized placebo-controlled vaccine efficacy trials, Gilbert and Hudgens <cit.>, henceforth GH, proposed a clinically relevant causal estimand called the Causal Effect Predictiveness (CEP) surface, using the principal stratification framework developed by <cit.> to assess whether and how treatment efficacy varies by subgroups defined by intermediate response endpoint principal strata. Procedures were developed for contrasting clinical risks under two treatment arms conditional on the pair of potential biomarker values under two treatment arms. The CEP surface assesses a causal effect of vaccine because the comparison groups are selected based on principal stratification, which is not subject to post-randomization selection bias. GH expressed the concept that studying the whole CEP surface is important that a more useful biomarker will have wide variability in the CEP surface thus is a strong effect modifier. Unfortunately, if no further assumptions are made, the CEP surface can not identified by the observed data because of the missing potential outcomes. <cit.> proposed two augmented trial design to solve the problem: baseline immunogenicity predictors (BIP) and closeout placebo vaccination (CPV). The BIP design uses baseline predictor(s) to infer the unobserved potential biomarker values, while the CPV design vaccinates placebo recipients who stay uninfected at the end of the follow-up and measures their immune response values, which are used in place of their biomarker values under vaccine. Examples of good baseline predictors are baseline biomarker measurements in trials where participants may enter the study with prior exposure and biomarker measurements at baseline reflects natural immunity arising from pre-trial exposure to the disease-causing pathogen. Furthermore, some baseline predictors may modify the CEP surface and contrasting clinical risks under each treatment assignment may depend jointly on those baseline predictors and the biomarker values measured at a fixed time after randomization. Therefore, it is of interest to estimate CEP in those baseline predictors defined subgroups. For example, an common question that emerges from dengue Phase III trials is whether it is the new biomarker response generated by the vaccine over the baseline value or the absolute biomarker value achieved following vaccination that predicts clinical treatment efficacy. Comparing the CEP surface within the baseline seropositive subgroup (defined as baseline biomarker value equal or above detection limit) to the baseline seronegative subgroup (defined as baseline biomarker value below detection limit) could provide important insights to this question. Previous methods allow this assessment if baseline predictors are measured in everyone. <cit.> studied a three-phase sampling design in which immune response is further measured among a subset of participants for whom the baseline predictors are available. However, additional complications in study design could happen in practice. For example, in dengue Phase III trials, the biomarker values at baseline were only measured in a fraction of those with the biomarker measured at the post-randomization time point. Our goal in this manuscript is to propose methods for a bivariate treatment effect modification analysis by biomarker-based principal strata and baseline covariates in general settings without requirements of a nested sub-sampling relationship between the immune response biomarker and baseline predictors, in other words, in the presence of non-monotone missingness, applicable to both the BIP-only design and the BIP+CPV design. The remainder of this article is organized as follows. In Section 2, we introduce the problem setting and propose an estimator for the CEP curve that accounts for effects of the baseline covariates together with the vaccine-induced biomarker in the risk model. In Section 3, we evaluate the finite-sample performance of the proposed estimator through extensive numerical studies. In Section 4, we present an analysis of two Phase 3 dengue vaccine efficacy trials using our proposed methods. Finally, Section 5 contains a discussion of our proposed methods, their applications in other settings, and possible areas of future research.§ METHODSConsider a study in which N subjects are independently and randomly selected from a given population of interest and are randomly assigned to either placebo or vaccine at baseline (time 0). Let Z=1 if subject is randomized to vaccine and Z=0 if subject is randomized to placebo. Let Q be a vector of baseline covariates used for modeling disease risk and Q can be partitioned into two part, X and B. X denotes baseline covariates recorded for everyone at baseline such as gender and country while B denotes baseline covariates that are only available in a subset of the N trial participants, such as baseline biomarker measurements. Trial participants are followed for the primary clinical endpoint for a predetermined period of time and let Y be the indicator of clinical endpoint event during the study follow-up period. At some fixed time τ >0 post randomization, an immune response endpoint, S, is measured. Because S must be measured prior to disease to evaluate its treatment effect modification, availability of S is conditional on remaining clinical endpoint free at time τ (denoted by Y^τ=0). If clinical endpoint occurs in the time interval [ 0, τ ] (Y^τ=1), then S is undefined and we set S=∗. In a CPV component is incorporated in the trial design, all or a fraction of placebo recipients who remain free of the clinical endpoint at the closeout of the trial are vaccinated and the immune response biomarker S_c is measured at time τ after vaccination. In addition, we consider cases where S is continuous and subject to "limit of detection" left censoring. The observable random variable S≡ max(S^*,c) where c is the limit of detection and S^* has a continuous cdf with Pr(S^*≤ c)>0.Similarly to S, S_c≡ max(S_c^*,c). If B denotes the baseline biomarker measurements, thenobservable random variable B≡ max(B^*,c) where B^* has a continuous cdf with Pr(B^*≤ c)>0. Let S(z),S^*(z), Y^τ(z), Y(z) be the potential outcomes if assigned treatment z, for z=0 or 1. We consider a general sampling framework where baseline covariates X and the clinical outcome data Y and Y^τ are measured for everyone, sampling probability of B depends on X and Z, and sampling probability of S(1) or S_c depends on Y, X and Z. The CEP surface is defined in terms of the clinical risks under each treatment assignment,risk_z(s_1, s_0)≡ P ( Y(z)=1|S(1)=s_1, S(0)=s_0, Y^τ(1)= Y^τ(0)= 0 ) for z=0,1. It conditions on the counterfactual pair (S(0), S(1))which forms a principal stratification and can be considered as an unobserved baseline characteristic of each subject. The latter condition Y^τ(1)= Y^τ(0)= 0 ensures that causal treatment effects on S are defined. With h(x,y) being a known contrast function satisfying h(x,y)=0 if and only if x =y, GH defined the CEP surface as CEP^risk(s_1, s_0)≡ h(risk_1(s_1, s_0),risk_0(s_1, s_0)). The marginal CEP curve, closely related to the CEP surface, is also of great value in studying biomarkers as effect modifiers. It contrasts the risks averaged over the distribution of S(0): mCEP^risk(s_1)≡ h(risk_1(s_1),risk_0(s_1)), where risk_z(s_1)≡ P ( Y(z)=1|S(1)=s_1, Y^τ(1)= Y^τ(0)= 0 ). A example of the marginal CEP curve is vaccine efficacy as a function of S(1), which is a causal estimand measuring the relative reduction in infection risk conferred by randomizing to vaccine versus placebo for different levels of S(1): VE(s_1)≡ 1-risk_1(s_1)/risk_0(s_1).In this manuscript, we propose methods to estimate the causal estimand for bivariate treatment effect modification analysis mCEP^risk(S(1),B)≡ h(risk_1(S(1),B),risk_0(S(1),B), applicable to both the BIP-only design and the BIP+CPV design based on an estimated likelihood approach in the presence of non-monotone missingness. Furthermore, in the special case where B denotes the baseline biomarker values, we also derive the estimator for the baseline seropositive mCEP curve (mCEP^risk(S(1),B>c)≡ h(risk_1(S(1),B>c),risk_0(S(1),B>c))) and the baseline seronegative mCEP curve (mCEP^risk(S(1),B=c)≡ h(risk_1(S(1),B=c),risk_0(S(1),B=c))).We make the common assumptions for randomized clinical trials of SUTVA (A1), ignorable treatment assignment (A2), and Equal early clinical risk: P ( Y^τ(1)=Y^τ(0))=1 (A3). These three assumptions reduce the number of missing potential outcomes and help with identifiability of our estimands. They have been used and discussed in details in previously literature (<cit.>,<cit.>, <cit.>,<cit.>). Henceforth, we drop the notation of Y^τ(1)=Y^τ(0) =0 and tacitly assume all probabilities condition on Y^τ(1)=Y^τ(0) =0. Furthermore, we assume the risk functions have a generalized linear model form: risk_z{ S(1),B,X }=g{β;S(1),B,Z,X } for some known link function g(· ), for z=0,1 (A4).In order to replace the unobservable S^*(1) among placebo recipients with the closeout measurement S^*_c, the following two assumptions are made for the BIP+CPV design only: (A5) Time constancy of immune response: For event-freeplacebo recipients, S^*(1)=S^* true+U_1, and S^*_c=S^* true+U_2, for some underlying S^* true and i.i.d. measurement errors U_1, U_2 that are independent of one another. (A6) No placebo subjects event-free at closeout experience the endpoint over the next τ time-units.Henceforth we consolidate the notation and let S^* be the potential outcome of S^* under treatment arm Z=1, either obtained during the standard trial follow-up for vaccine recipients or replaced by the CPV measurements for placebo recipients. We let S≡ max(S^*,c) and δ to be the indicator that S is measured. In addition, if B denotes the baseline biomarker values, we also replace a missing B with S(0) if it is available for placebo recipients based on (A3) and the next assumption (A7): (A7) B^*=B^* true+U_3, and S^*(0)=B^* true+U_4, for some underlying B^* true and i.i.d. measurement errors U_3, U_4 that are independent of one another.Henceforth, if B denotes the baseline biomarker values subject to detection limit: B= max(B^*,c), then we assume that B^* denotes the baseline biomarker values that could potentially being replaced by S^*(0). We let B≡ max(B^*,c) and δ_B to be the indicator that B is available.For the settings we consider in this article, { i: δ_i=1 } and { i: δ_Bi=1 } do not need to hold an inclusion relationship. In section <ref>, we discuss the special cases where δ=1 implies δ_B=1.Lastly, we assume observed data O_i≡ (Z_i,X_i,δ_i,δ_iS_i,δ_Bi,δ_BiB_i,Y^τ_i,Y_i)', i=1,...,n are independent and identically distributed (i.i.d). §.§ Risk Model Parameters EstimationWe propose an estimated likelihood estimator based on conditional likelihood for our risk model parameters β. Subjects with δ_Bi=δ_i=1 contribute to likelihood risk_Z_i(S_i,B_i,X_i; β)^Y_i(1-risk_Z_i(S_i,B_i,X_i; β))^1-Y_i. The likelihood contribution for subjects with δ_Bi=1 and δ_Si=0 is obtained by integrating risk_Z_i(· ,B_i,X_i; β) over the conditional cdf F^S|B,X. The contribution for subjects with δ_i=1 and δ_Bi=0 is obtained by integrating risk_Z_i(S_i,·,X_i; β) over the conditional cdf F^B|S,X. The contribution for subjects with δ_i=δ_Bi=0 is obtained by integrating risk_Z_i(·,·,X_i; β) over the conditional cdf F^(B,S)|X. Define nuisance parameter ν≡ ( F^B|X, F^S|B,X, F^(B|S,X ). Then the condition likelihood isL(β,ν)≡∏_i=1^nf(Y_i|Z_i,X_i,δ_Bi, δ_i,δ_BiB_i, δ_iS_i)where f(Y|Z,X,δ_B, δ,δ_BB, δ S)= { risk_Z(S,B,X; β)^Y(1-risk_Z(S,B,X; β))^1-Y}^δ_Bδ × { ( ∫ risk_Z(s_1,B,X; β) dF^S|B,X(s_1|B,X) ) ^Y . ×. ( 1-∫ risk_Z(s_1,B,X; β) dF^S|B,X(s_1|B,X) ) ^1-Y} ^δ_B (1-δ) × { ( ∫ risk_Z(S,b,X; β) dF^B|S,X(b|S,X) ) ^Y . ×. ( 1-∫ risk_Z(S,b,X; β) dF^B|S,X(b|S,X) ) ^1-Y} ^(1-δ_B)δ × { ( ∫∫ risk_Z(s_1,b,X; β) dF^S|B,X(s_1|b,X) dF^B|X(b|X) ) ^Y. ×.( 1-∫∫ risk_Z(s_1,b,X; β) dF^S|B,X(s_1|b,X) dF^B|X(b|X) ) ^1-Y} ^(1-δ_B)(1-δ) We consider the estimated likelihood approach by Pepe and Fleming (1991) <cit.> where consistent estimates of ν are obtained first and then L(β,ν̂) is maximized in β. Here we assume F^B|X and F^S|B,X have particular parametric distribution. For example, we might assume F^B|X is censored normal and F^S|B,X is also censored normal, with left-censoring of values below c. Then according to Bayes' theorem, we have f(B|S,X)=f(S|B,X)f(B|X)/∫ f(S|b,X)· f(b|X)db. We obtain the maximum likelihood estimator (MLE) for F^B|X using data from all individuals with B measured, { i: δ_Bi=1 }. For estimation of F^S|B,X=F^S|B,X,Z=1, we use data from vaccine recipients who have both S and B measured, with inverse probability weighting (IPW) used to account for biased sampling of S. Even if there is a CPV component in the study design, we can not use S for placebo recipients obtained during CPV because placebo recipients who are infected at study closeout have zero probability of obtaining S thus IPW is not applicable. The estimator of β is then derived as the maximizer of the estimated likelihood L(β,ν̂) and mCEP(S,B,X)=h(g{β;S,B,Z=1,X },g{β;S,B,Z=0,X } provides an estimate of bivariate treatment effect modification by S and B, adjusting for X. Standard errors for β can be estimated using a perturbation resampling technique. In essence, one can generate n random realizations of ϵ from a known distribution with mean of 1 and variance of 1 to create ℰ≡{ϵ_i, i=1,2,...,n}. Let L^(ϵ)(β,ν̂^(ϵ)) be a perturbed version of L(β,ν̂), where L^(ϵ)(β,ν)≡∏_i=1^nf(Y_i|Z_i,X_i,δ_Bi, δ_i,δ_BiB_i, δ_iS_i)·ϵ_i and ν^(ϵ) is the perturbed estimator of ν with ℰ being the weights. Then the perturbed estimator β^(ϵ) is a maximized of L^(ϵ)(β,ν̂^(ϵ)). In practice, one may obtain a variance estimator of β̂ based on the empirical variance of B realizations of β^(ϵ). In our simulation and example, we use B=500. §.§ Baseline Seropositive/Seronegative mCEP Curves In this section, we study the special case where B denotes the baseline biomarker values subject to detection limit, c, and derive the estimator for the marginal mCEP curve (mCEP(S)≡ h(risk_1(S),risk_0(S))), baseline seropositive mCEP curve (mCEP(S,B>c)≡ h(risk_1(S,B>c),risk_0(S,B>c))) and the baseline seronegative mCEP curve (mCEP(S,B=c)≡ h(risk_1(S,B=c),risk_0(S,B=c))). With some calculations, the risk functions in our estimands of interest can be expressed as: marginal risk function risk_Z(S,X)=∫_c-^∞risk_Z(S,b,X)dF^B|S,X(b|S,X); seropositive risk function risk_Z(S,B>c,X)=P(Y(Z)=1,B>c|S,X)/P(B>c|S,X)=∫_c+^∞risk_Z(S,b,X) dF^B|S,X(b|S,X) /P(B>c|S,X); and seronegative risk function risk_Z(S,B=c,X). All three risk functions can be estimated based on β̂ and ν̂.We consider situations where X is categorical with D levels: x_1, x_2,...,x_D. Then risk_Z(S)=∑_j=1^D risk_Z(S,x_j) · P(X=x_j|S) risk_Z(S,B>c) =∑_j=1^D risk_Z(S,B>c,x_j) · P(X=x_j|S,B>c) risk_Z(S,B=c)=∑_j=1^D risk_Z(S,B=c,x_j) · P(X=x_j|S,B=c). We model P(X|S), P(X|S, B>c), and P(X|S, B=c) using a multinomial logistic function with parameter γ and estimate γ by MLE. Because sampling of B and S depends on other phase-I variables such as Y, inverse prob. weighting (IPW) (Horvitz and Thompson, 1952) can be implemented.Appendix provides a detailed estimation procedure of mCEP(S), mCEP(S,B>c), and mCEP(S,B=c) for the case where F^B|X is assumed censored normal, F^S|B,X is assume censored normal, and the risk functions take the form risk_z{ S,B,X }=g{β;S,B,Z,X }=Φ(β_0+β_1Z+β_2S+β_3Z· S+β_4B+β_5Z· B+β_6X). A perturbation resampling method can be used to make simultaneous inference of marginal mCEP curve, baseline seropositive mCEP curve and baseline seronegative mCEP curve. To be specific, perturbed estimators β^(ϵ) and ν^(ϵ) are obtained based on ℰ. Then the corresponding perturbed estimators mCEP^(ϵ)(S), mCEP^(ϵ)(S,B>c), and mCEP^(ϵ)(S,B=c) are obtained by plugging in β^(ϵ) and ν^(ϵ) in equation <ref>, <ref>, and <ref>. Repeat this process B times to obtain B realizations of mCEP^(ϵ)(S), mCEP^(ϵ)(S,B>c), and mCEP^(ϵ)(S,B=c), and calculate the sample standard deviations σ̂_mCEP(S), σ̂_mCEP(S, B>c), σ̂_mCEP(S, B=C). 100(1-α)% pointwise confidence intervals can be constructed as mCEP(S)±𝒵_1-α/2σ̂_mCEP(S) mCEP(S, B>c)±𝒵_1-α/2σ̂_mCEP(S, B>c) mCEP(S, B=c)±𝒵_1-α/2σ̂_mCEP(S, B=c).And 100(1-α)% simultaneous confidence bands for S ∈ζ can be constructed as mCEP(S)±𝒬_1-ασ̂_mCEP(S) mCEP(S, B>c)±𝒬'_1-ασ̂_mCEP(S, B>c) mCEP(S, B=c)±𝒬”_1-ασ̂_mCEP(S, B=c),where 𝒵_1-α/2 is the 100(1-α/2)th percentile of N(0,1), 𝒬_1-α is the 100(1-α)th percentile of sup_S ∈ζ |√(n){mCEP^risk(ϵ)(S)- mCEP(S)}/σ̂_mCEP(S) |, and 𝒬'_1-α and 𝒬”_1-α defined similar to 𝒬_1-α with the mCEP estimator, mCEP perturbed estimator and standard error estimator replaced by its own version. Furthermore, simultaneous inference enables evaluation of the hypothesis testing of H_0: mCEP(S,B>c)=mCEP(S,B=c) for S ∈ζ. We first construct the simultaneous confidence band for mCEP(S, B>c) -mCEP(S, B=c). Let σ(mCEP(S, B>c) -mCEP(S, B=c)) denote the sample standard deviation of the perturbed estimates mCEP^(ϵ)(S, B>c) -mCEP^(ϵ)(S, B=c). Let 𝒬”'_1-α be the 100(1-α)th percentile ofsup_S ∈ζ |√(n){mCEP^(ϵ)(S,B>c)-mCEP^(ϵ)(S,B=c)-(mCEP(S,B>c)-mCEP(S,B=c) )}/σ (mCEP(S, B>c) -mCEP(S, B=c) ) | . Subsequently, the 100(1-α)% simultaneous confidence bands for mCEP(S, B>c) -mCEP(S, B=c), S ∈ζ is( l_α(S),u_α(S))≡mCEP(S, B>c) -mCEP(S, B=c) ±𝒬”'_1-ασ(mCEP(S, B>c) -mCEP(S, B=c)).The two-sided p-value for the testing H_0: mCEP(S,B>c)=mCEP(S,B=c) is defined as the minimum of α_1 and α_2 that satisfyinf_S ∈ζ u_α_1(S)=0, sup_S ∈ζ l_α_1(S)≤ 0 sup_S ∈ζ l_α_2(S)=0,inf_S ∈ζ u_α_2(S)≥0 .Note that at least one of α_1 and α_2 always exists.§ SIMULATION STUDIES Through simulation studies, we evaluate the finite-sample performance of our proposed estimators. Simulation data are generated with 10,000 subjects randomized to vaccine and placebo by a ratio of 2:1. Baseline covariate X was generated with a multinomial distribution to have four categories, 1, 2, 3, and 4 with corresponding probabilities of 0.25, 0.25, 0.25 and 0.25. X_2, X_3, and X_4 are dummy variables indicating category 2, 3, or 4, respectively. Baseline biomarker values B were generated from a normal distribution with mean of 1.38+0.93X_2+1.25X_3-0.25X_4 and standard deviation of 0.86. S were generated from a normal distribution with mean of 1.5+0.5B+0.2X_2-0.1X_3+0.4X_4 and standard deviation of 0.4, which indicates a correlation of 0.7 between S and B. Let the limit of biomarker value detection be 1. Simulated values of S and B less than 1 were set equal to 1. We assume a probit risk model of the clinical outcome Y conditional on S, B, Z, and X: P(Y=1|S,B,Z,X)=Φ(β_0+β_1Z+β_2S+β_3Z· S+β_4B+β_5Z· B+β_6X). We set (β_0, β_1, β_2, β_3, β_4, β_5, β_6) as (-0.50,0.16, -0.34, -0.21, -0.25, 0,(0.24,0.11,0.20 ) ) so that the probability of infection equals 0.04 in the placebo arm and 0.02 in the vaccine arm. These simulation parameters were chosen to reflect the characteristics of the two Phase 3 Dengue trials. To achieve a non-monotone sampling design, 35% of study participants have B retained. For the BIP-only design, S is set missing for all placebo recipients and retained in all cases and all subjects with B measured in the vaccine arm, that is { i: Z_i=1, Y_i=1 }∪{ i: Z_i=1, δ_Bi=1 }. For the BIP+CPV design, 70% of event-free placebo recipients are included in the CPV component and have S retained. Simulation results are based on 500 Monte-Carlo simulations and for each simulation 250 perturbation iterations are generated to construct point-wise confidence intervals and simultaneous confidence bands.We then evaluate the finite-sample performance of our proposed estimators for the marginal VE curve (VE (S=s_1 )), baseline seropositive VE curve (VE (S=s_1, B>c )), and baseline seronegative VE curve (VE (S=s_1, B=c )). Results are presented in Figure <ref> for BIP-only design and Figure <ref> for BIP+CPV design. The empirical coverage levels of the 95% simultaneous confidence bands from the perturbation methods are also reported as "simultaneous.cover" in Figure <ref> and <ref>. They demonstrate satisfactory performance of our proposed estimators, including nominal coverage probabilities of the confidence intervals given fixed s_1 values and simultaneous confidence band across all s_1 values. § APPLICATION TO THE CYD14 AND 15 TRIALSCYD14 (CYD15) is an observer-masked, randomized controlled, multi-center, phase 3 trial in five countries in the Asia-Pacific (Latin America) region where participants were randomized to receive three injections at month 0, 6, and 12. The primary goal is to assess vaccine efficacy against symptomatic, virologically confirmed dengue (VCD) occurring more than 28 days after the third injection. CYD14 achieved 56.5% efficacy (95% CI 43.8-66.4) and CYD15 achieved 60.8% efficacy (95% CI 52.0-68.0) in the per-protocol population. Concentrations of dengue neutralizing antibody titers to each of the four dengue serotype strains at month 13 were measured for all VCD cases and a subset of controls, of which only a fraction have their baseline titers measured. In this illustration, we applied our proposed method to data pooling across CYD14 and CYD15 9-16 year olds to assess how VE varied by Month 13 titers within baseline seropositive and seronegative subgroups. <cit.> provided the justification for pooling data across these two trial for data analysis purpose. We let S be the average of the log10-transformed neutralizing antibody titers to each of the four dengue serotypes at month 13 and B be the average of the log10-transformed titers at baseline. Baseline seropositive and seronegative subgroups are defined as B>1 and B=1. CYD14 and CYD15 hold a non-monotone sampling design. B is available for everyone in the immunogenicity subset ({ i: δ_B=1 }) and S is available for all vaccine recipients who are either in the immunogenicity subset or are cases ({ i:Z_i=1,δ_Bi=1 }∪{ i: Z_i=1, Y_i=1 }). These two sets do not have an inclusion relationship. X denotes participants' age and country categories, and Y is the indicator of VCD that took place between month 13 and end of follow up (month 25).Figure <ref> shows the estimated VE curve and 95% CIs and CBs based on 500 perturbation iterations. VE curves were similar for baseline seropositive and baseline seronegative subgroups, with estimated VE approximately 25% for vaccine recipients with no seroresponse at Month 13. For vaccine recipients with Month 13 average titers of 500 and 10,000, estimated VE was 79.3% and 97.3% for the baseline seropositive subgroup compared to 70.4% and 91.8% for the baseline seronegative subgroup, respectively. Furthermore, we tested the null hypothesis H_0: BL seropositive VE(S)=BL seronegative VE(S) for S ∈ range of month 13 average titer in vaccinees in the data using procedure provided in section 2.2, which gave a p-value of 0.35. This suggests that the seropositive VE curve was not significantly different from the seronegative VE curve, implying that it is not the new neutralization response generated by the vaccine over baseline value that predicts VE, but rather the absolute titer achieved following vaccination. See <cit.> for the reporting of the full analysis to a clinical audience.§ DISCUSSION In this article, we developed an estimated likelihood approach to evaluate the bivariate treatment effect modification analysis by biomarker-based principal strata and baseline covariates in general settings without requirements of a nested sub-sampling relationship between the immune response biomarker and baseline predictors suitable to both the BIP-only design and the BIP+CPV design. In our settings, the biomarker sampled set { i: δ_i=1 } and the baseline covariates sampled set { i: δ_Bi=1 } do not need to be a subset of the other. Dengue vaccine trials CYD14 and CYD15 are examples of this non-inclusion relationship. Our proposed method can apply to the special case where { i: δ_i=1 } is a subset of { i: δ_Bi=1 }. An example would be a three-phase sampling design when lab-assay-based baseline covariates are only measured from a subset of the trial participants due to high costs of acquiring lab assay and the vaccine-induced immune response is further measured among a subset of participants for whom the lab-assay-based baseline covariates are available. The phase 3 Zostavax Efficacy and Safety Trial (ZEST) adopted such a three-phase sampling design to study the effect of the Zostavax vaccine against varicella zoster virus (VZV) (<cit.>). Under this sampling framework, <cit.> proposed a semiparametric pseudo-score estimator based on conditional likelihood and also develop several alternative semiparametric estimated likelihood estimators when B is discrete. One can think of our work as an extension of <cit.> that our proposed method can incorporate broader sampling settings including the one in Huang (2017) and our baseline covariate B can be either discrete or continuous, possibly subject to detection limit left censoring. In general, our methods are applicable to intervention studies where a bivariate effect modification analysis is of interest where the bivariate is a post-randomization measurement and a baseline covariate, and measurements of one do not necessarily imply measurements of the other.§ SUPPLEMENTARY MATERIALSupplementary material is available online at <http://biostatistics.oxfordjournals.org>. § ACKNOWLEDGMENTSThe authors thank the participants, investigators, and sponsors of the CYD14 and CYD15 trials. Research reported in this publication was supported by Sanofi Pasteur and the National Institute of Allergy and Infectious Diseases (NIAID), National Institutes of Health (NIH), Department of Health and Human Services, under award number R37AI054165. The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH or Sanofi Pasteur. Conflict of Interest: None declared. biorefs | http://arxiv.org/abs/1710.09923v1 | {
"authors": [
"Yingying Zhuang",
"Ying Huang",
"Peter B. Gilbert"
],
"categories": [
"stat.ME"
],
"primary_category": "stat.ME",
"published": "20171026214232",
"title": "Evaluation of Treatment Effect Modification by Biomarkers Measured Pre- and Post-randomization in the Presence of Non-monotone Missingness"
} |
http://arxiv.org/abs/1710.10236v2 | {
"authors": [
"Benjamin Horowitz",
"Simone Ferraro",
"Blake D. Sherwin"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20171027164655",
"title": "Reconstructing Small Scale Lenses from the Cosmic Microwave Background Temperature Fluctuations"
} |
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[email protected] Department of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ UK Department of Physics and Astronomy, Macquarie University, 2109 NSW, Australia Department of Physics and Astronomy, Macquarie University, 2109 NSW, Australia Displayr Australia Pty Ltd,Glebe 2037 NSW, Australia The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to a further timescale with respect to which the coarse-graining time interval can be chosen. Different choices lead to different master equations that correspond to the local and global forms. These can be then understood as having different physical interpretations based on the role of the coarse-graining, as well as different limitations in application. Coarse-graining in the derivation of Markovian master equations and its significance in quantum thermodynamics C. Facer December 30, 2023 ==============================================================================================================§ INTRODUCTION The master equation has long been an essential tool in describing the dynamics of the reduced density operator of an open quantum system. This is particularly the case for systems described by Markovian master equations, equations which can be informally understood as describing a dynamic which possesses `no memory'. Themaster equation is known in this case to take a particular form <cit.>, typically referred to as the Lindblad form. However there are on-going issues that arise where the derivation of these equations from a microscopic model is concerned. Physically, a master equation describes the dynamics of a system coupled to some other, usually much larger system, typically the environment with which the system is invariably in interaction, and this usually modelled as a thermodynamic reservoir, and as such, the master equation is derived from a microscopic description of the system+reservoir. Except in a small handful of instances, of which an important example is that of harmonic oscillators coupled to a reservoir modelled as a collection of harmonic oscillators, the derivation of the exact master equation proves to be an intractable problem. However, fortunately, for many systems of on-going interest, approximations can be invoked yielding a master equation that is of the required Lindblad form. There are a number of ways that this master equation can be derived, though all of the derivations rely on there being a wide separation of typical timescales making up the dynamics, these timescales typically being the correlation time of the reservoir τ_c, the system evolution timescale τ_S, typically a system decay time, and system Bohr frequency timescales ω_S^-1, with τ_c≪ω_S^-1≪τ_S. One way or another, a coarse-graining, or smoothing of this dynamics over the very short time scales τ_c and ω_S^-1 plays an essential role in the derivation. In the early work of Bloch and Redfield <cit.> and later Cohen-Tannoudji et. al. <cit.>, the coarse-graining is initially done explicitly through the introduction of a coarse-graining interval Δ t which is constrained to satisfy τ_c≪Δ t≪τ_S. But in <cit.>, the role of explicit coarse-graining is bypassed and Born and Markov approximations introduced that nowadays are commonly implemented in derivations such as that presented in <cit.> where coarse-graining is not made explicit, though arguments based on timescales are used to introduce these approximations to simplify the exact master equation to a tractable form. Typically, a further approximation is required, the secular approximation, which can be understood as an averaging over terms in the master equation that are rapidly oscillating with a frequency ∼ω_S^-1, and is hence also a timescale based argument, leading to the required Lindblad form. Variations on the explicit coarse-graining approach have begun to appear increasingly often <cit.>. There are advantages in the latter method. Apart from anything else, the explicit coarse-graining method leads directly to a master equation of Lindblad form for essentially any choice of Δ t, i.e., the secular approximation is not required as a separate step, though the secular approximated form of the master equation will also follow for Δ t made sufficiently large <cit.>. Being able to choose Δ t in such a way emphasizes the point that the coarse-graining time interval over which the dynamics are smoothed is in fact a free parameter of the theory. Different choices of Δ t can be made, with an emphasis on, for instance, obtaining better approximations to the exact master equation <cit.>, or else aiming to consistently include interference (or cross-damping) terms <cit.>. The choice of time scale acquires greater significance when the evolution of the system has more than one time scale beyond the typical ones listed above. This situation is encountered in general when the system structure is such that there are dynamics internal to the system with their own well-defined time scale such as, for instance, for composite systems in which the internal coupling of the systems gives rise to a further time scale Ω^-1 where τ_c≪ω_S^-1≪Ω^-1≪τ_S. Such composite systems are currently of considerable interest in quantum thermodynamical applications, particularly when the individual subsystems are part of some kind of heat engine, and where each subsystem is coupled to separate thermodynamic reservoirs at different temperatures so that heat can flow from one reservoir to the other. In such instances, the question arises as to how to model the damping of the system(s), whether by the local or global approaches to deriving the Born-Markov-secular approximated form of the master equation <cit.>. Local coupling describes a model in which each reservoir is coupled only to the energy eigenstates of its associated subsystem, while global coupling describes a model in which each reservoir is coupled to the energy eigenstate of the combined system. The former has the unwanted property of predicting, for reservoirs all of the same temperature (or only one reservoir) a steady state which is not the expected Gibbs distribution, which can only be resolved by using the global approach, an issue which was first noted in the work of Carmichael and Walls <cit.> for coupled, damped harmonic oscillators, and subsequently dealt with for the damped Jaynes-Cummings model in <cit.>. But the global approach can lead to unexpected outcomes, e.g., <cit.> examine a circumstance in which there is no heat flow at all between reservoirs of different temperatures if the secular approximate forms of the master equation is used. The presence of an extra internal time scale also raises the possibility of not making the secular approximation (achieved by coarse-graining on a timescale Δ t≫Ω^-1), but by taking Δ t≪Ω^-1, referred to here as a partial secular approximation which will lead to a different but still Lindbladian master equations for the same set-up, but staying within the global approach. The impact of this choice on the expected heat flow between reservoirs of different temperatures if the appropriate master equation is used is one of the issues addressed in this paper. It can further be argued that Δ t can have a physical interpretation in terms of the temporal resolution of measurements made on the environment. The physical interpretations of the different forms of the master equation, apart from the fact that they are different approximations to the same underlying exact dynamics, is also investigated in this paper. The paper is structured as follows. In Section <ref>, the master equation derivation is presented in a way that leads to both the usual Bloch-Redfield result, and the coarse-grained version of <cit.>. Some comments on the Markov approximation, and a collisional model interpretation of the coarse-grained result, are also presented. In Section <ref>, the particular case of a system coupled to a bosonic reservoir is analysed, and the limiting forms for the full secular approximation and a partial secular aproximation for composite systems are introduced along with a proof that the partial secular approximated form for the master equation is still of Lindblad form. A measurement interpretation of the coarse-grained master equation is discussed in Section <ref> and the generalisation of some preceding results to the case of more than one independent reservoir is given in Section <ref>. Examples of composite systems are then given in Section <ref>.§ MASTER EQUATION DERIVATION Given a microscopic model of a system S interacting with a reservoir R as described by the total Hamiltonian of the combined system S⊕ R:H=H_S+H_R+V=H_0+Vthe evolved combined system density operator χ(t) is then given by χ(t)=U(t)χ(0)U^†(t) where U(t)=exp(-iHt/ħ) and the system density operator at time t given by ρ(t)=Tr_R[χ(t)].We will work in the interaction picture (indicated by overbars) so that the state of the combined system plus reservoir will evolve according toχ̅(t)=U̅(t,0)χ(0)U̅^†(t,0)where the time evolution operator in the interaction picture U̅(t,0)=e^iH_0t/ħe^-iHt/ħsatisfiesdU̅/dt=-i/ħV̅(t)U̅withV̅(t)=e^iH_0t/ħVe^-iH_0t/ħand the initial state must be taken to be the product state, χ(0)=ρ(0)⊗ρ_R(0).A refined version of the arguments of <cit.> are presented below, leading to a second order perturbative result that yields, in different limits, both the required coarse-grained master equation, as well as the Born-Markov master equation. §.§ Perturbative expansion of system density operator The derivation begins by obtaining an exact expression for the evolution of the system density operator over an interval (t,t+Δ t)ρ̅(t+Δ t)= Tr_R [χ̅(t+Δ t)] = Tr_R[U̅(t+Δ t,t)χ̅(t) U̅^†(t+Δ t,t)]where U̅(t+Δ t,t)=U̅(t,0)U̅(t+Δ t,0)^-1. We now note, as is done in Cohen-Tannoudji <cit.>, that χ̅(t) can be written χ̅(t)=ρ̅(t)⊗ρ̅_R(t)+[χ̅(t)-ρ̅(t)⊗ρ̅_R(t)] where the second term represents the contribution to χ̅(t) due to the correlations between the system and the reservoir at time t. But we can proceed further and separate out the corrections arising due to making the Born approximation. In this approximation it is assumed that, as far as the system is concerned, the state of the reservoir can be taken as being unchanged by the interaction, and further that any correlations that develop between the system and the reservoir over the time interval (0,t) are ignored. This can be implemented here by writingχ̅(t)= ρ̅(t)⊗ρ_R(0) +ρ̅(t)⊗[ρ_R(t)-ρ_R(0)]+[χ̅(t)-ρ̅(t)⊗ρ̅_R(t)] = ρ̅(t)⊗ρ_R(0)+χ̅_corr(t) in which χ̅_corr(t) is the contributions due to the correlations that develop between the system and the reservoir up to time t, plus corrections associated with changes to the state of the reservoir due to its coupling to the system, usually ignored under the Born approximation. Note that χ̅_corr does not give rise to a `correction' to the system density operator since Tr_R[χ̅_corr(t)]=0.If we now consider evolution for a further period Δ t, we could then trivially write χ̅(t+Δ t)=ρ̅(t+Δ t)⊗ρ_R(0)+χ̅_corr(t+Δ t), but this is not of much value. Instead, we can also writeχ̅(t+Δ t)= U̅(t+Δ t,t)ρ̅(t)⊗ρ_R(0)U̅^†(t+Δ t,t)+U̅(t+Δ t,t)χ̅_corr(t)U̅^†(t+Δ t,t)so thatρ̅(t+Δ t)= Tr_R[U̅(t+Δ t,t)ρ̅(t)⊗ρ_R(0)U̅^†(t+Δ t,t)]+Tr_R[U̅(t+Δ t,t)χ̅_corr(t)U̅^†(t+Δ t,t)].The second term on the right hand side, which is now non-zero for Δ t 0, is the contribution to the state of the system at time ρ̅(t+Δ t) due to the correlations/reservoir state present at time t. But it also represents an error that accumulates over the time interval (t,t+Δ t) if the exact density operator at time t is approximated by the product stateρ̅(t)⊗ρ_R(0).We now want to expand this expression to second order in the interaction V. In doing so, the separation given by Eq. (<ref>) makes it possible to implement the machinery of the Zwanzig-Nakajima projection operator method <cit.>, much used in deriving formally exact non-Markovian master equations. We first introduce the projection operators 𝒫,𝒬=1-𝒫 with 𝒫 defined by𝒫χ̅(t)=Tr_R[χ̅(t)]⊗ρ_R(0)=ρ̅(t)⊗ρ_R(0).We then find that χ̅_corr(t)=𝒬[χ̅(t)] so that, as already expected, Tr_R[χ̅_corr(t)]=Tr_R[𝒬[χ̅(t)]]=0. We then have, from Eq. (<ref>)ρ̅(t+Δ t) = Tr_R[U̅(t+Δ t,t) ρ̅(t)⊗ρ_R(0)U̅^†(t+Δ t,t)]+Tr_R[U̅(t+Δ t,t)𝒬[χ̅(t)]U̅^†(t+Δ t,t)].The usual procedure for deriving the equations of motion of 𝒫[χ̅] and 𝒬[χ̅] can be employed here. The equation of motion for χ̅(t)dχ̅/dt=-i/ħ[V̅(t),χ̅] =ℒ̅(t)χ̅has the formal solution χ̅(t)=U̅(t,0)χ̅(0)U̅^†(t,0) =𝒢̅(t,0)χ̅(0) where the free propagator for the system plus reservoir in the interaction picture is given by𝒢̅(t,t')=Texp(∫_t'^tℒ̅(t”)dt”)where T indicates time ordering. This yields the following equation of motion for 𝒬[χ̅(t)]:d𝒬[χ̅(t)]/dt=𝒬ℒ̅(t)𝒬[χ̅(t)]+𝒬ℒ̅(t)𝒫[χ̅(t)].On using the initial condition 𝒬[χ̅(0)]=𝒬[ρ(0)⊗ρ_R(0)]=0 the formal solution for 𝒬[χ̅(t)] is𝒬[χ̅(t)] =∫_0^t𝒢̅_𝒬(t,t_1) 𝒬[ℒ̅(t_1)𝒫[χ̅(t_1)]]dt_1where𝒢̅_𝒬(t,t') =Texp(∫_t'^t𝒬ℒ̅(t”)dt”). In terms of these propagators we can then write ρ̅(t+Δ t)= Tr_R[𝒢̅(t+Δ t,t)𝒫[χ̅(t)]]+Tr_R[𝒢̅(t+Δ t,t)∫_0^tdt_1𝒢̅_𝒬(t,t_1)𝒬[ℒ̅(t_1)𝒫[χ̅(t_1)]]]We are only going to consider contributions to second order in the interaction V, so we will make the approximations𝒢̅(t+Δ t,t)≈ 1+∫_t^t+Δ tdt_1ℒ̅(t_1)+∫_t^t+Δ tdt_2∫_t^t_2dt_1ℒ̅(t_2)ℒ̅(t_1)and𝒢̅_𝒬(t,t_1)≈ 1+∫_t_1^tdt”𝒬 ℒ̅(t”).Making use of the exact results Tr_R[ℒ̅(t)ρ_R(0)]=0 and Tr_R[𝒬[χ]]=0 as well as Eq. (<ref>) then givesρ̅(t+Δ t)-ρ̅(t)/Δ t≈ -1/ħ^21/Δ t∫_t^t+Δ tdt_2∫_t^t_2dt_1 Tr_R[[V̅(t_2),[V̅(t_1),ρ̅(t)⊗ρ_R(0)]]] -1/ħ^21/Δ t∫_t^t+Δ tdt_2∫_0^tdt_1 Tr_R[[V̅(t_2),[V̅(t_1),ρ̅(t_1) ⊗ρ_R(0)]]]This result is similar to that found in <cit.>, Eq. (IV D.7), though there the separation Eq. (<ref>) was not used, and a phenomenological argument was used to determine the form of the second term. There are now two directions that this result can be taken. §.§.§ No coarse-graining limit There is nothing in the above derivation that places, at this stage, any limit on Δ t other than being sufficiently small that a second order expansion in the interaction V is valid. In particular, we can now let Δ t→ 0, i.e., implying that there is no coarse-graining. The left hand side is just the derivative of ρ̅, the first term on the right hand side of Eq. (<ref>) vanishes and the second reduces todρ̅/dt=-1/ħ^2∫_0^tdt_1 Tr_R [[V̅(t),[V̅(t_1),ρ̅(t_1)⊗ρ_R(0)]]].This is the post-Born approximation result obtained in a typical derivation of the Bloch-Redfield master equation <cit.>. There is no explicit coarse-graining in time. The implicit coarse-graining of the subsequent Markov approximation in which ρ(t_1)≈ρ(t) then yieldsdρ̅/dt=-1/ħ^2∫_0^∞dt_1 Tr_R [[V̅(t),[V̅(t_1),ρ̅(t)⊗ρ_R(0)]]]with, if necessary, the secular approximation to follow to then yield the final Lindblad form of the master equation.§.§.§ Coarse-graining limit An alternate derivation as exemplified by that of Cohen-Tannoudji <cit.> makes the coarse graining time scale Δ t an explict part of the derivation of the master equation, and amounts to approximating the derivative of the interaction picture density operator by a coarse-grained version.dρ̅/dt≈Δρ̅/Δ t=ρ̅(t+Δ t)-ρ̅(t)/Δ ti.e., the instantaneous rate of change dρ̅/dt is effectively smoothed out on a time scale Δ t – the `coarse-grained' time scale – where τ_c≪Δ t≪τ_S. Coarse-graining in this manner was the starting point in the early work of Redfield <cit.>, but close attention to some of the underlying approximations was first done in <cit.> where it was shown that this coarse-grained derivative emerges naturally under conditions in which correlations embodied in χ̅_corr that develop between the system and the reservoir are negligible, and the Born approximation is valid.The explicit coarse-graining result follows if Δ t is kept non-zero, and t and Δ t chosen such that t,Δ t≫τ_c. The arguments of <cit.> can then be applied. The first term in Eq. (<ref>) will be of order 1/τ_S, while the second term, by virtue of the fact that the integrals are over non-overlapping intervals, will be of order (1/τ_S)·(τ_c/Δ t) and hence can be ignored relative to the first. The argument is outlined further in Appendix <ref>. This result now provides us with an approximate form for the coarse-grained time derivative Δρ̅/Δ t≈-1/ħ^21/Δ t∫_t^t+Δ tdt_2∫_t^t_2dt_1 Tr_R[[V̅(t_2),[V̅(t_1),ρ̅(t)⊗ρ_R(0)]]].The required coarse-grained master equation in the Schrödinger picture is then obtained by noting that dρ/dt=-i/ħ[H_0,ρ]+e^-iH_0t/ħdρ̅/dte^iH_0t/ħ.and we now approximate the exact time derivative on the right hand side by its coarse-grained approximation, in accordance with Eq. (<ref>), givingdρ/dt≈ -i/ħ[H_0,ρ]+e^-iH_0t/ħΔρ̅/Δ te^iH_0t/ħ ≈ -i/ħ[H_0,ρ] -1/ħ^21/Δ t∫_t^t+Δ tdt_2∫_t^t_2dt_1 Tr_R[[V̅(t_2-t),[V̅(t_1-t),ρ(t)⊗ρ_R(0)]]].A simple change of variable then yields the approximate resultdρ/dt≈-i/ħ[H_0,ρ] -1/ħ^21/Δ t∫_0^Δ tdt_2∫_0^t_2dt_1 Tr_R[[V̅(t_2),[V̅(t_1),ρ(t)⊗ρ_R(0)]]].This is an approximate equation for the exact density operator ρ(t). We can, instead, write an exact equation for the approximate density operator, which we could call, for instance, ρ_Δ t(t) to emphasize that it has been obtained by coarse-graining over a time interval Δ t. But, instead of using this cumbersome notation, we will instead simply continue to write ρ(t), which then satisfies the equationdρ/dt=-i/ħ[H_0,ρ] -1/ħ^21/Δ t∫_0^Δ tdt_2∫_0^t_2dt_1 Tr_R[[V̅(t_2),[V̅(t_1),ρ(t) ⊗ρ_R(0)]]]with the understanding that solving this equation will yield a coarse-grained approximate result for the exact ρ. This is the coarse-grained master equation as given by Cohen-Tannoudji <cit.>, Eq. (IV B.30), and derived elsewhere by similar arguments.As has been shown in a number of places <cit.>, manipulation of the double time integral enables this result to be rewritten asdρ/dt=-i/ħ[H_0+Δ H,ρ] -1/2ħ^21/Δ t∫_0^Δ tdt_2∫_0^Δ tdt_1 Tr_R[[V̅(t_2),[V̅(t_1),ρ(t) ⊗ρ_R(0)]]] with the energy shift Δ H given byΔ H=-i/2ħ1/Δ t∫_0^Δ tdt_2∫_0^t_2dt_1 Tr_R[[V̅(t_2),V̅(t_1)]ρ_R(0)].Hereinafter, the energy shift term will assumed to be sufficiently small as to be ignored.Defining𝒱(t_2,t_1)=∫_t_1^t_2V̅(t)dtthe dissipator in Eq. (<ref>) becomes𝒟[ρ]=-1/2ħ^21/Δ tTr_R[[𝒱(Δ t,0),[𝒱(Δ t,0),ρ(t)⊗ρ_R(0)]]]which is clearly of the required Lindblad form. The secular approximation as a separate step in the calculation is not needed to achieve this. Nevertheless, the secular approximated master equation can be regained here by a suitably large choice of Δ t, but the generality of the result Eq. (<ref>) implies that master equations not of the secular approximated form, but nevertheless still Lindblad, can be obtained, as discussed below in Section <ref>. §.§ Where is the Markov approximation? The general result Eq. (<ref>) leads to the two different forms for the master equation. In the first, Eq. (<ref>), the Markov approximation is yet to be implemented, and would be done so in the usual way <cit.>. In the second, equation Eq. (<ref>) is clearly Markovian, but the Born and Markov approximations have emerged as a consequence of the substitution Eq. (<ref>), showing that provided the coarse-graining time Δ t is made sufficiently large, the memory contributions implied by the correction term χ̅_corr can be neglected. Thus, the Markov nature of the master equation has emerged as a consequence of the coarse-grained averaging which has smoothed out the evolution of ρ(t) on a coarse-grained time-scale Δ t. §.§ Collisional model interpretation Returning to the expression Eq. (<ref>) in the Δ t≫τ_c limit, which can be written, with t=nΔ t, ρ̅_n=ρ̅(nΔ t), and 𝒢̅_n=𝒢̅(nΔ t,(n-1)Δ t) asρ̅_n+1⊗ρ_R(0)=𝒫𝒢_n+1ρ̅_n⊗ρ_R(0)which can be iterated and after tracing over the reservoir states givesρ̅_n=Tr_R[𝒢_n𝒫𝒢_n-1…𝒫𝒢_1[ρ(0)⊗ρ_R(0)]].This result has a ready interpretation. The system is exposed to the reservoir for an interval of time Δ t after which interaction ceases, and the reservoir returned to its original state ρ_R(0). Thereafter the system and reservoir in its initial state ρ_R(0) come into interaction again for a further period Δ t and so on. This is a picture reminiscent of the quantum collision models of open quantum systems currently attracting considerable attention, see e.g., <cit.> and references therein.§ COUPLING TO A BOSONIC THERMAL RESERVOIR We will now use the above result to look at the often encountered example of a system coupled to a reservoir of simple harmonic oscillators for which the interaction isV=BX where X is some system operator and B is the reservoir operatorB̅(t)=∫_0^∞g(ω)(b(ω)e^-iω t+b^†(ω)e^-iω t)dω.where [b(ω),b^†(ω')]=δ(ω-ω') and withH_R=∫_0^∞ħω b^†(ω)b(ω) dω.The above choice of model for the reservoir is much like that for the quantized electromagnetic field, and so excitations of the reservoir will be referred to below as photons, though it would probably be more appropriate to label them simply as `quanta'. The reservoir is assumed to be in the thermal equilibrium stateρ_R(0)=e^-H_R/kT/Tr_R[e^-H_R/kT]. The dissipative part of the master equation Eq. (<ref>) can then be written𝒟[ρ] =1/ħ^2Δ t∫_0^Δ tdt_2 ∫_0^Δ tdt_1G(t_2-t_1)[X̅(t_1)ρX̅(t_2) -12{X̅(t_2)X̅(t_1),ρ}].where the reservoir correlation function isG(t_2-t_1)= Tr_R[B(t_2)B(t_1)ρ_R] = ∫_0^∞g(ω)^2( (2n(ω)+1)cosω (t_2-t_1)-isinω (t_2-t_1)) dω, where the reservoir occupation number at frequency ω is given by the Planck formulan(ω)=(e^ħω/kT-1)^-1and where G(t) has the symmetry property G^*(t)=G(-t).The spectral density g(ω)^2 will be assumed to be Ohmic with a high frequency cutoff such that the correlation function G(t) will decay on a time scale τ_c. For the reservoir at zero temperature, τ_c will be the vaccuum correlation decay time τ_v. As usual, g(ω) will be assumed to be a sufficiently broad function that τ_v will be very short compared to the time scale τ_S characterizing the evolution of the system. For non-zero temperatures the correlation time τ_c will be determined by both τ_v and the correlation time of the thermal reservoir, τ_T∼ħ/2π kT, with τ_v dominating at high temperatures when τ_T≪τ_v and at low temperatures, but otherwise τ_c∼τ_T. This time scale τ_c will in all instances be assumed to be very short compared to time scales characterizing the evolution of the system, τ_v,τ_c≪ω_S^-1≪τ_S where ω_S is a typical Bohr frequency for the system. A more detailed analysis is to be found in <cit.>, based on the work of Carmichael <cit.> who also gives typical estimates of these time scales for optical systems, for which the inequalities are well-satisfied.Simplification of this result can be made in a standard way by expanding the system operator X̅(t) in the energy basis of the unperturbed Hamiltonian H_0:X̅(t)=e^iH_0t/ħXe^-iH_0t/ħ =∑_a,b^|a⟩⟨ b|⟨ a|X|b⟩ e^i(ω_a-ω_b)t.Now introduce the eigenoperatorsX_m=∑_a,b^|a⟩⟨ b|⟨ a|X|b⟩δ_ω_m,ω_a-ω_bwhich are such that [H_0,X_m]=ħω_mX_m. By construction, if m n, then ω_mω_n. In fact, we can order the frequencies such that ω_m>ω_n if m>n, and we can further adopt the convention that ω_n≶ 0 if n≶ 0 and ω_0=0.The expression for X̅(t) can now be writtenX̅(t)=∑_m=-N^NX_me^iω_mt =∑_m=-N^NX^†_me^-iω_mtfrom which follows that X^†_m=X_-m and ω_m=ω_-m.The master equation now becomes, ignoring energy level shifts,dρ/dt=-i/ħ[H_S,ρ]+∑_m,n^γ_mn(Δ t) [X_mρ X_n^†-1/2{X_n^† X_m,ρ}]where the matrix elements γ_mn(Δ t) can be expressed asγ_mn(Δ t)=1/ħ^2Δ t∫_0^Δ tdt_2∫_0^Δ tdt_1 G(t_2-t_1)e^iω_m t_1e^-iω_n t_2.From this expression, and from G(t)=G^*(-t), we immediately see that γ_mn(Δ t)=γ_nm^*(Δ t) so that the γ_mn(Δ t) are elements of a Hermitean matrix which can be readily shown to be positive semidefinite, as expected from the general result Eq. (<ref>). Thus we have a legitimate Lindblad equation for any value of Δ t satisfying τ_S≫Δ t≫τ_c. However, it is only in a couple of physically distinct limiting cases that, for clear physical reasons, these elements become effectively insensitive to the value of Δ t: the secular approximation limit, and in the circumstance in which the system is a composite system for which there are internal time scales for the interaction of the various component parts of the system. §.§ Limiting cases for γ_mn To arrive at the required expressions for γ_mn, we make the change of variable t=t_2+t_1 and τ=t_2-t_1. The expression for γ_mn(Δ t) becomes γ_mn(Δ t)= 1/ħ^2∫_0^Δ tdτ[G(-τ)e^i(ω_m+ω_n)τ/2+G(τ)e^-i(ω_m+ω_n)τ/2]e^iω_mnΔ te^-iω_mnτ/2-e^iω_mnτ/2/iω_mnΔ tThis result assumes a simple form when |ω_mn|Δ t≪1, (which includes ω_mn=0) and when |ω_mn|Δ t≫1. For a given value of Δ t and for any ω_mn such that |ω_mn|Δ t≪1, and given that Δ t≫τ_c so that the upper limit on the integral can be taken to be +∞, then we can approximate Eq. (<ref>) byγ_mn(Δ t)= 1/ħ^2∫_-∞^+∞G(τ)e^-i(ω_m+ω_n)τ/2dτ-1/Δ t1/ħ^2∫_0^∞[G(-τ)e^i(ω_m+ω_n)τ/2+G(τ)e^-i(ω_m+ω_n)τ/2]τ dτThe first term in Eq. (<ref>) will turn out to be the dominant term and will define the long term evolution timescale τ_S while, as is shown in Appendix <ref>, the second term in Eq. (<ref>) is ∼τ_c/(Δ t τ_S). Thus we have γ_mn(Δ t)∼τ_S^-1(1-τ_c/Δ t) so we can neglect the second term for Δ t≫τ_c and writeγ_mn(Δ t)=1/ħ^2∫_-∞^+∞ G(τ)e^-i(ω_m+ω_n)τ/2dτ, |ω_mn|Δ t≪1.Finally, if |ω_mn|Δ t≫1, we have γ_mn(Δ t)=0. So, in summary, we have, with , τ_c≪Δ t≪τ_Sγ_mn(Δ t)= 1/ħ^2∫_-∞^+∞G(τ)e^-i(ω_m+ω_n)τ/2dτ, |ω_mn|Δ t≪1 0, |ω_mn|Δ t≫ 1 In particular we note that if m and n are of opposite signs, then |ω_mn|∼ω_S, so if Δ t is chosen such that Δ t≫ω_S^-1 it follows that γ_mn=0 for m and n of opposite signs. This tells us that there will be no contributions to the master equation of the form X_mρ X_n^† where m and n are of opposite signs. For instance, a term such as X_nρ̅X_-n^† will, in the interaction picture, oscillate rapidly as exp(2iω_nt), and are hence averaged over to zero here. Thus eliminated are all terms that are eliminated in the secular approximation as it is usually applied.These results are insensitive to the choice of Δ t, provided the inequalities in Eq. (<ref>) are satisfied. If Δ t≫|ω_mn|^-1, the dissipative dynamics (i.e., not the system dynamics) have been smoothed over oscillations of frequency ω_mn, and if Δ t≪|ω_mn|^-1, there is no smoothing of oscillations of frequency ω_mn. For ω_mn values that do not satisfy the inequalities in Eq. (<ref>), γ_mn will still be perhaps strongly dependent on the value of Δ t. The interesting case however, is if, for a given system, the values of ω_mn are such that a value of Δ t can be chosen to satisfy either |ω_mn|^-1≪Δ t≪τ_S for all ω_mn, the usual (full) secular approximation limit, or τ_c≪Δ t≪|ω_mn|^-1≪τ_S for some ω_mn, dubbed the partial secular approximation below, where it is discussed further. §.§ Full secular approximation The secular approximation as it is usually defined requires that the coarse-graining time interval Δ t be chosen so that Δ t≫|ω_mn|^-1 for all frequency differences ω_mn=ω_m-ω_n. As mentioned above, as the frequencies ω_m can be both positive and negative, this condition also includes the requirement that Δ t≫|ω_m|^-1 for all ω_m, generically written as Δ t≫ω_S^-1. In that case, the only terms that will survive are those for which ω_m=ω_n, i.e. only the diagonal terms γ_nn(Δ t) will be non-zero and the master equation Eq. (<ref>) will look likedρ/dt=-i/ħ[H_S,ρ]+∑_n^γ_nn[X_nρ X_n^†,-12{X_n^† X_n,ρ}].If we make use of the expression for G(τ) from Eq. (<ref>):G(τ)=∫_0^∞g(ω)^2[(ħω/2kT) cosωτ-isinωτ]dωso, with γ(ω)=2π g(ω)^2/ħ^2 and (ħω/2kT)=2n(ω)+1 we getγ_nn(Δ t)=γ(|ω_n|)(n(|ω_n|)+θ(-ω_n))where θ(x) is the unit step function. Typically we will ignore the dependence of γ(ω) on ω for the typical range of values of ω_n, i.e., set γ(ω)=γ and we haveγ_nn(Δ t)= γ n(|ω_n|)+γθ(-ω_n)which are the usual results for energy level damping. For such a master equation, with γ_nn as given by Eq. (<ref>), the steady state is known to be the expected Gibbs distribution <cit.>ρ(∞)=e^-H_S/kT/Z. §.§ Partial secular approximation For certain systems, the possibility exists of making more than one choice of Δ t, each still consistent with the constraints of either |ω_mn|^-1≫ or ≪Δ t of Eq. (<ref>). This arises if the frequency differences ω_mn break up into blocks of widely different frequencies, and will occur in cases in which the system can itself be considered as being made up ofinteracting subsystems. The result of such an interaction is to introduce splitting in the energy level structure of the non-interacting systems. For an interaction weak in the sense that this splitting, Ω say, will be much less than the typical transition frequencies ω_S, there can arise frequency differences ω_mn∼Ω, thereby introducing a new distinct time scale ∼Ω^-1 into the dynamics of the system, this, roughly speaking, being the time scale of exchange of energy between the subsystems. This then suggests the possibility of choosing a coarse-graining time scale ω_S^-1≪Δ t≪Ω^-1 which is sufficiently short to `capture', within the master equation, the time dependence associated with this energy exchange dynamics, or ω_S^-1≪Ω^-1≪Δ t which will lead to the secular approximation result of Section <ref>.So here we will suppose that the frequency differences |ω_mn| will be either ≲Ω, or else ≫Ω, and note that if ω_m and ω_n have opposite signs, then |ω_mn|∼ω_S, so such differences belong to the second group. This establishes a new time scale with respect to which the coarse-graining can be imposed, and which leads to a master equation which is insensitive to the choice of Δ t, provided any such choice satisfies the earlier stated conditions. Some simple models illustrating the situation will be presented below.Physically, what is being done here is to smooth the dynamics on a time scale such that any oscillations at a frequency Ω are either smoothed over, Δ t≫Ω^-1, which is the secular approximation case, or not, Δ t≪Ω^-1, which is below referred to as the partial secular approximation.Given the conditions outlined above satisfied by the frequency differences ω_mn, we can now choose Δ t to be such that Δ t≪|ω_mn|^-1 for all pairs of frequencies ω_m,ω_n such that |ω_mn|≲Ω, but Δ t≫|ω_mn|^-1 for all pairs of frequencies ω_m,ω_n such that |ω_mn|≫Ω. In this case, certain of the off-diagonal elements γ_mn(Δ t) will be non-zero, and from Eq. (<ref>) and using the expression for G(τ) in Eq. (<ref>), are given byγ_mn(Δ t)=γ n(|ω_m+ω_n|/2) +γθ(-ω_m-ω_n), |ω_mn|≲Ω,ω_S^-1≪Δ t≪Ω^-1.Finally, noting that for m and n of opposite sign ω_mn∼ω_S, and with γ_mn = 0 for Δ t≫|ω_mn|^-1∼ω_S^-1, the partial secular master equation will take the formdρ/dt=-i/ħ[H_S,ρ] +∑_m,n^θ_mnγ_mn[X_mρ X_n^†-12{X_m^† X_n,ρ}] where θ_mn=1 if mn>0 and =0 if mn<0.§.§.§ Lindblad form retained It might be expected in light of the approximations made leading to Eq. (<ref>) that the positivity of the γ matrix will no longer be guaranteed, and that the master equation will not be of Lindblad form. However, by virtue of the fact that the occupation number n(|ω_m+ω_n|/2) is evaluated at the mean of the two frequencies ω_m and ω_n, it turns out that positivity is in fact retained. Positivity amounts to the requirement that∑_m,n^a_m^*γ_mn(Δ t)a_n≥ 0for all vectors 𝐚. Thus we have to evaluate∑_m,n^ a_m^* [n(|ω_m+ω_n|/2)+θ(-ω_m-ω_m)]a_n.Since, for m and n of opposite sign ω_mn∼ω_S, and with γ_mn=0 for Δ t≫ω_S^-1, this condition means that the sum will separate into two contributions where both m and n are positive, and where they are both negative. Thus we can write∑_m,n^a_m^*γ_mn(Δ t)a_n = ∑_m,n>0^a_m^*n((ω_m+ω_n)/2)a_n+∑_m,n<0^a_m^*[n((ω_m+ω_n)/2)+1]a_nWritingn(ω)=1/e^ħω/kT-1 =∑_p=0^∞e^-(p+1)ħω/kT we find that∑_m,n^ a_m^*γ_mna_n = ∑_p=0^∞[|∑_m>0^a_me^-(p+1)ħω_m/2kT|^2 +|∑_m<0^a_me^-pħω_m/2kT|^2]≥ 0 for any choice of 𝐚, and hence we conclude that the matrix γ is positive. Thus the master equation for the partial secular case, with the elements of the matrix γ defined by the approximate result Eq. (<ref>), will still be Lindblad.§.§.§ Equilibriation vs thermalization The presence of the off diagonal elements γ_mn can result in the coherences of the system steady state not vanishing in the H_S energy basis. Consequently the steady state will not, in general, be the expected Gibbs distribution. While this might be perceived as a thermodynamic failing of the partial secular approximated master equation, it has been pointed out by Subaşı et al <cit.> (and see also articles cited therein, <cit.>) that a master equation cannotnecessarily be expected to yield the thermal equilibrium state at infinite time, the latter only emerging in the limit of vanishing system-reservoir interaction strengthlim_γ→0lim_t→∞ρ(t)=e^-H_S/kT/Tr_S[e^-H_S/kT]In an example shown below in Section <ref>, in the limit of γ→0, it is found that these coherences do vanish, so that in the limit of vanishingly small system-reservoir coupling, the expected Gibbs steady state distribution is regained. § MEASUREMENT INTERPRETATION OF COARSE-GRAINED MASTER EQUATION The above results indicate that different choices of Δ t will result in different Lindblad master equations. This outcome can be looked on as representing different levels of approximation to the exact master equation and hence to the exact system density operator. However, it can be argued that the different choices of Δ t represent different outcomes due to a physical choice, that of the temporal resolution of observations made on the reservoir. To arrive at this perspective, we need to write the master equation in terms of jump operators that represent the change in the number of photons in the reservoir. §.§ Coarse-grained derivative in terms of increments in total photon numberFirst consider the coarse-grained derivative Eq. (<ref>) written in the interaction picture and ignoring energy shift terms, Δρ̅/Δ t= -1/2ħ^21/Δ t∫_t^t+Δ tdt_2∫_t^t+Δ tdt_1 Tr_R[[V̅(t_2),[V̅(t_1),ρ̅(t) ⊗ρ_R(0)]]].Since the reservoir in a thermal state is diagonal in the energy basis, it is straightforward to show that we can replace the interaction V̅(t) byV̅(t)→V̅_ϕ(t)=X̅(t)B̅_ϕ(t)=∫_0^∞g(ω)(b(ω)e^-i(ω t+ϕ) +b^†(ω)e^i(ω t+ϕ))dωfor any value of ϕ.If we make use of the Heisenberg equation of motion for total photon number N, N=∫_0^∞b^†(ω)b(ω) dω.we find thatdN/dt=ħ^-1XB_-π/2.and hence, correct to second order in the interaction, we can then write in the Heisenberg pictureV̅(t)≈ V_-π/2(t)=X(t)B_-π/2(t)=ħdN(t)/dtand hence to second order in the system reservoir coupling∫_t^t+Δ tV̅(t_1)dt_1 =ħ(N(t+Δ t)-N(t)) =ħΔ N.Thus we find thatΔρ̅/Δ t= -1/2Δ tTr_R [[Δ N,[Δ N,ρ̅(t) ⊗ρ_R(0)]]] This equation exposes in a formal manner an already familiar physical picture, namely that provided the reservoir is in a energy-diagonal state, the open system dynamics is driven by the gain and loss by the system of quanta from and to the reservoir. In particular we can also determine the kinds of quantum jumps that the coarse-graining method has introduced. The coarse-grained derivative as can be writtenρ̅(t+Δ t) = ρ̅(t) -i/ħ[H̅_cρ̅ -ρ̅H̅_c^†]Δ t+ Tr_R [Δ Nρ̅(t) ⊗ρ_R(0)Δ N]The expression for the non-Hermitean Hamiltonian H_c can be read off from Eq. (<ref>). It is not analysed here, but can be immediately understood as being the `no-jumps' contribution to the stochastic evolution of the sate of the system. The second term describes a quantum jump, the nature of which will be determined below.The jump term of Eq. (<ref>) can be rewritten asTr_R[Δ Nρ̅⊗ρ_R(0)Δ N]=Tr_R[Δ Nρ̅⊗ρ_R(0)Δ N]/Tr_SR[Δ Nρ̅⊗ρ_R(0)Δ N] P(t+Δ t,t)Δ t where the factorP(t+Δ t,t)Δ t=Tr_SR[Δ Nρ̅⊗ρ_R(0)Δ N]/Δ tΔ tcan be understood as the probability of a jump occuring in the time interval (t,t+Δ t) which suggests that within the coarse-grained dynamics, the evolution of the system, by virtue of its interaction with the reservoir, is driven by jump processes which involve the loss or gain of photons to and from the reservoir over the coarse graining time interval Δ t, with these jumps described byρ̅(t)→ρ̅(t+Δ t)=Tr_R[Δ Nρ̅⊗ρ_RΔ N]/Tr_SR[Δ Nρ̅⊗ρ_RΔ N].This is a non-selective jump in that it can be rewritten in terms of jumps that separately lead to a gain or a loss of a photon from the reservoir. To see this, we can separate B(t) into its positive and negative frequency parts:B(t)=B^(+)(t)+B^(-)(t)withB^(+)(t)=∫_0^∞g(ω)b(ω)e^-iω tdω=B^(-)†(t).We can then write Eq. (<ref>) asdN/dt=dN^(+)/dt+dN^(-)/dtso that Eq. (<ref>) isρ̅(t)→ρ̅(t+Δ t)= Tr_R[Δ N^(+)ρ̅(t)⊗ρ_RΔ N^(-)]+Tr_R[Δ N^(-)ρ̅(t)⊗ρ_RΔ N^(+)]where the cross terms such as Tr_R[Δ N^(-)ρ̅⊗ρ_RΔ N^(-)] will vanish by virtue of the diagonal nature of ρ_R(0). The first term then represents a jump in which the reservoir loses quanta in the time interval Δ t, the second in which it gains quanta. That this gain or loss takes place over an interval of time suggests that it can be modelled in terms of a measurement process that monitors the change in the number of quanta in the reservoir over the time interval Δ t. This is given in the next Section. §.§ Measurement interpretation We can give this result a measurement intepretation along the lines of the definition of work proposed by <cit.>. Assume that at the beginning of each interval Δ t the reservoir is coupled to an auxiliary system 𝒜 spanned by a set of states {|n⟩,n∈Z}, initially prepared a state |0⟩ via an entangling interaction U_I such that for a reservoir state an eigenstate of the number operator N with n quanta, |ϕ_n⟩ say, thenU_I^†(|ϕ_n⟩⊗|0⟩) =|ϕ_n⟩⊗|-n⟩.The system and reservoir then interact over the time interval (t,t+Δ t) after which time the inverse entangling operation U_I acts. The state of the auxiliary system is then ρ_A=Tr_SR[U_I𝒢̅(t+Δ t,t) [U^†_Iρ(t)⊗ρ_R(0) ⊗|0⟩⟨ 0|U_I]U_I^†]. Introducing projection operators P_n on the Hilbert space of the reservoir such thatP_n|ϕ_n⟩=|ϕ_n⟩, P_mP_n=P_nδ_mn,∑_m=0^∞P_m=1and using [P_m,ρ_R(0)]=0 we find that this becomesρ_A=∑_m,n^|n-m⟩⟨ n-m|P(n|m)whereP(n|m)=Tr_SR[P_n𝒢̅(t+Δ t,t) [ρ(t)⊗ρ_R(0)P_m]]is the probability that n quanta are measured in the reservoir at time t+Δ t given that m were measured there at time t. Thus, the probability that k quanta have been added to the reservoir in this time interval will beP(k)= ∑_m=0^∞P(m+k|m) = ∑_m=0^∞Tr_SR[𝒢̅(t+Δ t,t) [ρ(t)⊗ρ_R(0)P_m]P_m+k]. Substituting for 𝒢̅(t+Δ t,t) to second order, Eq. (<ref>), we then findP(k)= (1-ħ^-2Tr_SR[(𝒱^(-)𝒱^(+)+𝒱^(+)𝒱^(-))ρ(t)⊗ρ_R(0)])δ_k0+ħ^-2Tr_SR[𝒱^(-)𝒱^(+)ρ(t)⊗ρ_R(0)]δ_k1 +ħ^-2Tr_SR[𝒱^(+)𝒱^(-)ρ(t)⊗ρ_R(0)]δ_k,-1 where𝒱^(±)=∫_0^Δ tX̅(t)B̅^(±)(t)dt.Thus this probability P(k) breaks up, not unexpectedly, into three contributions, the first where there is no loss or gain by the reservoir and is to be associated with the non-Hermitean term (the `no jump' contribution to the stochastic evolution) of Eq. (<ref>), and not analysed here, while the remaining two terms are the jump contributions assocaited with the reservoir gaining or losing a qanta in the time interval Δ t.§ MORE THAN ONE RESERVOIR In order to be able to study heat transport between reservoirs of different temperatures, the above formalism has to be generalised to take into account the possibility of the system being in contact with more than one reservoir, these reservoirs not necessarily being at the same temperature. In such a case, the interaction with the reservoirs will take the formV=∑_p^B_pX_p.However, provided the reservoirs are independent systems, which means[B_m,B_n]=0 m nand that the reservoirs are all prepared in a thermal state, not necessarily of the same temperatureρ_R_p=e^-H_R_p/kT_p/Z_pso that Tr_R_p[ρ_R_p]=0, the dissipative part of the master equation Eq. (<ref>) will become 𝒟[ρ]=∑_p^𝒟_p[ρ]=1/ħ^2Δ t∑_p^∫_0^Δ tdt_2∫_0^Δ tdt_1G_p(t_2-t_1)[X̅_p(t_1)ρX̅_p(t_2)-12{X̅_p(t_2)X̅_p(t_1),ρ}] withG_p(t_2-t_1)=Tr_Rp[B̅_p(t_2)B̅_p(t_1) ρ_R_p]which can then be evaluated as for the single reservoir case. Examples of this situation are examined in the following Section.§ ILLUSTRATIVE EXAMPLES Below we will give two examples in which the system has an internal time scale Ω^-1. There then arises two choices of how the coarse-graining interval Δ t can be chosen, either much larger or much smaller that Ω^-1, leading to two different master equations for the same system. The differences in the two master equations, and their physical meaning is studied below. Particular attention is paid to the predictions for the full and partial secular approximation results in the case in which Ω is treated as a parameter that can undergo a quasi-static reduction towards zero. In both cases a global approach is adopted in the sense that the reservoir(s) are coupled to the delocalised energy eigenstates of the composite system. §.§ Damped coupled qubits Consider a system consisting of a pair of interacting qubits, with one or both qubits coupled to a thermal reservoir, as treated in, for instance <cit.>, with the coupling between the qubits being allowed to vary from some initially strong value, to zero. The model could be used, for instance, to study the work done by allowing the coupling between the qubits to decrease to zero (which would be equivalent, e.g., to `separating the qubits' if the coupling is modelled as dipole interaction). The first instance, one isolated and the other coupled to a thermal reservoir, is discussed here; the generalisation to both qubits coupled to separate reservoirs is considered further below.The qubit pair Hamiltonian isH_S =12ħω_S(σ_1z+σ_2z) +ħΩ(σ_1+σ_2- +σ_1-σ_2+).and the eigenstates and eigenvalues of H_S are2 |u⟩= |e_1⟩|e_2⟩,E_u=ħω_S |±⟩= 1/√(2)(|e_1⟩|g_2⟩±|g_1⟩|e_2⟩),E_±=±ħΩ |l⟩= |g_1⟩|g_2⟩,E_l=-ħω_S.Assuming the coupling to the reservoir is via qubit 2, the interaction is given by V=Bσ_2x where the reservoir operator B is as defined earlier, Eq. (<ref>). We require σ_2x in the interaction pictureσ̅_2x(t)=e^iH_St/ħσ_2x e^-iH_St/ħ=∑_n=-2^2e^iω_ntX_nwhere the transition frequencies ω_n and associated operators X_n with ω_-n=-ω_n and X_n^†=X_-n are4ω_1= ω_S-Ω,X_1= 1/√(2)(|u⟩⟨ +|-|-⟩⟨ l|) ω_2= ω_S+Ω,X_2= 1/√(2)(|+⟩⟨ l|+|u⟩⟨ -|)and where, for completeness, we could also define X_0=0,ω_0=0. These operators can also be written in terms of individual qubit operators asX_n=12(1_1 ⊗σ_2+-(-1)^nσ_1+⊗σ_2z), n=1,2.The master equation Eq. (<ref>) will then be determined by the choice of the coarse-graining interval that arises in the calculation of the γ_mn.§.§.§ Full secular approximation In this case, coarse graining is done on a time scale Δ t given by Δ t≫Ω^-1. Only the diagonal elements of the matrix γ survive and are given by, with n_±=n(ω_S±Ω)γ = [ γ_-2,-2 γ_-2,-1 …; γ_-1,-2 γ_-1,-1 …; ⋮ ⋮ ⋮ ] = γ[ n_++1 0 0 0; 0 n_-+1 0 0; 0 0 n_- 0; 0 0 0 n_+ ].The master equation isdρ/dt= -i/ħ[H_S,ρ] +(n_++1)γ[X_2^†ρX_2-12{X_2X_2^†,ρ}]+(n_-+1)γ[X_1^†ρX_1-12{X_1X_1^†,ρ}]+n_-γ[X_1ρX_1^†-12{X_1^†X_1,ρ}]+n_+γ[X_2ρX_2^†-12{X_2^†X_2,ρ}].The equilibrium steady state solution of this can be readily shown to be given by the expected Boltzmann distribution of energy level probabilities for a system in thermal equilibrium at temperature T, that isρ(∞)=Z(Ω)^-1e^-H_S/kTwith the partition function Z(Ω) given by Z(Ω)=Tr[e^-H_S/kT] =2[cosh(ħω_S/kT) +cosh(ħΩ/kT)].The steady state reduced density operators ρ_n(∞) for each qubit n=1,2 readily follows and are given byρ_n(∞)= Z(Ω)^-1[(e^-ħω_S/kT +cosh(ħΩ/kT))|e_n⟩⟨ e_n|.+.(e^ħω_S/kT +cosh(ħΩ/kT))|g_n⟩⟨ g_n|].i.e., the density operators for the two qubits are identical. In the spirit of the quantum trajectory formalism, we can consider the kinds of quantum jumps induced by this master equation. It is most easy to see this if we assume that the coupling between the two qubits is weak in the sense that Ω≪ω_S. In this case, provided also that Ω≪ kT/ħ, we can neglect the Ω dependence of n_±, whilst retaining the Ω dependence of the Hamiltonian H_S. We find in this limit, with n_±→ n, that the master equation collapses todρ/dt= -i/ħ[H_S,ρ]+γ(n+1)[σ_2-ρσ_2+ -12{σ_2+σ_2-,ρ}]+γ n[σ_2+ρσ_2- -12{σ_2-σ_2+,ρ}]+γ(n+1)[σ_2zσ_1-ρσ_1+σ_2z -12{σ_1+σ_1-,ρ}]+γ n[σ_2zσ_1+ρσ_1-σ_2z -12{σ_1-σ_1+,ρ}]which indicates that both qubits will undergo jumps, i.e., this is still a delocalized master equation.If this result were to be used to study the behaviour of the system in the limit of the coupling between the two qubits being `turned off', Ω→0, a difficulty arises. Specifically, for the steady state, we find that the density operator for the combined system factorises:ρ(∞)=ρ_1(∞)⊗ρ_2(∞)with, from Eq. (<ref>)ρ_n(∞)=Z(0)^-1(e^-1/2ħω_S/kT|e_n⟩⟨ e_n|+e^1/2ħω_S/kT|g_n⟩⟨ g_n|)i.e., both qubits settle into the canonical state for temperature T, which is the expected equilibrium state of a pair of non-interacting qubits immersed in a common reservoir of temperature T. It is difficult to understand this result if only the second qubit is coupled to the reservoir and there is no interaction between the qubits. The problem is, of course, that the master equation is not valid in this limit. The master equation is derived under the condition that the coarse-graining time interval Ω^-1≪Δ t≪τ_S, which clearly cannot be satisfied as Ω approaches zero.§.§.§ Partial secular approximation In this case, coarse graining is on a time scale ω_S^-1≪Δ t≪Ω^-1. All the diagonal elements γ_nn will be as in Eq. (<ref>), while the off-diagonal elements will survive. The γ matrix becomesγ = [ γ_-2,-2 γ_-2,-1 …; γ_-1,-2 γ_-1,-1 …; ⋮ ⋮ ⋮ ] =γ[ n_++1 n+1 0 0; n+1 n_-+1 0 0; 0 0 n_- n; 0 0 n n_+ ]with n_±≡ n(ω_S±Ω) as before and n≡ n(ω_S). The master equation (expanded out in detail in Appendix <ref>) is given by dρ/dt =-i/ħ[H_S,ρ] +(n_++1)γ[X_2^†ρX_2-12{X_2X_2^†,ρ}]+(n+1)γ[X_2^†ρX_1-12{X_1X_2^†,ρ}] +(n+1)γ[X_1^†ρX_2-12{X_2X_1^†,ρ}]+(n_-+1)γ[X_1^†ρX_1-12{X_1X_1^†,ρ}] +n_-γ[X_1ρX_1^†-12{X_1^†X_1,ρ}]+nγ[X_2ρX_1^†-12{X_1^†X_2,ρ}] +nγ[X_1ρX_2^†-12{X_2^†X_1,ρ}]+n_+γ[X_2ρX_2^†-12{X_2^†X_2,ρ}].which by the general result Eq. (<ref>) is a Lindblad master equation. This master equation differs from the secular approximation form by the presence of cross-terms which, in this case, mean that the populations and coherences do not decouple, unlike what is found for the full secular approximation form for the master equation. Some of the consequences of this can be seen in the steady state. The only non-vanishing elements of ρ that survive at steady state are the diagonal elements ρ_uu, ρ_++, ρ_–, ρ_ll, and the off-diagonal elements ρ_+- and ρ_-+ and are given by ρ_uu= e^-ħω_S/kT/Z(Ω) +p_0(-1/4(n_++n_-+1)+(n+12)n_++n_-+12/(2n_-+1)(2n_++1)) ρ_++= e^-ħΩ/kT/Z(Ω) +p_0(1/4(n_++n_-+1)+(n+12)n_–n_++1/2/(2n_++1)(2n_-+1)) ρ_–= e^ħΩ/kT/Z(Ω) +p_0(1/4(n_++n_-+1)+(n+12)n_+-n_-+1/2/(2n_++1)(2n_-+1)) ρ_ll= e^ħω_S/kT/Z(Ω) +p_0(-1/4(n_-+n_++1)-(n+12)n_++n_-+3/2/(2n_-+1)(2n_++1)) where Z(Ω) is as in Eq. (<ref>), andp_0= ρ_+-+ρ_-+ =1-(2n+1)(n_++n_-+1)/(2n_-+1)(2n_++1)/R-(2n+1)^2 n_++n_-+1/(2n_-+1)(2n_++1) R= 16Ω^2/(n_-+n_++1)γ^2+n_++n_-+1which, unfortunately, does not seem to allow itself to be simplified any further, andρ_+--ρ_-+=-4iΩ/(n_++n_-+1)γ(ρ_+-+ρ_-+).So, along with the usual Boltzmann distribution contribution to the populations of the eigenstates of H_S, there is a contribution in each case due to the coherence term p_0, so this result is clearly not a canonical thermal state for the combined qubits. However, in the limit of vanishingly small system-reservoir coupling, γ→0, we find that p_0→ 0 and we have the same result as earlier, Eq. (<ref>), the canonical Boltzmann probabilities, as expected from the general result Eq. (<ref>).What is of interest is the reduced density operators for the two qubits. We saw earlier that these reduced states were identical for the secular form of the master equation, Eq. (<ref>), but that that result is not acceptable as it gives the incorrect Ω→0 limit. Here, the reduced states are ρ_n = Z(Ω)^-1[(e^-ħω_S/kT+cosh(ħΩ/kT) +12p_0(Z(0)-(-1)^nZ(Ω))) |e_1⟩⟨ e_1|.+.(e^ħω_S/kT+cosh(ħΩ/kT) -12p_0(Z(0)-(-1)^nZ(Ω))) |g_1⟩⟨ g_1|], n=1,2 So the reduced states of the two qubits are clearly not the same, the origin of this being, of course, the fact that the steady state density operator for the combined qubit system is not diagonal in the energy basis. So, to preserve this expected asymmetry between the qubit states, the steady state cannot be diagonal, i.e., it cannot be a Gibbs state. This result is enough to drive home the fact that the partial secular approximation result is closer to the exact density operator result, and that the diagonal density thermal equilibrium result is only achieved in the limit of vanishingly small system-reservoir interaction.Once again, we can consider the kinds of quantum jumps induced by this master equation. This can be most clearly seen if, as before, we consider the limit of weak coupling between the two qubits, Ω≪ω_S. We can then neglect the Ω dependence of n_±, whilst retaining the Ω dependence of the Hamiltonian H_S. We find, on using Eq. (<ref>), the master equation collapses todρ/dt = -i/ħ[H_S,ρ] +(n+1)γ[σ_2-ρσ_2+ -12{σ_2+σ_2-,ρ}]+nγ[σ_2+ρσ_2- -12{σ_2-σ_2+,ρ}].so the only quantum jumps taking place are of qubit 2; the dissipative term describes dissipation of qubit 2 only. This is the expected form of a local master equation, i.e., in which the reservoir is coupled solely to the local energy eigenstates of the qubit coupled to the reservoir. This result was not found when the same limit is taken for the full secular approximation form for the master equation.If we take the limit Ω→0 in H_S then there is no time independent steady state, in general, unless qubit 1 is initially in a mixture of eigenstates of H_S. In this case, the result isρ_1 = p_e(0)|e_1⟩⟨ e_1| +p_g(0)|g_1⟩⟨ g_1| ρ_2= Z_0^-1[e^-ħω_S/2kT|e_2⟩⟨ e_2| +e^ħω_S/2kT|g_2⟩⟨ g_2|] Z_0= e^-ħω_S/2kT+e^ħω_S/2kTwith ρ=ρ_1⊗ρ_2. In other words, the steady state is that in which the qubit in contact with the reservoir ends up in the canonical state, while the qubit outside the reservoir can be found in its initial (diagonal) state.§.§.§ Both qubits damped An obvious generalization of the above example is the case in which both qubits are damped by separate reservoirs, not necessarily of the same temperature. If qubit 1 is coupled to a reservoir of temperature T_h and qubit 2 to a reservoir of temperature T_c, with T_h≥ T_c the required master equations in the secular and partial secular approximations for independent reservoirs (see Section <ref>) follow directly from those given above Eq. (<ref>) and Eq. (<ref>) by simply adding on the dissipator associated with qubit 1. This dissipator can be obtained from that for qubit 2 by the simple substitution |-⟩→-|-⟩ in the dissipative contributions to the equations for the individual matrix elements. The damping rates will be γ_c and γ_h and can be assumed to be unequal in general. The previous example then corresponds to the choice of γ_h=0,γ_c=γ. The transition frequencies of each qubit are assumed to be identical at ω_S.Confining our attention to the steady state and for the reservoirs at equal temperatures, we readily find that the full secular approximation form of the master equation predicts a Gibbsian equilibrium state for the system. In contrast, for the partial secular master equation, there is coupling between the system populations and coherences provided that γ_hγ_c, so at steady state there are non-zero coherences in the energy basis, and the populations are not the expected Boltzmann distribution.For reservoirs at different temperatures, there will be a steady state heat current J between the reservoirs that turn out to be quite different for the secular and partial secular cases. Letting 𝒟_h[ρ] be the dissipators for qubit 1 coupled to the hot reservoir and 𝒟_c[ρ] the dissipator for qubit 2 coupled to the cold reservoir, the heat current between the reservoirs will be given byJ=Tr_S[H_S(𝒟_h-𝒟_c)[ρ]].We will only concern ourselves with the steady state in the limit of Ω≪ω_S so that, provided also that Ω≪ kT/ħ, we can replace n_p(ω_S±Ω)≈ n_p(ω_S), p=c,h. So with γ̅= (γ_h+γ_c)/2, Δ n= (γ_hn_h(ω_S)-γ_cn_c(ω_S))/γ̅ n̅= (γ_hn_h(ω_S)+γ_cn_c(ω_S))/2γ̅ Δγ= (γ_h-γ_c)/γ̅we find that in the secular approximation case, at steady state, ρ_+-=0 and ρ_–=ρ_++ and the steady state heat current J_sec is given byJ_sec=γ̅ħω_S[Δ n(ρ_ll-ρ_uu)-Δγ(ρ_uu+ρ_++)]which reduces toJ_sec=2γ_cγ_hħω_S(n_h-n_c)/(γ_h+γ_c)(2n̅+1).It is clearly the case that the secular approximate case remains at a constant non-zero value as Ω→0 which is physically unacceptable. The origin of this behaviour can be traced, once again, to the fact that the secular master equation is invalid in the limit of vanishing Ω.For the partial secular case we find on using the Ω≪ω_S limiting form for the dissipators that, on using Eq. (<ref>), the master equation collapses to the local form of the master equationdρ/dt= -i/ħ[H_S,ρ]+(n_c+1)γ[σ_1-ρσ_1+-12{σ_1+σ_1-,ρ}]+n_cγ[σ_1+ρσ_1--12{σ_1-σ_1+,ρ}]+(n_h+1)γ[σ_2-ρσ_2+-12{σ_2+σ_2-,ρ}]+n_hγ[σ_2+ρσ_2--12{σ_2-σ_2+,ρ}] and the heat current becomesJ_parsec=γ̅ħω_S[Δ n(ρ_ll-ρ_uu)-12Δγ(2ρ_uu+ρ_–+ρ_++)-(2n̅+1)(ρ_+-+ρ_-+)]Contributions due to non-vanishing coherences ρ_+- now appear. This, in addition to the fact that the steady state populations are not the same as in the secular case as the coherences also contribute there, leads to a different result J_parsec =J_sec4Ω^2/(2n_c+1)(2n_h+1)γ_hγ_c+14[(2n_h+1)γ_h-(2n_c+1)γ_c](γ_h-γ_c)+4Ω^2 in which case the heat current vanishes for Ω→ 0 as ought to be expected. In the limit when Ω^2≫(2n_h+1)(2n_c+1)γ_nγ_c, i.e., when the energy separation of the |+⟩ and |-⟩ states is much larger than their linewidths (but still ≪ω_S), the partial secular and full secular heat currents agree. §.§ The tunnelling qubit model We shall now apply the above formalism to another example in which there is two clear choices for how the coarse-graining can be invoked, leading to two master equations that would appear to have two different physical interpretations. The model is that of a qubit that can tunnel between two potential wells, e.g., as might be realised with a pair of nearby quantum dots. It is further assumed that the two potential wells are immersed in independent thermal reservoirs. These reservoirs are in general not at the same temperature, so that the tunnelling process will enable the transport of energy (heat) from one reservoir to the other.The system Hamiltonian isH_S= H_A⊗1_T+1_A⊗ H_T = 1/2ħω_Sσ_z+1/2ħΩ(|l⟩⟨ r|+|r⟩⟨ l|)where the energy eigenstates of H_A are |e⟩ and |g⟩, with σ_z=|e⟩⟨ e|-|g⟩⟨ g|, and where |l⟩ and |r⟩ are the position eigenstates of the atom, at the site of the left hand and right hand potential wells respectively. It will further be assumed that Ω≪ω_S i.e., that the tunnelling rate will be very much slower that the transition frequency of the qubit.Note that this is not the case of two independent subsystems coming into interaction: there is no coupling between `system' A and `system' T, and the notion of a `local' and a `global' form for the master equation becomes ill-defined. Nevertheless it is meaningful to consider the full secular and partial secular limits of the master equation, and to show that once again, the full secular approximation yields results that are invalid when the tunnelling rate Ω becomes very small.Setting|±⟩=1/√(2)(|l⟩±|r⟩) the eigenstates and eigenvalues of H_S are H_S|e,±⟩=12ħ(ω_S±Ω)|e,±⟩ H_S|g,±⟩=-12ħ(ω_S∓Ω)|g,±⟩The interaction with the reservoirs is given by V=B_lP_lσ_x+B_rP_rσ_x=B_lX_l+B_rX_rwhere P_n=|n⟩⟨ n|, n=l,r and where [B_l,B_r]=0 i.e., the reservoirs are independent in which case the analysis of Section <ref> carries through, with the dissipation expressed as the sum of dissipators for each reservoir independently.We require the time dependence in the interaction picture of the system operators X̅_l(t)=P̅_l(t)σ̅_x(t) and X̅_r=P̅_r(t)σ̅_x(t) expressed in terms of the associated set of eigenoperators, X_lm and X_rm whereX̅_p(t)=∑_m^X_pme^iω_mt, p=l,r.We shall do this for P̅_l(t)σ̅_x(t), the other following by inspection. If we now put Σ_+=|+⟩⟨ -| with Σ_-=Σ_+^† we haveP̅_l(t)=1/2[1+e^iΩ tΣ_++e^-iΩ tΣ_-]while σ̅_x(t)=σ_-e^-iω_St+σ_+e^iω_St and soσ̅_x(t)P̅_l(t) =∑_m=-3^3X_lme^iω_m twith the individual elements X_lm and frequencies ω_m, with X_l,-m=X^†_l,m and ω_-m=-ω_m,given by[ X_l1; X_l2; X_l3 ] =1/2[ σ_+Σ_-;σ_+; σ_+Σ_+ ] and [ ω_1; ω_2; ω_3 ] = [ ω_S-Ω; ω_S; ω_S+Ω ].with the understanding that X_l0≡ 0.In a similar way, the corresponding result for the coupling to the right hand reservoir followsσ̅_x(t)P̅_r(t)=∑_m=-3^3X_rme^iω_mt where[ X_r1; X_r2; X_r3 ] =1/2[ -σ_+Σ_-; σ_+; -σ_+Σ_+ ].The dissipator here will then consist of two contributions, as given by Eq. (<ref>) with the elements γ_pmn(Δ t) given by Eq. (<ref>). A matrix of values of ω_mn=ω_m-ω_n (excluding the row ω_0n and column ω_m0) is useful to get an overview the frequency differences:[ ω_-3,-3 ω_-3,-2 …; ω_-2,-3 ω_-2,-2 …; ⋮ ⋮ ⋮; ]=[0 -Ω-2Ω-2ω_S-2ω_S-Ω -2ω_S-2Ω;Ω0 -Ω-2ω_S+Ω-2ω_S-2ω_S-Ω; 2ΩΩ0 -2ω_S+2Ω-2ω_S+Ω-2ω_S; 2ω_S 2ω_S-Ω2ω_S-2Ω0 -Ω-2Ω; 2ω_S-Ω 2ω_S 2ω_S+ΩΩ0 -Ω;2ω_S+2Ω 2ω_S+Ω 2ω_S 2ΩΩ0 ]. For Ω≪ω_S there appears two distinct time scales, ∼Ω^-1 and ∼ω_S^-1, from which we can construct the possible values for the matrix of γ_pmn(Δ t) values, depending on the choice of coarse-graining. Two cases are of interest: Δ t≫Ω^-1 and Ω^-1≫Δ t≫ω_S^-1, the first corresponding to the full secular approximation, the second to a partial secular approximation.§.§.§ Full secular approximation In the full secular case, Δ t≫Ω^-1, only the γ_pnn(Δ t) are non-zero, i.e., from Eq. (<ref>), γ_pnn(Δ t)=γ(n_p(|ω_n|)+θ(-ω_n)), p=l,rwith, for reservoirs at temperatures T_p, p=l,rn_p(ω)=(e^ħω/kT_p-1)^-1.The master equation becomesdρ/dt=-i/ħ[H_S,ρ]+𝒟_l[ρ]+𝒟_r[ρ]where the dissipators are given by𝒟_l[ρ]+𝒟_r[ρ]=14γ(n_l(ω_S)+1) [σ_-Σ_-ρΣ_+σ_+ -12{Σ_+σ_+σ_-Σ_-,ρ}] +14γ(n_l(ω_S)+1) [σ_-ρσ_+ -12{σ_+σ_-,ρ}] +14γ(n_l(ω_S-Ω)+1) [σ_-Σ_+ρΣ_-σ_+ -12{Σ_-σ_+σ_-Σ_+,ρ}] +14γ n_l(ω_S-Ω) [σ_+Σ_-ρΣ_+σ_- -12{Σ_+σ_-σ_+Σ_-,ρ}] +14γ n_l(ω_s) [σ_+ρσ_- -12{σ_-σ_+,ρ}] +14γ n_l(ω_S+Ω) [σ_+Σ_+ρΣ_-σ_- -12{Σ_-σ_-σ_+Σ_+,ρ}] +(l→ r, Σ_±→-Σ_±)These equations do not predict any coupling between the populations and the coherences. The latter damp to zero, leaving a steady state density operator diagonal in the energy basis. In the simplest instance of the two reservoirs being at the same temperature, T_l=T_r=T it is straighforward to show that the steady state population distribution is the expected Boltzmann distributionρ_nn=e^-E_n/kTZ^-1, n=1… 4.The coarse-graining underlying the derivation of the master equation has its consequences when one considers the quantum trajectory unravelling. In particular, for a jump unravelling, and focusing on the post-jump positional state of the tunnelling qubit when the qubit ends up in its ground state, having emitted a photon into one or the other of the reservoirs, the unnormalised post jump positional state of the qubit will be given by, once again assuming for clarity Ω≪ω_S so that, provided also that Ω≪ kT_l/ħ, n_l(ω_S±Ω)≈ n_l(ω_S) and similarly for n_r,Σ_-ρ_eeΣ_++ρ_ee+Σ_+ρ_eeΣ_- = (|-⟩⟨ -|+|+⟩⟨ +|)(ρ_11+ρ_22)+|+⟩⟨ -|ρ_12+|-⟩⟨ +|ρ_21At steady state the coherences ρ_12 and ρ_21 will be zero in which case the normalised post jump state is12(|-⟩⟨ -|+|+⟩⟨ +|)a mixed state with an equal probability of finding the qubit in either the symmetric or antisymmetric positional states. (There is a slight bias towards the lower energy state |-⟩ if the full frequency dependencies of n_l and n_r are taken into account.) Thus, the jump provides no information concerning the position (either left or right quantum well) of the qubit after the emission has occurred. The tunnelling timescale Ω^-1 is much larger than the coarse-graining timescale Δ t, so during the time Δ t the qubit oscillates many times between the left and right hand reservoirs, so the position of the qubit after the emission occurs cannot be resolved. This can be understood as being due to an uncertainty Δ t in the instant at which the photon has been emitted into the reservoir.For unequal temperatures, T_l T_r, there is a heat currentJ=Tr_S[H_S(𝒟_l-𝒟_r)[ρ]]between the reservoirs, mediated by the tunnelling qubit. In the system energy eigenstate basis, Eq. (<ref>), this current can be shown to be J_secular = 14ħγ[(ω_S+Ω)Δ n(ω_S+Ω)+ω_SΔ n(ω_S)](ρ_g-,g--ρ_e+,e+) +14ħγ[ω_SΔ n(ω_S)+(ω_S-Ω)Δ n(ω_S-Ω)](ρ_g+,g+-ρ_e-,e-). with Δ n(ω)=n_l(ω)-n_r(ω). This is a complex expression for arbitrary Ω, but assumes a much simpler form if it is assumed that Ω≪ω_S so that, provided also that Ω≪ kT_l/ħ, n_l(ω_S±Ω)≈ n_l(ω_S)≡ n_l and similarly for n_r. In this case it is found that J_secular=ħω_Sγ(n_l-n_r)/2(n_l+n_r+1).Of note here is that this does not vanish in the limit of Ω→0, i.e., there is still a heat current present although there is no tunnelling. This unphysical outcome is a consequence of the fact that the coarse-graining condition Δ t≫Ω^-1 on which basis the master equation was derived will fail for vanishing Ω. §.§.§ Partial secular approximation For the partial secular case ω_S^-1≪Δ t≪Ω^-1, from Eq. (<ref>) γ_pmn(Δ t) =γ(n_p(|ω_m+ω_n|/2)+θ(-ω_m-ω_n))with ω_mand the X_l,m given by Eq. (<ref>) and the X_r,m by Eq. (<ref>). Substituting into the expression Eq. (<ref>) for the dissipator yields a very complex expression presented in Appendix <ref> which differs from the full secular master equation by the presence of 24 extra terms. Of these terms 16 contribute only if the temperatures of the two reservoirs are not equal. If the reservoirs are at the same temperature, in spite of the presence of 8 extra terms in the master equation as compared to the secular master equation, the populations evolve independently of the coherences, and approach the expected canonical Boltzmann distributions at steady state. However, the master equation simplifies dramatically under the approximation n_l(ω_S±Ω)≈ n_l(ω_S) and similarly for n_r, for Ω≪ω_S, provided also that Ω≪ kT/ħ, in which case the master equation reduces to dρ/dt= -i/ħ[H_S,ρ] +γ(n_l+1)[σ_-P_lρ P_lσ_+ -12{σ_+σ_-P_l,ρ}] +γ n_l[σ_+P_lρ P_lσ_- -12{σ_-σ_+P_l,ρ}]+(l→ r).A quantum jump interpretation of this equation follows from extracting the jump terms from this master equation, given byγ(n_l+1)σ_-P_lρ P_lσ_+γ n_lσ_+P_lρ P_lσ_-+(l→ r).The projection operators P_l and P_r clearly show that for a jump in the qubit state accompanied by emission or absorption will also project the qubit into either the left or the right potential well. The coarse-graining time interval Δ t is now much shorter than the tunnelling time, so that when a jump occurs, the temporal resolution is such that the position of the qubit after the jump can be resolved as being either on the left or the right. This is to be contrasted with what was seen in the secular approximation case, where the temporal resolution implied by the choice of Δ t cannot resolve the qubits position.Further consequences of the choice of coarse-graining timescales can be seen by examining the heat current J, given by Eq. (<ref>). For the partial secular master equation the steady state current is J_parsec= 14ħγ[(ω_S+Ω)Δ n(ω_S+Ω)+ω_SΔ n(ω_S)](ρ_g-,g--ρ_e+,e+) +14ħγ[ω_SΔ n(ω_S)+(ω_S-Ω)Δ n(ω_S-Ω)](ρ_g+,g+-ρ_e-,e-) +ħγ[(ω_S+12Ω)n̅(ω_S+12Ω)+(ω_S-12Ω)n̅(ω_S-12Ω)]Re[ρ_g+,g-] -ħγ[(ω_S+12Ω)(n̅(ω_S+12Ω)+1)+(ω_S-12Ω)(n̅(ω_S-12Ω)+1)]Re[ρ_e+,e-] where we note that coherences in the energy basis now contribute. This is a complex expression for arbitrary Ω, but assumes a much simpler form if it is assumed that Ω≪ω_S in which case this expression becomesJ_parsec=J_secularΩ̃^2/1+(2n_l+1)(2n_r+1)+Ω̃^2which, in sharp contrast to the full secular result, Eq. (<ref>), vanishes as the tunneling rate Ω→ 0. Note that the secular and partial secular results come into agreement when the tunnelling rate becomes large. § CONCLUSIONSThe derivation of the master equation by a coarse-grained approach was revisited with attention focussed on firstly the conditions for coarse-graining to be applied, and secondly the importance of timescales in determining the possible forms of the master equation. The consequences of this are observed in the case of a composite system with a well-defined internal time scale that is damped by coupling with one or more reservoirs. This is the scenario of on-going interest in studying the global versus local approaches for deriving master equations for such systems. Difficulties with the global secular approximation form found by others, in particular the erroneus prediction of heat currents in circumstances when none should arise, are reproduced here, but are shown to be resolvable by a change in the choice of coarse-graining timescale. It was also argued that the choice of timescales can be given a measurement interpretation that can be directly related under some circumstances to the time resolution of the kinds of quantum jumps that are predicted by the different master equations. The authors wish to acknowledge useful conversations with Steve Barnett, Sarah Croke, Alexei Gilchrist, and Thomas Guff.apsrev4-1.bst § ESTIMATION OF TIMESCALES We wish to provide estimates for the correction terms arising in the coarse-graining result Eq. (<ref>) and in the calculation of γ_mn(Δ t) in Eq. (<ref>). We first note that in the integral1/ħ^2∫_-∞^∞ G(τ)e^i(ω_m+ω_n)τ/2dτ.as G(τ) is assumed to decay on a timescale τ_c≪ω_S^-1, we can approximate this by1/ħ^2∫_-∞^∞ G(τ)e^i(ω_m+ω_n)τ/2dτ≈1/ħ^2∫_-∞^∞ G(τ)dτ∼τ_S^-1Similarly we can write1/ħ^21/Δ t∫_0^∞[G(-τ)e^i(ω_m+ω_n)τ/2+G(τ)e^-i(ω_m+ω_n)τ/2]τ dτ ≈1/ħ^21/Δ t∫_0^∞[G(-τ)+G(τ)]τ dτNormalising this expression by1/ħ^21/Δ t∫_0^∞[G(-τ)+G(τ)]dτ=(Δ t τ_S)^-1we have1/ħ^21/Δ t∫_0^∞[G(-τ)e^i(ω_m+ω_n)τ/2+G(τ)e^-i(ω_m+ω_n)τ/2]τ dτ ≈τ_S^-1∫_0^∞[G(-τ)+G(τ)]τ dτ/∫_0^∞[G(-τ)+G(τ)]dτ∼τ_c/τ_Swhere we have taken the ratio of integrals in Eq. (<ref>) as an estimate of the temporal width τ_c of the correlation function G(t), with this result now leading to Eq. (<ref>).If we now turn to the correction term in the coarse-grained result Eq. (<ref>), an interaction of the form V=BX as analysed in Section <ref> will lead to structures of the form of Eq. (<ref>) but with changed time limits on the integrals:∫_t^t+Δ tdt_2∫_0^tdt_1 G(t_2-t_1)[X̅(t_1)ρX̅(t_2)-… ]With G(t) having a temporal width ∼τ_c, any oscillating factors from X̅(t) in this expression can be replaced by unity, so we effectively have to deal with the integral over G(t_2-t_1) only. A change of variable then gives∫_0^Δ tdt_2∫_0^tdt_1G(t_2+t_1).If we further make use of the fact that G(t) has a width ∼τ_c, and requiring t,Δ t are both ≫τ_c, we can allow the upper limits of the double integral to approach infinity. We do this here by introducing the Fourier transform G̃(ω) of G(t) to enable us to write∫_0^Δ tdt_2∫_0^tdt_1G(t_2+t_1) =lim_ϵ→0∫_-∞^+∞dωG̃(ω)(∫_0^∞dte^-iω t-ϵ tdt)^2 =∫_0^∞G(τ)τ dτ∼τ_c/τ_SSo the correction term in Eq. (<ref>) will be ∼τ_c/(Δ t τ_S) which leads to this term being negligible compared to the first term in that expression (of order τ_S^-1) for Δ t≫τ_c. § PARTIAL SECULAR MASTER EQUATION FOR TWO QUBIT MODEL The partial secular master equation for the two qubit model studied in Section <ref> for each qubit coupled to separate reservoirs in general at different temperatures T_h and T_c, and with different damping rates γ_h and γ_c is given bydρ/dt=-i[12ω_S(σ_z1+σ_z2)+Ω(σ_1+σ_2-+σ_1-σ_2+)]+𝒟_sec[ρ]+𝒟_nonsec[ρ]where 𝒟_sec are contributions that appear only in the secular approximation form of the master equation, while 𝒟_nonsec are further terms that appear in the partial secular form. In terms of the mean decay rate γ̅=(γ_h+γ_c)/2 and using the notationn̅(ω)= (2γ̅)^-1(γ_hn_h(ω)+γ_cn_c(ω)) Δ n(ω)= γ̅^-1(γ_hn_h(ω)-γ_cn_c(ω))and defining L_ab=|a⟩⟨ b|, the secular approximation term 𝒟_sec[ρ] can be written𝒟_sec[ρ]= (n̅(ω_S+Ω)+1)γ̅[L_l+ρ L_+l+L_-uρ L_u--12{L_uu+L_++,ρ}]+(n̅(ω_S-Ω)+1)γ̅[L_u+ρ L_+u+L_-lρ L_l--12{L_+++L_ll,ρ}]+n̅(ω_S-Ω)γ̅[L_+uρ L_u++L_l-ρ L_-l-12{L_–+L_uu,ρ}]+n̅(ω_S+Ω)γ̅[L_+lρ L_l++L_u-ρ L_-u-12{L_–+L_ll,ρ}]+12Δ n(ω_S+Ω)γ̅[L_u+ρ L_l-+L_-lρ L_+u-L_l+ρ L_u--L_-uρ L_+l]+12Δ n(ω_S-Ω)γ̅[L_+uρ L_-l+L_l-ρ L_u+-L_+lρ L_-u-L_u-ρ L_l+].There is no coupling between the populations and the coherences in these secular contributions. However, this is not the case for the non-secular contributions, given by𝒟_nonsec[ρ]= (n̅(ω_S)+1)γ̅[L_l+ρ L_u++L_+uρ L_+l-L_-uρ L_-l-L_l-ρ L_u-]+n̅(ω_S)γ̅[L_+lρ L_+u+L_u+ρ L_l+-L_u-ρ L_l--L_-lρ L_-u]+12Δ n(ω_S)γ̅[L_l+ρ L_-l+L_l-ρ L_+l+L_+lρ L_l-+L_-lρ L_l+.. -L_u+ρ L_-u-L_u-ρ L_+u-L_+uρ L_u--L_-uρ L_u+].which couple the populations to the coherences ρ_+- and ρ_-+.The particular case of only one qubit coupled to a reservoir is obtained by setting γ_h=0 and γ_c=γ.§ PARTIAL SECULAR MASTER EQUATION FOR TUNNELLING QUBIT MODEL The partial secular master equation for the tunnelling qubit model, with n̅(ω)=(n_l(ω)+n_r(ω))/2 and Δ n(ω)=n_l(ω)-n_r(ω) is presented below, with the dissipator broken into three distinctive contributions:dρ/dt=-i[12ω_Sσ_z+12Ω(|l⟩⟨ r|+|r⟩⟨ l|),ρ]+𝒟_sec[ρ]+𝒟_coh1[ρ]+𝒟_coh2[ρ]where the secular contributions to the dissipator are𝒟_sec[ρ] = 12γ(n̅(ω_S+Ω)+1) [σ_-Σ_-ρΣ_+σ_+ -12{Σ_+σ_+σ_-Σ_-,ρ}] +12γ(n̅(ω_S)+1) [σ_-ρσ_+ -12{σ_+σ_-,ρ}] +12γ(n̅(ω_S-Ω)+1) [σ_-Σ_+ρΣ_-σ_+ -12{Σ_-σ_+σ_-Σ_+,ρ}] +12γn̅(ω_S-Ω) [σ_+Σ_-ρΣ_+σ_- -12{Σ_+σ_-σ_+Σ_-,ρ}] +12γn̅(ω_S)[σ_+ρσ_- -12{σ_-σ_+,ρ}] +12γn̅(ω_S+Ω)[σ_+Σ_+ρΣ_-σ_- -12{Σ_-σ_-σ_+Σ_+,ρ}]while the partial secular approximation introduces extra contributions that give rise to coherences in the spatial degrees of freedom of the qubit, but do not couple the coherences to the populations,𝒟_coh1= +12γ(n̅(ω_S)+1)σ_-Σ_-ρΣ_-σ_+ +12γ(n̅(ω_S)+1)σ_-Σ_+ρΣ_+σ_+ +12γn̅(ω_S)σ_+Σ_+ρΣ_+σ_- +12γn̅(ω_S)σ_+Σ_-ρΣ_-σ_- +14γΔ n(ω_S+12Ω) [σ_-Σ_-ρσ_+ -12{σ_+σ_-Σ_-,ρ}] +14γΔ n(ω_S+12Ω) [σ_-ρΣ_+σ_+-12{Σ_+σ_+σ_-,ρ}]as well as further terms that also give rise to contributions to the coherences, but are present only if the reservoirs are at different temperatures, in which case coupling between the coherences and the populations does occur:𝒟_coh2= 14γΔ n(ω_S+12Ω) [σ_-Σ_-ρσ_+ -12{σ_+σ_-Σ_-,ρ}] +14γΔ n(ω_S+12Ω) [σ_-ρΣ_+σ_+-12{Σ_+σ_+σ_-,ρ}] +14γΔ n(ω_S-12Ω) [σ_-ρΣ_-σ_+ -12{σ_+Σ_-σ_-,ρ}] +14γΔ n(ω_S-12Ω) [σ_-Σ_+ρσ_+ -12{σ_+σ_-Σ_+,ρ}] +14γΔ n(ω_S-12Ω)[σ_+ρΣ_+σ_- -12{Σ_+σ_-σ_+,ρ}] +14γΔ n(ω_S-12Ω)[σ_+Σ_-ρσ_- -12{σ_-σ_+Σ_-,ρ}] +14γΔ n(ω_S+12Ω)[σ_+Σ_+ρσ_- -12{σ_-σ_+Σ_+,ρ}] +14γΔ n(ω_S+12Ω)[σ_+ρΣ_-σ_- -12{Σ_-σ_-σ_+,ρ}]. | http://arxiv.org/abs/1710.09939v2 | {
"authors": [
"J D Cresser",
"C Facer"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20171026230736",
"title": "Coarse-graining in the derivation of Markovian master equations and its significance in quantum thermodynamics"
} |
⟩⟨d Instytut Fizyki imienia Mariana Smoluchowskiego,Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, Poland Instytut Fizyki imienia Mariana Smoluchowskiego,Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, PolandNational Centre for Nuclear Research, ul.Hoża 69, PL-00-681 Warsaw, Poland Instytut Fizyki imienia Mariana Smoluchowskiego,Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, PolandMark Kac Complex Systems Research Center, Uniwersytet Jagielloński, ulica Profesora Stanisława Łojasiewicza 11, PL-30-348 Kraków, PolandTime crystals are quantum many-body systems which, due to interactions between particles, are able to spontaneously self-organize their motion in a periodic way in time by analogy with the formation of crystalline structures in space in condensed matter physics. In solid state physics properties of space crystals are often investigated with the help of external potentials that are spatially periodic and reflect various crystalline structures. A similar approach can be applied for time crystals, as periodically driven systems constitute counterparts of spatially periodic systems, but in the time domain. Here we show that condensed matter problems ranging from single particles in potentials of quasi-crystal structure to many-body systems with exotic long-range interactions can be realized in the time domain with an appropriate periodic driving. Moreover, it is possible to create molecules where atoms are bound together due to destructive interference if the atomic scattering length is modulated in time. Time crystal platform: fromquasi-crystal structures in time to systems with exotic interactions Krzysztof Sacha December 30, 2023 =======================================================================================================Although crystals have been known for years, time crystals sound more like science fiction than a serious scientific concept. In 2012 Frank Wilczek initiated new research area by suggesting that periodic structures in time can be formed spontaneously by a quantum many-body system <cit.>. While the original Wilczek idea could not be realized because it was based on a system in the ground state <cit.>, it turned out that spontaneous breaking of discrete time translation symmetry and self-re-organization of motion of a periodically driven quantum many-body system was possible <cit.>. This phenomenon was dubbed "discrete time crystals" <cit.> and it was already realized experimentally <cit.>, for review see <cit.>. Apart from the realization of spontaneous breaking of discrete time translation symmetry, periodically driven quantum systems can be also employed to model crystalline structures in time in a similar way as external time-independent spatially periodic potentials allow one to model space crystals <cit.>. It should be stressed that driven systems with crystalline properties in time do not require external spatially periodic potentials. Crystal structures in time emerge due to periodic driving provided it is resonant with the unperturbed motion of a system. It is possible to investigate Anderson localization <cit.> in the time domain <cit.> or many-body localization caused by temporal disorder <cit.>. In the following we show that proper manipulation of higher temporal harmonics of a periodic perturbation is a perfect tool to engineer a wide class of condensed matter systems in the time domain including many-body systems with exotic interactions. Moreover, it is possible to create molecules where atoms are bound together via disordered potentials.Let us begin with a classical single particle system in one-dimension (1D) described by the Hamiltonian H_0(x,p). If the motion of a particle is bounded it is convenient to perform a canonical transformation to the so-called action-angle variables <cit.>. Then, H_0=H_0(I) where the momentum (action) I is a constant of motion and the canonically conjugate angle θ changes linearly with time, i.e. θ(t)=Ω t+θ(0) where Ω(I)=dH_0(I)/dI is a frequency of periodic evolution of a particle. Assume we turn on a periodic driving of the form H_1=λ h(x)f(t) where f(t+2π/ω)=f(t)=∑_kf_ke^ikω t and λ determines the strength of the driving. The spatial part of H_1 can be expanded in a Fourier series h(x)=∑_n h_n(I)e^inθ. A particle will be resonantly driven if the period of its unperturbed motion is equal to an integer multiple of the driving period, i.e. ω=sΩ(I_0) where s is an integer number and I_0 is a resonant value of the action. In order to analyze motion of a particle in the vicinity of a resonant trajectory, i.e. for I≈ I_0, it is convenient to switch to the moving frame, Θ=θ-ω/st, and apply the secular approximation <cit.>. It results in the effective time-independent Hamiltonian, H_ eff=P^2/2m_ eff+λ V_ eff(Θ), where P=I-I_0, the effective mass 1/m_ eff=d^2H_0(I_0)/dI_0^2 and the potential V_ eff(Θ)=∑_n h_ns(I_0)f_-ne^insΘ. If the second order corrections are negligible, which can be easily monitored <cit.>, H_ eff provides an exact description of particle motion in the vicinity of a resonant trajectory. The effective Hamiltonian, H_ eff, indicates that a resonantly driven system behaves like a particle on a ring, i.e. 0<Θ≤ 2π, with a certain effective mass and in the presence of a time-independent effective potential V_ eff(Θ). If there are many non-zero Fourier components of h(x), a proper choice of temporal Fourier components of f(t) allows one to create a practically arbitrary effective potential. Indeed, any potential on a ring can be expanded in a series V_ eff(Θ)=∑_nd_ne^inΘ and in order to realize it we can choose the fundamental 1:1 resonance (s=1) and periodic driving with the Fourier components f_-n=d_n/h_n(I_0). If s>1, a potential energy structure is duplicated s times. For s≫ 1, V_ eff(Θ) allows one to reproduce condensed matter problems where a particle can move in a potential with s identical wells of arbitrary shape and with periodic boundary conditions. Before we illustrate our idea with an example we have to address two issues. Firstly, so far our approach was classical but we would like to deal with quantum systems. In order to obtain a quantum description one can either perform quantization of the effective Hamiltonian, i.e. (P,Θ)→(P̂,Θ̂), or apply a quantum version of the secular approximation from the very beginning <cit.>. Both approaches lead to the same results. Eigenstates of the effective Hamiltonian in the moving frame correspond to time-periodic Floquet eigenstates of the original Floquet Hamiltonian, H_F=H_0+H_1-iħ∂_t, in the laboratory frame <cit.>. Second issue: what is the relation of the class of problems we consider with time crystals? Space crystals are related to periodic arrangement of particles in space. If we take a snapshot of a space crystal at some moment in time (t=const.), then we can observe a crystalline structure in space. Switching to time crystals the role of time and space is exchanged. We fix position in the configuration space (x=const.), i.e. we choose location for the detector, and ask if the probability of clicking of a detector behaves periodically in time. We have shown that in the frame moving along a classical resonant orbit, Θ=θ-ω/s t, we obtain an effective Hamiltonian which can describe a solid state problem. Such a crystalline structure in Θ is reproduced in the time domain if we return to the laboratory frame, as the relation between Θ and t is linear. Thus, if we locate a detector close to a classical resonant trajectory, the probability of detection of a particle as a function of time reproduces crystalline structure described by means of H_ eff in the moving frame. In Refs. <cit.> it was proven that stable orbits of classical dissipative systems can reveal quasi-crystal tiling in time. We will show that quantum properties of quasi-crystals in time can be investigated, see also <cit.>. In condensed matter physics quasi-crystals are systems which do not have any minimal part which appears periodically in space. Nevertheless, two or more unit cells are not placed randomly because a d-dimensional quasi-crystal can be constructed as a slice through a 2d-dimensional periodic crystal <cit.>. We will focus on the d=1 case when 1D quasi-crystal structure can be constructed as a cut through a 2D square lattice. The cut with the line whose gradient is the golden ratio generates the Fibonacci quasi-crystal which can be also constructed with the help of the so-called inflation rule <cit.>: B→ BS and S→ B where B and S denote, e.g., big and small wells, respectively, of a potential energy of a single particle. Successive application of the inflation rule shows the process of growing of the quasi-crystal, i.e. B→ BS→ BSB→ BSBBS→ BSBBSBSB→….In order to illustrate how to realize the Fibonacci quasi-crystal in the time domain experimentally, let us consider e.g., a particle which bounces on a vibrating mirror in the presence of a gravitational field <cit.> in a 1D model.In the coordinate frame vibrating with the mirror, the mirror is fixed but the gravitation strength oscillates in time. Then the Hamiltonian of the system, in gravitational units <cit.>, reads H=p^2/2+x+λ xf(t) where f(t)=∑_kf_ke^ikω t and λ/ω^2 is related to the amplitude of the mirror vibration. The secular approximation leads to the previously derived effective Hamiltonian with m_ eff=-π^2/ω^4 and h_n=-(-1)^n/n^2ω^2 if the 1:1 resonance condition (s=1) is fulfilled. A proper choice of f(t) allows one to realize the effective potential V_ eff(Θ) that reproduces any finite Fibonacci quasi-crystal. In Fig. <ref> we show what kind of driving leads to a quasi-crystal with the total number of big and small potential wells given by the seventh Fibonacci number. Transport properties in the quasi-crystal that can be analysed with the help of the effective Hamiltonian in the frame moving along the 1:1 resonant orbit will be observed in the time domain in the laboratory frame. Now we will demonstrate that periodically driven many-body systems allows for realization of solid state problems with exotic interactions. Let us illustrate this idea with ultra-cold atoms bouncing on a mirror which oscillates harmonically with frequency ω. If the s:1 resonance condition is fulfilled, the single-particle effective Hamiltonian in the moving frame reads H_ eff=P^2/2m_ eff+V_0cos(sΘ) where m_ eff=-π^2s^4/ω^4 and V_0=-λ(-1)^s/ω^2.Let us assume s≫ 1 and V_0 sufficiently big so that in the quantum description eigenvalues of H_ eff form well separated energy bands and eigenstates are Bloch waves e^ikΘv_k(Θ) where v_k(Θ+2π/s)=v_k(Θ). Note that for a fixed position in the laboratory frame, θ=const, the periodic Bloch waves character emerges in time, e^ik(θ-ω t/s)v_k(θ-ω t/s), with the period 2π/ω. Due to the negative effective mass m_ eff, the effective Hamiltonian H_ eff is bounded from above, not from below. Therefore, the first energy band possesses the highest energy. For simplicity, let us restrict ourselves to the first band and choose as the basis in the corresponding Hilbert subspace, the Wannier states w_j=w(Θ-j2π/s) where j denotes at which site of the effective potential a Wannier function is localized <cit.>. In the laboratory frame the Wannier states w_j(x,t) describe localized wavepackets moving along the resonant trajectory. We assume the normalization ∫_0^s2π/ωdt⟨ w_j|w_j⟩=1. Thus, s sites of the effective potential in the moving frame correspond to s Wannier wavepackets evolving in the laboratory frame. The width of the first energy band of H_ eff is determined by J=-2∫_0^s2π/ωdt⟨ w_i+1|H_ eff|w_i⟩ which is an amplitude of nearest neighbour tunnelings.In ultra-cold atomic gases interactions are described by the contact Dirac-delta potential, g_0δ(x), where g_0 is determined by the atomic scattering length which can be modulated in time by means of a Feshbach resonance <cit.>. We will see that these contact interactions between atoms can result in exotic long-range interactions in the effective description of the resonantly driven many-body system (effective long-range interactions in the phase space crystals <cit.> have been considered in <cit.>, see also <cit.>). For example in the case of bosonic particles, when we restrict ourselves to the Hilbert subspace spanned by Fock states |…,n_j,…⟩, where n_j is the number of atoms occupying a mode w_j, we obtain a many-body effective Hamiltonian of the Bose-Hubbard form, Ĥ_ eff=-J/2∑_⟨ i, j⟩â_i^†â_j+1/2∑_i,jU_ij â_i^†â_j^†â_jâ_i,where the bosonic operators â_j annihilate particles in modes w_j's and U_ij=∫_0^s2π/ωdtg_0(t)u_ij(t) with u_ij(t)=2∫_0^∞ dx|w_i|^2|w_j|^2 for i j and u_ii=∫_0^∞ dx|w_i|^4<cit.>, where we assume that the atomic scattering length g_0(t) can be modulated in time. The Hamiltonian (<ref>) is valid provided the interaction energy per particle is always smaller than the energy gap between the lowest and first excited energy bands of the single-particle system. A given interaction coefficient U_ij is determined mostly by g_0(t) at the moment when the corresponding wavepackets overlap. Suitable modulation of the scattering length g_0(t) allows us to shape the interactions in (<ref>). In order to perform a systematic analysis one can apply the singular value decomposition of the matrix u_ij(t) where (i,j) and t are treated as indices of rows and columns, respectively. Left singular vectors tell us which sets of interaction coefficients U_ij can be realized, while the corresponding right singular vectors give the recipes for g_0(t). In Fig. <ref> we present an example of the interaction coefficients and the corresponding function g_0(t). In this example the magnitude of the interactions of a particle located at a given site with other particles located at the same or distant sites is nearly the same, but their repulsive or attractive character changes in an oscillatory way. Time is a single degree of freedom and it is hard to imagine multidimensional time crystals. However, we will see that resonantly driven systems can reveal properties of 2D or 3D space crystals in the time domain. Let us begin with a single particle bouncing between two mirrors that oscillate harmonically in two orthogonal directions with frequency ω, see Fig. <ref>— generalization to the 3D case is straightforward. The single particle Hamiltonian reads H=p_x^2+p_y^2/2+x+y+λ xcosω t+λ ycos(ω t+φ) where φ is the relative phase of the mirror oscillations. Assuming that for each of the two independent degrees of freedom the s:1 resonance condition is fulfilled we obtain (in terms of the action-angle variables and in the moving frame Θ_j=x,y=θ_j-ω/st) the effective Hamiltonian, H_ eff=P_x^2+P_y^2/2m_ eff+V_0[cos(sΘ_x)+cos(sΘ_y)], which describes a particle in a 2D square lattice. For s≫ 1, eigenstates of H_ eff are Bloch waves e^i(k_xΘ_x+k_yΘ_y)v_k_x(Θ_x)v_k_y(Θ_y). When we a fix position in the laboratory frame, i.e. we fix θ_x and θ_y, periodic character of Bloch waves emerges in time, e^i(k_xθ_x+k_yθ_y-(k_x+k_y)ω t/s)v_k_x(θ_x-ω t/s)v_k_y(θ_y-ω t/s). Different fixed values of θ_x and θ_y allows us to observe in the time domain different cuts of the square lattice described by H_ eff. We restrict ourselves to the first energy band of H_ eff and define the Wannier state basis W_ j=w_x(Θ_x-j_x2π/s)w_y(Θ_y-j_y2π/s) where j=(j_x,j_y) denotes at which site of the effective potential a Wannier function is localized. In the laboratory frame the Wannier states W_ j(x,y,t) describe localized wavepackets moving along resonant trajectories. The shape of the trajectories depends on the relative phase φ of the mirrors oscillations. For φπ/2 there are s different trajectories in the configuration space and s wavepackets W_ j(x,y,t) moving along each of them, see Fig. <ref>. Thus, s^2 sites of the effective potential in the moving frame correspond to s^2 Wannier wavepackets evolving in the laboratory frame. Switching to the many-body case we obtain, for ultra-cold bosons, a 2D version of the Hamiltonian (<ref>). If the scattering length is not modulated in time, i.e. g_0(t)=const., the on-site interactions are dominant and the system reproduces, in the moving frame, a 2D squared lattice problem with on-site interactions <cit.>. If we locate detectors at different positions in the laboratory frame, the time dependence of the probabilities of detection reflects cuts of the 2D square lattice, see Fig. <ref>. Finally, let us show that periodic driving allows one to create a molecule where Anderson localization is responsible for the binding of two atoms. Assume that two atoms move on a ring and their scattering length is modulated in time employing a Feshbach resonance so that the Hamiltonian of the system reads H=p_1^2+p_2^2/2+2πλδ(θ_1-θ_2)f(t) where λ is a constant, f(t)=∑_k 0f_ke^ikω t and θ_1,2 denote positions of the atoms on the ring. If the first atom is moving in the clockwise direction with momentum p_1≈ω, and the other in the anticlockwise direction with p_2≈ -ω, then the secular approximation results in H_ eff=P_1^2+P_2^2/2+λ V_ eff(Θ_1-Θ_2) in the moving frame, i.e. Θ_1=θ_1-ω t and Θ_2=θ_2+ω t. Interactions between atoms are described by the effective potential V_ eff=∑_nf_-2ne^in(Θ_1-Θ_2) whose shape can be engineered at will by a suitable choice of the Fourier components f_k of the periodic driving. For example if f_k=1/√(k_0)e^iφ_k for |k|≤k_0/2 and zero otherwise, where φ_k=-φ_-k are random variables chosen from a uniform distribution, the atoms interact via the effective disordered potential characterized by the correlation length √(2)/k_0 and the standard deviation λ. Then, eigenstates ψ(Θ_1-Θ_2) of H_ eff are Anderson localized around different values θ_0 of the relative coordinate <cit.>, i.e. |ψ|^2∝ e^-|Θ_1-Θ_2-θ_0|/l_0, provided the localization length l_0≪ 2π— within the Born approximation l_0=Ek_0^2/πλ^2, which is valid when λ^2/k_0^2≪ E≪k_0^2/4, where E is energy of the system in the moving frame <cit.>. Hence, we are dealing with a situation where two atoms are bound together not by attractive interactions but due to destructive interference, i.e. due to Anderson localization phenomenon induced by disordered mutual interactions <cit.>.If atoms are identical bosons (fermions), an eigenstate must be symmetric (antisymmetric) under their exchange. This symmetry is easily restored because we can exchange the role of the atoms. That is, the first atom can move in the anticlockwise direction, p_1≈ -ω, while the other one in the clockwise direction, p_2≈ω. Consequently, proper Floquet eigenstates for bosons or fermions, in the laboratory frame, read ψ(θ_1-θ_2-2ω t)±ψ(θ_2-θ_1-2ω t).Experimental demonstration of two atoms bound due to destructive interference seems straightforward if atoms are prepared in a toroidal trap <cit.>. In summary, we have shown that a wide class of condensed matter problems can be realized in the time domain if single-particle or many-body systems are resonantly driven. It opens up unexplored territory for investigation of condensed matter physics in time and for the invention of noveltime devices because time is our new ally. As an example we have demonstrated that periodic driving allows one to realize molecules where atoms are bound together not due to attractive mutual interactions but due to destructive interference.Support of the National Science Centre, Poland via Projects No. 2016/20/W/ST4/00314 (K.G.) and No. 2016/21/B/ST2/01095 (K.S.) is acknowledged. 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Lin, journalThe European Physical Journal Dvolume53, pages127 (year2009), ISSN issn1434-6079, <http://dx.doi.org/10.1140/epjd/e2009-00116-7>.§ SUPPLEMENTAL MATERIAL In this Supplemental Material we present an analysis of the validity of the effective Hamiltonian approach for two examples: a particle bouncing on a vibrating mirror and formation of a molecule where two atoms are bound together due to destructive interference. However, we begin with a short introduction to the classical and quantum description of periodically driven systems. If the Hamiltonian of, e.g., a single-particle system in one-dimensional space depends explicitly on time, H(x,p,t), the energy is not conserved. In the classical description one can extend the phase space of the system by defining t as an additionaldimension of the configuration space and p_t=-H as the conjugate momentum <cit.>. Then, the new Hamiltonian in such an extended phase space reads H=H(x,p,t)+p_t and the motion of a particle is parametrized by some fictitious time τ. Because dt/dτ=∂ H/∂ p_t=1, t and τ are essentially identical. The new Hamiltonian does not depend explicitly on τ and thus it is conserved as τ evolves.In the quantum description there are no energy eigenstates because the energy is not conserved. However, if a particle is periodically driven, H(x,p,t+2π/ω)=H(x,p,t), we can look for a kind of stationary states of the form of ψ_n(x,t)=e^-iE_ntv_n(x,t) where v_n(x,t+2π/ω)=v_n(x,t). Indeed, substituting ψ_n(x,t) in the time-dependent Schrödinger equation we obtain the eigenvalue problem, (H(x,p,t)-i∂_t)v_n(x,t)=E_nv_n(x,t),where v_n(x,t) fulfills periodic boundary conditions in time <cit.>.Thus, one may define a new Hamiltonian (so-called Floquet Hamiltonian) H_F=H-i∂_t whose eigenstates (called Floquet states) are time-periodic and form a complete basis in the Hilbert space of the system at any time t<cit.>. The Floquet Hamiltonian can be considered as the quantized version of the classical Hamiltonian H in the extended phase space where p_t→-i∂_t. In the Letter we show different periodically driven systems whose Floquet states possess properties of condensed matter systems in the time domain. In order to identify systems with such properties and find suitable parameters we begin with the classical approach, obtain effective Hamiltonians and then switch to the quantum description. §.§ Particle bouncing on a vibrating mirror Let us consider a particle which bounces on a periodically vibrating mirror in the presence of the gravitational force <cit.>. In the frame moving with the mirror, the mirror does not vibrate but the gravitational potential changes periodically in time. The Hamiltonian of the system, in gravitational units, reads H=H_0+H_1 withH_0(x,p)=p^2/2+x,H_1(x,t)=λ xf(t),where f(t+2π/ω)=f(t) and λ determines the strength of the perturbation. For λ=0, the classical description of the system becomes very simple if we apply the canonical transformation to the so-called action-angle variables (I,θ)<cit.>. Then, H_0=H_0(I)=1/2(3π I)^2/3 where the action I (new momentum) is a constant of motion and the conjugate angle θ (new position variable) changes linearly in time. Now, θ(t)=Ω t+θ(0) where Ω(I)=dH_0(I)/dI is the frequency of a periodic motion of the unperturbed particle <cit.>. The portrait of the (θ,I) phase space is very simple because it consists of straight lines corresponding to different values of I=constant. When λ 0, the phase space structure changes around resonant values of I. Let us focus on f(t)=∑_kf_ke^ikω t where f_k's correspond to the time quasi-crystal structure presented in Fig. 1 of the Letter. In the present Supplemental Material in Fig. <ref>(a)-<ref>(b) we show a stroboscopic picture of the (Θ,P) space obtained by integration of the classical equations of motion generated by the Hamiltonian (<ref>) where Θ=θ-ω t and P=I-I_0 with I_0 corresponding to the 1:1 resonance condition between the driving force and the unperturbed particle motion, i.e. ω=Ω(I_0). We have chosen I_0=1 and two different values of λ. If λ is sufficiently small the phase space portrait around elliptical islands does not reveal chaotic motion and we may expect that the first order secular approximation <cit.> is able to perfectly describe motion of a particle close to the resonant value of I. The Hamiltonian (<ref>) in the frame moving along the resonant orbit, i.e. in the Θ=θ-ω t and I variables, readsH = H_0(I)-ω I+λ f(t)∑_n h_n(I)e^in(Θ+ω t),where h_0=(π I/√(3))^2/3 and h_n=(-1)^n+1/n^2(3I/π^2)^2/3 for n 0. In the moving frame, Θ and I are slow variables if P=I-I_0≈ 0. Then, averaging (<ref>) over fasttime variable and performing Taylor expansion around the resonant value of I lead to the effective secular Hamiltonian,H_ eff=P^2/2m_ eff+λ∑_n h_n(I_0)f_-ne^inΘ,where m_ eff=-π^2/ω^4. The phase space portraits generated by (<ref>) are shown in Fig. <ref>(c)-<ref>(d) for the values of λ corresponding to the exact portraits presented in Fig. <ref>(a)-<ref>(b). In the case when λ=0.01, the effective Hamiltonian results and the exact data are identical. Thus, the motion of a particle in the quasi-crystal potential predicted by the effective Hamiltonian is reproduced by the full classical dynamics provided λ is sufficiently small.In order to switch to quantum effective description we can either perform quantization of the classical effective Hamiltonian (<ref>) or apply quantum version of secular approximation for the Hamiltonian (<ref>) <cit.>. Let us first discuss the former approach. Classical equations of motion possess the scaling symmetry which implies that by a proper rescaling of the parameters and dynamical variables of the system we obtain the same behavior as presented in Fig. <ref> but around arbitrary value of I_0 1<cit.>. That is, when we redefine ω'=I_0^-1/3ω and λ'=λ we can use the results presented in Fig. <ref> if we rescale p'=I_0^1/3p, x'=I_0^2/3x and t'=I_0^1/3t. In the quantum description the scaling symmetry is broken because the Planck constant sets a scale in the phase space,[x,p]=i ⇒ [x',p']=i/I_0.For I_0≫ 1, the quantized version of the effective Hamiltonian (<ref>), i.e. when P→ -i∂/∂Θ, provides perfect quantum description of the resonant behavior of the system. The same quantum results can be obtained by applying the quantum secular approach <cit.> which yieldsn'|H_ eff|n =(E_n-nω)δ_nn'+λ n'|x|n f_n-n',where |n's are eigenstates of the unperturbed (λ=0) system and 2^1/3E_n are zeros of the Airy function <cit.>. Equation (<ref>) has been obtained by switching to the moving frame, with the help of the unitary transformation Û=e^in̂ω t, and by averaging the Hamiltonian over the short time scale 2π/ω<cit.>. §.§ Molecule formation due to destructive interference Let us consider two atoms which move on a ring and interact via a contact potential,H=p_1^2+p_2^2/2+2πλδ(θ_1-θ_2)f(t),where f(t)=∑_k 0f_ke^iω t with f_k=1/√(k_0)e^iφ_k for |k|≤k_0/2 and zero otherwise. φ_k=-φ_-k are random variables chosen from the uniform distribution.If the momenta of atoms fulfill p_1≈ω and p_2≈ -ω, then in the moving frame,Θ_1=θ_1-ω t,P_1=p_1-ω, Θ_2=θ_2+ω t,P_2=p_2+ω, the secular approximation results in the effective HamiltonianH_ eff=P_1^2+P_2^2/2+λ∑_nf_-2ne^in(Θ_1-Θ_2),which describes two particles interacting via a disorder potential. For a suitable choice of the system parameters, two atoms can be bound together due to Anderson localization. The effective Hamiltonian does not depend on ω but its validity does. For a given λ, the effective Hamiltonian (<ref>) is always valid if ω is sufficiently large as the second order corrections are proportional to λ^2/ω^2. This is illustrated in Fig. <ref> where we compare an eigenstate of (<ref>) with the corresponding Floquet eigenstate of the full Hamiltonian (<ref>) for two different values of ω. Thus, for sufficiently large ω, higher order terms neglected in the first order secular approximation do not modify the Anderson localization phenomena predicted by the effective Hamiltonian (<ref>). 4 natexlab#1#1 bibnamefont#1#1 bibfnamefont#1#1 citenamefont#1#1 url<#>1 urlprefixURL [Lichtenberg and Lieberman(1992)]Lichtenberg1992sauthorA. LichtenbergandauthorM. Lieberman, titleRegular and chaotic dynamics, Applied mathematical sciences (publisherSpringer-Verlag, year1992), ISBN isbn9783540977452, <https://books.google.pl/books?id=2ssPAQAAMAAJ>.[Buchleitner et al.(2002)Buchleitner, Delande, and Zakrzewski]Buchleitner2002sauthorA. Buchleitner, authorD. Delande, andauthorJ. Zakrzewski, journalPhysics reportsvolume368, pages409 (year2002), <http://www.sciencedirect.com/science/article/pii/S0370157302002703>.[Shirley(1965)]Shirley1965sauthorJ. H.Shirley, journalPhys. Rev.volume138, pagesB979 (year1965), <https://link.aps.org/doi/10.1103/PhysRev.138.B979>.[Berman and Zaslavsky(1977)]Berman1977sauthorG. BermanandauthorG. Zaslavsky, journalPhysics Letters Avolume61, pages295 (year1977), ISSN issn0375-9601, <http://www.sciencedirect.com/science/article/pii/0375960177906181>. | http://arxiv.org/abs/1710.10087v3 | {
"authors": [
"Krzysztof Giergiel",
"Artur Miroszewski",
"Krzysztof Sacha"
],
"categories": [
"cond-mat.quant-gas",
"cond-mat.dis-nn",
"cond-mat.mes-hall",
"physics.atom-ph",
"quant-ph"
],
"primary_category": "cond-mat.quant-gas",
"published": "20171027114455",
"title": "Time crystal platform: from quasi-crystal structures in time to systems with exotic interactions"
} |
Comparison of Efficiencies of some Symmetry Tests around an Unknown Center Bojana Milošević[[email protected]] , Marko Obradović [[email protected]] Faculty of Mathematics, University of Belgrade, Studenski trg 16, Belgrade, Serbia=================================================================================================================================================================================== In this paper, some recent and classical tests of symmetry are modified for the case of an unknown center. The unknown center is estimated with its α-trimmed mean estimator. The asymptotic behavior of the new tests is explored. The local approximate Bahadur efficiency is used to compare the tests to each other as well as to some other tests. keywords: U-statistics with estimated parameters; α-trimmed mean; asymptoticefficiencyMSC(2010):62G10, 62G20 § INTRODUCTION The problem of testing symmetry has been popular for decades, mainly due to the fact that many statistical methods depend on the assumption of symmetry. The well-knownexamples are the robust estimators of location such as trimmed means that implicitly assume that the data come from a symmetric distribution. Another example are the bootstrap confidence intervals, that tend to converge faster whenthe corresponding pivotal quantity is symmetrically distributed.Probably the mostfamous symmetry tests are the classical sign and Wilcoxon tests, as well as the tests proposed following the example of Kolmogorov-Smirnov and Cramer-von Mises statistics (see. <cit.>). Other examples of papers on this topic include <cit.>. All of these symmetry tests are designed for the case of a known center of distribution. They share many nice properties such as distribution-freeness under the null symmetry hypothesis.There have been many attempts to adaptthese tests to the case of an unknown center. Some modified Wilcoxon tests can be foundin <cit.> and a modified sign test in<cit.>.There also exist some symmetry tests originally designed for testing symmetry around an unknown center.The famous √(b_1) test is one of the examples. Some other tests have been proposed in <cit.>.The goal of our paper is to compare some symmetry tests around an unknown center. Instead of commonly used power comparison(see e.g. <cit.>), here we compare the tests using the local asymptotic efficiency. We opt for the approximate Bahadur efficiency since it is applicable to asymptotically non-normally distributed test statistics. The Bahadur efficiency of symmetry tests has been considered in, among others, <cit.>. Consider the setting of testing the null hypothesis H_0:θ∈Θ_0 against the alternative H_1:θ∈Θ_1. Let us suppose that for a test statistic T_n, under H_0, the limitlim_n→∞P{T_n≤ t}=F(t), where F is non-degenerate distribution function, exists.Further, suppose that lim_t→∞t^-2log(1-F(t))=-a_T/2, and that the limit in probability P_θ lim_n→∞T_n=b_T(θ)>0, exists for θ∈Θ_1.The relative approximate Bahadur efficiency with respect to another test statistic V_n ise^∗_T,V(θ)=c^∗_T(θ)/c^∗_V(θ),wherec^∗_T(θ)=a_Tb_T^2(θ) is the approximate Bahadur slope of T_n. Its limit when θ→ 0 is called the local approximate Bahadur efficiency.The tests we consider may be classified into two groups according to their limiting distributions: asymptotically normal ones; and those whose asymptotic distribution coincides with the supremum of some Gaussian process.For the first group of tests, the coefficient a_T is the inverse of the limiting variance. For the second, it is the inverse of the supremum of the covariance function of the limiting process (see <cit.>). § TEST STATISTICSMost of considered tests are obtained by modifying the symmetry tests around known location parameter.Let X_1,..,X_n be an i.i.d. sample with distribution function F. The tests are appliedto the sample shifted by the value of the location estimator. Typical choices of location estimators are the mean and the median. Here we take a more general approach, using α-trimmed means μ(α)=1/1-2α∫_F^-1(α)^F^-1(1-α)xdF(x),0<α<1/2,including their boundary cases μ(0) – the mean and μ(1/2) –the median. The estimator is μ(α)=1/1-2α∫_F_n^-1(α)^F_n^-1(1-α)xdF_n(x),0<α<1/2. In the case of α=0 and α=1/2, the estimators are the sample mean and the sample median, respectively.The modified statistics we consider in this paper are: * Modified sign test S=1/n∑_i=1^nI{X_j-μ(α)>0}-1/2; * Modified Wilcoxon test W=1/n2∑_1≤ i<j≤ n I{X_i+X_j-2μ(α)>0}-1/2; * Modified Kolmogorov-Smirnov symmetry test KS=sup_t|F_n(t+μ(α))+F_n(μ(α)-t)-1|; * Modified tests based on the Baringhaus-Henze characterization (see <cit.>) BH^I =1/n n2∑_i_3=1^n∑_𝒞_2(1/2 I{|X_i_1-μ(α)|<|X_i_3-μ(α)|}+1/2 I{|X_i_2-μ(α)|<|X_i_3-μ(α)|}- I{|X_2;X_i_1,X_i_2-μ(α)|<|X_i_3-μ(α)|});BH^K =sup_t>0|1/ n2∑_𝒞_2(1/2 I{|X_i_1-μ(α)|<|X_i_3-μ(α)|}+1/2 I{|X_i_2-μ(α)|<|X_i_3-μ(α)|}- I{|X_2;X_i_1,X_i_2-μ(α)|<t})|; * Modified tests based on the Ahsanullah's characterization (see <cit.>) NA^I(k) =1/n nk∑_i_k+1=1^n∑_𝒞_k( I{|X_1;X_i_1,…,X_i_k-μ(α)|<|X_i_k+1-μ(α)|}- I{|X_k;X_i_1,…,X_i_k-μ(α)|<|X_i_k+1-μ(α)|});NA^K(k) =sup_t>0|1/ nk∑_𝒞_k( I{|X_1;X_i_1,…,X_i_k-μ(α)|<t}- I{|X_k;X_i_1,…,X_i_k-μ(α)|<t})|; * Modified tests based on the Milošević-Obradović characterization (see <cit.>) MO^I(k) =1/n n2k∑_i_2k+1=1^n∑_𝒞_2k( I{|X_k;X_i_1,…,X_i_2k-μ(α)|<|X_i_2k+1- μ(α)|}- I{|X_k+1;X_i_1,…,X_i_2k-μ(α)|<|X_i_2k+1-μ(α)|});MO^K(k) =sup_t>0|1/ nk∑_𝒞_2k( I{|X_k;X_i_1,…,X_i_2k-μ(α)|<t}- I{|X_k+1;X_i_1,…,X_i_2k-μ(α)|<t})|, where X_k;X_i_1,...X_i_m stands for the kth order statistic of the subsample X_i_1,...,X_i_m, and 𝒞_m={(i_1,…, i_m) : 1≤ i_1 < ⋯ < i_m ≤ n}.Note that NA^I(2) and MO^I(1) coincide. The same holds for NA^K(2) and MO^K(1).Among the tests originally intended for testing symmetry around an unknown mean, the most famous isclassical √(b_1) test,based on the sample skewness coefficient, with test statistic √(b_1)=m̂_3/s^3,where m̂_3 is thesamplethird central moment and s is sample standard deviation. The test is applicable if the sample comes from a distribution withfinite sixth moment.We also consider the class of tests based on so-called Bonferoni measure. In <cit.> the following test statistic is proposed: CM=X̅-M̂/s,where M̂ is the sample median and s is the sample standard deviation. Similar tests are proposed in <cit.> and <cit.>, with the following statistic γ=2(X̅-M̂) MGG=X̅-M̂/J,whereJ=√(π/2)1/n∑_i=1^n|X_i-M̂|. These tests are applicable if the sample comes from a distribution with finite second moment.For the supremum-type tests, we consider large values of test statistic to be significant. For others, which are asymptotically normally distributed,we consider large absolute values of tests statistic to be significant.§ BAHADUR APPROXIMATE SLOPES We can divide the considered test statistics into three groups based on their structure: * non-degenerate U-statistics with estimated parameters; * the suprema of families of non-degenerate (in sense of <cit.>) U-statisticswith estimated parameters; * other statistics with limiting normal distribution. Since we are dealing with U-statistics with estimated location parameters, we shall examine their limiting distribution using the techniquefrom <cit.>. With this in mind, we give the following definition. Let 𝒰 be the family of U-statistics U_n(μ), with bounded symmetric kernel Φ(·;μ), that satisfy the following conditions: * EU_n(μ)=0; * √(n)U_n converges to normal distribution whose variance does not depend on μ; * For K(μ,d), a neighbourhood of μ of a radius d, there exists a constant C>0, such that, if μ'∈ K(μ,d) thenE(sup_μ'∈ K(μ,d)|Φ(·;μ')-Φ(·;μ)|)≤ Cd. All our U-statistics belong to this family due to unbiasedness, non-degeneracy and uniform boundnessof their kernels. Since our comparison tool is the local asymptotic efficiency, we are going to consider alternatives close to some symmetric distribution.Therefore, we define the family of close alternatives that satisfy some regularity conditions (see also <cit.>). Let 𝒢={G(x;θ)} be the family of absolutely continuous distribution functions, with densities g(x;θ), satisfying the following conditions: * g(x;θ) is symmetric around some location parameterμ if and only if θ=0; * g(x;θ) is twice continuously differentiable along θ in some neighbourhood of zero; * all second derivatives of g(x;θ)existand are absolutely integrable for θ in some neighbourhood of zero. For brevity, in what follows, we shall use the following notation: F(x):=G(x,0);f(x):=g(x,0);H(x):=G'_θ(x,0);h(x):=g'_θ(x,0). The null hypothesis of symmetry can now be expressed as: H_0:θ=0. To calculate the local approximate slope (<ref>), we need to find the variance of the limiting normal distribution under the null hypothesis,as well as the limit in probability under a close alternative. We achieve this goal usingthe following two theorems. Let X=(X_1,X_2...,X_n) be an i.i.d. sample from an absolutely continuous symmetric distribution, with distribution function F. Let U_n(μ) with kernel Φ(X;μ) be a U-statistic of order m from the family 𝒰; and let μ(α), 0≤α≤ 1/2, be the α-trimmed sample mean (<ref>). Then √(n)U_n(μ(α)) converges in distribution to a zero mean normal random variable with the following variance: σ^2_U,F(α) =m^2(∫_-∞^∞φ^2(x)f(x)dx+2/(1-2α)^2(∫_-∞^∞φ(x)f'(x)dx)^2·(∫_0^q_1-αx^2f(x)dx+α(q_1-α)^2) +4/1-2α(∫_-∞^∞φ(x)f'(x)dx)·(∫_0^q_1-αφ(x)xf(x)dx+ q_1-α∫_q_1-α^∞φ(x)f(x)dx)),for 0<α<1/2, whereφ(x)=E(Φ(𝕏;μ)|X_1=x) is the first projection of the kernelΦ(𝕏;μ)on a basic observation, and q_1-α=F^-1(1-α) is the (1-α)th quantile of F. In the case of boundary values of α, the expression above becomes: σ^2_U,F(0) =m^2(∫_-∞^∞φ^2(x)f(x)dx+ 2(∫_-∞^∞φ(x)f'(x)dx)^2(∫_0^∞x^2f(x)dx) +4(∫_-∞^∞φ(x)f'(x)dx)·(∫_0^∞φ(x)xf(x)dx )), and σ^2_U,F(1/2) =m^2(∫_-∞^∞φ^2(x)f(x)dx+1/4f^2(0)(∫_-∞^∞φ(x)f'(x)dx)^2 +2/f(0)(∫_-∞^∞φ(x)f'(x)dx)·(∫_0^∞φ(x)f(x)dx)). Proof. We prove only the case 0<α<0.5. The rest are analogous and simpler. Notice that μ̂(α) has its Bahadur representation <cit.> μ̂(α)-μ(α)=1/n∑_j=1^nψ_α,F(X_j)+R_n,whereψ_α,F(x)=1/1-2α∫_α^1-αt-I{x<F^-1(t)}/f(F^-1)(t)dtis the influence curve of μ(α), and √(n)R_n converges in probability to zero.Using the multivariate central limit theorem for U-statistics we conclude that the joint limitingdistribution of√(n)U_n and√(n)(μ̂(α)-μ(α)) is bivariate normal 𝒩_2(0,Σ), whereΣ=([ m^2∫_-∞^∞φ^2(x)dF(x) m∫_-∞^∞ψ_α,F(x)φ(x)dF(x); m∫_-∞^∞ψ_α,F(x)φ(x)dF(x)∫_-∞^∞ψ^2_α,F(x)dF(x) ]). Therefore, the conditions 2.3 and 2.9B of <cit.> are satisfied. Hence we have √(n)U_n(μ(α))d→𝒩(0,σ^2_U,F(α)) where σ^2_U,F(α)=[1, A]^TΣ[1, A],andA=E_γΦ(·;μ)^'_μ |_μ=γ=m∫_-∞^∞φ(x)f'(x)dx. Under the same assumptions as in Theorem <ref>, the limit in probability of the modified statistic U_n(μ(α)) under alternative g(x;θ)∈𝒢 is b(θ,α)=m∫_-∞^∞φ(x)(h(x)+μ'_θ(0,α)f'(x))dx·θ+o(θ), where μ'_θ(0,α)=1/1-2α(-q_1-α(H(q_1-α)+H(-q_1-α))+ ∫_-q_1-α^q_1-αxh(x)dx), for 0<α<1/2, μ'_θ(0,1/2)=-H(0)/f(0), and μ'_θ(0,0)=∫_-∞^∞xh(x)dx.Proof.Let L(x;θ) be the likelihood function of the sample. Using the law of large numbers for U-statistics with estimated parameters (see <cit.>),the limit in probability of U_n(μ(α)) is b(θ,α) =∫_-∞^∞Φ(x-μ(θ,α))L(x;θ)dx=∫_-∞^∞Φ(x)L(x+μ(θ,α);θ)dx=∫_-∞^∞Φ(x)∏_i=1^n g(x_i+μ(θ,α);θ)dx. The first derivative with respect to θ at θ=0 is b'(0,α) =∫_-∞^∞Φ(x)[∑_j=1^n(μ'(0,α)f'(x_j)+h(x_j))∏_i≠ ji=1^nf(x_i)]dx=m∫_-∞^∞φ(x)((μ'(0,α)f'(x)+h(x))dx.Expanding b(θ,α) in the Maclaurin series we complete the proof. In the case of supremum-type statistics, the following two theorems, analogous to the Theorem <ref> and Theorem <ref>, are used. Let X=(X_1,X_2...,X_n) be an i.i.d. sample from an absolutely continuous symmetric distribution, with function F. Let {U_n(μ;t)} be a non-degenerate family of U-statistics of order m with kernel Φ(·;t), that belong to the family 𝒰; and let μ(α), 0≤α≤ 0.5, be the α-trimmed sample mean (<ref>). Then the family {U_n(μ(α);t)} is also non-degenerate with the variance function σ^2_U,F(α;t) =m^2(∫_-∞^∞φ^2(x;t)f(x)dx+ (∫_-∞^∞φ(x;t)f'(x)dx)^22/(1-2α)^2·(∫_0^q_1-αx^2f(x)dx+α(q_1-α)^2) +4/1-2α(∫_-∞^∞φ(x;t)f'(x)dx)·(∫_0^q_1-αφ(x;t)xf(x)dx+ q_1-α∫_q_1-α^∞φ(x;t)f(x)dx)), for 0<α<1/2. In the case of boundary values of α, we have σ^2_U,F(0;t) =m^2(∫_-∞^∞φ^2(x;t)f(x)dx+ 2(∫_-∞^∞φ(x;t)f'(x)dx)^2(∫_0^∞x^2f(x)dx) +4(∫_-∞^∞φ(x;t)f'(x)dx)·(∫_0^∞φ(x;t)xf(x)dx+ )), and σ^2_U,F(1/2;t) =m^2(∫_-∞^∞φ^2(x;t)f(x)dx+1/4f^2(0)(∫_-∞^∞φ(x;t)f'(x)dx)^2 +2/f(0)(∫_-∞^∞φ(x;t)f'(x)dx)·(∫_0^∞φ(x;t)f(x)dx)). Moreover, √(n)sup|U_n(μ(α);t)| converges in distribution to the supremum of a certain centered Gaussian process. Proof. The asymptoticbehaviour of√(n)U_n(μ(α);t) for a fixed t is established in Theorem <ref>. From <cit.> we have U_n(μ̂(α);t)d= U_n(μ;t)+η_0(μ̂(α);t)+R'_n,where √(n)R'_np→0 and η_0(μ;t)=EU_n(μ;t).Next, using the mean value theorem and the Bahadur representation (<ref>), we getη_0(μ̂(α);t)) = ∂/∂μη_0(μ(α);t)1/n∑_j=1^nψ_α,F(X_j)+R”_n, where √(n)R”_np→0.Combining (<ref>) and (<ref>) we get √(n)U_n(μ̂(α);t) d=√(n)(U_n(μ;t)+η_0(μ(α);t)+∂/∂μη_0(μ(α);t)1/n∑_j=1^nψ_α,F(X_j))+√(n)(R'_n+R”_n)=√(n)U^*_n(μ(α);t)+√(n)R_n. U^*_n(μ(α);t) is asymptotically equivalent to a family of U-statistics with symmetrized kernel Ξ(𝐗;μ(α),t)=1/m∂/∂μη_0(μ(α);t)∑_j=1^mψ_α,F(X_j)+Φ(𝐗;μ(α),t). Using the result from <cit.>, we have that √(n)U^*_n converges in distribution to a zero mean Gaussian process. Then, since√(n)R_n converges to zero in probability, using the Slutsky theorem for stochastic processes <cit.>, we complete the proof.Under the same assumptions as in Theorem <ref>, the limit in probability of the modified statistic sup_t|U_n(μ(α);t)|, under alternativeg(x;θ)∈𝒢, is b(θ,α)=msup_t|∫_-∞^∞φ(x;t)(h(x)+μ'_θ(0,α)f'(x))dx|·θ+o(θ), Proof.The limit in probability of U_n(μ(α);t) for a fixed t is established in Theorem <ref>.Denote η(μ;t)=E_θ(U_n(μ;t)) and let η(μ(α);t) be its estimator.From<cit.> we have that U_n(μ(α);t)-η(μ;t)=U_n(μ;t)-η(μ;t)+η(μ(α);t)-η(μ;t)with probability one. Then usingGlivenko-Cantelli theoremforU-statistics <cit.>we complete the proof. Finally, the following two theorems give us the Bahadur approximate slopes of the tests based on the Bonferoni measure and √(b_1), respectively. Let (X_1,X_2...,X_n) be an i.i.d. sample with distribution function G(x,θ)∈𝒢.Then the Bahadur approximate slopesof test statisticsCM, γ, and MGG are equal to c(θ) =(∫_-∞^∞xh(x)dx+H(0)/f(0))^2/σ^2+1/4f^2(0)-τ/f(0)·θ^2+o(θ^2),θ→ 0, where σ^2=∫_-∞^∞x^2f(x)dx and τ=2∫_0^∞xf(x)dx. Proof. We shall prove the theorem in the case of statistic CM. The other cases are completely analogous.Denote D=X̅-M̂. Notice that D is ancillary for the location parameter μ. Hence, we may suppose that μ=0.Using the Bahadur representation we haveD=1/n∑_i=1^n(X_i- sgn(X_i)/2f(0)) + R_n,where √(n)R_n converges to zero in probability. Using the central limit theorem we have that the limiting distribution of √(n)D, when n→∞ isnormal with zero mean and the varianceVar(X_1- sgn(X_1)/2f(0))=σ^2+1/4f^2(0)-τ/f(0). Using the Slutsky theorem we obtain that the limiting distribution of √(n) CM is zero mean normal with the variance (σ^2+1/4f^2(0)-τ/f(0))σ^-1. Next, using the law of large numbers and the Slutsky theorem, we have that the limit in probability under a close alternativeG(x,θ)∈𝒢 isb(θ)=m(θ)-μ(θ)/σ(θ),where m(θ), μ(θ) and σ(θ) are the mean, the median and the standard deviation, respectively. Expanding b(θ) in the Maclaurin series,and combining with (<ref>) into (<ref>), we complete the proof. Let (X_1,X_2...,X_n) be an i.i.d. sample with distribution function G(x,θ)∈𝒢.Then the Bahadur approximate slope of test statistic√(b_1) is c(θ) =(∫_-∞^∞ x^3h(x)dx-3σ^2∫_-∞^∞ xh(x)dx)^2/m_6-6σ^2m_4+9σ^6·θ^2+o(θ^2),θ→ 0, where σ^2=∫_-∞^∞x^2f(x)dx, and m_j is the jth central moment of F. The proof goes along the same lines as in the previous theorem, so we omit it here. § COMPARISON OF THE TESTS Since no test is distribution free, we need to choose the null variance in order to calculate the local approximate slope. Since we deal with the alternatives close to symmetric, it is natural to choose the closest symmetric distribution for the null. §.§ Null and Alternative Hypotheses We consider the normal, the logistic and the Cauchy asnull distributions. Using Theorem <ref> we calculated the asymptotic variances of all our integral-type statistics, as well as the suprema of variance functions of the supremum-type statistics. In Figure <ref> we present the limiting variances of some integral-type statistics as function of the trimming coefficient α. It can be noticed thatfor some values of α the variances are very close to each other.This "asymptotic quasi distribution freeness" might be of practical importance providingan alternative to standard bootstrap procedures.For each null distribution, we consider two types of closealternatives from 𝒢: * a skew alternativein the sense of Fernandez-Steel <cit.>, with the density g(x;θ)=2/1+θ+1/1+θ(g(x/1+θ;0)I{x<0}+g((1+θ)x;0)I{x≥ 0}) * a contamination alternative with the density g(x;θ)=(1-θ)g(x;0)+θ g(x-1;0). Another popular family of alternatives are the Azzalini skew alternatives (see <cit.>). However, in the case of the skew-normal distribution, all our test have zero efficiencies, while in the skew-Cauchy case, theBahadur efficiency is not defined. Hence, these alternatives are notsuitablefor comparison, and we decided not to include them. §.§ Bahadur equivalence It turns out that some tests have identical Bahadur approximate slopes. With this in mind, we say that two tests are Bahadur equivalent if their Bahadur local approximate slopes coincide. It is then sufficient to consider just one representative from each equivalence class for comparison purposes.We have the following Bahadur equivalence classes: * BH^I ∼ MO^I(1) ∼ NA^I(2) ∼ NA^I(3) for all 0≤α≤ 1/2; * BH^K ∼ MO^K(1) ∼ NA^K(2) ∼ NA^K(3) for all 0≤α≤ 1/2; * CM ∼γ∼ MGG ∼ S(μ̂(0)); * KS(μ̂(α))∼ S(μ̂(α))(up to a certain value of α).The first three equivalence classes can be easily obtained from the expressions for the corresponding Bahadur approximateslopes. For the fourth equivalence, notice that the term which is maximized in the KSstatistic, for t=0, is twice the absolute value of the S statistic.So, these tests are equivalentwhenever both the supremum of the asymptotic variance and the supremum of the limit in probability, are reached for t=0(see Figure <ref> as an example). This is the case for small α, from zero up to certain point that depends on the underlying null and alternative distributions. §.§ Discussion Using Theorems <ref>-<ref> we calculated the local approximate Bahadur slopes for all statistics, all null and allalternative distributions.Taking into account the Bahadur equivalence, we choose the following tests: BH^I and BH^K; NA^I(4) and NA^K(4)(denoted as NA-I and NA-K); MO^I(2) andMO^K(2) (denoted as MO-I and MO-K); S; W; and KS.For the convenience in presentation, we display the Bahadur approximate indices graphically as functions of α. We also present the indices of CM and √(b_1) (denoted as b1). Since they are notfunctions of α, we show them ashorizontal lines.It is visible from all the figures that in the case of the integral-type statistics, the efficiencies vary significantly with α.In particular, for all the tests exist a value of α for which they have zero efficiencies. On the other hand, the supremum-type tests are muchless sensitive to the change of α. The exception is the classical KS test which is by its definition inefficient for α=0.5.A natural way to compare tests, for a fixed null distribution, would be to compare the maximal values of their Bahadurindices overα∈ [0,0.5].It can be noticed that, in most cases, the integral-type tests outperform the supremum-type ones. The only exception is the contamination alternative to theCauchy distribution. This is in concordance with the previous results (see e.g. <cit.>).In the case of normal distribution, the best of all tests are √(b_1), and W for α=0. In the case of the contamination alternative,MO^I(2) test for α=0 is also competitive.As far as the logistic distribution is concerned, NA^I(4) and BH^I are most efficient. In the case of the Cauchy null, the situation is different. The tests CM and √(b_1) are not applicable, and neither are the othertests for α=0. Also, the “order” of the tests is much differentfor the two considered alternatives. In case of the Fernandez-Steelalternative, the best tests areMO^I(2), NA^I(4) and BH^I, while in thecase of the contamination alternative, MO^K(2) test is the most efficient. As a conclusion, it is hard to recommend which test, and for which α, is the best to use in general, when the underlyingdistribution is completelyunknown. The integral-type tests for small values of α could be the right choice, but they could also be calamities. In contrast, the supremum-type tests with α close or equal to 0.5 are quite reasonable, and, most importantly, never a bad choice. § ACKNOWLEDGEMENT We would like to thank the referees for their useful remarks. This work was supported by the MNTRS, Serbia underGrant No. 174012 (first author). plain | http://arxiv.org/abs/1710.10261v2 | {
"authors": [
"Bojana Milošević",
"Marko Obradović"
],
"categories": [
"stat.ME",
"stat.CO",
"62G20, 62G10"
],
"primary_category": "stat.ME",
"published": "20171027110228",
"title": "Comparison of Efficiencies of Symmetry Tests around Unknown Center"
} |
e-mail: [email protected] Institute for Nuclear Research, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 7a, Moscow, 117312 Russia MEPhI National Research Nuclear University, Kashirskoe sh. 31, Moscow, 115409 Russia e-mail: [email protected] Institute for Nuclear Research, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 7a, Moscow, 117312 Russia e-mail: [email protected] Institute for Nuclear Research, Russian Academy of Sciences, pr. 60-letiya Oktyabrya 7a, Moscow, 117312 Russia Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090 Russia Previously, it has been established that axion dark matter (DM) is clustered to form clumps (axion miniclusters) with masses M∼10^-12M_⊙. The passages of such clumps through the Earth are very rare events occurring once in 10^5 years. It has also been shown that the Earth's passage through DM streams, which are the remnants of clumps destroyed by tidal gravitational forces from Galactic stars, is a much more probable event occurring once in several years. In this paper we have performed details calculations of the destruction of miniclusters by taking into account their distribution in orbits in the Galactic halo. We have investigated two DM halo models, the Navarro-Frenk-White and isothermal density profiles. Apart from the Galactic disk stars, we have also taken into account the halo and bulge stars. We show that about 2-5% of the axion miniclusters are destroyed when passing near stars and transform into axion streams, while the clump destruction efficiency depends on the DM halo model. The expected detection rate of streams with an overdensity exceeding an order of magnitude is 1-2 in 20 years. The possibility of detecting streams by their tidal gravitational effect on gravitational-wave interferometers is also considered. Destruction of Axion Miniclusters in the Galaxy I. I. Tkachev December 30, 2023 =============================================== § INTRODUCTION Although dark matter (DM) accounts for ≃27% of the mass of the Universe, its nature still remains unknown. As yet undetected elementary particles are considered as a probable candidate, and a number of specific candidate particles, such as, for example, neutralinos, sterile neutrinos, or axions, have been proposed in theoretical works. The axion field was initially introduced in particle physics to explain the absence of CP violation in strong interactions. The quanta of this field, axions, turned out to be promising candidates for DM particles. Although the axions are expected to have small masses, they belong to the type of cold DM, because their production mechanism is nonthermal; they were not in chemical and kinetic equilibrium with the cosmic plasma.The existence of DM clumps consisting of axions, axion miniclusters, was predicted in <cit.>. The clumps are formed due to strong fluctuations of the axion field in various regions of space on the horizon scale at the epoch when the axion oscillations began. The fraction of DM in the form of axion miniclusters is f_ mc∼1.A new effect that could increase the chances of detecting the axion DM passing through the Earth was presented in <cit.>. It was shown in <cit.> that although the passage of a whole clump is an extremely rare event, some of the clumps in the Galactic halo are destroyed when interacting with stars, and DM streams with a large overdensity from the destroyed clumps can be observed in ground-based detectors approximately once in 20 years. However, a simplified calculation was performed in <cit.>. In particular, the orbital motion of clumps and their distribution in orbits were disregarded.The orbits of clumps in the Galactic halo are not circular but, as a rule, eccentric. The clumps have some distribution in their orbital parameters, with their orbits undergoing precession (see <cit.>, <cit.>). If a clump or its stream is now passing through the Solar system, then it could previously pass closer to the Galactic center, where the number density of stars is larger and the destruction probability is higher. Therefore, it is necessary to consider the passages through the Galactic disk not only in the solar neighbourhood, as was done in <cit.>, but also at other distances throughout the entire life history of the clump in the Galactic halo. The goal of this paper is to perform such a calculation. The calculation technique used here is similar to that applied in <cit.>, <cit.>, <cit.>. In this paper we also take into account the halo and bulge stars, which contribute noticeably to the destruction of clumps.The DM density and velocity in the Galactic halo in the solar neighbourhood are largely fixed by the observational data on the distribution and motion of stars, because DM and baryons move in the same gravitational potential. Nevertheless, there exists some freedom in the choice of halo parameters, and the observational data are compatible with various DM halo models. To ascertain the dependence of the final results on the halo model, we will perform calculations for two distributions of DM particles and DM clumps in the halo in their orbital parameters: the Navarro-Frenk-White and isothermal density profiles. The Galactic halo model turned out to affect noticeably the result.In this paper we use the characteristic parameters of clumps from <cit.> in our calculations: the clump mass M=10^-12M_⊙, the mean density ρ̅=140ρ_ eqΦ^3(1+Φ), where Φ are the initial entropy density perturbations in the medium of axions (for more details, see <cit.>), at Φ=1 the clump radius is then R=2.3×10^12 cm, while the DM particle velocity dispersion in the clump is v_ mc≃(GM/R)^1/2≃7.6 cm s^-1 (v_ mc/c≃2.5×10^-10). We will choose the clump density profile in accordance with the theory of multistream instability <cit.>:ρ_ int(r)=3-β/3 ρ̅(r/R)^-β.where β=1.7-1.9 (below in the calculations we set β=1.8).§ DESTRUCTION OF CLUMPS AND ADIABATIC PROTECTION A DM clump is destroyed if the net change in its internal energy ∑(Δ E)_j) after one or more gravitational interactions with disk stars or field is comparable to the binding energy of the clump |E|:∑_j(Δ E)_j∼|E|,where the summation is over the successive gravitational interactions. In reality, the destruction occurs not at once but there is a gradual mass loss predominantly through the tidal stripping of outer DM layers <cit.>. In this case, the central dense cores of the clumps can survive <cit.>.If the internal revolution frequencies of DM particles in their orbits inside the clump ω = v_ mc/R are much higher than the characteristic frequency of the external tidal force τ^-1 (the reciprocal of the passage time through the disk or the reciprocal of the passage time of the impact parameter l to the star), then the influence of the tidal force weakens significantly. This effect is characterized by the so-called adiabatic correction or the Weinberg correction A(a), where a=ωτ, that is defined as the ratio of the energy change in the real case to the energy change calculated in the impulse approximation <cit.>. The following approximate formula was found in <cit.>:A(a)=(1+a^2)^-3/2. The Galactic disk consists of stars and gas. The destruction of clumps includes both the interaction with the collective disk field and the interaction with individual stars that happen to be near the clump trajectory in the halo. In the former case, the tidal gravitational field is produced not only by the disk stars but also by the gas-dust clouds in the disk, i.e., by the total gravitating disk mass.Let us first consider the interaction of clumps with the collective gravitational field of the disk when τ≃ H_d/v, where H_d∼ 500 pc is the Galactic disk halfthickness and v∼200 km s^-1 is the characteristic passage velocity. Whereas for neutralino clumps a∼1 <cit.>, for axion clumps a∼200. Thus, in the latter case, the clump destruction effect is suppressed approximately by three orders of magnitude. The estimate made in the impulse approximation shows that the destruction time of axion miniclusters by the collective disk field is ∼10^15 years. If the adiabatic correction is taken into account, then this time increases to ∼10^18 years. Thus, this destruction channel is inefficient, and during the passage of axion clumps through the disk only the interactions with individual stars are important, while the gravitational shocks by the collective disk field play no role.During the interactions with individual stars τ∼ l/v. Let us take the maximum impact parameter at which the clump is destroyed in a single star flyby <cit.> as l:(l_*/R)^4=4(5-2β)/3(5-β)Gm_*^2/MRv_ rel^2,where v_ rel is the relative velocity of the clump and the star, and m_* is the mass of the star. For the typical parameters of clumps given at the end of the Introduction and at β≃2, v_ rel=200 km s^-1, m_*∼ M_⊙ we obtain l_*∼500R and a∼10^-4, i.e., in typical cases, during the interactions with stars the adiabatic correction is unimportant and will be disregarded below. § DESTRUCTION OF CLUMPS IN GRAVITATIONAL COLLISIONS WITH STARS In <cit.> the characteristic destruction time of a clump as it moves in a medium of stars with number density n_* and mass m_* was found for the case of l_*/R>1:t_*=|E|/Ė=1/4π n_*m_*[3(5-β)/(5-2β)M/GR^3]^1/2.Note that at l_*/R>1 the destruction time (<ref>) does not depend on the relative velocity of the clump and the stars.The survival probability of some specific clumpP_1=e^-∫ dt/t_*depends on its trajectory; therefore, we will first determine the parameters of the clump trajectories in the Galactic halo needed for the subsequent discussion in a general form. §.§ The Trajectories of Clumps in the Galactic Halo Let us denote the orbital angular momentum of a clump by J. The equation of the trajectory for r(t) is thenMṙ^2=2[E_ orb-U(r)]-J^2/Mr^2,where U(r) is the potential energy of the clump in the halo. In what follows, we will consider an isotropic distribution of clump orbits in the halo with a total orbital energy E_ orb (not to be confused with the internal energy E) distribution function F(E_ orb) when the relation to the halo density ρ_ H(r) is given by the expressions <cit.>ρ_ H(r) = 2^5/2π∫^0_U(r)√(E_ orb-U(r))F(E_ orb)dE_ orb,F(E_ orb) = 1/2^3/2π^2d/dE_ orb∫_r(E_ orb)^∞dr/√(E_ orb-U(r))dρ_ H(r)/dr,where the function r=r(E_ orb) is found from the equation U[r(E_ orb)]=E_ orb.For the convenience of our subsequent discussion, let us introduce the following dimensionless variables:ξ=r/R_c,ρ̃(ξ)=ρ_ H^ (r)/ρ_0,y=J^2/2MU_0R_c^2, ε=E_ orb/U_0, ψ=U/U_0,where U_0, ρ_0, and R_c are some characteristic values of the gravitational potential, the density, and the radius in a specific DM halo model. In what follows, we will choose these parameters so that U_0=4π Gρ_0R_c^2M.The equation of the clump trajectory for the azimuthal angle ϕ(ξ) isdϕ/dξ=y^1/2/ξ^2√(ε-ψ(ξ)-y/ξ^2).The equation for the extreme points of the orbit ṙ^2=0 will be written asy/ξ^2=ε-ψ.In our calculations we then numerically find the roots of this equation ξ_ min and ξ_ max. Twice the time of clump motion from ξ_ min to ξ_ maxT_c(ε,y)=1/√(2π Gρ_0)∫_ξ_ min^ξ_ maxdξ/√(ε-ψ(ξ)-y/ξ^2)is not equal to the orbital period, because the orbital precession should be additionally taken into account. The precession angle in the time T_ c/2 isϕ̃=y^1/2∫_ξ_ min^ξ_ maxdξ/ξ^2√(ε-ψ(ξ)-y/ξ^2)-π,and ϕ̃<0. Therefore, the true period (the revolution around the Galactic center through 2π) isT_ t=T_ c(1+ϕ̃/π)^-1. Below we consider the clumps whose orbits are currently passing through the Solar system at a distance r=r_⊙=8.5 kpc from the Galactic center. Denote p=cosθ, where θ is the angle between the radius vector of the clump r⃗ and its velocity v⃗ in the solar neighborhood. The dimensionless parameter y characterizing the angular momentum of the clump can then be found from the expressiony=(1-p^2)ξ^2[ε-ψ(ξ)],where we should set ξ=r_⊙/R_c.§.§ Destruction of Clumps by Disk StarsIn the lifetime of the Galaxy t_ G≃10^10 years a clump experiences N≃ t_ G/T_ t double crossings of the Galactic disk, with the crossing points every time being shifted by angles |ϕ̃| due to the precession effect.The survival probability (<ref>) of some specific clump contains the integral∫ m_*n_* dt≃∑∫m_* n_* dl/v,where the summation is over the successive Galactic disk crossings in the time t_ G, while the integration is over one specific crossing. This integral is expressed via the mass surface density σ_s of the stellar component of the Galactic disk,∫m_* n_* dl/v=σ_s/ v_z,while the distribution of stars in masses m_* does not enter into the result. The clump velocity component along the normal to the disk is written asv_z=J/rsinγ,where γ is the angle between the normal to the orbital plane and the normal to the Galactic disk plane. The surface density of the stellar component of the Galactic disk at point r of its crossing by the clump is given by the expressionσ_s(r)=M_d/2π r_0^2 e^-r/r_0,where M_d=3×10^10M_⊙ and r_0=4.5 kpc, so that σ_s(r_⊙)=35M_⊙ pc^-2. Here, we take into account the fact that the stars constitute only part of the total disk mass. The normalization σ_s(r_⊙)=35M_⊙ pc^-2 corresponding to the stars was taken from <cit.> (page 635).The orbital precession effect facilitates considerably the calculation of the sum in Eq. (<ref>), because the clump successively crosses the disk at equal angular intervals at all radii between the minimum and maximum radial distances of the orbit owing to the precession. Therefore, we approximately calculate the sum as follows:∑_i=1^Nσ_s(r)r≃1/|ϕ̃|∫σ_sR_cξ dϕ≃R_c/|ϕ̃|∫_ξ_ min^ξ_ maxσ_s(ξ)ξdϕ/dξdξ2t_ G/T_ t, Let us rewrite (<ref>) asP_1=exp{-λ/Φ^3/2(1+Φ)^1/2sinγ},whereλ=2(2π)^1/2(5-2β/5-β)^1/2G^1/2t_ GS/T_ t|ϕ̃|(140ρ_ eq)^1/2U_0^1/2, S=∫_ξ_ min^ξ_ maxdξσ_s(ξ)/ξ√(ε-ψ(ξ)-y/ξ^2).Using (<ref>), we obtain the fraction of destroyed clumps in the solar neighbourhood:P =1-∫_0^1dp∫_0^sinαdcosγ∫_ψ(ξ)^0dε[ε-ψ(ξ)]^1/2F(ε)P_1, /∫_0^1dp∫_0^sinαdcosγ∫_ψ(ξ)^0dε[ε-ψ(ξ)]^1/2F(ε),where we should substitute ξ=r_⊙/R_c and α≈π/2. §.§ Destruction of Clumps by Halo and Bulge Stars Outside the Galactic disk there are stars of the spherical subsystems: these are halo and bulge stars (plus stars in globular clusters, which we disregard). The number density of stars in the halo at a distance r>3 kpc from the Galactic center isn_h,*(r)=(ρ_⊙/m_*) (r_⊙/r)^3,where we took ρ_⊙=10^-4 M_⊙/pc^3 as an estimate. Note, however, that an order of magnitude larger value, ρ_⊙=10^-3 M_⊙/pc^3, is obtained in some studies (see <cit.> and references in <cit.>, <cit.>). However, the authors of <cit.> point out that even ρ_⊙=10^-4 M_⊙/pc^3 should be considered as an upper limit for the density of stars in the halo.The number density of stars in the bulge at a distance r=1-3 kpc <cit.> isn_b,*(r)=(ρ_b/m_*)exp[ -(r/r_b)^1.6],where ?ρ_b=8M_⊙/pc^3 and r_b=1 kpc.For our calculations we will need to sum the clump energy change over the orbital period or, which is mathematically equivalent, to average t^-1_* over the clump trajectory in the halo:⟨ t_*^-1⟩=R_c/ T_c√(2/U_0)∫_ξ_ min^ξ_ maxt_*^-1dξ/√(ε-ψ(ξ)-y/ξ^2).The survival probability of a single clump isP_1=e^-t_ G⟨ t_*^-1⟩=exp{-λ/Φ^3/2(1+Φ)^1/2},where, in the case under consideration,λ=2(2π)^1/2(5-2β/5-β)^1/2R_c m_*G^1/2t_ GS/T_ c(140ρ_ eq)^1/2U_0^1/2, S=∫_ξ_ min^ξ_ maxdξ n_s(ξ)/√(ε-ψ(ξ)-y/ξ^2).Owing to the presumed spherical symmetry of the halo and the bulge, the expression for the fraction of destroyed clumps is simplified:P =1-∫_0^1dp∫_ψ(ξ)^0dε[ε-ψ(ξ)]^1/2F(ε)P_1 /∫_0^1dp∫_ψ(ξ)^0dε[ε-ψ(ξ)]^1/2F(ε) .§ DESTRUCTION OF CLUMPS IN THE NAVARRO-FRENK-WHITE HALO MODEL Let us first calculate the destruction of axion miniclusters for the Navarro-Frenk-White density profileρ_ H(r)=ρ_0/(r/R_c)(1+r/R_c)^2,where ρ_ H(r_⊙)=0.3 GeV/cm^3 and R_c=20 kpc. It should be noted that a density profile close in shape to the Navarro-Frenk-White profile was obtained in the analytical model <cit.>.The halo density in dimensionless variables isρ̃(ξ)=1/ξ(1+ξ)^2.Choosing U_0=4π Gρ_0R_c^2, we find the gravitational potential in dimensionless variables:ψ(ξ)= -log(1+ξ)/ξ. The distribution function F(ε) for the profile (<ref>) was approximated in <cit.> by the expressionF(ε)=F_1(-ε)^3/2(1+ε)^-5/2[-ln(-ε)/(1+ε)]^qe^P,where F_1=9.1968×10^-2, P=∑_ip_i(-ε)^i, (q,p_1,p_2,p_3,p_4)=(-2.7419, 0.3620, -0.5639,-0.0859, -0.4912). Then,ρ̃(ξ)=4π√(2)∫_ψ(ξ)^0dε[ε-ψ(ξ)]^1/2F(ε). The fraction of clumps in the solar neighbourhood destroyed in their collisions with stars (<ref>) found by numerically calculating all of the integrals in it is indicated by the circles in Fig. <ref> for various values of Φ. If the quantity in the exponent in (<ref>) is much smaller than unity in absolute value, then we can expand the exponential into a series e^x≈1+x and take the integral (<ref>) over γ analytically and the remaining integrals numerically. This allows the functional dependence on Φ to be separated out. The result of such a calculation isP = 6.6×10^-3/Φ^3/2(1+Φ)^1/2and is indicated in Fig. <ref> by the solid line. It can be seen that, in this case, there is some difference between the exact and approximate expressions.Our calculation of the destruction by halo and bulge stars using Eq. (<ref>) is indicated by the triangles in Fig. <ref>. If the exponential in (<ref>) can be expanded, then, as above, we approximately obtainP^(s)=1.8×10^-2/Φ^3/2(1+Φ)^1/2.This quantity is indicated in Fig. <ref> by the solid line. It can be seen that at Φ≥1 the quantity (<ref>) serves as a good approximation to the exact numerical result.The total fraction of destroyed axion miniclusters, including their destructions by disk, halo, and bulge stars, is indicated by the squares in Fig. <ref> and is satisfactorily described by the sum of Eqs. (<ref>) and (<ref>).Comparing (<ref>) with Eqs. (3.3) from <cit.>, we see that the numerical calculation performed in this paper gives approximately a factor of 3 smaller fraction of destroyed clumps if only the destruction by Galactic disk stars is taken into account. The difference between the results is explained by the fact that, in reality, the orbits of clumps in the halo are noncircular and a predominant fraction of the clumps crossing the orbit of the Solar system today spent most of the time at a distance from the Galactic center larger than the distance from the center to the Sun (as was suggested in <cit.>). The disk crossings in the outer region of the Galaxy, where the disk has a lower surface density, exert a smaller destructive effect on the clumps. However, additional destruction is caused by halo and bulge stars, which, as a result (in the sum with (<ref>)), leads to an increase in the total fraction of destroyed clumps by 25%. Thus, the final result turns out to be close to the result of <cit.>, where only the Galactic disk stars were taken into account and no halo and bulge stars were considered. § THE ISOTHERMAL DENSITY PROFILE To ascertain how the result obtained depends on the Galactic halo model, let us perform calculations similar to the previous ones, but for the isothermal density profile of the Galactic haloρ_ H(r)=1/4πv_ rot^2/Gr^2,where v_ rot=(GM_ H/R_ H)^1/2, R_ H≃200 kpc, and ρ(r)=0 at r>R_ H. We choose R_c = R_ H; in this case, U_0 = v_ rot^2 and the potential in dimensionless variables (<ref>) isψ(r)=log(ξ). Using (<ref>) for the profile (<ref>) with the boundary at r=R_ H, we obtainF(ε)=1/2^5/2π^3ev_ rot^1/2/GM^3/2R_ H^2 F(ε),wheref(ε)=√(2π) e^-2ε+2erf[√(-2ε)]+e^2/√(-ε).Note that this distribution function does not reproduce the isothermal profile exactly.The fraction of clumps in the solar neighbourhood destroyed in their collisions with stars (<ref>) found by numerically calculating all of the integrals in it is indicated by the dots in Fig. <ref> for various values of Φ. If the quantity in the exponent in (<ref>) is much smaller than unity in absolute value, thenP ≃1.3×10^-2/Φ^3/2(1+Φ)^1/2.This result is indicated in Fig. <ref> by the solid line.The results of our calculation of the destruction by halo and bulge stars using Eq. (<ref>) are indicated in Fig. <ref> by the triangles. If the exponential in (<ref>) can be expanded, then, as above, we approximately obtainP^(s)=5.3×10^-2/Φ^3/2(1+Φ)^1/2.This quantity and the total quantities for the isothermal density profile are shown in Fig. <ref>. § OBSERVATIONAL CONSEQUENCES§.§ Detection of Streams in Axion Detectors We calculate the expected detection rate of streams in ground-based detectors in the same way as was done in <cit.>. According to <cit.> (with the correction coefficient 3/2 in Eq. (4.5) from <cit.>), the frequency of stream-crossing events isdν=3P_ mc(Φ)[P(Φ)+P^(s)(Φ)]a(Φ)/2τ(Φ)A^3dAdΦ,Here, P_ mc(Φ) is the distribution of axion miniclusters in perturbations Φ, a(Φ) is the overdensity in the ministream with respect to the mean DM density in the Galactic halo in the solar neighbourhood ρ_ H(r_⊙) in the case where the minicluster is destroyed immediately after the Galactic disk formation (for more details, see <cit.>), A is the real overdensity in the ministream, and τ(Φ)=2R/v is the passage time of the Earth through the ministream cross section. To obtain the detection rate of bursts N(>A)=ν(A)Δ t with a density amplification larger than A in an observation time Δ t, it is necessary to integrate (<ref>) over ν from max(A,a(Φ)) to ρ̅(Φ)/ρ_ H(r_⊙) and over all Φ. The result of our calculations is indicated in Fig. 3 by the lower and upper lines for the Navarro-Frenk-White and isothermal density profiles, respectively.Thus, we see that there is a dependence of the results on the Galactic halo model. For the Navarro-Frenk-White profile the destruction of clumps by the disk is approximately half as efficient as that for the singular isothermal halo. For the destruction by halo stars the isothermal profiles gives an almost a factor of 3 larger value. §.§ On the Possibility of Detecting Streams by the LISA Detector If a stream passes through the Solar system, then its gravitational field will act on gravitational-wave interferometers. The relative length of the interferometer arm Δ l/l will change under the tidal gravitational force from the stream. It is interesting to consider such an action on the planned LISA interferometer, which is expected to have a very high sensitivity, ∼2×10^-18. The signals in the interferometer will be in the form of single pulses. The pulse structure in three directions will be strictly synchronized with the signals in the ground-based axion detectors. Therefore, based on the pattern of the pulses, it will be possible to prove almost unambiguously the passage of a stream and to ascertain its velocity direction and overall structure. The possibility of detecting compact objects with masses 10^14-10^20 g using LISA was pointed out in <cit.>, <cit.>, <cit.>, where primordial black holes, asteroids, or massive DM objects were considered as compact objects. In contrast to these papers, in our case, it is necessary to consider a noncompact mass distribution in the form of an elongated stream.We model the stream by a straight thin thread of length L=v_ mct, where v_ mc is the internal velocity dispersion in the clump and t is the time elapsed since the clump destruction. The gravitational field of the stream at distance r from the axis is theng=2GM/rL.If l∼5×10^11 cm is the interferometer arm length (in the new eLISA project the arm length was reduced to 1×10^11 cm), then the tidal acceleration isΔ g∼2GM/r^2Ll,while the change of the arm length in the stream passage time Δ t∼ r/v_ rel isΔ l∼Δ g (Δ t)^2/2,where v_ rel∼200 km s^-1. The relative change of the arm isΔ l/l∼GM/v_ rel^2v_ mct∼3×10^-19at t∼5×10^9 years. The quantity (<ref>) does not depend on r and is comparable to the LISA sensitivity. Slow streams with a lower v_ rel will act on the detector more efficiently, but their number is also smaller. Axion streams will produce additional “noise” in space-borne detectors. The same expression for Δ l/l as (<ref>) is also obtained if the passage of the interferometer arm inside a stream is considered.To assess more accurately the prospects for the detection of axion streams by gravitational-wave interferometers, we will take into account the detector noise distribution. In our calculation we follow the method described in <cit.>. Let r_ min be the minimum distance from the axis of the passing stream to the center of the segment connecting the two interferometer mirrors. We will assume that during the passage the detector is always outside the stream. The tidal gravitational acceleration that the interferometer arm experiences isa(t)=2GMl/L[r_ min^2+(tv_ rel)^2],while its Fourier spectrum isa(f)=∫_-∞^+∞dte^2π ifta(t)=2GMl/Lv_ relr_ mine^-2π r_ minf/v_ rel.If the detection is based on the optimal filtering method, then for the square of the signal-to-noise ratio we haveρ^2_ SN=4∫_0^+∞dfa^2(f)/S^2(f),where for the LISA detector S≃ a_0≃6×10^-13 cm s^-2 Hz^-1/2. Assuming for the estimate that S=a_0=const, we obtainρ^2_ SN=4π G^2M^2l^2/a_0^2L^2v_ relr_ min^3.To detect a stream with a given ρ_ SN, it is necessary that the stream pass at a distance no larger than r_ min from the detector. We numerically obtainr_ min=6×10^12(l/5×10^12)^2/3× (a_0/6×10^-13)^-2/3(Δ t/5×10^9)^-2/3× (v_ rel/200)^-2/3(ρ_ SN/0.05)^-2/3 ,With these normalization values the detection rate of streams will beRate=π r_ min^2v_ relf_ mcρ_ DMP/Mv_ mcΔ t/R∼0.1,wheref_ mc∼1 is the fraction of DM in the form of axion miniclusters and P∼0.02 is the minicluster destruction probability calculated in previous sections.Let us first consider the case with a single LISA-type detector. If we assume in (<ref>) that the signal-to-noise ratio is ρ_ SN∼5, as is commonly assumed for a single detector, and choose l=5×10^11 cm, then (<ref>) is smaller than the stream radius approximately by two orders of magnitude. The detection rate would be ∼10^-5 yr^-1. Thus, next-generation detectors, in which the interferometer arm l is larger than that in LISA by one and a half or two orders of magnitude and the noise a_0 is lower, are needed for the detection of streams with an acceptable rate. Allowance for the detector passage inside a stream and for the distribution in v_ rel, probably, will not change greatly the result. In the new eLISA project the detector noise at low frequencies is very large <cit.>; at a characteristic frequency of 10^-5 Hz we have a_0≃1.5×10^-11 cm s^-2 Hz^-1/2. Therefore, in comparison with LISA, the detection rate of streams will be lower approximately by two more orders of magnitude. The dependence of the result on the distribution of clumps in velocities and directions is also interesting in the problem of the detection of clumps by gravitational-wave interferometers, but these questions are beyond the scope of this paper.However, we may consider the detection of streams in two detectors, if there are two or more orbiting interferometers of the next (after LISA) generation, by the coincidence method and the characteristic signal shape. Suppose that the interferometer arm is an order of magnitude larger than was planned in LISA. In this case, a variant with ρ_ SN<1 is admissible, which was chosen in the normalization coefficients in (<ref>) and (<ref>). In this case, one might expect an acceptable detection rate from the viewpoint of real observations. § CONCLUSIONSThe particles being lost by a clump during its gravitational interactions with stars form a stream behind the clump being destroyed. In this way the bulk of the mass or the entire mass of the clump can pass into the stream. Since the area of the stream is larger than that of the clump by several orders of magnitude, the Earth's passage through the stream is a much more probable event than its passage through the whole clump. Therefore, allowance for the streams is of great and, possibly, fundamental importance for the experiments aimed at directly detecting axion DM particles, as was shown in <cit.>.In this paper we performed a calculation similar to that in <cit.>, but with allowance made for two additional effects. First, we took into account the fact that the orbits of clumps in the halo are noncircular and precess and that throughout its life history a clump could cross the Galactic disk at different distances from the center and could pass through halo regions with different number densities of stars. We considered two halo models, the Navarro-French-White profile and an isothermal sphere, and showed that the destruction in the second model is approximately a factor of two or three more efficient. Thus, the halo model affects noticeably the result. This influence is related to a different distribution of DM clumps in their orbital parameters.The Navarro-Frenk-White profile was obtained in the numerical simulations of galaxies without including the baryonic component. The cooling of baryons and their settling to the halo center must lead to a deepening of the potential well and an additional increase of the DM density in the central halo region. Therefore, it is possible that the isothermal profile corresponds better to the real one, because in it the density concentration at the center is larger than that in the Navarro-Frenk-White halo.Second, we took into account the destruction of clumps by Galactic halo and bulge stars. This effect increases the overall destruction efficiency. As is easy to show, the destruction of clumps during their pair interactions with one another is several orders of magnitude less efficient than that during the interactions of clumps with halo stars.As a result, we obtained the distribution of the rate of stream-crossing events as a function of overdensity. For example, we found that at an overdensity A>10 one might expect 1-2 events in 20 years.The prospects for the detection of streams from destroyed clumps with gravitational-wave interferometers look realistic only for detectors of the next (after LISA) generation or in the case of several LISA-type detectors and using a detection technique based on the coincidence method at a signal-to-noise ratio much less than unity in one detector.We are grateful to D. Levkov and A. Panin for the useful discussions. This study was supported by the Russian Science Foundation (project no. 16-12-10494). 99KolTka94 E. W. Kolb and I. I. Tkachev, Phys. Rev. D 50, 769 (1994); arXiv:astro-ph/9403011.TinTkaZio16 P. Tinyakov, I. Tkachev, K. Zioutas, JCAP 01, 035 (2016); arXiv:1512.02884 [astro-ph.CO].BerDokEro07 V.S. Berezinsky,V.I. DokuchaevandYu.N.Eroshenko, JCAP 07, 011 (2007); arXiv:astro-ph/0612733.BerDokEro08 V. Berezinsky,V. Dokuchaev,Yu. Eroshenko, Phys. Rev. D. 77, 083519 (2008); arXiv: 0712.3499 [astro-ph].BerDokEro06 V. Berezinsky,V. Dokuchaevand Yu. Eroshenko, Phys. Rev. D. 73, 063504 (2006); arXiv:astro-ph/0511494.ufn1 A.V. Gurevich and K.P. Zybin, Sov. Phys. — JETP 67, 1 (1988); Sov. Phys. — JETP 67, 1957 (1988); Sov. Phys. — Usp. 165, 723 (1995).gnedin2 O.Y. Gnedin and J.P. Ostriker, Astrophys. J. 513, 626 (1999).TayBab01 J.E. Taylor and A. Babul, Astrophys. J. 559, 716 (2001).DieKuhMad J. Diemand, M. Kuhlen, P. Madau, Astrophys. J. 667, 859 (2007).ZTSH H.S. Zhao , J. Taylor, J. Silk and D. Hooper, arXiv:astro-ph/0502049v4.GoeGneMooDieSta07 T. Goerdt et al., Mon. Not. Roy. Astron. Soc. 375, 191 (2007).Wein1 M.D. Weinberg, Astron. J, 108, 1403 (1994).SikTkaWan96 P. Sikivie, I.I. Tkachev, Y. Wang, Phys. Rev. D. 56, 1863 (1997);arXiv:astro-ph/9609022.Edd16A.S. Eddington, Mon. Not. Roy. Astron. Soc., 76, 572 (1916).Wid00 L.M. Widrow, Astrophys. J. Supp. 131, 39 (2000).KuiGil89 K. Kuijken, G. Gilmore, Mon. Not. Roy. Astron. Soc. 239, 605 (1989). ColOst81 J.A.R.Caldwell, J.P. Ostriker, Astrophys. J. 251, 61 (1981). MarSuch L. S. Marochnik and A. A. Suchkov, The Galaxy (Nauka, Moscow, 1984) [in Russian].MDSQ2 B. Moore, J. Diemand, J. Stadel, T. Quinn; arXiv:astro-ph/0502213.LauZylMez R. Launhardt, R. Zylka and P. G. Mezger, Astron. Astrophys. 384, 112 (2002).LISAPBH N. Seto, A. Cooray , Phys. Rev. D 70, 063512 (2004).LISAAsteroid P. Tricarico, Class. Quantum Grav. 26, 085003 (2009).LISADM A.W. Adams, J.S. Bloom, arXiv:astro-ph/0405266.Amaetal12 P. Amaro-Seoane et al., Class. Quantum Grav. 29, 124016(2012); arXiv:1202.0839 [gr-qc]. | http://arxiv.org/abs/1710.09586v2 | {
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"Yu. N. Eroshenko",
"I. I. Tkachev"
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[fitpaper=true,pages=-]sparse-v7 | http://arxiv.org/abs/1710.09854v1 | {
"authors": [
"Jianqiao Wangni",
"Jialei Wang",
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"Tong Zhang"
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"published": "20171026182643",
"title": "Gradient Sparsification for Communication-Efficient Distributed Optimization"
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High-precision radial velocities and two alternative spectral indicators The CARMENES survey is a high-precision radial velocity (RV) programme that aims to detect Earth-like planets orbiting low-mass stars.We develop least-squares fitting algorithms to derive the RVs and additional spectral diagnostics implemented in the SpEctrum Radial Velocity Analyser (SERVAL), a publicly available python code.We measured the RVs using high signal-to-noise templates created by coadding all available spectra of each star.We define the chromatic index as the RV gradient as a function of wavelength with the RVs measured in the echelle orders. Additionally, we computed the differential line width by correlating the fit residuals with the second derivative of the template to track variations in the stellar line width.Using HARPS data, our SERVAL code achieves a RV precision at the level of 1 m/s. Applying the chromatic index to CARMENES data of the active star YZ CMi, we identify apparent RV variations induced by stellar activity. The differential line width is found to be an alternative indicator to the commonly used full width half maximum.We find that at the red optical wavelengths (700–900 nm) obtained by the visual channel of CARMENES, the chromatic index is an excellent tool to investigate stellar active regions and to identify and perhaps even correct for activity-induced RV variations.Institut für Astrophysik, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, [email protected] de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía s/n, 18008 Granada, SpainCentro Astronómico Hispano-Alemán de Calar Alto (CSIC–MPG), Observatorio Astronómico Calar Alto s/n, Sierra de los Filabres-04550 Gérgal (Almería), SpainInstituto de Astrofísica de Canarias, Vía Láctea s/n, 38205 La Laguna, Tenerife, SpainDepartamento de Astrofísica, Universidad de La Laguna, 38206 La Laguna, Tenerife, SpainDepartamento de Astrofísica, Centro de Astrobiología (CSIC–INTA), ESAC, Camino Bajo del Castillo, 28691 Villanueva de la Cañada, Madrid, SpainThüringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, GermanyHamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, GermanyLandessternwarte, Zentrum für Astronomie der Universität Heidelberg, Königstuhl 12, 69117 Heidelberg, GermanyMax-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, GermanyDepartamento de Astrofísica y Ciencias de la Atmósfera, Facultad de Ciencias Físicas, Universidad Complutense de Madrid, 28040 Madrid, SpainInstitut de Ciències de l'Espai (IEEC-CSIC), Can Magrans s/n, Campus UAB, 08193 Bellaterra, SpainSpectrum radial velocity analyser (SERVAL) M. Zechmeister1A. Reiners1P. J. Amado2M. Azzaro3F. F. Bauer1V. J. S. Béjar4,5J. A. Caballero6E. W. Guenther7H.-J. Hagen8S. V. Jeffers1A. Kaminski9M. Kürster10R. Launhardt10D. Montes11J. C. Morales12A. Quirrenbach9S. Reffert9I. Ribas12W. Seifert9L. Tal-Or1V. Wolthoff9 Received / Accepted ===============================================================================================================================================================================================================================================================================§ INTRODUCTION The radial velocity (RV) method has been a very successful technique to discover and characterise stellar companions and exoplanets. There are many algorithms to compute RVs, which can be grouped depending on their complexity and choices for the model of the reference spectra, number of model parameters, and statistics; these algorithms range from simple cross-correlation with binary masks <cit.>, to least-squares fit with coadded templates <cit.>, and finally to least-squares fitting with modelling of line spread functions <cit.>. Recently, Gaussian processes have also been proposed to derive RVs <cit.>. The algorithm choice is influenced by instrument type (e.g. those with stable point spread functions) and the need for accuracy (e.g. fast and flexible reduction directly at the telescope, generality) or high precision <cit.>. While most algorithms compute radial velocities in the wavelength or pixel domain, there are also methods that work in the Fourier domain (e.g. ) and can efficiently disentangle double-lined spectroscopic binaries (KOREL; ). A review can be found in <cit.>.In this work we present our methods to measure RVs and spectral diagnostics, which are implemented in a Python programme called SERVAL (SpEctrum Radial Velocity AnaLyser). It aims for highest precision with stabilised spectrographs. Therefore, we employ forward modelling in pixel space to properly weight pixel errors, and we reconstruct the stellar templates from the observations themselves to make optimal use of the RV information inherent in the stellar spectra <cit.>.The SERVAL code was developed as the standard RV pipeline for CARMENES, which consists of two high resolution spectrographs covering the visible and near-infrared wavelength ranges from 0.52 to 1.71 μm <cit.>. The CARMENES consortium regularly obtains spectra for about 300 M dwarfs to search for exoplanets. The spectra are processed, wavelength calibrated, and extracted by a pipeline called CARACAL <cit.>. Using CARMENES and HARPS data we validate the performance of SERVAL, which is publicly available [<www.github.com/mzechmeister/serval>]. § RADIAL VELOCITY ALGORITHM Our RV computation aims for highest precision and is based on least-squares fitting. <cit.> demonstrated that this approach can yield more precise RVs than the cross-correlation function (CCF) method. This is because the RV precision depends on both the signal-to-noise ratio of the data and the match of the model to the data. With a proper model of the observations, the RV information, i.e. each local gradient, is optimally weighted <cit.> and outliers can be detected.Our very general concept is to decompose simultaneously all observations into (1) a high signal-to-noise template F(λ); (2) RV shifts v_n for observation number n; and (3) multiplicative background polynomials p(λ) to account for flux variations, e.g. wavelength-dependent fiber coupling losses due to imperfect correction of the atmospheric dispersion. Therefore, our forward model f(λ) for the observed flux isf(λ)=p(λ)· F(λ'(λ,v)) ,where the Doppler equation λ'(λ,v) is given in Eq. (<ref>).We denote some n=1 ...N observations taken at times t_n each consisting of flux measurements f_n,i at pixel i with calibrated wavelengths λ_n,i and flux error estimates ϵ_n,i. For cross-dispersed echelle spectrographs the measurements are carried out in several echelle orders o (so a more explicit indexing is f_n,o,i). The weighted sum of residuals isχ^2=∑_n,i[f_n,i-f(λ_n,i)]^2/ϵ_n,i^2 =∑_n,iw_n,i[f_n,i-p(λ_n,i,a)· F(λ'(λ_n,i,v_n),b)]^2with weights w_n,i=1/ϵ_n,i^2, the polynomial coefficients a, and the coefficients b describing the template.However, this simultaneous approach is hardly feasible in practice because of the large amount of data (spectra, orders, pixels, N× O× K) and large number of parameters. Therefore, we perform the decomposition sequentially and iteratively. First, the observed spectrum with the highest signal-to-noise is taken as a reference to shift all observations into this reference frame and to coadd them to a high signal-to-noise template F (Sect. <ref>). With this new template the radial velocities are recomputed while the coefficients of the background polynomial are fitted simultaneously (Sect. <ref>). This one iteration is usually sufficient <cit.>. Before we detail the two steps in Sect. <ref> and <ref>, we define the model and prepare the data.§.§ Model details, data, and input preparation Given a template F(λ') of the emitting source, a spectral feature at reference wavelength λ' in F appears at λ in the observation f, i.e. f(λ)=F(λ'). The measured wavelength λ depends on the radial velocity of the moving source, i.e. the star, and on the velocity of the moving observer on Earth <cit.>.To eliminate the contribution from Earth's motion, we transform the measured wavelengths on Earth to the solar system barycentre via the barycentric correction (cf. Eq. (25) in with λ_meas=λ and λ_meas,true=λ_B)λ_B =λ(1+z_B) , where z_B is the total redshift due to barycentric motion of the observer. To compute the barycentric correction, SERVAL uses the Fortran based code of <cit.> [<http://sirrah.troja.mff.cuni.cz/ mary/> ], which we modified to account for the proper motion of the star. Moreover, we correct for secular acceleration <cit.>, which is in particular important for high proper motion stars and requires the parallax of the star. For ultimate precision in the barycentric correction, the code from <cit.> with 1 cm/s accuracy can be used instead, which includes relativistic corrections [This requires internet access (<http://astroutils.astronomy.ohio-state.edu/exofast/barycorr.html>) or an installation of IDL. ]. The stellar radial motion v=cz redshifts the wavelength (cf. Eq. (7) in )λ_B=λ'(1+z) ,where c is the speed of light. Therefore, the equation to Doppler shift the template with radial velocity parameter v isλ'(λ,v) = λ1+z_B/1+v/c .With the approximation 1/1+v/c≈1-v/c and barycentric corrected wavelengths, one could write this as λ'=λ_B·(1-v/c)as, for example, in <cit.>. The approximation is accurate to 1 m/s over a velocity range of 17.3 km/s, i.e. well sufficient for the small amplitudes of exoplanets. But since there is actually no need for this approximation, we employ the exact equation.The SERVAL code has the option to perform the computation in logarithmic wavelengths. Then Eq. (<ref>) becomeslnλ_B=lnλ'+ln(1+v/c) .For small shifts the Doppler shift is linear, lnλ_B≈lnλ'+v/c. This approximation is accurate to 1 m/s over a velocity range of 24.5 km/s and should be avoided in high RV precision work in particular in the barycentric correction in Eq. (<ref>).We assume that the spectrum is continuous and use cubic spline interpolation to evaluate the template F at any wavelength λ, which is needed for the forward modelling in Eq. (<ref>).Furthermore, we generate and propagate a bad pixel map to flag pixels with saturation, significant negative flux (f_i,n<-3ϵ_i,n), significant deviation in the fitting (outliers), or contamination by tellurics and sky emission lines. The default telluric mask flags atmospheric features deeper than 5%. We also check the fits header for the correct observing mode (e.g. accuracy and efficiency mode in HARPS, dark time), available drift measurements, and suitable signal-to-noise ratios. Those cases are flagged and, when required, excluded in the analysis.§.§ Least-squares RVs For the determination of the RV, we optimise Eq. (<ref>) with respect to the polynomial coefficients a and the RV shift v. The model is linear in a, but the RV parameter v makes the fit non-linear. There are several algorithms, e.g. Levenberg-Marquardt or bisection, to solve non-linear least-squares problems. We use, in analogy to the CCF method, a direct stepping through the velocity parameter space v with a default step size of Δ v=100 m/s as a compromise between oversampling the resolution element (~km/s) and computational speed. At fixed velocity v_k, the Doppler-shifted template F_i,k=F(λ_i,v_k,b) is evaluated at each pixel i, and χ^2(v_k) is obtained from a simple linear least-squares fit for aχ_k^2=∑_iw_i[f_i-p(λ_i,a)· F_i,k]^2 .In the form χ^2=∑_iF_i^2w_i[f_i/F_i-p(λ_i,a)]^2 and with the substitution F_i^2w_i→ w_i and f_i/F_i→ f_i, standard library routines for polynomial fitting can be applied (division by zero flux F_i needs to be handled). We include the polynomial in the model. <cit.> argued that this “would couple the flux normalisation coefficients to Doppler factor in a nonlinear fashion” in their least-squares algorithm with iterative differential corrections and, therefore, they applied it to the data. However, they do not correspondingly re-adjust the errors as implied in our derivation here. Since in practice the variations in the polynomial function should be only a few percent, their approximation is feasible. The χ^2(v) function sampled at (v_k,χ_k^2) is explored for its global minimum. Around its minimum, the χ^2 function takes a parabolic shape and the first derivative vanishes, (χ^2)'=0. The Taylor expansion of the χ^2 function at the minimum to second order isχ^2(v+δ v)≈χ_min^2+(χ^2)'δ v+1/2(χ^2)”δ v^2=χ_min^2+1/2(χ^2)”δ v^2 .Since we use cubic interpolation for the template, the second derivative of the template F (and of χ^2) is continuous. Then a parabolic interpolation through the minimum of the χ^2(v) function and the two adjacent neighbours provides a refined estimate for v and an error estimate for v can be obtained from the parabola curvature. The parabola minimum is located at the point,v=v_m-Δ v/2·χ_m+1^2-χ_m-1^2/χ_m-1^2-2χ_m^2+χ_m+1^2 ,where m indexes the discrete global minimum of χ_k^2 (see Eq. (10.2.1) inthat is specialised here for a uniform grid, Δ v=v_k-v_k-1, or Eq. <ref> in Appendix <ref>).The uncertainty of v is estimated from Eq. (<ref>) with the Δχ^2(δ v=ϵ_v)=χ^2-χ_min^2=1 criterion along with the curvature from Eq. (<ref>)ϵ_v^2=21/(χ^2)”=2Δ v^2/χ_m-1^2-2χ_m^2+χ_m+1^2 .This value might be rescaled with χ_red^2 to account for under- or overdispersion of the fit.Finally, a statistical analysis of the fit is performed. The χ^2 and χ_red^2 are computed. Linear residuals r_i=f_i-f_mod,i exceeding the threshold|r_i|>κϵ_i√(χ_red^2) ,where the clipping value κ is typically 3...5, are flagged as outliers (e.g. by setting their weights to zero w_i=0) and excluded in a repeated fitting. The same can be formulated with normalised residuals χ_i=r_i/ϵ_i|χ_i|>κ√(χ_red^2) .Figure <ref> illustrates the best fit between one observation and a template.Each order is fitted separately and finally a weighted mean for the radial velocities v_o from Eq. (<ref>) with errors ϵ_v_o from Eq. (<ref>) over all orders o is computed (see also Fig. <ref>)v=∑ϵ_v_o^-2v_o/∑ϵ_v_o^-2with the error estimateϵ_v=√(1/∑ϵ_v_o^-2·1/N_o-1∑(v_o-v)^2/ϵ_v_o^2)=wrms/√(N_o-1) ,where wrms is the weighted root mean square and N_o is the number of orders.§.§ Spectra coadding At first glance, coadding spectra seems straightforward. However, the stellar spectra are Doppler shifted by potential Keplerian orbits and by the barycentric motion of Earth. Thus, spectra from different epochs are sampled at different wavelength footpoints. Hence naive coadding would require either some kind of resampling or interpolation of the observations onto a common wavelength grid (e.g. ), which leads to difficulties in applying the data uncertainties, or calculating some form of bin means, which ignores the local gradient over the bin width. We point out that bin means are zeroth-order B-splines.We carry out the “co-adding“ with a uniform cubic basic spline (B-spline) regression to the normalised data. We emphasise here the term regression, which means least-squares fit and should be distinguished from spline interpolation and smoothing splines. There are several benefits of this approach: (1) B-spline regression is a linear least-squares method, and, therefore, it is fast; (2) there is no need to interpolate the data; (3) data point uncertainties can be easily taken into account; (4) robust statistics for outlier detections can be obtained with kappa-sigma clipping; and (5) a spline function is a direct outcome and consistent with our input for the forward model.Given v_n for the N observations from Sect. <ref>, we recompute the polynomials p_n,i for each order (now with the mean RV v_n) to normalise the data (f_n,i/p_n,i) and calculate the Doppler-shifted wavelengths with Eq. (<ref>). We factor p_n,i^2 in Eq. (<ref>) and write it in the formχ^2=∑_n,ip_n,i^2w_n,i[f_n,i/p_n,i-F(λ'_n,i,b)]^2 .Now the task is to find all the coefficients b_k of the spline with K knots, so that the residuals are minimal. The knots of the template are positioned in a uniform grid in (logarithmic) wavelengths λ_k (lnλ_k) and f_k (or B-spline coefficient b_k, respectively). By default, the number of knots K is similar to the number of data points per spectrum in the order, i.e. we have about one knot per pixel. Depending on the needs, the sampling could also be decreased to obtain a smoother template (e.g. noisy observations or fast rotators) or increased to obtain subpixel sampling (many observations). It can be inferred from Eq. (<ref>) that the normalised error estimates are ϵ_n,i/p_n,i.Equation (<ref>) is a linear least-squares problem due to the linearity of the coefficients in the B-spline <cit.> used to describe the templateF(x,b)=∑ b_kB_k(x) ,where B are the basic functions. Figure <ref> shows the normalised and Doppler-shifted data and the best spline fit.However, caution is needed when there are large gaps in the data, i.e. regions where we have no information to say anything about the template (i.e. no constraints for b_k). Possible solutions are (1) splitting the spline fit at those points; (2) fitting penalised splines; or (3) heavily down-weight, but not fully reject, the flagged data points (outliers, tellurics), which caused the gaps. We chose the last option (down-weighting) for tellurics. This means that we include regions heavily contaminated by telluric lines in coadding, while in RV measurements they are totally excluded.For each knot we estimate an error for the knot value as well as the number of contributing good points. We calculate the errors in the knot values by first estimating the error in the B-spline coefficients asϵ_b_k=1/√(∑_iw_iB_k(x_i))and then through error propagationϵ_f_k=1/6ϵ_b_k-1^2+4/6ϵ_b_k^2+1/6ϵ_b_k+1^2 .This simplified estimate does not use the covariance matrix, since calculating the inverse matrix would be very time consuming.To increase the robustness against outliers, a few (≤3) kappa-sigma clipping iterations (κ=5) are performed for the coadding (similar as in Eq. (<ref>)).Problems can arise with this coadding method when the pixel phase coverage is small because of a small barycentric range owing to close observations or high ecliptic latitude. Such small coverage may lead to ringing in the spline function. Reverting to simple bin means (zeroth order B-splines) or using penalised splines (p-splines, as a generalisation of B-splines; ) may help in such situations. Also sharp (undersampled) features, such as cosmics and emission lines, may lead to ringing features. The SERVAL code offers the option to use p-splines, but the choice for the value of penalty/smoothing parameter in each particular case will require some user testing or cross-validation algorithms. We also point out the Gaussian process approach of <cit.>, where the variance/smoothness parameter is a hyperparameter, and that B-splines and p-splines can be considered as a special case of Gaussian processes <cit.>.SERVAL also offers the possibility to use external templates, e.g. from other observed stars of similar spectral type or synthetic spectra. The latter may be useful to derive absolute RVs and line indices (Sect. <ref>). § SPECTRAL DIAGNOSTIC AND ACTIVITY INDICATORS §.§ Chromatic RV indexIn Eq. (<ref>) we compute a simple weighted average for the RV over the orders. However, since the echelle orders are related to wavelength, we can also try to get some information about wavelength dependency for instance by simply extending the model to a straight linev(o)=α+βlnλ_o ,where λ_o is a representative wavelength of echelle order o (e.g. lnλ_o=⟨lnλ_o,i⟩). Via a χ^2-fit we obtain a best fit estimate for the slope parameter β, which we call chromatic index. Its unit is velocity per wavelength ratio e (Neper, symbol Np). Figure <ref> illustrates the slope definition for two observations of the active M dwarf YZ CMi. The parameter α is considered as a nuisance parameter. Using the mean velocity v from Eq. (<ref>) and the substitution α=v-βlnλ_v, we can re-parametrise Eq. (<ref>) asv(o)=v+βlnλ_o/λ_v ,where λ_v is the wavelength at which the slope intersects the weighted mean RV. This allows us to report an effective wavelength, which is in particular useful when comparing data from instruments covering different wavelengths.In the definition of the chromatic index in Eq. (<ref>), we chose lnλ, the natural logarithm of wavelength λ, as independent variable. While other choices like wavelength λ or order o, which is basically equivalent to reciprocal wavelength λ^-1 (i.e. frequency), lead to qualitatively very similar results for β, our choice gives the same weights to each velocity element (resolution element) and is also useful when comparing data from a variety of instruments. Moreover, the wavelength range of 1 Np is typically covered by high resolution echelle spectrographs, i.e. the values of the slopes are of the order of the RV change over all echelle orders.§.§ Differential line width (dLW) A feature of the CCF method is that the CCF can be interpreted as a mean stellar line profile that is actually convolved with a kind of kernel. By analysing the CCF shape, for example by fitting a Gaussian function, we can obtain information about the moments, among them the centre of the Gaussian (first moment, i.e. RV) and the width (second moment, i.e. full width half maximum, FWHM). To obtain information about asymmetries (third moment, skewness), often bisectors are used (e.g. ).We study if there is an analogous set of parameters that can be associated with least-squares fitting. Since the CCF is basically a χ^2 function, one could consider fitting a Gaussian function to the χ^2 function. However, we argue that the shape (curvature, asymmetry) of the χ^2 function is formed by the signal-to-noise of the observation, masking of outliers and tellurics, and also the simultaneous fit of the background polynomial.In the following, we suggest another approach. We review a simple method to measure differential RVs as implied in <cit.>. Figure <ref> (left) shows that when a Gaussian function is slightly displaced, the residuals, i.e. the difference between both curves, are correlated with the first derivative of the Gaussian function. In fact, this is actually the definition of the first derivative. From the scaling factor, which was already applied in Fig. <ref>, we can derive the RVg(x+v/c)-g(x)≈v/cg'(x) .Now we replace on the left side the function difference by the residuals r_i between (drifted) data and (flux-scaled) template, and on the right side the derivative of the Gaussian function by the derivative of the template. The derivative must be with respect to velocity, i.e. f'=df/dlnλ=λdf/dλ. The relation becomesr_i=v/cf'(x_i) .In simple terms, we scale the first derivative to the residuals. The best scaling factor v can be derived with a linear least-squares fit weighted with flux uncertainties σ_i. In Sect. <ref>, on the other hand, the RV shift was already applied by actually minimizing this correlation.Now the question is whether we can find a similar relation using the first or higher derivatives to check for variations in the line width. Therefore, we vary the line width of the Gaussian. In one variant we keep the line height constant (Fig. <ref>, middle) and in another variant we keep the same area under the curve (Fig. <ref>, right). For both cases we plot again the scaled residuals and compare these residuals with squared first derivative (middle) and second derivative (right), respectively.The case of unit area (right panel) results in balanced positive and negative deviations, which is more similar to the situation we are facing after a least-squares fit of the RV (Sect. <ref>). Indeed, we prove in Appendix <ref> that the mean value of the second derivative is zero (the same is true for least-squares fit residuals) and thatg(x,σ+δσ)-g(x,σ)≈σ·δσ· g”(x) .This relation is in a strict sense valid only for Gaussian functions, but it suggests simply scaling the second derivative of the template (f”=d^2f/dlnλ^2=λ^2d^2f/dλ^2) to the residualsr_i=σ·Δσ/c^2f”(x_i) .Hence we have to assume that the stellar lines are Gaussian-like shaped. The relation also holds when we build up a spectrum by adding Gaussian lines at other positions and with different strengths due to the linearity. Then this also implies that it is applicable to blended lines. However, the lines should have all the same width. We point out that similar assumptions are made when fitting a Gaussian profile to the CCF.The scaling factor Δσ carries information about line width changes and we compute its value via a weighted linear least-squares fit resulting indLW≡σΔσ=c^2∑ w_if_i”r_i/∑ w_if_i”^2and estimate its uncertainty through error propagation asϵ_σΔσ=c^2√(1/∑ w_if_i”^2) .The dLW is computed in each order and finally averaged similar to the RVs in Eq. (<ref>) and (<ref>). The SERVAL programme propagates the uncertainties of the spectra which are mostly photon and readout noise. On top of this there can be other noise sources from the instrument, such as focus or resolution change, or observation, such as line broading due to barycentric motion during the exposure. Therefore, even quiet stars with presumable stable intrisic line shapes may show excess variations in line width indicators.Since our template is a cubic B-spline, we can easily calculate the second derivative f” at each position of the spectrum. Also the third derivative would be possible and the possibility of defining a differential alternative to the bisector span in the CCF would appear very straightforward, but this is not yet implemented. Of course, the uncertainty in the higher order moments increase.Owing to our differential approach, our width indicator σΔσ has also a differential nature, and, therefore, we call it differential line width (dLW). Its unit is m^2/s^2 if the derivative is calculated in velocity or logarithmic wavelength scale. In analogy to differential RVs, in which an additive offset remains with respect to to absolute RVs, the offset now becomes multiplicative in the differential width (second moment). The σΔσ indicatoris sensitive to regions where the second derivative is large, such as the cores of spectral lines (cf. Fig. <ref>, right panel).Variations in the spectral line width can be intrinsic to the star, for example pulsation or activity, or of instrumental origin in the form of focus change and smearing due to the barycentric motion during exposure. In any case, correlation of a RV signal with line width variations would argue against a planet hypothesis.§.§ Line indices The SERVAL code also provides indices in the form of time series data for a number of spectral lines (e.g. Ca ii H&K, Hα, Na i D, and Ca ii IRT). Before measuring the line indices, we need to find the line positions, i.e. the absolute RVs. These can be obtained by measuring the RV of one spectrum against an absolute reference (e.g. a PHOENIX spectrum).Following <cit.>, the indices are computed asI=⟨ f_0⟩/0.5·(⟨ f_1⟩ +⟨ f_2⟩ ) ,where ⟨ f_0⟩ is the mean flux around the line centre and ⟨ f_1⟩ and ⟨ f_2⟩ are the mean fluxes over reference regions.As an example, we choose for Hα (λ_air=6562.8Å) the region [-40,+40] km/s and as reference [-300,-100] km/s and [+100,+300] km/s. We apply a larger core region compared to <cit.>, who only applied it to the inactive star GJ 699, to collect all Hα emission for most of the M dwarfs, while for most active stars an even wider range could be considered at the cost of decreasing index precision.We estimate the error in the mean fluxes using the data error ϵ_iϵ_⟨ f⟩=1/N√(∑ϵ_i^2) .This choice is not unique, but it preserves some information about the signal-to-noise ratio in the spectrum; in case of pure photon noise ϵ_i=√(f_i), it satisfies ϵ_⟨ f⟩=√(1/N⟨ f⟩). From the stand point of least squares, the standard error of the mean would be calculated via the standard deviation as 1/√(N)√(1/N-1∑(f_i-⟨ f⟩ )^2) and for a weighted mean as √(1/∑ϵ_i^-2) (cf. Eq. (<ref>)). However, when modelling a line such as Hα with a mean, i.e. we fit a box, we should be aware of a large model mismatch with this simplistic model making those error estimates misleading. Indeed, one could consider more tailored models, for example the Mount Wilson S-index employs a triangular shape <cit.>.Finally, we simply propagate the errors from Eq. (<ref>) through Eq. (<ref>) to an uncertainty for the line indexϵ_I=I√(ϵ_0^2/⟨ f_0⟩ ^2+ϵ_1^2+ϵ_2^2/⟨ f_1⟩ ^2+⟨ f_2⟩ ^2) .Sometimes (pseudo-) equivalent widths (pEW) are preferred instead of line indices. We outlined in <cit.> that pEWs are closely related to line indices, which might be interpreted as “pseudo line heights”. Indeed, both quantities measure the zero moment (area, integrated flux) and condense it into one parameter. This simple estimate is sufficient for studies on the activity level of stars and temporal variations. However, the true line shape is more complex (e.g. self-absorption features) and needs to be described by more moments/parameters.We have also implemented a differential version for spectral line indices where the mean fluxes in Eq. (<ref>) are replaced by the scaling factor between the observation and the coadded template. This method aims for optimal weights and highest precision and would be also applicable in case of missing data points, for example due to cosmics or tellurics, but we recommend this only for inactive stars. § RESULTS In this section we evaluate the performance of SERVAL using its RV precision, the dLW, and the chromatic index. To achieve this, we use data from HARPS and CARMENES. The HARPS instrument and its data reduction software has demonstrated long-term 1 m/s precision since its installation in 2003 <cit.>. Each of the points to be tested are evaluated for (1) a stable and inactive M dwarf (Barnard's star), (2) a G4 dwarf with strong correlations between FWHM and RVs (ζ^1 Ret), and (3) two very active M dwarfs (YZ CMi and GJ 3379). The time-series measurements are shown in Figs. <ref>–<ref>.§.§ TargetsGJ699 Barnard's star is an inactive M4V star. For comparison we use the same 22 HARPS observations as previously used in <cit.>. The spectra were obtained between 2007-04-04 and 2008-05-02. All spectra were secured without simultaneous drift measurements and two spectra [2007-04-09T09:51:56.457 and 2007-04-10T09:55:45.767 ] were taken during twilight. ζ^1 Ret This G4 dwarf was found by <cit.> to have a strong correlation between RV and FWHM and hence we selected it as an example for dLW analysis. We analysed 61 HARPS measurements taken during 24 nights over a time span of 1401 days. Typical FWHM values of 7.1 km/s indicate that the star is not a slow rotator. YZ CMi This star (GJ 285, Karmn J07446+035) is a very active M4.5 dwarf with significant rotation measured in several works (vsin i∼5 km/s, log(L_Hα/L_ bol)=-3.48; ; ), a photometric period of 2.78 d <cit.>, an average magnetic field of the order of 3–4 kG, and local field strengths up to 7 kG <cit.>. We obtained 45 CARMENES VIS observations spanning 480 days (from 2016-01-08 to 2017-05-01). GJ 3379 This star (G 099-049, Karmn J06000+027) is an active M4.0 dwarf with significant rotational broadening (vsin i=(7.4±0.8) km/s, ). We found 16 HARPS spectra in the ESO archive and obtained 14 spectra with CARMENES for GJ 3379. We only used this star to compare the chromatic index between HARPS and CARMENES wavelength ranges.§.§ Criterion 1: Radial velocity precisionGJ 699 Figure <ref> (left) compares the RVs derived with SERVAL (rms=1.30 m/s) and the HARPS DRS pipeline (rms=1.54 m/s), while <cit.> reported a dispersion of rms=1.24 m/s for the same dataset. Secular acceleration is subtracted in all cases. This demonstrates the capability of SERVAL to achieve precise RVs at the 1 m/s level. ζ^1 Ret The RVs have an rms of 10 m/s (Fig. <ref>). Since there is activity-induced jitter, we cannot evaluate the individual RV performance with this star. Still, the RV difference between SERVAL and DRS CCF has an rms of 1.2 m/s and points to 1 m/s performance of SERVAL for solar-like stars as well. YZ CMi The RVs show an rms of 85 m/s (Fig. <ref>, left). The variation takes place on short time scales (days) with a significant periodicity of 2.776 d with an amplitude of 120 m/s (Fig. <ref> and a less significant one-day alias at 1.556 d and 100 m/s). This period also appears in the Hα-line equivalent width but at low significance. §.§ Criterion 2: dLWGJ 699 The comparison between dLW and FWHM, drawn on the left and right axes, respectively, is shown as a time series in the middle panel of Fig. <ref>. Both indicators have relatively similar error bars and vary significantly; we adopted ϵ_FWHM=2.35ϵ_RV. Using the relation FWHM=2.35σ for a Gaussian, we would expect Δ FWHM=FWHM·Δ FWHM/FWHM=2.35^2/FWHM·σΔσ. This theoretical prediction is shown as a blue line in Fig. <ref> (top left panel). We see that dLW values scatter around this prediction, but the large uncertainty for the best fitting slope shows that the dLW and FWHM correlate poorly for GJ 699. Surprisingly, we find a much better correlation of dLW with FWHM divided by the contrast indicator, which is the relative amplitude of the Gaussian fit (Fig. <ref>, top right). This is currently not well understood, but it likely reminds us that dLW and FWHM are not fully tracking the same effect. Indeed, the dLW indicator assumes that the area of the Gaussian-shaped line, i.e. the product of FWHM and contrast, remains constant, while a Gaussian fit to the CCF decomposes both parameters simultaneously. Hence, we could argue that additional contrast variations (on top of contrast ∝ FWHM^-1) could lead to biased dLW measurements. Those contrast variations could be induced by imperfect background subtraction or have stellar origin, albeit we believe the latter is less likely for this quiet star. Alternatively, the poor dLW-FWHM correlation for GJ699 could point to a bias when fitting a Gaussian function to the CCF of an M dwarf, which is known to exhibit prominent side lobes (see e.g. Fig. 7 in ). Neither dLW nor FWHM correlate with RVs.ζ^1 Ret The time series of dLW and FWHM already indicates a similar behaviour (Fig. <ref>, middle). Both, FWHM and dLW, correlate with RV (Fig. <ref>). There is also a direct correlation between dLW and FWHM; still the correlation improves again when dividing the FWHM by the contrast (Fig. <ref>, bottom panels).YZ CMi The dLW (Fig. <ref>, middle) reveals a 2.776 d period that we have already found in the RVs. The correlation between dLW and RV is not linear (Fig. <ref>, bottom and top), but we find that our observations follow a well-defined path when we colour code the rotational phase with this period. The one dLW outlier is an observation with an Hα flare. The spectrum has the highest Hα core emission and noticable rising Hα wings compared to the other spectra. Likely, the flare adds continuum flux over the full spectrum, leading primarily to line contrast variations. The flare event is not noticable in RV and chromatic index, although the point has maximum RV and minimum chromatic index.§.§ Criterion 3: chromatic RV IndexGJ 699 The chromatic index (Fig. <ref>, right) does not vary significantly. This is expected for an inactive and slowly rotating star. ζ^1 Ret The chromatic index (Fig. <ref>) shows long-term variations that seem to be anti-correlated with the RV long-term variations, which could be an activity cycle. However, the chromatic index does not follow the short-term variations of RV and dLW. Therefore, the chromatic index apparently has a complementary role to line width indicators.YZ CMi In the top panel of Fig. <ref>, we show the chromatic index as a function of RV for our CARMENES observations of YZ CMi; both parameters are clearly anti-correlated. We call this correlation and the corresponding slope chromaticity κ. RV variations of ±150 m/s correspond to variations of the chromatic index of ∓450 m/s/Np. Of course, due to the anti-correlation, the periodograms for chromatic index and RV have a similar shape and the 2.776 d period is also found in the chromatic index. For this period, we colour code our data in Fig. <ref> according to their rotational phase. One can clearly follow how the individual observations are distributed throughout the rotational phase of YZ CMi.The chromatic index-RV correlation demonstrates that the dominating effect in the RV variations of YZ CMi is wavelength dependent, which is consistent with the hypothesis that they are caused by co-rotating features such as active regions on the surface of the star. The wavelength-dependent temperature contrast between quiet and active regions is expected to cause a variation of the RVs as a function of wavelength (e.g. Fig. 12 in ). Active regions (or spots) that are somewhat cooler (or hotter) than the photosphere show less contrast to the photosphere at longer wavelengths. The chromatic index attempts to capture this dependence on wavelength. The advantage of the chromatic index over chromospheric emission (Ca ii H&K, Hα) is that it comes with a directional sense, i.e. it can be positive and negative. The chromatic index is directly connected to RV because it is calculated from the same line profile deformation.A negative chromaticity, as in case of YZ CMi, means that the RV scatter decreases towards redder wavelength. This amplitude decrease can been seen in Fig. <ref> and is predicted with spot simulations <cit.>. The slope in the correlation plot (Fig. <ref>, top) provides information about spot temperatures. Moreover, we identify a separation between the values of the chromatic index at rotational phases 0.5 and 1, which would not be the case if the relation was simply linear and is probably related to convective blue shift. A deeper interpretation of these effects requires more detailed observations and goes beyond the scope of this paper.§.§ Chromatic index with CARMENES and HARPS Using the example of YZ CMi, we demonstrated that the chromatic index in the wavelength range of CARMENES VIS (550–970 nm) is a powerful tool for understanding RV variations induced by stellar activity. A closer look at the individual RVs per wavelength (Fig. <ref>) shows that a fairly tight relation between RV and wavelength exists for the wavelength range 600–920 nm, but the five spectral orders short of λ=590 nm fail to follow this trend. We observe this behaviour in the other exposures of YZ CMi, too. The significance of this effect for YZ CMi and other stars needs more detailed investigation. There are two plausible explanations for the lack of correlation at shorter wavelengths: first,the contrast grows too high in that active regions no longer contribute to the observed spectrum in a significant way; and second that the intensity of spectral features at short wavelengths also differs dramatically between active and quiet regions <cit.>.A possible way to investigate whether the chromatic index effect continues down to shorter wavelengths is to determine RVs for individual spectral orders of spectra taken with HARPS. This was attempted for the active M4.5 dwarf AD Leo by <cit.>. Interestingly, the activity-induced RV amplitude reported there grows with wavelength (positive chromaticity κ), but the effect is more than an order of magnitude weaker than what we observe in the CARMENES data of YZ CMi. Additionally, the two RV amplitudes shown for AD Leo by <cit.> that overlap with the CARMENES wavelength range in fact show a much steeper wavelength dependence with the same negative sign as observed in the CARMENES data for YZ CMi. The comparison between AD Leo and YZ CMi, however, remains inconclusive because we might just be seeing very different effects in different stars.In order to make a more useful comparison of the HARPS and CARMENES wavelength ranges, we looked for targets that were observed with both instruments, albeit not simultaneously. One of the few targets available for this exercise is GJ 3379. Radial velocities and line indicators are shown in Fig. <ref>. The datasets were taken at very different times. The RVs from HARPS spectra exhibit an rms of 74 m/s while the CARMENES RVs show an rms of 21 m/s. The intrinsic uncertainties of the individual RV measurements are of the order of a few m/s in both cases and negligible for this comparison.We calculated the chromatic index in both datasets and show the correlations between chromatic index and RV in Fig. <ref>. For our CARMENES data, chromatic index and RVs are correlated. The relation can be described by a linear fit with a gradient of -2.6 Np^-1(±21%). In the HARPS data, we find no correlation between chromatic index and RV, although the scatter in RV is higher than in the CARMENES data.This comparison seems to indicate that the RVs of GJ 3379 show larger scatter at bluer wavelengths and that the RV variations at longer wavelengths are correlated with the chromatic index, while those at bluer wavelengths are not. However, an alternative explanation is that the RV jitter of the star changed during the course of a potential activity cycle; the HARPS and CARMENES data are separated in time by four years. One way to test this scenario is to compare RVs from the same wavelength range in both datasets; HARPS and CARMENES overlap in the range 550–690 nm. Unfortunately, this wavelength range is too short to draw definite conclusions about the chromatic index, and we have already shown above that the slope is most useful at longer wavelengths. Nevertheless, we find a correlation slope between chromatic index and RV of about -1 Np^-1. Although the scatter around this relation is relatively high, HARPS and CARMENES data points are consistent with each other. The correlation between chromatic index and RVs is much weaker than for the full CARMENES wavelength range and the correlation itself is marginally significant. On the other hand, the RVs calculated from the limited wavelength range are precise enough to compare the two time series; in this case we find that the rms of CARMENES RVs (30 m/s) is significantly lower than the HARPS rms (77 m/s). It is unlikely that this effect is caused by differences between the two instruments because for both we estimate the internal uncertainties to be much smaller and comparable with each other (4 m/s). Thus, for now the case remains undecided. We find that GJ 3379 shows reduced activity jitter during the epoch observed with CARMENES, while the RV rms was high when it was observed with HARPS. Nevertheless, the CARMENES RVs show a correlation with chromatic index when the full CARMENES wavelength range is taken into account. We expect that such a correlation would still be visible when GJ 3379 exhibits larger RV variations, but simultaneous observations of the HARPS and CARMENES wavelength ranges are required to settle this question. § SUMMARY We have presented our concept to derive high-precision RVs and have provided details on the creation of a template. Following an iterative and sequential approach we first derive approximate RVs measured against an observed spectrum and then improve the template by co-adding all observed spectra and recompute the RVs. The spectra are “coadded” via a weighted least-squares regression with a cubic B-spline.The SERVAL code yields an output consisting of time series for high-precision RVs and a number of spectral indicators useful for further diagnostics as well as the high signal-to-noise templates partly cleaned from telluric contamination taking advantage of barycentric shifts. We find a similar performance when comparing the RV results from SERVAL and the HARPS-DRS pipeline using data for GJ 699 and ζ^1 Ret. An additional example can also be found in <cit.>, where an older version of SERVAL was applied to ~150 τ Cet spectra obtained during one night. This demonstrates that SERVAL produces RVs at a 1 m/s precision level.Furthermore, we motivated a definition to study differential changes in the spectral line widths. We scale the second derivative of the template to the residuals from the fit for the best RVs and have shown that it can be useful in practice. This differential indicator nicely fits in our self-consistent framework of least-squares fitting (for differential RVs) without the need for external templates. Of course, this comes at the price that the differential results have no external accuracy, i.e. the line width remains unknown. In contrast, unbiased and accurate line width measurements for M dwarf spectra with their overall blended lines can be obtained with least-squares deconvolution <cit.>, but those are more complicated algorithms that require accurate line positions as input.The chromatic index, while very simple in its definition, turns out to be a very powerful indicator to identify stellar activity in RV signals. We have shown that it correlates very well with RVs for the active M dwarf YZ CMi. Hence it will help us to identify signals that are associated with the intrinsic stellar variations rather than variations induced by Keplerian motion, to lower the detection limit for active and inactive stars because it can be used to remove and disentangle the activity contribution from the RVs, and to better understand the physics of the stars, such as spot temperature or convective blue shift. The chromatic index summarises in one value a first-order effect of wavelength dependence in the RVs and is well suited for an efficient analysis. Still, a more detailed investigation in smaller wavelength ranges might further our understanding of the wavelength dependence of stellar radial velocities <cit.>. A large wavelength range, such as that offered by CARMENES, is essential for this.We thank J. Zhao, Y. Thiele, E. Nagel, L. Nortmann, and G. Anglada-Escudé for software testing and helpful discussions. M. Z. acknowledges support from the Deutsche Forschungsgemeinschaft under DFG RE 1664/12-1 and Research Unit FOR2544, project no. RE 1664/14-1. I. R. acknowledges support by the Spanish Ministry of Economy and Competitiveness (MINECO) and the Fondo Europeo de Desarrollo Regional (FEDER) through grant ESP2016-80435-C2-1-R, as well as the support of the Generalitat de Catalunya/CERCA programme. V. Wolthoff acknowledges funding from the DFG Research Unit FOR2544, project no. RE 2694/4-1. VJSB are supported by grant AYA2015-69350-C3-2-P from the Spanish Ministry of Economy and Competiveness (MINECO). The UCM, CAB, and IAA team members acknowledges support by the Spanish Ministry of Economy and Competitiveness (MINECO) from projects AYA2016-79425- C3-1,2,3-P. CARMENES is an instrument for the Centro Astronómico Hispano-Alemán de Calar Alto (CAHA, Almería, Spain). CARMENES is funded by the German Max-Planck-Gesellschaft (MPG), the Spanish Consejo Superior de Investigaciones Científicas (CSIC), the European Union through FEDER/ERF funds, and the members of the CARMENES Consortium (Max-Planck-Institut für Astronomie, Instituto de Astrofísica de Andalucía, Landessternwarte Königstuhl, Institut de Ciències de l'Espai, Insitut für Astrophysik Göttingen, Universidad Complutense de Madrid, Thüringer Landessternwarte Tautenburg, Instituto de Astrofísica de Canarias, Hamburger Sternwarte, Centro de Astrobiología and Centro Astronómico Hispano-Alemán), with additional contributions by the Spanish Ministry of Economy, the German Science Foundation (DFG) through the Major Research Instrumentation Programme and DFG Research Unit FOR2544 “Blue Planets around Red Stars”, the Klaus Tschira Stiftung, the states of Baden-Württemberg and Niedersachsen, and by the Junta de Andalucía.aa§ PARABOLIC INTERPOLATION We want to derive the interpolating parabola y=ax^2+bx+c through the three points (x_i,y_i), (x_i-1,y_i-1), and (x_i+1,y_i+1). The points should be equidistant, i.e. x_i+1-x_i=Δ x and x_i-x_i-1=Δ x. Now we centre points to (x_i,y_i) and the three transformed points are (0,0), (-Δ x,Y_-1) and (Δ x,Y_+1). Inserting into the parabola equation leads to a system of three equations0 =cY_-1=aΔ x^2-bΔ xY_1=aΔ x^2+bΔ x .The solution isa =Y_-1+Y_1/2Δ x^2=y_i-1-2y_i+y_i+1/2Δ x^2b =Y_1-Y_-1/2Δ x=y_i+1-y_i-1/2Δ xc =0 .Reforming the parabola equation to y=y_i+a(x-x_i+b/2a)^2-(b/2a)^2+c, the extremum of the parabola is atx_c=x_i-b/2a=x_i-Δ x/2·y_i+1-y_i-1/y_i-1-2y_i+y_i+1and the second derivative at x_c isy”=2a=y_i-1-2y_i+y_i+1/Δ x^2 , which is the definition of the finite second derivative. § DERIVATIVES OF THE GAUSSIAN FUNCTION We start with a Gaussian function with unit amplitude. Their first and second derivatives (with respect to x) are given byf(x) =exp(-1/2x^2/σ^2)f'(x) =-x/σ^2exp(-1/2x^2/σ^2)=-x/σ^2f(x)f”(x) =1/σ^2(x^2/σ^2-1)f(x).We seek to understand how the residuals should appear when we change the width while keeping the amplitude. The derivative of Eq. (<ref>) with respect to parameter σ can be written as∂ f(x)/∂σ=x^2/σ^3exp(-1/2x^2/σ^2)=x^2/σ^3f(x) =σf'(x)f'(x)/f(x) .When we are interested in the residuals while conserving the area under the curves (Fig. <ref>, right panel), we have to include the normalising factor 1/σg(x) =1/√(2π)σexp(-1/2x^2/σ^2) .The relation of g'(x) and g”(x) to g(x) is the same as for f(x). However, the derivative with respect to σ is∂ g(x)/∂σ=1/σ(x^2/σ^2-1)g(x) =σ g”(x) .This suggests that we correlate the second derivative of the spectrum with the residuals when we want to infer differential variations in the width.The mean value of the second derivative is∫_-∞^∞g”(x)dx=.g'(x)|_-∞^∞=0 . | http://arxiv.org/abs/1710.10114v1 | {
"authors": [
"M. Zechmeister",
"A. Reiners",
"P. J. Amado",
"M. Azzaro",
"F. F. Bauer",
"V. J. S. Béjar",
"J. A. Caballero",
"E. W. Guenther",
"H. -J. Hagen",
"S. V. Jeffers",
"A. Kaminski",
"M. Kürster",
"R. Launhardt",
"D. Montes",
"J. C. Morales",
"A. Quirrenbach",
"S. Reffert",
"I. Ribas",
"W. Seifert",
"L. Tal-Or",
"V. Wolthoff"
],
"categories": [
"astro-ph.IM",
"astro-ph.EP",
"astro-ph.SR"
],
"primary_category": "astro-ph.IM",
"published": "20171027130729",
"title": "Spectrum radial velocity analyser (SERVAL). High-precision radial velocities and two alternative spectral indicators"
} |
1 Department of Physics and Astronomy, University of California, Los Angeles, Los Angeles, CA 90095, [email protected] School of Physics and Astronomy,Tel Aviv University, Ramat Aviv, Israel3 Department of Physics, New Mexico Institute of Mining and Technology, Socorro, NM 878014National Radio Astronomy Observatory, Soccorro, NM 87801 USA5Instituto Nacional de Astrofísica, Óptica y Electrónica, Puebla, México C. P. 728406Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 USA We report ALMA observations of ^12CO(3-2) and ^13CO(3-2) in the gas-poor dwarfgalaxy NGC 5253.These 0.3(5.5 pc) resolution imagesreveal small, dense molecular gas clouds that are located inkinematically distinct, extended filaments. Some of thefilaments appear to be falling into the galaxy and may be fueling its current star formation. The most intense CO(3–2) emission comes from the central ∼ 100 pc region centered on the luminousradio-infrared HII region known as the supernebula.The CO(3–2) clumps within the starburst regionare anti-correlated with Hα on ∼5 pc scales, but are well-correlated with radio free-free emission.Cloud D1, which enshrouds the supernebula, has a high ^12CO/^13CO ratio, as does another cloud within the central 100 pc starburst region, possibly because the clouds are hot.CO(3–2) emission alone does not allow determination of cloud masses as molecular gas temperature and column density are degenerate at the observed brightness, unless combined with otherlines such as ^13CO. § INTRODUCTION Super star clusters (SSCs) are the birth places of most massive stars; these immense concentrations of hot stars have the potential to have enormous impact on their host galaxies. Understanding the formation and evolution of SSCs is critical to understanding the energy budget,feedback, and regulation of star formation in galaxies and the environments that produce them. NGC 5253 is a local<cit.> dwarf spheroidal galaxy undergoingan intense starburst that has created a large population of massive star clusters over the course of a Gyr <cit.>. NGC 5253 has a stellar mass of only ∼ 1.5 × 10^8 <cit.>, and a dark matter mass that is likely 8-9 times higher <cit.>.This is a galaxy with dispersion-dominated kinematics and little rotation <cit.>.The abundant atomic hydrogen gas, comparable to the stellar mass at ∼2-3×10^8<cit.>, resides largely outside the galaxy, infilaments andstreamers extending to ∼3-5 kpc (∼ 2-3 optical radii). The dominant radio/IR HII region in NGC 5253, the “supernebula,” is excited by a massive, young SSC <cit.>. The supernebula has anionizing rate of N_Lyc = 3.3 × 10^52 s^-1 <cit.>, corresponding to an IR luminosity of ∼ 5× 10^8 L_⊙ <cit.>, contributing to a total galactic infrared luminosity of 1.6×10^9 L_⊙<cit.>. Many investigators find that there are two massive clusters in the nucleus <cit.>, one associated with the supernebula and one with thepeak;however, extinction within the central starburst region is very high, making it difficult to match radio, infrared, and optical observations <cit.>.Abundant molecular gas associated with such a luminous HII region in the ultracompact stage is expected, but the CO(1-0) and (2-1) emission is weak.Roughly three-quarters of this CO emission, as well as the HI, is found along the minor axis, outside the central regions <cit.>. Previous observations of CO(3–2)from the Submillimeter Array (SMA) indicate that the star formation in the CO cloud (Cloud D) near the supernebula is very efficient, having converted>50% of its gas mass into stars <cit.>.The unexpectedly bright CO(3–2) emission in the central starburst region,as compared to lower J CO lines, indicates that this central cloud, Cloud D <cit.>, iswarm, T∼ 300 K.However, the ∼4×2 beam of the SMA imagesdoes not separate the dense gas at thecluster scale. With higher resolution and sensitivity, the Atacama Large Millimeter/Submillimeter Array (ALMA) canimage NGC 5253at the ∼5 pc cluster scale. We have been able to image the structure of its remarkable starburst.Here we present 2015 ALMA observations of the J=3–2 rotational transition of CO in NGC 5253with 0.3 (∼5.5 pc) resolution.§ OBSERVATIONS NGC 5253was observed in ALMA Band 7as a Cycle 2 (Early Science) program (ID = 2012.1.00105.S, PI = J. Turner)on 2015 4 and 5 June. Two fields, each with an 18 field-of-view, centered on 13:39:56.62 -31.38.33.5 and 13:39:55.91 -31.38.26.5 (ICRS)were observed simultaneously with 8383 seconds on both sources.The uv range covers ∼24.5-900 kλ; the largest structures sampled are 4-8. Spectral resolution of 244.14 kHz, or 1per channel resolves the CO(3–2) lines. Bandpass, flux, and phasewere calibrated with J1427-42064, Titan,and J1342-2900, respectively. Absolute flux density calibration is estimated to be better than 10% in Band 7 Cycle 2 <cit.>.Data calibration was performed using the code produced by the Joint ALMA Observatories with CASA (Common Astronomy Software Applications).Continuum emission was subtracted in the (u,v) plane before creating line maps. The beam size is 0.33 x 0.27atp.a.=-89.8^o for ^12CO(3–2) and 0.33 x 0.29 at p.a.=89.0^o for ^13CO(3–2). The conversion to brightness is such that 1 K corresponds to ∼10-17 mJy, depending on source geometry.The RMS in a single 1 channelis 2.7 mJy/bm for^12CO(3-2) and 3.2 mJy/bm for ^13CO. Integrated intensity images (MOM0) were created by summing emission in all channels that was greater than ± 1σ; for the intensity-weighted mean velocity map (MOM1); the cutoff was± 5σ. The single dish integrated flux density reported by <cit.> with the 22 CSO beam was 3.4 K km/s, corresponding to ∼155–170 Jy km s^-1, depending on assumed source geometry. The integrated flux density recovered in the lower resolution SMA synthesis map of <cit.> was 110 ± 20 Jy km s^-1. We detect ≈50% of the expected integrated flux density the SMA maps in the form of individual clumps; the remaining flux density is extended and much has been resolved out because of the lack of short spacings in the array. Although the individual dense clouds that emit CO(3-2) are not likelyto be large, collections of such clouds with low(≲10%)volume filling may lie below even the ALMA sensitivity limit.§ THE ^12CO(3–2) LINE: HOW IS THE DENSE GAS IN NGC 5253 DISTRIBUTED?§.§ The Dense Gas is Clumpy and the Clumps Form Filaments TheCO(3-2) integrated line intensity, S_CO=∫I_COdv,inNGC 5253 is shown in Figure <ref>, superimposed on an archivalHST/ACS image of the galaxy. The J=3 state of CO has an excitation energy of 33K, requiring a critical density of n≳ 20,000 cm^-3 for excitation, thus the J=3 state of COtends to trace warm and dense star-forming molecular clouds <cit.>. Within the ∼ 30 by 60 region covered by the two ALMA pointings, CO is detected over an area ≈20or 370 pc. We identify five regions, labeled in Figure <ref>, whose properties are given in Table <ref>.In addition to“Cloud D” <cit.> which has the brightest CO(3–2) clumps,there isthe “Streamer,” which lies along the optical dust lane to thesoutheast, “Cloud E,” as observed in <cit.>, northwest of the center, “Cloud F” from <cit.> to the southwest, andfinally, the “Southern filament,” a group of clumps due south of the center. Individual filaments have characteristic widths of only a few clumps, ∼0.5, but typical lengths of ∼5or 90 pc.Individual clumps at these densities have gravitational collapse free-fall timescales of t_ff≲3.6×10^5 yr.Previous observations of NGC 5253established that lower J CO emission is located primarily in the Streamer <cit.>, following the prominent dust lane to the east along the minor axis of the galaxy.Optical emission lines, probably from the surface of the streamer, are also present in this feature<cit.>. At the 10 times higher spatial resolution of the ALMA imagesthe Streamersplits up into clumps roughly a beamsize (∼0.3, 5 pc) in extent or less, arranged in filamentary structures that are ∼ 25–200 pc in extent.The properties and kinematics of the clumps within the filaments are discussed in this section.Individual clouds in the central starburst region, within Cloud D, are discussed furtherin 5.§.§ The Clumps Form Filamentary Structures that are Distinct in Velocity Kinematic information is important in interpreting the molecular gas structure in NGC 5253. The systemic velocity of the galaxy is not well defined. HI observations show a systemic velocity of ∼ 400 <cit.>.However, the HI lies in the outskirts of the galaxy, is asymmetric, and is affected by large infalling streamers<cit.>.The Hα images of <cit.> show a central velocity of ∼385 .Since Hα is more closely associated with the stellar distribution, this is probably closer to the systemic velocity. Our ALMA CO(3–2) intensity-weighted mean velocity (MOM1) image is shown in Figure 2. This image reveals that the median CO radial (radio definition, barycentric) velocity within the central 100 pc is ∼380 . We adopt 380 as the central velocity, although it is not clear how our v_sys relates to the stellar velocity. The filamentary structures identified in the integrated intensity imageare not only spatially but kinematically distinct. The Streamer consistently runs∼ 40–20 redward of the systemicvelocity v_sys∼ 380 .Cloud F is ∼ 20–30 blueward,and the Southern filament is ∼50 to the red.Clouds D, in the central region, andE are closer tothe systemic velocity.Thus, each filament is distinctly defined kinematically and spatially, formed of gas that is sufficiently dense to form stars. The HI cloud surrounding NGC 5253 is also composed of filaments. The CO filaments of the Streamer have been identified as forming the inner portion of HI filaments <cit.>.The other CO(3–2) filaments may also form the inner portion of more extended HI filaments.Based on radial velocityalone it cannot be determined if red- or blue-shifted emission is inflowing or outflowing. However,many of the filaments correspond to areas of extinction in visible images, indicating that they are located on the near side of the galaxy.Since both the Streamer and the Southern filament are redshifted and foreground, they are thus falling toward NGC 5253. On this basis, it has previously been proposed that the starburst in NGC 5253 has been fueled by accretion from this dust lane<cit.>.By contrast Cloud F to the southwest of the central starburst is blueshifted. Cloud Fdoes not lie in an area of obvious extinction. Thus, we propose that Cloud F represents an infalling filament on the far side of the galaxy.There is no clear velocity structure within individual filaments, which seem to have coherent velocities. In particular, we find no evidence of acceleration in the gas along the Streamer, as would be expected if it were in free-fall towards the galaxy. Simple models of gravitational free-fall, assuming acentrally-concentrated mass structure,predict acceleration of the gas towards the center that is not clearly visible in the data shown in Figure <ref>. The lack of evidence for acceleration was also noted by <cit.> in the associated HI filament.We note, however, thatthe gas could be infalling nonetheless. Gas experiencing drag can fall towards a central mass at constant velocity.§.§ Conditions in the Filament Clumps Theclumps typically have diameters of about a beam, 5 pc, with ^12CO(3–2) integrated line intensities of 0.1-0.5 Jy km s^-1. Clumps have been individually identified using cprops,a clump finding code designed to identify clumps using velocity and spatial information <cit.>. Their individual characteristics will be published elsewhere<cit.>. Masses for the larger clumps and features arelisted in Table 1.To obtain clump masses from the CO emission, we use the CO conversion factor, X_CO, which relates CO(1–0) integrated line intensities to H_2 gas masses. CO(1–0) is very weakin NGC 5253 <cit.>.Thus, CO(1–0) integrated line intensities are estimated from the CO(3–2) using the line ratio,R_31=I_CO32/I_CO10=7, observed in II Zw 40<cit.>, a similar low-metallicity star-forming galaxy. The II Zw 40 value is appropriate sincethe deep ALMA CO(3–2) and CO(1–0)images used for that calculationhad matched beams of ∼0.3, similar to the beam here, in a galaxy with a similar enrichment and metallicity environment.This ratio corresponds to optically thick, slightly subthermal gas.However, the unknown value of R_31 in NGC 5253 is a source of systematic uncertainty in the gas masses.For a conversion factor of X_CO = 4.7 × 10^20 cm^-2 (K km^-1)^-1, the CO(3–2) integrated line intensities correspond to clump masses of 3.3-35×10^4 M_⊙ (the largest clumps are listed in Table 1).The conversion factor mass may overestimate the masses in Clouds D1 and D2, which are likely tocontain significant stellar mass<cit.>.These individualdense clumps by themselves are not large enough to form the large, ∼ 10^4-10^6 ,clusters that are currently seen in NGC 5253 <cit.>. We estimate thatindividual clouds in the filaments have internal pressures P/k=nT ≳ 2-4× 10^5 cm^-3K for temperatures of ∼20K, and higher for Cloud D, which appears to be warmer.This valueis higher than typicalambient pressures in the interstellar medium,P/k=2250 cm^-3K <cit.>, of our Galaxy.§ DENSE GAS AND STAR FORMATION IN NGC 5253The central ∼150 pc region surrounding the supernebula in NGC 5253<cit.> contains anextraordinary concentration of young stars.Here we use ALMA CO(3-2) images and previous lower resolution observations of CO(2–1) and CO(3–2) to understand the properties of dense gas and related star formation. The CO(3–2) integrated line intensity is shown with two tracers of star formation, Hα and radio continuum, in Figures <ref> and <ref>. TheCO(3–2) clumps in the ALMA image and the emission in NGC 5253 do not show an obvious correlation (Figure <ref>). In fact on ∼5 pc scales these properties appear to beanti-correlated, with CO(3–2) appearing in dust lanes in the Hα image. The correspondance of CO with dust lanes is not surprising: CO forms at A_V ≳ 2 <cit.>, corresponding to anoverall cloud column ofA_V≳ 4. The bright central HII region for which NGC 5253 is so well known <cit.>is centered on the heavily extincted supernebula <cit.>.Loops and arcs in the Hα suggest that the supernebula is responsible for not only large scale leakage of photons,but also possibly winds.Both effects suggest patchiness of extinction in the supernebula core, consistent with the kinematics and brightness of Cloud D1 <cit.>. Radio continuum emission traces ionizing photons through free-free emission; like , it traces emission measure, but is far weaker than .Radio continuum is the most reliable tracer of star formation in regions of high extinction.In Figure <ref>,a VLA 6 cm image <cit.> is overlaid onthe ALMA CO(3–2) map.Except for an isolated supernova remnantto the east, the remaining 6 cm emission isthermal free-free emission from regions.The ∼ 2 resolution of this VLA image is much lower than the ALMA images, butan excellent spatial correlation of the free-free continuum with CO(3–2) is apparent. The dominant radio continuum source in this image is the supernebula,associated with Cloud D1. Radio continuum peaks are alsoassociated with Clouds F and the Southern Filament, located ∼ 140-180 pc to the south and west of the supernebula, as well as clouds to the northwest within the Cloud D supernebula complex. The Streamer alsoshows radio continuum emission, presumably from the same gas producing the“ionization cone" seen in [OIII] λ5007, Hα, [SII] and [SIII]λλ6716, 9069<cit.>. The relation of gas, star formation, and molecular gas depletion time can be estimated from the radio continuum and CO(3–2).Becausefree-free emission is weak and difficult to detect at subarcsecond scales,we quantify the relation of the 6 cm free-free and CO using the VLA mapswith larger 1-2 (20-40 pc) beams <cit.> and images ofCO(2–1) from the Owens Valley Millimeter (OVRO) Array<cit.>for gas masses. The Streamer has the brightest emission in low J CO images, and contains most of themolecular mass in NGC 5253.The estimated gas mass of the streamer based on CO(2-1) is ∼ 1.7× 10^6 <cit.>.The gas surface density based on their 9.6 × 4.8 beam is ∼ 90 pc^-2,comparable to surface densities inferred forGMCs in the Galaxy <cit.>. Using Starburst99 models and a full Kroupa IMF, the relation between young stellar mass and Lyman continuum rate is log [(M/M_⊙)/(N_Lyc/s^-1)] = -46.6, with N_Lyc=1.25×10^50 s^-1 (T/10^4 K)^-0.507(ν/ 100 GHz)^0.118D_Mpc^2S_3.3mm(mJy). For a mean star formation timescaleof 5 Myr for the star formation, and for a radio continuum flux of ∼1 mJy forCloud C <cit.>,which contains most of the Streamer emission, the estimated mass depletion timescale is τ_dep=M_gas/SFR≈0.3 Gyr for the Streamer. We estimate that this number is uncertain to factors of 3–5, due to the unknown star formation timescale and uncertain gas mass.The central 1 (∼ 18.4 pc) region surrounding the supernebula, Cloud D, is forming stars at a rate of SFR∼0.1 M_⊙ yr^-1, <cit.>. CO(2–1) images indicate that M_H_2∼8×10^5 M_⊙ within Cloud D. If this gas is all associated with the current star formation, the gas depletion timescale is τ_SF∼8 Myr. However, these ALMA maps reveal that Cloud D actually consists of many clumps (5), and it is unclear if all of them are related to the currently forming massive cluster. If some of this gas is not associated with the current star formation as suggested by the gas kinematics <cit.>,then this is an upper limit to τ_SF. The molecular gas depletion time in the Streamer, τ_dep= 0.3 Gyr, isless than the median value of 0.8 Gyrfor local low metallicity galaxiesas determined by <cit.>, butwithin the observed spread. While CO-dark H_2 may be partly responsible for the low value, presumably the same holds for the general low-metallicity sample as a whole. It may also be that the ionizedgas of the Streamer is caused by uv photons from the nuclear starburst, and not from star formation within the streamer, which would increase the value of τ_dep. Cloud D has a very short molecular gas depletion time of τ_dep= 8 Myr, a factor of ∼100 times smaller than the typical low metallicity galaxy in the <cit.> sample. We caution that the central gas mass is uncertain, and the star formation rate depends on our assumption of 5 Myr for the age; the latter is probably uncertain to factor two, although there is an additional uncertainty in the spread in star formation age. However, these uncertainties of a factor of a few are not expected to be large enough to explain the two order of magnitude shorter depletion time in Cloud D1. This star formation is unusually efficient.<cit.> point out that individual GMCs in the Galaxy do not obey a Kennicutt-Schmidt law; the star formation rate surface density varies more than two orders of magnitude among Galactic GMCs. They propose that theSchmidt-Kennicutt law for galaxies <cit.> isa function of relative dense gas fraction and beam dilution on kpc scales.With these high resolution ALMA imageswe resolve individual GMCs in NGC 5253,and thus it is not surprising that we see aspread in star formation surface densities similar to that observed in Galactic GMCs. However by Galactic standards, Cloud D isextraordinarily efficient at forming stars.§ ANALYSIS OF ^13CO IN THE CLUMPS WITHIN CLOUD DImages of these ^12CO(3–2) and ^13CO(3–2)for the central ∼ 100 pc region corresponding to Cloud Dare shown in Figure <ref>. With the high, ∼5 pc, resolution of the ALMA images, the central Cloud D detectedin CO(3–2) images from the Submillimeter Array <cit.>,breaks into a number of smaller clouds with diameters of ∼ 5-20 pc. Thelarger CO structuresare labeled and their characteristicsare detailed in Table <ref>. ^13CO(3–2) emission provides valuable additional information on the optical depth of the CO and the gas mass in the central star-forming region of NGC 5253. All cloudsin the central region are at least weakly detected in ^13CO. ^13CO(3–2) emission is also detected for the brighter clouds in the Streamer, not shown here; but this emission is very faint ≲2σ. Clouds D3, D4, D5, D6, and Cloud F have ^12CO(3-2)/^13CO(3-2) flux ratios of ∼ 6-8.These values are somewhat lower, than the value of ∼13 observed in Orion <cit.>. We infer that, as in Orion, these clouds are optically thick in ^12CO(3-2).In fact, half of the measured 2.4 Jy km s^-1 of ^13CO(3-2) emission arises inCloud D4, an apparently massive cloud in the central region located about 15-20 pc south and west of the supernebula thatdoes not host obviousstar formation and whichhasan optically thick ^12CO/^13CO ratio of 7.Cloud D4 also appears in H_2 λ2.12μm images of the region <cit.>, suggesting the presence of shocks (turbulence) or fluorescence; given the extinction implied by the CO, it is likely that the H_2emission arises on the near surface of the cloud. The highest^12CO(3–2)/^13CO(3–2) ratio is ∼40, for Cloud D1, the small (r∼2.8 pc) cloud coincident with the giant central HII region known as thesupernebula. This value is uncertain because the ^13CO(3–2) is so weak, but is unlikely to be <20. Star formation may stillbe ongoing within D1, which appears to coexist with the∼ 2.5 × 10^5 M_⊙ super star cluster and HII region <cit.>.Based on the high^12CO(3–2)/^12CO(2–1) ratios of lower resolution maps,<cit.> suggested that the largerCloud D is optically thin.The ^12CO(3–2)/^13CO(3–2) ratio for D1 is consistent with the suggestion. This situation would be unusual; Galactic GMCs areoptically thick in these lines.Cloud D1may beoptically thin because the gas is hot <cit.> and the molecules spread out over the rotational ladder. At 300K, the partitionfunction is 100, such that only 1% of the CO molecules will be in the ground state and ∼3% in J=1.It is also possible that Cloud D1 couldalso have selective photodissociation or that the ^12CO(3–2) and ^13CO(3–2) could probedifferent physical environments due to temperature gradients in cloud surfaces due to the presence of strong radiation fields in the cluster environment.Cloud D1 is also bright inH_2 λ 2.12 μm emission <cit.>;given that Cloud D1 is likely to consist of multiple star-forming clumps, the H_2 emission may arise on the surfaces of theseindividual clumps. (However Clouds D4 and D6, which areoptically thick, are also bright in the H_2 line.)Cloud D2, located ∼40 pc to the northeast of Cloud D1,may also be optically thin, with a ^12CO/^13CO ratio of ∼12, whichis higher than average for this galaxy. As this cloud appears in radio continuumemission at cm wavelengths (Beck+18, private communication) andin H_2 λ 2.12 μm emission<cit.>,Cloud D2 appears similar to Cloud D1, if smaller in scale. Thus, Cloud D2 appears to be a young, compact cluster that has warmmolecular gas, although significantly smaller than themassive cluster within D1. If ^13CO(3-2) is optically thin in some clouds, we can use the ^13COintegrated line intensities to estimate optically thin masses. These masses are given inTable 1.For Cloud D1 we adopt a temperature of 300 K, based on RADEX models of the CO(3–2)/CO(2–1) line ratio <cit.> and for clouds besides D1 we adopt 20K. The optically thin masses are not very sensitive to temperature for this line until temperatures exceed ∼100K, although if the clouds are actually very warm, these masses will be underestimates.If we adopt a Galactic [CO]/[H_2] ratio of 8.5× 10^-5, and a[^12CO]/[^13CO] abundance ratio of 40, the minimum indicated by the Cloud D1 line ratio, we obtain cloud masses M_Cl∼0.6-11× 10^3 M_⊙ (Table 1), with the most massivecloud being the apparently quiescent Cloud D4). The masses based on ^12CO(3-2), assuminga conversion factor (“^12CO Masses") and thosebased on ^13CO(3-2)assuming optically thin emission (“^13CO Masses")differ by an order of magnitude. There are many possible reasons for the differences. For optically thin clouds, D1, D2,thevalues of [CO]/[H_2] and the [^13CO]/[^12CO ]abundance ratios are sources of systemic uncertainty, since they may differ significantly from Galactic values in this low metallicity galaxy. For optically thick clouds, the conversion factor cannot account for clouds that are unusually warm; or that have significant stellar mass contained within the cloud, such as Clouds D1 andD2; or for clouds that havesignificant CO-dark gas, although this is not likely to be a major mass contributorin these dense clouds<cit.>.The CO(3-2) critical density isaligned with the gas densities expected for star-forming gas. Even the highest estimates of cloud mass indicate that thegas masses of these clouds are fairly small. The largest cloud, D4, has anestimated mass of 3.4× 10^5 M_⊙, a relatively low mass considering the ∼ 10^5 M_⊙ star clusters that NGC 5253 has been forming. The current suite of clouds does not seem capable of forming a massive super star cluster, unless mergers of clouds take place, perhaps along accreting filaments, and if the star formation is extraordinarily efficient. § SUMMARY AND CONCLUSIONS We present new ALMA images of ^12CO(3–2) and ^13CO(3–2) emission from the nearby (3.8 Mpc) dwarf galaxy NGC 5253.The spatial resolution is ∼0.3, 10 times smaller in diameter than the best previous maps.The images reveal dense (∼ 20,000 cm^-3) molecular clumps ∼5 pc in diameter with estimated masses of a few thousand M_⊙. While atomic and ionized filaments are a feature of NGC 5253 on the largest scales <cit.>,these observations are the first to detect clear, kinematically distinct filaments in the molecular gas.The inner 200 pc of NGC 5253 holds four filaments, composed ofsmall (∼5 pc diameter) clouds. These filaments extend to ∼ 50-150 pc in length. The Streamer filament, detected previously in CO(1-0) and CO(2-1) inlower resolution images, is aligned along the prominent dust lane to the east of the galaxy. The Streamer isred-shifted with respect to the systemicvelocity, and thus appears to befalling intothe galaxy. Also redshifted, and foreground, thus apparently infalling, is the weaker Southern filament to the south of the galaxy center. Cloud F, a filament southwest of the center, is blue-shifted with respect to the galaxy but shows no apparent extinction, and may be on the far side of the galaxy. We propose that the CO filaments form the inner position of HI filaments that are falling into the galaxy from outside. This model particularly explains the Streamer along the prominent dust lane (redshifted) and Cloud F (blueshifted, no apparent foreground extinction).The clearest filaments have length-to-width aspect ratios of ≈20.Dense clumps in the filamentsare likely to be overpressured relative to ambient gas, and probably gravitationally supported; they may be likelysites of future star formation.The central cloud surrounding the supernebula, Cloud D, previously detected as a bright CO(3–2) source, breaks up into multiple clouds. Some of these clouds are optically thin in CO(3-2), as indicated byhigh ^12CO/^13CO flux ratios of ∼ 12–40, higher than the average cloud ratio of ∼6-8.Cloud D1 is coincident with the giant HIIregion and supernebula; its flux ratio of ^12CO/^13CO∼40 suggests that the [^12CO]/[^13CO] ratio in NGC 5253 is 40. Cloud D2 appears to contain a smaller young cluster. We propose that Clouds D1 and D2 are most likely optically thin in CO(3–2) because the molecular gas is hot,due to the proximity ofyoung and massive star clusters, although itis also possible that optical depth effects in the presence of high temperature gradients could produce the high observed ratios.These clouds are likely to be brighter than would be expected based on their gas columns alone. From radiocontinuum flux densities and lower J CO lines,wederive average gas depletion timescales of 0.3 Gyr for the Streamer and 8Myr for Cloud D1 coincident with the supernebula. The masses we obtain for the individual clumps from the ALMA observations, from assumption of optically thin emission, for the thin clouds, and from ^12CO(3–2) emission, for the thick clouds, differ by about an order of magnitude. Uncertainties in the H_2 masses are due to unknown [CO]/[H_2] for this low metallicity, high radiation environment,unknown CO(3–2)/CO(1–0)ratio, and uncertainty in the conversion factor applicability. However, even the highest estimates of mass for the clumps, ∼ 10^4-10^5 M_⊙, aresmall compared to the masses of the star clusters that are currently forming. This suggests that filament mergers and/or highly efficient star formation is taking place in NGC 5253. Further high resolution CO observations may clarify these issues and allow for improved masses. Even with these uncertainties, it appears that the star formation in the central Cloud D1 associated with the supernebula is extraordinarily efficient.lcccccccc Source, ^12CO and ^13CO Fluxes, Ratio, and Masses^** 0pt 1 SourceICRS RA^aICRS Dec^av_sys^b^12CO(3-2)^c^13CO(3-2)^c^12CO/^13CO^12CO Mass^d^13CO Mass^e 13^h39^m-3138 Jy Jy ratio10^3 M_⊙10^3 M_⊙D1 55.96524.36 3852.40.06±0.0540^+30_-20 100 3.0 D256.1223.1 3940.80.07±0.0212±4330.6 D356.0322.9 3951.70.28±0.046±1702.6D455.9125.0 4028.41.2±0.077±0.4 34511D555.7925.6 3840.870.12±0.027±1 361.1 D655.8827.0 4091.80.25±0.047±1 742.3 Squiggle (D7) 56.0 27 4034.10.5±0.1 9±2 170 4.4Center Sum of D Clouds 55.95 25 385 20 2.4±0.1 8±0.5823 24.7Streamer^f 56.5 33 4308.79 0.86±0.210±2 360 8.1Cloud E 55.5 19 40012.550.8±0.216±4 510 7.5 Cloud F55.831 368 6.82 0.9±0.1 8±1280 8.2 Southern Filament 55.9 3674132.450.54±0.1 5±1 100 5.1Overall Sum 56.3 3038051 5.59 2073 53.6 **All measurements are made from velocity-selected, primary beam corrected MOM0 integrated flux density maps. As ^13CO emission is weak, ^12CO was used to define velocity regions for each cloud, and all flux was summed for that velocity range. aPrecision in coordinates depends on the compactness andstructure of the source. b± 5cUncertainties in ^12CO flux density are10% from flux density calibration uncertaintiesunless otherwise stated. Uncertainties in ^13CO flux density, which is mostly signal-to-noise limited,are computed from the individual channel rms times√(N_chanN_beam).dMass derived assuming optically thick emission, using a conversion factor of α_CO=4.7×10^20 cm^-2 (K kms^-1)^-1, and an estimated CO(3–2)/CO(1–0) ratio of 7. This mass will overestimate the cloud mass for optically thin, warm clouds andfor clouds in which enclosed stellar mass is significant. Mass includes He. eMass derived assuming optically thin ^13CO flux density and 20 K, a Galactic value of [CO]/[H_2]=8.5× 10^-5 and an isotopic ratio of 40, except for Cloud D1, where T=300 K is assumed.The Rayleigh-Jeans approximation is not used.This mass will underestimate the true cloud mass for optically thick emission orif the cloudsare warmer than 20 K.Mass includes He. f All integrated line intensities, particularly ^12CO, are subject to the missing short spacing data, and thereforethe inability of thisimage to reconstruct structures >4 in sizeThis will not affect small clouds such as the Cloud D individual clumps.The Streamer, however, appears to be missing flux on the whole. See text.This paper makes use of the following ALMA data: ADS/JAO.ALMA# 2012.1.00105.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory (NRAO) is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.Support for this work was provided by the NSF through award GSSP SOSPA2-016 from the NRAO to SMC and grant AST 1515570 to JLT, and by the UCLA Academic Senate through a COR seed grant. The authors wish to thank the W. M. Goss, P. T. P. Ho, and the anonymous referee for their helpful comments. Facilities: ALMA.[Alonso-Herrero et al.(2004)]2004ApJ...612..222AAlonso-Herrero, A., Takagi, T., Baker, A. J., et al. 2004, , 612, 222 [Atherton et al.(1982)]1982MNRAS.201..661AAtherton, P. D., Taylor, K., Pike, C. D., et al. 1982, , 201, 661[Bendo et al.(2017)]2017arXiv170706184BBendo, G. J., Miura, R. E., Espada, D., et al. 2017, arXiv:1707.06184[Bisbas et al.(2015)]2015ApJ...803...37B Bisbas, T. G., Papadopoulos, P. 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"authors": [
"S. Michelle Consiglio",
"Jean L. Turner",
"Sara Beck",
"David S. Meier",
"Sergiy Silich",
"Jun-Hui Zhao"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20171027180800",
"title": "ALMA CO(3-2) Observations of Star-Forming Filaments in a Gas-Poor Dwarf Spheroidal Galaxy"
} |
Intrinsic diameter control under mean curvature flow]Diameter and curvature control under mean curvature flow[ Panagiotis Gianniotis Robert Haslhofer December 30, 2023 ========================================== We prove that for the mean curvature flow of two-convex hypersurfaces the intrinsic diameter stays uniformly controlled as one approaches the first singular time. We also derive sharp L^n-1-estimates for the regularity scale of the level set flow with two-convex initial data. Our proof relies on a detailed analysis of cylindrical regions (-tubes) under mean curvature flow. The results are new even in the most classical case of mean convex surfaces evolving by mean curvature flow in ℝ^3. § INTRODUCTIONA family of hypersurfaces {M_t^n⊂ℝ^n+1}_t∈ [0,T) evolves by mean curvature flow if the normal velocity at each point is given by the mean curvature vector. By classical theory (see e.g. <cit.>), given any closed embedded initial hypersurface M_0^n⊂ℝ^n+1, there exists a unique smooth solution defined on a maximal time interval [0,T). The maximal existence time T<∞ is characterized bylim_t↗ Tmax_M_t A=∞ ,where |A| denotes the norm of the second fundamental form.One naturally wonders to what extend one can control the geometry of the hypersurfaces – curvature integrals, intrinsic diameter, etc – as one approaches the first singular time (more generally, one can pose these questions also beyond the first singular time in the setting of mean curvature flow with surgery and level set flow, respectively):[Curvature control] Can one control the curvature integrals ∫_M_tA^p dμ as the flow approaches the first singular time? [Diameter control] Can one control the intrinsic diameter as the flow approaches the first singular time? The two questions are in fact tightly related. For example, by a result of Topping <cit.> we have the estimate(M_t,d_t) ≤ C_n ∫_M_tH^n-1dμ. To obtain diameter control, one thus might try to derive uniform L^n-1-bounds for the mean curvature.For n>2, the situation where there is some hope to get uniform L^n-1-bounds concerns the evolution of two-convex hypersurfaces, i.e. when the sum of the smallest two principal curvatures λ_1+λ_2 is positive.[One of course cannot hope to get L^n-1-bounds assuming only mean convexity. Indeed, consider the case that M_0≈ S^n-2_r× S^2_R is a very thin rotationally symmetric torus, i.e. r is very small. Under the flow these small (n-2)-spheres will degenerate to points, and it is easy to see that lim_t↗ T∫_M_t H^n-1dμ = ∞.] This curvature condition arises naturally in the work on mean curvature flow with surgery by Huisken-Sinestrari <cit.> (see also <cit.>), and its main feature is that it excludes generalized cylinders ℝ^j× S^n-j with more than one ℝ-factor (i.e. j≥ 2) as potential blowup limits.For n=2, the notion of two-convexity boils down to the simpler notion of mean convexity, i.e. positive mean curvature, or in other words the assumption that the flow is moving inwards.It has been proved by Head <cit.> and Cheeger-Haslhofer-Naber <cit.> that for the mean curvature flow of two-convex hypersurfaces one has∫_M_tA^n-1-dμ≤ C(M_0,)<∞,for any >0. Motivated by this result, it is natural to conjecture:[R.H. thanks John Head for introducing him to these conjectures during a visit to the Courant Institute in 2011. While we unfortunately don't know the precise history, the conjectures have certainly been discussed among experts well before 2011, c.f. Perelman's bounded diameter conjecture for 3d Ricci flow <cit.>.] If {M_t⊂ℝ^n+1}_t∈ [0,T) is a mean curvature flow of two-convex closed embedded hypersurfaces, then∫_M_tA^n-1dμ≤ C,for some constant C<∞ depending only on the geometric parameters of the initial hypersurface.If {M_t⊂ℝ^n+1}_t∈ [0,T) is a mean curvature flow of two-convex closed embedded hypersurfaces, then diam(M_t,d_t) ≤ C,for some constant C<∞ depending only on the geometric parameters of the initial hypersurface. Note that by the result of Topping <cit.> an affirmative answer to Conjecture <ref> would imply an affirmative answer to Conjecture <ref>.Conjecture <ref> states that one can get rid of thein the curvature estimate (<ref>). The question of whether or not one can actually get rid of thein estimates like (<ref>) is a very delicate question, that depends on the fine structure of singularities and regions of high curvature.For comparison, it is useful to look at related elliptic questions. In recent impressive work <cit.>, Naber-Valtorta improved the known L^3- estimates for the gradient of minimizing harmonic maps to sharp L^3_weak-estimates. The simple example f(x)=xx in dimension three, shows that for minimizing harmonic maps the L^3_weak estimate actually cannot be replaced by an L^3 estimate; cf. also the related work <cit.>.Having discussed the subtleties of sharp integral estimates, let us now emphasize that our approach for the solution of Conjecture <ref> and Conjecture <ref> actually proceeds in the opposite order. Namely, we first prove that we can control the intrinsic diameter: If {M_t⊂ℝ^n+1}_t∈ [0,T) is a mean curvature flow of two-convex closed embedded hypersurfaces, then(M_t,d_t)≤ C,for a constant C=C(α,β,γ,𝒜)<∞, which only depends on certain geometric parameters of the initial hypersurface M_0 (see Section <ref>). The diameter bound from Theorem <ref> is new even in the classical case of mean convex surfaces evolving in ℝ^3 (note that for n=2 mean convex surfaces are automatically β-uniformly two-convex with β=1).We then use Theorem <ref> to derive sharp integral estimates for the second fundamental form. More precisely, we actually obtain sharp integral estimates for the regularity scale of the level set flow with two-convex initial data. Recall from <cit.>, if ℳ={M_t}_t∈ [0,T) is a weak solution of the mean curvature flow (here we only consider the mean convex case, where Brakke solutions and level set solutions are known to be equivalent) then the regularity scale r_ℳ(x,t) at a point (x,t)∈ℳ is defined as the supremum of 0≤ r≤ 1 such that M_t'∩ B_r(x) is a smooth graph for all t-r^2<t'<t+r^2 and such thatsup_|x'-x|<r, |t'-t|<r^2 r |A|(x',t')≤ 1.Obviously |A|(x,t)≤ r_ℳ(x,t)^-1, but of course a bound for r_ℳ(x,t)^-1 captures geometric control on a whole parabolic neighborhood of definite size as opposed to just information at the single point (x,t).If ℳ={M_t⊂ℝ^n+1}_t∈ [0,T) is a level set flow with two-convex initial data, then we have the sharp estimate∫_M_t r_ℳ(x,t)^-(n-1) dμ_t(x)≤ C,for a constant C=C(α,β,γ,𝒜)<∞, which only depends on certain geometric parameters of the initial hypersurface M_0 (see Section <ref>). Since |A|(x,t)≤ r_ℳ(x,t)^-1, Theorem <ref> immediately implies a sharp integral estimate for the second fundamental form:If ℳ={M_t⊂ℝ^n+1}_t∈ [0,T) is a level set flow with two-convex initial data, then we have the sharp curvature estimate[Note that M_t is n-rectifiable with multiplicity one for all t, and that the second fundamental form A is defined classically μ_t-almost everywhere.]∫_M_t |A|^n-1 dμ_t ≤ C,for a constant C=C(α,β,γ,𝒜)<∞, which only depends on certain geometric parameters of the initial hypersurface M_0 (see Section <ref>). In particular, Theorem <ref> and Corollary <ref> crucially improve the regularity estimates from Head <cit.> and Cheeger-Haslhofer-Naber <cit.>.Let us now sketch the main ideas for our proof of Theorem <ref>.To get some intuition, imagine first the potential scenario of the so-called fractal tube, as illustrated in Figure <ref>. The worry is that necks might slowly but steadily change their axes as one moves over an uncontrolled number of scales, and arrange themselves in a Koch snowflake like way, to make the intrinsic diameter arbitrarily large.In a recent breakthrough <cit.>, Colding-Minicozzi proved a Łojasiewicz inequality for the mean curvature flow. Their Łojasiewicz inequality implies, roughly speaking, that cylindrical regions can actually only tilt by a controlled amount, provided one can ensure a priori that one is very close to a cylinder at all scales in consideration. Colding-Minicozzi applied the Łojasiewicz inequality to prove uniqueness of cylindrical tangent flows <cit.>, to derive sharp results about the singular set of mean curvature flow with generic singularities <cit.>, and also to derive related sharp results for the arrival time function of a mean convex flow <cit.>. In particular, if the flow is two-convex then its space-time singular set is contained in finitely many compact embedded Lipschitz curves together with a countable set of points. This of course immediately rules out the example of an “exact fractal tube", i.e. a fractal tube as above that becomes singular exactly on a fractal curve.While the Łojasiewicz inequality from <cit.> has been applied many times to derive results about the singular set, it hasn't been applied yet in a more quantitative way to derive results about high curvature regions. Note that a bound for the size of the singular set by itself, doesn't yield any control for the diameter. For example, one could imagine the scenario of an “approximate fractal tube" that looks Koch-like shortly before the first singular time, but then only becomes singular with a neck pinching off at one single point.The main challenge in proving Theorem <ref> is thus to ensure a priori that we can locate enough cylindrical regions propagating all the way from a microscopic scale to a macroscopic scale of definite size, to which we can apply the Łojasiewicz inequality. We achieve this as follows:Given a two-convex mean curvature flow ℳ={M_t⊂ℝ^n+1}_t∈ [0,T), and a time t̅< T, we want to derive a bound for diam(M_t̅,d_t̅) depending only on certain geometric parameters of the initial hypersurface M_0. Using in particular the canonical neighborhood theorem from <cit.>, we first argue (see Proposition <ref> and Proposition <ref>) that it is enough to control the quantityL(M_t̅):=sup{diam(N,d_t̅)|N⊂ M_t̅ is an -tube with H>H̅ on N}.Here, H̅ denotes a very large curvature scale, that in particular is much larger than the curvature scale H_can from the canonical neighborhood theorem. An -tube is a subset N⊂ M_t̅ diffeomorphic to S^n-1×ℝ such that each x∈ N lies on the central sphere of a very strong -neck at time t̅. Roughly speaking, having a very strong -neck means that after rescaling to unit curvature one sees an almost round shrinking cylinder in a very large space-time neighborhood. The precision parameterenters in the rigorous definitions, replacing the informal words “almost" and “very large" (see Section <ref> for the precise definitions). Moreover, we consider a central curve γ parametrized by arc-length such that N is covered by -necks centered at p∈γ and axis parallel to ∂_s γ(p).The central task is to understand how the flow arrived at the tube N at time t=t̅. We show that there is a uniform τ>0 such that around any p∈γ and at any time t∈ [t̅- τ,t̅] the flow has a strong _1-neck (here ≪_1≪ 1) centered at p of radius √(2(n-1)(t_p-t)), where t_p essentially determines the scale of the neck and only depends on p. To this end, first recall that by the canonical neighborhood theorem all points with H≥ H_can have a precise geometric description. The canonical neighborhoods are modeled on so-called (α,β)-solutions (see Section <ref>), which in turn look neck-like away from cap-like pieces of controlled size. Inspired by <cit.>, we prove a backwards-stability result for necks in (α,β)-solutions (Proposition <ref>). Roughly speaking, the result says that if one sees a neck in an (α,β)-solution then going far enough back in time one sees an even more precise neck, and in particular not a cap. Combining the neck-stability result with a continuity argument and the assumption that we started off with a very stong -tube, we arrive at the conclusion that around any p∈γ one sees _1-necks of the right radius for a uniform amount τ backwards in time.Having done the above, we can finally apply the Łojasiewicz inequality from <cit.> to conclude that the total tilt of every neck in N, when followed via normal motion from time t̅ back to time t̅-τ, is in fact small (Proposition <ref>). Since at time t̅-τ all necks have macroscopic size, and since overlapping necks must have aligned axes, this gives the desired diameter control.To prove Theorem <ref>, we first generalize Theorem <ref> to the setting of mean curvature flow with surgery in the framework of <cit.>. The important point is that the diameter bound is independent of the surgery parameters, see (<ref>). We then combine this with the canonical neighborhood theorem <cit.> to establish uniform L^n-1-bounds for the mean curvature under mean curvature flow with surgery, see (<ref>).Given a fixed two-convex initial condition M_0, we then consider a sequence {M_t^i} of flows with surgery where the surgery parameters degenerate suitably. It follows from the work of Head <cit.> and Lauer <cit.>, that for i→∞ the sequence of flows with surgery converges to the level set flow. The convergence is in the space-time Hausdorff sense, and also in the sense of varifolds for every time. By lower-semicontinuity, this implies L^n-1-control for the mean curvature of level set flow with two-convex initial data. Finally, by the local curvature estimate <cit.>, this can be upgraded to L^n-1-control for the regularity scale.This article is organized as follows: In Section <ref>, we introduce our notation and summarize the needed preliminaries. In Section <ref>, we carry out the reduction of the problem to the case of high curvature tubes. In Section <ref>, we prove the key propositions that have been outlined above and combine them to prove Theorem <ref>. Finally, in Section <ref>, we generalize the diameter bound to the setting of mean curvature flow with surgery and prove Theorem <ref>.Acknowledgments. P.G. has been supported by a Fields-Ontario Postdoctoral Fellowship. R.H. has been supported by NSERC grant RGPIN-2016-04331, NSF grant DMS-1406394 and a Connaught New Researcher Award. We thank the Fields Institute and the University of Toronto for providing an excellent research environment. We thank Nick Edelen, John Head, Mohammad Ivaki, Bruce Kleiner, and Felix Schulze for useful discussions.§ NOTATION AND PRELIMINARIES§.§ Mean curvature flowLet M_0⊂ℝ^n+1 be a closed embedded hypersurface. Then there exist a unique smooth evolution by mean curvature flow ℳ={M_t}_t∈ [0,T) defined on a maximal time interval [0,T). The maximal existence time T<∞ is characterized by lim_t↗ Tmax_M_t A=∞. A closed embedded hypersurface M⊂ℝ^n+1 is called 2-convex, ifλ_1+λ_2>0for all p∈ M, where λ_1≤λ_2≤…≤λ_n are the principal curvatures, i.e. the eigenvalues of the second fundamental form A. In particular, every 2-convex domain has positive mean curvatureH>0.We also need more quantitative notions of 2-convexity and embeddedness: A 2-convex hypersurface M is called β-uniformly 2-convex, ifλ_1+λ_2≥β Hfor all p∈ M. An embedded hypersurface M⊂ℝ^n+1 is called α-noncollapsed if it has positive mean curvature and each p∈ M admits interior and exterior balls of radius at least α/H(p). By <cit.>, all the above properties are preserved under the flow.We also assume that H≤γ initially.To keep track of the constants, we put =(α,β,γ) and say that M_0 is -controlled. By compactness, every 2-convex embedded hypersurfaces M_0 is -controlled for some parameters α,β>0,γ<∞. Some estimates will also depend on a bound 𝒜 for the initial area (recall that area is decreasing under the flow). While clearly most constants also depend on the dimension n, we usually don't explicitly indicate this in our notation. §.§ Necks and tubesIn this section, we summarize our basic terminology about necks and tubes.We say that an embedded hypersurface M^n⊂ℝ^n+1 has an -neck with center p and radius r if the rescaled surface r^-1 � (M-p) is -close in C^⌊ 1/⌋ in B_1/(0) to a round cylinder S^n-1×ℝ (up to rotation) with center 0 and radius 1.As in <cit.>, we say that a mean curvature flow ℳ has a strong -neck with center p and radius r at time t_0, if the parabolically rescaled flow {r^-1 (M_t_0+r^2 t -p)}_t∈(-1,0] is -close in C^⌊ 1/⌋ in B_1/(0) × (-1, 0] to the evolution of a round cylinder S^n-1×ℝ (up to rotation) with center 0 and radius 1 at t = 0.We say that ℳ has a very strong -neck with center p and radius r at time t_0, if in the above definition (-1,0] can be replaced by (-2𝒯,0], where 𝒯=𝒯(2_1,12_1,α,β) denotes the constant from Proposition <ref>.[Here, _1 is a certain quality parameter, that will be fixed in Section <ref>.]A subset N⊂ M_t̅ is called an -tube, if it is diffeomorphic to a cylinder and each x∈ N lies on the central sphere of a very strong -neck at time t̅ (more precisely, the slice of a strong -neck at its maximal time).Finally, as in <cit.>, for each -tube we can find an -approximate central curve γ. In particular, for each p∈γ the vector ∂_sγ(p) determines the axis of the -neck centered at p (recall that the axis of an -neck is well defined up to errors of order ). §.§ Canonical neighborhoods We recall the canonical neighborhood theorem from Haslhofer-Kleiner <cit.> in the special case of smooth flows without surgeries.[In this special of smooth flows without surgeries the proof of the canonical neighborhood theorem simplifies quite a bit. In fact, Theorem <ref> follows directly from the global convergence theorem for α-noncollapsed flows <cit.>.] The general case of the canonical neighborhood theorem for mean curvature flow with surgery will only be needed at the end, and its discussion will thus be deferred until Section <ref>. For every _can>0, there exist a constant H_can(_can)=H_can(,_can)<∞ with the following significance. If ℳ is a smooth mean curvature flow with -controlled initial data, then any (x,t)∈ℳ with H(x,t)≥ H_can(_can) is _can-close to a β-uniformly 2-convex ancient α-noncollapsed mean curvature flow. For brevity we refer to “β-uniformly 2-convex ancient α-noncollapsed mean curvature flows" simply as (α,β)-solutions. We recall that being _can-close is understood in a scale invariant sense. The conclusion in Theorem <ref> thus means that the flow ℳ' which is obtained from ℳ by shifting (x,t) to the origin and parabolically rescaling by H^-1(x,t) is _can-close in C^⌊ 1/_can⌋ in B_1/_can(0)× (-_can^-2,0] to an (α,β)-solution. The structure of (α,β)-solutions has been analyzed in <cit.>. In particular, (α,β)-solutions are always convex and look neck-like away from caps of controlled size (in a scale invariant sense, as always). §.§ Łojasiewicz-Simon inequalityFor a hypersurface M ⊂ℝ^n+1 the Gaussian surface area is defined byF(M)=(4π)^-n/2∫_M e^-|x|^2/4 dℋ^n.The entropy λ is defined as supremum of the Gaussian area over all centers and scales:λ(M)=sup_a>0,b∈ℝ^n+1F(a M-b).By Huisken's monotonicity formula <cit.> the entropy is nonincreasing under mean curvature flow. Thus, the entropy of all hypersurfaces under consideration will be bounded above by some constant Λ=Λ(α,γ,𝒜). Also recall that if {M_t⊂ℝ^n+1}_t<0 moves by mean curvature flow, then Σ_s=1√(-t)M_t, s=-log(-t), moves by rescaled mean curvature flow∂_s x= H+12x^⊥.The rescaled mean curvature flow is the negative gradient flow of the F-functional.There exist constants K,R̅<∞, _L>0, μ<1 (depending only on the dimension and an upper bound for the entropy) such that if {Σ_s}_s∈ [t-1,t+1] is a rescaled mean curvature flow such that B_R̅∩ M_s is for each s a C^2,α graph with norm at most _L over the cylinder Z=S^n-1_√(2(n-1))×ℝ, then|F(Σ_t)-F(Z)|^1+μ≤ K ( F(Σ_t-1)-F(Σ_t+1)).To see that (<ref>) is indeed a discrete Łojasiewicz-Simon gradient inequality, it helps to rewrite the right hand side usingF(Σ_t-1)-F(Σ_t+1)=∫_t-1^t+1|∇_Σ_s F|^2 ds.§ REDUCTION TO THE CYLINDRICAL CASE Let ℳ={M_t}_t∈ [0,T) be a 2-convex flow with -controlled initial condition, and area bounded by 𝒜. Let t̅<T. We want to show that diam(M_t̅)≤ C, for some C<∞ depending only onand 𝒜. §.§ Reduction to the high curvature case We will argue first that it is enough to estimate the length of geodesics that stay entirely in regions of high curvature. To this end, assume that H̅ is a constant with H̅≥ H_can(_can), where the precision parameters >0 and _can=_can(,)>0 are small enough as in <cit.>, and consider the quantityD(M_t̅):=sup{ℓ(γ) |γ is a minimizing geodesic in (M_t̅,d_t̅), and H>2H̅ along γ}.There exists a constant N̅=N̅(α,𝒜,H̅)<∞ such thatdiam(M_t̅)≤N̅+ (N̅+1)D(M_t̅). Let γ:[0,L]→ (M_t̅,d_t̅) be a minimizing geodesic parametrized by arclength. Choose a maximal collection s_1,…,s_N∈ [0,L] such that H(γ(s_i))≤ 2H̅ and |s_i-s_j|≥ 1. Consider then the geodesic balls B_i with center γ(s_i) and radius 1/2.We claim that the curvature is uniformly bounded in small balls centered at γ(s_i) of definite size (depending on H̅ and ): To this end, first observe that the α-noncollapsing and the bound H≤γ at the initial time imply that |A|≤√(n)γ/α at t=0. By standard doubling estimates there is a constant T_1=T_1()>0 such that if t̅≤ T_1 then |A|≤ 2√(n)γ/α on the whole hypersurface. If t̅> T_1 using that H(γ(s_i))≤ 2H̅ we can apply the local curvature estimate <cit.> to get curvature control in a ball centered at γ(s_i) of definite size.By the above and the α-noncollapsing we get a lower boundℋ^n(B_i)≥ c(,H̅)>0.Since the balls B_i are disjoint, this implies thatN≤𝒜/c=:N̅.Now restricting γ to the connected components of [0,L]∖⋃_i=1^N (s_i-12,s_i+12) gives at most N̅+1 minimizing geodesics with the additional property that H>2H̅ along them. The assertion follows.§.§ Applying the canonical neighborhood theorem Consider the quantityL(M_t̅):=sup{diam(N,d_t̅)|N⊂ M_t̅ is an -tube with H>H̅ on N}.There exists a constant C=C(,,H_can)<∞ such thatD(M_t̅)≤ C+C L(M_t̅). Let γ be a minimizing geodesic in (M,d_t̅) with H>2H̅ along γ. We can apply the canonical neighborhood theorem (Theorem <ref>) at each point (γ(s),t̅). If the canonical model at some point is a convex solution of controlled geometry, then γ has controlled length and we are done. In all other cases, arguing as in the proof of <cit.> we see that γ must be contained in an -tube, possibly capped at one or both ends, or with its ends identified. The caps have diameter at most C()H_can^-1, and thus contribute at most a controlled amount to the length of γ. Similarly, in the case that the ends of the -tube are identified, we can also simply ignore a piece of small size, to reduce it again to the case of an -tube. The assertion follows. § ESTIMATING THE LENGTH OF CYLINDRICAL REGIONS§.§ Small axis tilting under a priori assumptionsIn this section, we consider mean curvature flows under the a priori assumption that they are close to cylinders along a large – possibly uncontrolled – number of scales. We assume that the space-time center point is fixed, but a priori the axis is allowed to be different at every scale. Using methods from <cit.>, we show that the total tilt of the axis is in fact small. For all _0>0 there exists an _1=_1(_0,Λ)>0 with the following significance. Let ℳ be a mean curvature flow with entropy bounded by Λ, and suppose there are p∈ℝ^n+1 and t_0<t_1<t_∗, such that for all t∈[t_0,t_1], ℳ has a strong _1-neck with center p and radius √(2(n-1)(t_∗-t)) at time t. Then there exists a v∈ℝ^n+1 such that for all t∈[t_0,t_1], ℳ has a strong _0-neck with center p, radius √(2(n-1)(t_∗-t)) at time t, and axis in the fixed direction v.Our argument is a variant of the one from Colding-Minicozzi <cit.>. Consider the rescaled mean curvature flow Σ_s=1/√(t_∗-t)M_t, s=-log(t_∗-t). If _1 is small enough (depending on ε_L=_L(Λ), R̅=R̅(Λ)) then the hypothesis of Theorem <ref> are satisfied. Hence,|F(Σ_s) - F(Z)|^1+μ≤ K(F(Σ_s-1) - F(Σ_s+1)),for every s∈[s_0,s_1], where s_0=⌈ - log(t_∗-t_0) +1⌉, s_1=⌊ -log(t_∗-t_1)-1⌋.Applying the discrete Łojasiewicz lemma (Lemma <ref>) for the function f(s)=F(Σ_s)-F(Z) we infer that for every >0 there exists an =(,Λ)>0 such that∑_j=s_0+1^s_1 (F(Σ_j) - F(Σ_j-1))^1/2 <,provided _1<. Using the Cauchy-Schwarz inequality and the fact that the rescaled mean curvature flow is the negative gradient flow of F, this implies∫_s_0^s_1 ∫_Σ_s| H⃗+12x^⊥ | e^-|x|^2/4/(4π)^n/2 dℋ^nds ≤Λ^1/2∑_j=s_0+1^s_1( ∫_j-1^j∫_Σ_s| H⃗+12x^⊥ |^2 e^-|x|^2/4/(4π)^n/2 dℋ^nds )^1/2= Λ^1/2∑_j=s_0+1^s_1 (F(Σ_j) - F(Σ_j+1))^1/2 <Λ^1/2.Choosing _1=_1(_0,Λ) small enough, we can make the time-integral of the weighted L^1-norm of H⃗+12x^⊥ as small as we want, and the assertion follows. §.§ Backwards stability of necks Inspired by <cit.>, we prove the following neck-stability result for (α,β)-solutions: There is a δ_neck>0 such that for all δ_0,δ_1≤δ_neck there is a 𝒯=𝒯(δ_0,δ_1,α,β)<∞ with the following property.Suppose that ℳ is an (α,β)-solution that has a δ_0-neck with center p and radius √(2(n-1)) at time -1. Then for all t∈ (-∞, -𝒯] the flow ℳ̂ which is obtained from ℳ by shifting (p,0) to the origin and parabolically rescaling by |t|^-1/2 is δ_1-close to a round shrinking cylinder S^n-1×ℝ (up to rotation) that becomes extinct at time 0. In particular, for all t∈ (-∞, -𝒯] the unrescaled flow ℳ has a strong δ_1-neck with center p and radius √(2(n-1)|t|) at time t.We first fix small enough constants δ_neck>0 and c>0 (depending only on the dimension) such that the Gaussian density (<ref>) satisfiesF(M_-1)≥λ(S^n)+c,whenever ℳ has a δ_0-neck with center 0 and radius √(2(n-1)) at time -1.If the conclusion of the theorem didn't hold, then we could find a sequence ℳ^i of (α,β)-solutions with a δ_0-neck with center 0 and radius √(2(n-1)) at time -1, but such that the flows ℳ̂^i which are obtained from ℳ^i by parabolically rescaling by |t_i|^-1/2 are not δ_1-close to a round shrinking cylinder, for some sequence t_i→ -∞.Consider Huisken's monotone quantity based at (0,0), namelyΦ (M_t^i)=∫_M_t^i1/(4πt)^n/2e^-|x|^2/4|t|dℋ^n (t<0).Since for each fixed i, the flow ℳ̂^i has a blowdown-limit (ancient soliton) which must be either a plane, a round shrinking sphere, or a round shrinking cylinder with only one ℝ-factor (see <cit.>), we get the upper boundΦ (M_t^i)≤λ(S^n-1). After passing to a subsequence, we can find t̃_̃ĩ∈ (t_i,-1) with t̃_̃ĩ/t_i→ 0 such that |Φ(M^i_t)-Φ(M^i_t̃_i)|< 1/ifor t∈[A_i t̃_i,A_i^-1t̃_i], where A_i→∞. Let ℳ̃^i be the sequence of flows that is obtained from ℳ^i by parabolically rescaling by |t̃_i|^-1/2. After passing to a subsequence we can pass to a limit ℳ̃^i→ℳ̃^∞ which must be must be either a flat plane, a round shrinking sphere, or a round shrinking cylinder with only one ℝ-factor (see again <cit.>). The plane and the sphere are excluded by (<ref>). In particular, we see that Φ (M^i_t̃_i)→λ(S^n-1). Thus, for i large enough Φ(M^i_t) is almost constant on the interval (-∞,t̃_i). Consequently, after passing to another subsequence we can pass to a limit ℳ̂^i→ℳ̂^∞ which must be a round shrinking cylinder with one ℝ-factor (see again <cit.>); this gives the desired contradiction. §.§ Conclusion of the argument Let ℳ={M_t}_t∈ [0,T) be a 2-convex flow with -controlled initial condition, and area bounded by 𝒜.Let t̅<T. We want to show that diam(M_t̅)≤ C, for some constant C=C(,𝒜)<∞.The following argument depends on various quality parameters for necks. The logical order for choosing these parameters is that one first fixes a small constant _0 (depending only on the dimension), and a large factor Q<∞ (e.g. Q=10), and then successively determines suitable quality parameters _1, , and _can (where _can≪≪_1 ≪_0) by reading the argument backwards.Let N⊂ M_t̅ by an -tube with H>H̅=Q H_can on N. By the reduction from Section <ref>, namely by Proposition <ref> and Proposition <ref>, it is enough to estimate the length of N.Let γ:[0,L]→ℝ^n+1 be an -approximate central curve for N (see Section <ref>), parametrized by arclength. We want to establish an upper bound for L, depending only onand 𝒜. Suppose L>2 (otherwise we are done), and consider the truncated curve γ|_[1,L-1] (which we still denote by γ) and the corresponding truncated -tube around it (which we still denote by N).Given any p∈γ, let x∈ N be a point on the central sphere of the -neck centered at p of radius r_p at time t̅. Note that |x-p|≤(1+)r_p. Also note that, since H(x)>H̅, forsmall enough we have r_p≤ 2(n-1) (Q H_can)^-1.Set t_p=t̅+12(n-1) r_p^2, and let τ:=116H_can^-2. We claim that for every p∈γ and every t∈ [t̅-τ,t̅], the flow ℳ has a strong _1-neck with center p and radius r(t)=√(2(n-1)(t_p-t)),at time t, providedand _can are small enough. Note that the axis is a priori allowed to change when going from scale to scale.Suppose towards a contradiction that t_0∈ [t̅-τ,t̅] is the largest time such that ℳ does not have a strong _1-neck with center p and radius r(t_0) at time t_0.Since every point in an -tube belongs to the final time slice of a very strong -neck, we immediately see that t_0<t̅. More precisely, given _1 let 𝒯= 𝒯(2_1,12_1,α,β) be the constant from Proposition <ref> (backwards stability). Since there is a very strong -neck centered at p of radius r(t̅) at time t̅, then for every t satisfying t-r(t)^2≥t̅-2𝒯 r(t̅)^2, there is a strong _1-neck centered at p of radius r(t) at time t. At time t=t_0 the inequality (<ref>) must be violated, in other wordst_p - t_0/t_p-t̅> 4(n-1)𝒯+1/2(n-1)+1≥3/2𝒯,where we tacitly assume that 𝒯 is large enough (depending only on the dimension). Since obviously t_p - t_0/t_p-t_0=1, by the intermediate value theorem we can find a t_1∈ (t_0,t̅] such that t_p- t_0/t_p-t_1=3/2𝒯. Since t_1∈ (t_0,t̅], the flow ℳ has an _1-neck centered at p of radiusr(t_1) at time t=t_1. Hence, there is a point x_t_1∈ M_t_1 on the central sphere of that neck, which satisfies in particularn-1(1+_1)r(t_1)<H(x_t_1)<(1+_1)(n-1)r(t_1),and|x_t_1-p| < (1+_1)r(t_1). Using (<ref>), (<ref>), and our choice of Q and τ we see thatr^2(t̅-τ)= r_p^2+2(n-1)τ≤(n-1)^2/4H_can^-2,which together with (<ref>) and the obvious inequality r(t_1)≤ r(t̅-τ), implies thatH(x_t_1)≥ H_can.Thus, by Theorem <ref> (canonical neighborhoods) and (<ref>) the flow ℳ̂ that is obtained from ℳ by shifting (p,t_1) to the origin and parabolically rescaling by H^-1(x_t_1) is _can-close in C^⌊ 1/ _can⌋ in B_1/_can-2n(0)× (-_can^2,0] to an (α,β)-solution 𝒩.For _can small enough, the (α,β)-solution 𝒩 has a 2_1-neck with radius (n-1) and center 0 at time 0. ByProposition <ref> (backwards stability) it follows that 𝒩 has a strong 12_1-neck centered at 0 of radius √(2(n-1)(s+(n-1)/2) ) at time -s, provided s≥ (n-1)(𝒯-1)/2. Putting things together, it follows that if _can≪_1 (depending only on ) then ℳ has a strong 34_1-neck centered at p of radius r(t_0)=√(2(n-1)(t_p-t_0)) at time t_0; a contradiction. This proves the claim.Now we can apply Proposition <ref> (small tilt) with _1=_1(_0,Λ) to conclude that for every p∈γ there exists O_p∈SO_n+1 such that for all t∈ [t̅-τ,t̅] we have that M_t is _0-close to the cylinder Z_p=p+O_p(S^n-1_√(2(n-1)(t_p-t))×ℝ), with fixed axis, in B__0^-1√(2(n-1)(t_p-t))(p).Finally, if p_1,p_2∈γ are points with |p_1-p_2|<14_0^-1√(τ), then the associated cylinders Z_p_1,Z_p_2 have substantial overlap at time t=t̅-τ, hence ||O_p_1 -O_p_2||=O(ε_0) and min{t_p_1/t_p_2, t_p_2/t_p_1}=1+O(ε_0). Thusd_γ(p_1,p_2)≤ (1+O(_0))|p_1-p_2|,i.e. the intrinsic distance along γ between any two points p_1,p_2∈γ with |p_1-p_2|<14_0^-1√(τ) is controlled by (1+O(_0)) times their extrinsic distance. Thus, the intersection of γ with any ball of radius 18_0^-1√(τ) is O(_0)-close to a linear segment. Since M_0 is contained inlarge ball of radius R=R(,𝒜)<∞, we conclude that the length of γ is bounded by some constant depending only onand 𝒜. This finishes the proof of Theorem <ref>.§ PROOF OF THEOREM <REF>In this final section, we will prove Theorem <ref>. We use the framework of mean curvature flow with surgery from <cit.>. Thus, for us a mean curvature flow with surgery is a (,δ,ℍ)-flow as defined in <cit.>. Recall in particular that =(α,β,γ) denotes the control parameters for the two-convex initial hypersurface M_0, that δ specifies the quality of the surgeries, and that the three curvature scales ℍ=(H_trig,H_neck,H_th) are used to specify more precisely when and how surgeries (and/or discarding) are performed.The main existence theorem <cit.> and the canonical neighborhood theorem <cit.>, tell us that for any -controlled initial hypersurface M_0, there exists an (,δ,ℍ)-flow starting at M_0 such that all points with H≥ H_can(, _can) possess canonical neighborhoods, provided δ is small enough, and H_th as well as the ratios H_trig/H_neck, H_neck/H_th are large enough (depending only on ).We make the following two minor adjustments compared to <cit.>. First, we work with `very strong' necks instead of `strong' necks. This is just a cosmetic change, that only slightly alters some constants. Second, and more importantly, in the line before <cit.> instead of selecting an arbitrary minimal collection of disjoint δ-necks that separates the thick part and the trigger part, we select an `innermost' collection of such δ-necks, i.e. we impose the additional condition that∑_p∈𝒥̂_jdist(p,{H=H_trig})is minimal. If the surgeries are performed this way, then on the discarded components we haveH ≥ c H_neckfor some c=c()>0.The argument from the previous sections (taking into account the one additional case that canonical neighborhoods can now also be modelled on the evolution of a standard cap preceeded by the evolution of a round shrinking cylinder) shows that every connected component M_t̅'⊂ M_t̅ satisfiesdiam(M_t̅',d_t̅)≤ C(,𝒜)<∞for all t̅≥ 0, where d_t̅ denotes the intrinsic distance on M'_t̅.Moreover, by the canonical neighborhood theorem, the nature of the surgeries, and the α-noncollapsing, the number of connected components is uniformly bounded by some N=N(,𝒜,H_thick)<∞.We want to show that for any t̅≥ 0 we have the estimate∫_M_t̅' H^n-1 dμ_t̅≤ C(,𝒜)<∞. To this end, write M'_t̅=M_t̅^low∪ M_t̅^high, with M_t̅^low={x∈ M'_t̅: H(x)≤H̅} and M_t̅^high={x∈ M'_t̅: H(x)> H̅}, where H̅≫ H_can is a large but fixed constant as in the previous section.Using the canonical neighborhood theorem, we can decompose M_t̅^high into the union of -tubes, caps, standard caps, and compact solutions of controlled geometry. It is easy to see that the latter three only contribute a controlled amount to the integral in (<ref>). Hence, let M^tubes_t̅⊂ M_t̅^high be the union of the remaining -tubes, with curvature larger than H̅. Note that at this point we allow the -tubes to have identified ends and combine tubes with their ends attached. We can then separate the connected components of M_t̅^tubes in two types: -tubes {T^low_i} with curvature less than 2H̅ and -tubes {T^high_i} that intersect the set {H>2H̅}. Since we obviously have∫_M_t̅∩{H≤ 2H̅} H^n-1 dμ_t̅≤(2H̅)^n-1𝒜,it remains to estimate∫_⋃_i T^high_i H^n-1 dμ_t̅. Now, we can write any -tube T^high as the union of a maximal collection of -necks N_i centered at p_i of radius r_i at time t=t̅, such that any two balls B_r_i/5(p_i), B_r_j/5(p_j) are disjoint. We can then estimate∫_T^high H^n-1dμ_t≤∑_i ∫_N_i H^n-1dμ_t ≤ c ∑_i r_i ≤ c L,where L is the length of the -tube N and c=c(n)<∞ is a numerical constant. Since the intrinsic diameter of each connected component is uniformly bounded, the integral (<ref>) is bounded by some constant C=C(,𝒜)<∞.To bound the integral (<ref>), we need to control the number of tubes in the union. Assume without essential loss of generality that M_t̅^low≠∅ (thecase M_t̅^low= ∅ can be handled easily). Note that each tube T^high has a point x with H(x)=32H̅. This implies that x belongs to an -neck that is entirely contained in T^high since its curvature is approximately 32H̅, soeach tube T^high_i contributes a definite amount of area. Since the total area of M_t̅ is bounded by 𝒜, this suffices to control the number of such tubes by a constant that depends only onand 𝒜, and concludes the proof of (<ref>). Now, in order to prove the regularity estimate for the level set flow, fix the initial hypersurface M_0, and consider a sequence (,δ,ℍ^j)-flows starting at M_0, with H_neck^j→∞, but H^j_thick bounded. By the above, we have the uniform estimate∫_M_t^j H^n-1dμ_t ≤ C(,𝒜). By the work of Head <cit.> and Lauer <cit.> (see also <cit.>; in particular, note that equation (<ref>) ensures that the curvature of the discarded components goes to infinity) for j→∞ the flows with surgery {M_t^j} converge to the level set flow {M_t} with initial condition M_0. The convergence is both in the space-time Hausdorff sense and also in the varifold sense for every time. By lower-semicontinuity of L^p-norms of the mean curvatureunder varifold convergence, we thus infer that the level set flow satisfies∫_M_t H^n-1dμ_t ≤ C(,𝒜) for every t≥ 0. Finally, by the local curvature estimate <cit.> we have r_ℳ^-1≤ C()H, and the assertion of Theorem <ref> follows.§ DISCRETE ŁOJASIEWICZ LEMMAFor every >0, there exists a δ=δ(,K,μ)>0 with the following significance. Suppose that f:{0,1,…, T}→ℝ is a non-increasing function such that for some K<∞ and μ<1 it holds that|f(t)|^1+μ≤ K( f(t-1) - f(t+1))for t= 1,… ,T-1, and suppose that |f|≤δ. Then∑_j=1^T(f(j) - f(j-1))^1/2≤. For most parts of the proof we only assume |f|≤ 1. We will impose the condition that |f|≤δ is actually small towards the end.Let t_0∈{0,1,…,T} be the smallest integer with the property f(j)< 0 for every t_0<j ≤ T. If t_0>0, then as in <cit.>, there is a C=C(K,μ)<∞ such thatf(t)≤ C t^-1/μ,for every t∈ [0,t_0]. Moreover, for p∈ (1,1/μ) and any j_0∈ [1,t_0], using the Cauchy–Schwarz inequality we obtain(∑_j=j_0^t_0 (f(j) - f(j+1))^1/2)^2≤(∑_j=j_0^t_0 (f(j) - f(j+1)) j^p) ∑_j=j_0^t_0 j^-p.Estimating the right-hand side of (<ref>) as in <cit.> we find b_1=b_1(δ,K,μ)<∞ such that if b_1≤ j_0≤ t_0 then∑_j=j_0^t_0 (f(j) - f(j+1))^1/2<ε/4.In case that t_0<T, we also consider the function f̃(t):=-f(T-t). This function f̃:[0, T-t_0]→ [0,∞) is non-increasing and satisfiesf̃(t)^1+μ=|f(T-t)|^1+μ ≤ K( f(T-t-1)- f(T-t+1) )=K( f̃(t-1) -f̃(t+1)),and |f̃|≤ 1. As above, we can find b_2=b_2(δ,K,μ)<∞ such that if b_2≤ j_0≤ T-t_0-2 then∑_j=t_0+1^T-j_0-1 (f(j) - f(j+1))^1/2=∑_k=j_0^T-t_0-2 (f̃(k) - f̃(k+1))^1/2<ε/4.Together with (<ref>) this implies that for b=max{b_1,b_2+1}∑_j=b^T-b (f(j)-f(j+1))^1/2<ε/2,tacitly assuming that T>2b (otherwise there is not much to prove, see the next sentence). Since only 2b-2 terms of (<ref>) missing from (<ref>), and each is bounded by (f(j)-f(j+1))^1/2<(2δ)^1/2, we conclude that (<ref>) holds, provided we choose δ small enough. amsplain 10andrews1 B. Andrews, Noncollapsing in mean-convex mean curvature flow, Geom. Topol. 16 (2012), no. 3, 1413–1418.BH_mod2con S. Brendle and G. Huisken, A fully nonlinear flow for two-convex hypersurfaces, arXiv:1507.04651 (2015).BH_surgery3d S. Brendle and G. Huisken, Mean curvature flow with surgery of mean convex surfaces in R^3, Invent. Math. 203 (2016), no. 2, 615–654.BHH R. Buzano, R. Haslhofer, and O. Hershkovits, The moduli space of two-convex embedded spheres, arXiv:1607.05604 (2016).CHN J. Cheeger, R. Haslhofer, and A. Naber, Quantitative stratification and the regularity of mean curvature flow, Geom. Funct. Anal. 23 (2013), no. 3, 828–847.CM T. Colding and W. Minicozzi, Uniqueness of blowups and Lojasiewicz inequalities, Ann. of Math. (2) 182 (2015), no. 1, 221–285.CM_AT1 , Differentiability of the arrival time, Comm. Pure Appl. Math. 69 (2016), no. 12, 2349–2363.CM_AT2 , Regularity of the level set flow, arXiv:1606.05185 (2016).CM_sing , The singular set of mean curvature flow with generic singularities, Invent. Math. 204 (2016), no. 2, 443–471.EE N. Edelen and M. Engelstein, Quantitative stratification for some free-boundary problems, arXiv:1702.04325 (2017).HK R. Haslhofer and B. Kleiner, Mean curvature flow of mean convex hypersurfaces, Comm. Pure Appl. Math. 70 (2017), no. 3, 511–546.HK_surgery , Mean curvature flow with surgery, Duke Math. J. 166 (2017), no. 9, 1591–1626.Head J. Head, On the mean curvature evolution of two-convex hypersurfaces, J. Differential Geom. 94 (2013), no. 2, 241–266.Huisken84 G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), no. 1, 237–266.Huisken_monotonicity , Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299.HuiskenPolden G. Huisken and A. Polden, Geometric evolution equations for hypersurfaces, Calculus of variations and geometric evolution problems (Cetraro, 1996), Lecture Notes in Math., vol. 1713, Springer, Berlin, 1999, pp. 45–84.HS_surg G. Huisken and C. Sinestrari, Mean curvature flow with surgeries of two-convex hypersurfaces, Invent. Math. 175 (2009), no. 1, 137–221.KL_singular B. Kleiner and J. Lott, Singular Ricci flows I, arXiv:1408.2271 (2014).Lauer J. Lauer, Convergence of mean curvature flows with surgery, Comm. Anal. Geom. 21 (2013), no. 2, 355–363.NV2 A. Naber and D. Valtorta, The Singular Structure and Regularity of Stationary and Minimizing Varifolds, arXiv:1505.03428 (2015).NV3 , Stratification for the singular set of approximate harmonic maps, arXiv:1611.03008 (2016).NV A. Naber and D. Valtorta, Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps, Ann. of Math. (2) 185 (2017), no. 1, 131–227.perelman_entropy G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159, 2002.Topping_diameter P. Topping, Relating diameter and mean curvature for submanifolds of Euclidean space, Comment. Math. Helv. 83 (2008), no. 3, 539–546.WangYM Y. Wang, Quantitative stratification of stationary Yang-Mills fields, arXiv:1610.00351 (2016).Department of Mathematics, University of Toronto,40 St George Street, Toronto, ON M5S 2E4, Canada E-mail: [email protected], [email protected] | http://arxiv.org/abs/1710.10347v1 | {
"authors": [
"Panagiotis Gianniotis",
"Robert Haslhofer"
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"published": "20171027220317",
"title": "Diameter and curvature control under mean curvature flow"
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Semiconducting DEOQ dynamics in presence of magnetic and charge noisesE. FerraroM. De MichielisCNR-IMM, Unit of Agrate Brianza, Via C. Olivetti 2, 20864 Agrate Brianza (MB), [email protected] M. Fanciulli CNR-IMM, Unit of Agrate Brianza, Via C. Olivetti 2, 20864 Agrate Brianza (MB), ItalyDipartimento di Scienza dei Materiali, University of Milano Bicocca, Via R. Cozzi 55, 20125 Milano, Italy Semiconducting double-dot exchange-only qubit dynamics in presence of magnetic and charge noises E. FerraroM. FanciulliM. De Michielis Received: date / Accepted: date ================================================================================================The effects of magnetic and charge noises on the dynamical evolution of the double-dot exchange-only qubit (DEOQ) is theoretically investigated. The DEOQ consisting of three electrons arranged in an electrostatically defined double quantum dot deserves special interest in quantum computation applications. Its advantages are in terms of fabrication, control and manipulation in view of implementation of fast single and two qubit operations through only electrical tuning. The presence of the environmental noise due to nuclear spins and charge traps, in addition to fluctuations in the applied magnetic field and charge fluctuations on the electrostatic gates adopted to confine the electrons, is taken into account including random magnetic field and random coupling terms in the Hamiltonian. The behavior of the return probability as a function of time for initial conditions of interest is presented. Moreover, through an envelope-fitting procedure on the return probabilities, coherence times are extracted when model parameters take values achievable experimentally in semiconducting devices.03.67.Lx, 73.21.La, 03.65.Yz § INTRODUCTIONConfinement of electron spins in solid state architectures represents a fruitful platform for universal quantum computation as witnessed by several experimental <cit.> and theoretical <cit.> proposals. The approaches developed range from quantum dot (QD) <cit.> to donor-atom nuclear or electron spins <cit.>. The reasons that make semiconductor nanostructures based qubits an attractive scenario for technological applications are due to their relatively long coherence times, the easy manipulation and fast gate operations. Thanks to the compatibility with the existing semiconductor electronics industry the DEOQ is directly scalable <cit.> . In the framework of QDs qubits several architectures have been proposed based on single <cit.>, double <cit.> and triple <cit.> QDs, implemented in III-V compounds such as GaAs <cit.>, and group IV element like Si <cit.> but also in InSb <cit.> nanostructures. With the aim to devise an architecture capable to assure the best compromise among fabrication, tunability, fast gate operations, manipulability and scalability, double-dot exchange-only qubit (DEOQ) has been proposed <cit.>, demonstrated <cit.> and constantly developed <cit.>. Exchange interactions between adjacent spins suffice for all one- and two-qubit operations <cit.>. The fidelity in the realization of quantum gates is deeply influenced by the unavoidable environmental noise mainly due to two different sources of disturbance that cause decoherence. One contribution to decoherence comes from the magnetic field noise due to the nuclear spins in the host material, in addition to fluctuations in the applied magnetic field needed to remove spin degeneracy of quantum states. We point out that DEOQ is protected against global magnetic fluctuations since it is a decoherence-free subspace qubit. This means that it is only affected by local fluctuations, such as the Overhauser field. The second major disturbance is represented by the charge noise that originates from charge fluctuations on nearby impurities that act as traps or on the electrostatic gates adopted to confine the electrons, that affects the exchange couplings between the spins. Magnetic noise is very sensitive to the host material under consideration. For this reasons it is of minor entity in Si thanks to the presence of stable isotopes with zero nuclear spins while it becomes considerable for example in GaAs compounds, where its effect could be considerable reduced through dynamical decoupling techniques and nuclear polarization. Nevertheless when Si is taken into account, charge traps and fluctuations in the magnetic field remain two issues to face.The aim of the present work is to develop a theoretical study on the effects of both magnetic and charge noises on the dynamics of the DEOQ. The main hypothesis is based on an alternative approach with respect to the usual quantum computing gate operations in which an accurate control over timing and duration of pulses is required in order to perform the desired one or two-qubit gate operations <cit.>. In our study we consider the natural evolution in time of the DEOQ as if the control parameters, that are the external gate voltages, are always turned on and two different sources of noise, namely the charge and magnetic noise, are included. In other words we are studying the "always on" configuration and this choice is taken in order to ease the comparison between our results and the experimental ones due to the simple (constant) inputs to be applied by the experimentalist. Note the clear difference with respect to the study presented in Ref. <cit.> where time-variant gate sequences are calculated to obtain rotations along x̂ and ẑ axes of the Bloch sphere. Having a clear and complete picture of how the different physical mechanisms that give birth to environmental noise on the system affect the dynamics and quantify them represents a fundamental step to progress in the development of a qubit technology.The paper is organized as follows. Section 2 is devoted to the presentation of the effective Hamiltonian model for the DEOQ and to the derivation of the closed analytical form for the unitary evolution operator that accounts for the dynamical evolution starting from an arbitrary initial condition. Sect. 3 contains the main results about the noise analysis in which two different initial conditions are considered and an estimation on the coherence times is given. Finally in Sect. 4, some concluding remarks are summarized.§ DOUBLE-DOT EXCHANGE-ONLY QUBIT MODELThis Section presents the effective Hamiltonian model that accounts for the quantum behaviour of the DEOQ in the regime of low energy excitations. The DEOQ arises from a new elaboration of the triple-dot exchange-only qubit proposed in Ref. <cit.> in which the architectures is based on the fabrication of two QDs instead of three. In particular three electrons are distributed during the operations between the two QDs, with at least one electron in each. In Fig. <ref> a schematic representation is showed.It represents a promising compromise between high speed and simple fabrication for solid state implementations of single qubit and two qubits quantum logic gates. The Schrieffer-Wolff effective Hamiltonian that describes in a simple and compact form the qubit by combining a Hubbard-like model with a projector operator method is derived in Ref. <cit.>. As a result, the Hubbard-like Hamiltonian is transformed into an equivalent expression that is the sum of contributions in which each term represents the exchange interaction between each pair of electrons involved, beyond the Zeeman term. The final closed form for the effective Hamiltonian in ħ units is given byH=1/2E^z(σ_1^z+σ_2^z+σ_3^z)+1/4j'σ_1·σ_2+1/4j_1σ_1·σ_3+1/4j_2σ_2·σ_3, where σ_i (i=1,2,3) are the Pauli operators referring to each electron and E^z=gμ_BB^z is the Zeeman energy associated to the magnetic field B lying in the ẑ direction with g the electron g-factor and μ_B the Bohr magneton. The exchange couplings j', j_1 and j_2, whose explicit expressions can be found in Ref. <cit.>, include the effects of dot tunneling, dot bias and both on-site and off-site Coulomb interactions. Each coupling term contains the ferromagnetic direct exchange between the two electrons from their Coulomb interactions and the anti-ferromagnetic superexchange. Consequently, in principle the value of each coupling can be either positive or negative and it depends strictly on the values of the parameters. However the superexchange term is usually larger than the direct one, leading to positive values for j_1 and j_2 <cit.>. In general, the coupling constants are tunable thanks to the control on the tunneling couplings which can be provided by external gates and on the inter-dot bias voltage. On the contrary, the Coulomb energy as well as the intra-dot bias voltage, directly linked to the exchange coupling j' <cit.>, are geometry dependent and they cannot be easily tuned. These approximations reflect the realistic conditions in which a QD is operated. The effective Hamiltonian is projected in the eigenspace spanned by the logical basis introduced in Ref. <cit.>. To encode the DEOQ we restrict to the two-dimensional subspace of three-spin states with spin quantum numbers S=1/2 and S_z=-1/2. S and S_z represent the total angular momentum state and its projection along ẑ respectively. We point out that only states with the same S and S_z can be coupled by spin independent terms in the Hamiltonian. The logical basis {|0⟩,|1⟩} is constituted by singlet and triplet states of a pair of electrons, for example the pair in the left dot, in combination with the single angular momentum state of the third spin, localized in the right dot, through appropriate Clebsh-Gordan coefficients. This means that the logical states chosen are finally expressed in this way|0⟩≡|S⟩|↓⟩,|1⟩≡√(1/3)|T_0⟩|↓⟩-√(2/3)|T_-⟩|↑⟩where |S⟩, |T_0⟩ and |T_-⟩ are respectively the singlet and triplet states|S⟩=|↑↓⟩-|↓↑⟩/√(2),|T_0⟩=|↑↓⟩+|↓↑⟩/√(2),|T_-⟩=|↓↓⟩and |↑⟩ and |↓⟩ denote a single state electron with spin-up and spin-down respectively. Explicit calculations of the matrix elements of the Hamiltonian in this basis give as final resultH= ([-E^z/2-3/4j'-√(3)/4(j_1-j_2);-√(3)/4(j_1-j_2) -E^z/2+1/4j'-1/2(j_1+j_2) ]).The effective Hamiltonian just derived represents the starting point to successfully analyze the dynamical behavior of the system.§.§ Analytical closed form of the evolution operatorThe evolution operator U(t)=e^-iHt associated to the Hamiltonian (<ref>) in the 2× 2 logical basis {|0⟩,|1⟩} can be recast in a compact analytical closed form exploiting the Cayley-Hamilton theorem. The matrix exponential is expressible as a polynomial of order n-1 as given by the following identitye^-iHt=s_0(t)I_2+s_1(t)(-iH),where I_2 is the 2× 2 identity matrix and s_0(t)=λ_1 e^λ_2 t-λ_2 e^λ_1 t/λ_1-λ_2, s_1(t)=e^λ_1 t-e^λ_2 t/λ_1-λ_2.The coefficients λ_1 and λ_2 are the eigenvalues of the Hamiltonian (<ref>) conveniently multiplied by the imaginary factor -i. Eq.(<ref>) after algebraic manipulations has been recast into the following form e^-iHt=e^λ_1tI_2+e^λ_1t-e^λ_2t/λ_1-λ_2(-iH-λ_1I_2).Equation (<ref>) constitutes a fundamental result exploitable for studying the dynamical behaviour of the DEOQ starting from an arbitrary initial condition and in the always on configuration.§ NOISE ANALYSIS AND COHERENCE TIMESIn this Section the dynamical behaviour of the DEOQ is investigated when different source of noises are included.Starting from the return probability of finding the qubit in a given logical state when the initial condition is imposed and the system is free to evolve in time, the disorder averaged return probability for two different initial conditions of interest considering both the magnetic disorder and the exchange one is calculated. DEOQ is protected against global magnetic fluctuations and, as it is evident in Eq.(<ref>), the Zeeman term only provides a global energy shift. For this reason the magnetic noise is included in our analysis by local magnetic field fluctuations acting between the two QDs. It is assumed that the disturbance obeys to a Gaussian distribution f_δ E(δ E) with zero mean and standard deviation √(2)σ_E. In an analogous manner the exchange couplings j_1 and j_2 follow a Gaussian distribution f_j_i(j_i), i=1,2, restricted to non-negative values with mean j_0i and standard deviation σ_j_i. The intra-dot exchange coupling j' as we have pointed out in Section 2 is fixed by the geometry of the system and not tunable from external gates.Given an arbitrary quantity P, that in the framework of our work is represented by the return probability, its disorder average is defined, following an analogous approach developed in <cit.>, by [P]_α=∫_0^+∞∫_0^+∞∫_-∞^+∞dj_1dj_2d(δ E)f_δ E(δ E)f_j_1(j_1)f_j_2(j_2)Pwheref_δ E(δ E)=1/2σ_E√(π)e^-(δ E)^2/4σ_E^2andf_j_i(j_i)=1/σ_j_i√(2π)2/1+erf(j_0i/σ_j_i√(2))e^-(j_i-j_0i)^2/2σ_j_i^2,with i=1,2. In the following the results are presented in correspondace to two initial conditions of interest when several values of σ_E and σ_j_1=σ_j_2≡σ_j are considered. §.§ Initial condition |ψ(0)⟩=|0⟩Let's initialize the qubit in the logical state |ψ(0)⟩=|0⟩ and follow the dynamical evolution looking at the return probability P_|0⟩(t) when subjected to the noise. After some algebraic manipulations that involve the evolution operator (<ref>) previously defined applied to the initial condition fixed, the probability P_|0⟩(t) is finally given byP_|0⟩(t)=1-4C^2/(A-B)^2+4C^2sin^2(β t),where A=E^z/2+3/4j',B=E^z/2-1/4j'+1/2(j_1+j_2),C=√(3)/4(j_1-j_2)andβ=√((A-B)^2+4C^2)/2.The multiple integral for [P_|0⟩(t)]_α obtained inserting Eq.(<ref>) into Eq.(<ref>) must be determined numerically. The parameters chosen for the following numerical calculation are given in function of j_0, that can be extracted doing specific calculations through a simulator based on spin density functional theory and mainly depends on the geometrical parameters and on the material of the physical qubit under investigation <cit.>. With this choice our results are completely general and easily exportable to whatever physical context once that j_0 is calculated. The parameters fixed in all the following figures are the mean values for the two Gaussian distributions followed by the exchange couplings j_1 and j_2 that are respectively j_01=0.5 j_0, j_02=1.5 j_0, the fixed intra-dot exchange coupling j'=0.5 j_0 and the external magnetic field E^z=10 j_0. Fig. <ref> reports the results when σ_E is varied in correspondence to different values of σ_j. It is shown that the return probability oscillates around and decays to a steady-state value. The decay rate as expected is set by the disorder strength quantified by the parameters σ_E and σ_j and increases when they increase. The steady-state value also experiences sensible modifications when magnetic disorder is included. Moreover, in panel (a), the dynamical behavior with zero noise, i.e. σ_E=σ_j=0 (black line), is reported. Those oscillations are not the common Larmor nor Rabi ones, due to the fact that the rotation is not performed along the ẑ axis nor an axis in the x̂-ŷ plane of Bloch sphere. Fig. <ref> reports the results when σ_j is varied in correspondence to two fixed values of σ_E. In both cases, charge noise is very effective in dampening the oscillations.In the following starting from the return probabilities just derived the intrinsic coherence time T_2^∗ is estimated. From an experimental point of view, a common way to define T_2^∗ is trough a Ramsey type experiment <cit.>. Such experiment requires a precise sequence starting with a π/2 pulse that bring the |0⟩ state to the superposition 1/√(2)(|0⟩+|1⟩), then the qubit is free to evolve for a time T and finally a second π/2 pulse is applied. In this way the coherence time T_2^∗ is extracted by fitting the decay of the oscillation. Our approach is instead based on the "always on" configuration, that means that the external parameters of control, i.e. gate voltages, are always turned on <cit.> during the qubit evolution. This definition is based on constant amplitudes of the control signals so it is independent on their detailed time behavior. In particular the figure of merit is represented by the dimensionless quantity j_0 T_2^∗. It appreciates the number of coherent oscillations shown by the return probability before it decays. In order to proceed with this analysis, the numerical results just presented are exploited. The main idea is to extract the values of the coherence times from the return probabilities through an envelope-fitting procedure. Firstly the envelope of the return probability is derived, then we look for a curve of the formP(t)=P(0)+(1-P(0))e^-(t/T_2^∗)^αthat closely approximates the envelope function previously derived.Fig. <ref> shows some explicative examples of the decaying [P_|0⟩(t)]_α curves in correspondence to different noise parameters and highlights the good fit of Eq.(<ref>) (red lines) to the upper envelope. The fitting parameters obtained give an estimation on the dimensionless coherence times as reported in the left part of Fig. <ref> for different values of σ_j.The clear trend of j_0T_2^∗ shows that it decreases increasing either type of disorder. The magnetic noise affects lesser the oscillatory behaviour of the return probability, that on the contrary is deeply influenced by the entity of the charge noise. A function that gives an immediate measure of the coherence of the qubit in an intuitive and clear way is the quality factorQ=exp(-1/j_0 T_2^∗),that represents the exponential decay factor for the return probability over 1/j_0. In the right part of Fig. <ref> a two-dimensional map in which the dependence of Q on the charge noise σ_j/j_0 and on the magnetic noise σ_E/j_0 is reported. When both the sources of noise are suppressed, i.e. σ_j=σ_E=0, the maximum of Q corresponding to the value of 1 is found. As it can be seen, Q factor is strongly reduced by charge noise whereas is less sensitive to magnetic noise.The results obtained in normalized units are very general and mainly depends on j_0. This parameter strongly depends on the geometry and on the nature of the material composing the qubit and in order to convert the normalized units in physical ones it is necessary to explicate it. We consider a DEOQ achieving a reasonable value of j_0=1 μ eV <cit.> and including this value in our analysis it is possible to extract a range of physical coherence times for the DEOQ in presence of environmental noise that goes from units to tens of j_0T_2^∗ that corresponds to tens up to hundreds of ns. In Fig. <ref> a 2D map in logarithmic scale of the resulting coherence times is presented when realistic physical units are considered for Si and GaAs. In this figure, vertical lines highlighting the typical values σ_E=3 neV and 100 neV for Si and GaAs respectively <cit.> are added to mark the minimum σ_E due to background nuclear spins in each material. §.§ Initial condition |ψ(0)⟩=1/√(2)(|0⟩+|1⟩)In an analogous manner this subsection reports the results obtained when the qubit is prepared in an initial condition that is a superposition of the two logical states |ψ(0)⟩=1/√(2)(|0⟩+|1⟩), for which the analytical expression for the return probability of finding the DEOQ in the logical state |0⟩ is given byP_|0⟩(t)=1/2[1+4C(A-B)/(A-B)^2+4C^2sin^2(β t)],where the parameters involved A,B,C and β are defined in Eqs.(<ref>) and (<ref>).Once again the expected oscillatory behaviour is reproduced with oscillations gradually reduced when noise increases. We notice that the steady state reached by the qubit, that is larger in correspondence to smaller values of σ_E (Fig. <ref>), is more susceptible to the magnetic disorder with respect to the previous case in which the initial condition is the |0⟩ logical state. As done in Fig.<ref>(a), Fig. <ref>(a) displays also the dynamical evolution when zero noise is considered (black line).Fig. <ref> reports the results on the coherence time when σ_j is varied in correspondence to two fixed values of σ_E.Figs. <ref> and <ref> present the effect of charge and magnetic noises on j_0T_2^∗ (left) and on Q factor (right). As it can be seen, charge and magnetic noises affect roughly in the same way the coherence time. In addition, it is worth noting that in Figs. <ref> and <ref> a more detrimental effect of the noise on the coherence time is observed with respect results reported in Figs. <ref> and <ref>. While both types of noise suppress T_2^∗, charge noise is more effective at producing decoherence in the system than magnetic noise. The results presented cover wide parameter regimes for both magnetic and charge noise strengths because the two noise sources have not comparable magnitudes in semiconductors. Our results imply that a given amount of charge noise overall has a more detrimental effect than an equal amount of magnetic noise. However, magnetic noise is usually much stronger than charge noise in GaAs, and so this is a problem in this material. Moreover such noise can not be reduced by isotopic purification in GaAs due to the fact that both Ga and As have no stable isotopes with zero nuclear spins. Isotopic purification, on the converse, is exploitable in Si since it can reduce the presence of magnetic isotopes of Si. Another strategy to reduce the effects of magnetic noise that works for all the considered materials is simply to increase the average exchange coupling j_0. In fact, the disorder averaged probability depends only on σ_E/j_0 and σ_j/j_0, where σ_E remains constant when j_0 changes and σ_j is linear in j_0 <cit.>.§.§ Comparing coherence times in semiconducting DEOQFig. <ref> collects the most significant results obtained in the previous sections for two initial conditions of interest, the pure logical state |0⟩ (filled marks) and the linear superposition of the two states of the logical basis 1/√(2)(|0⟩+|1⟩) (open marks). Addressing three specific values for σ_E we report the dependence of T_2^∗ in function of σ_jwhen j_0=1 μeV. The values addressed for the minimum magnetic noise due to nuclear spins in the host materials are σ_E=0 eV, 3 neV and 100 neV, that correspondto the nuclear free isotope ^28Si, the Si and the GaAs, respectively.For each host material the simulated coherence time of DEOQ decreases when σ_E increases, compressing all curves to similar values when σ_j is greater than 0.1 μeV. When σ_j is lower than 0.05 μeV, the coherence time of DEOQ in GaAs is evidently lower than those of Si and ^28Si whereas ^28Si shows a T_2^∗ - in the μs range - higher than natural Sionly for a smaller σ_j= 0.003 μeV. As a result, an implementation of a DEOQ in ^28Si should be preferred to Si only if sufficiently small values of charge noise can be reached. All those considerations holds for both the initial conditions under study. If the same level of charge noise is assumed, Si, as well as ^28Si, outperforms GaAs in terms of noise resilience but there are still challenges associated with Si as a material platform. In fact, Si has charge carriers with an higher effective mass than GaAs which require the fabrication of smaller dots to confine single electrons and, moreover, the valley degeneracy of energy levels which impedes the addressing of two nondegenerate levels needed to define the qubit base states.§ CONCLUSIONSThe DEOQ dynamical evolution in the "always on" configuration is derived. The purpose of investigating and quantifying how the dynamics is affected by external environmental noise is pursued in view of interesting and fruitful applications that the DEOQ will have in quantum computation. The disorder-average probabilities are calculated starting from two different initial conditions of interest adopting a closed and compact analytical form for the evolution operator that we have derived. The oscillatory dynamical behavior in presence of both magnetic and charge noises is shown and it is demonstrated that the oscillation decay strongly depends on the entity of the disturbance on the qubit. The number of coherent oscillations shown by the return probability before it decays is quantified adopting an envelope-fitting procedure on the disorder-average probabilities through which an estimation of the intrinsic coherence times is obtained. Our results demonstrate that DEOQ is more sensitive to charge noise than magnetic noise and that DEOQ implemented in Si outperforms GaAs qubit in terms of noise resilience. Moreover, only if charge noise is sufficiently small an implementation of DEOQ in ^28Si provides an effective increase of coherence time with respect to Si.§ ACKNOWLEDGMENTSThis work has been funded from the European Union's Horizon 2020 research and innovation programme under grant agreement No 688539.spphys | http://arxiv.org/abs/1710.10032v2 | {
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"published": "20171027085310",
"title": "Semiconducting double-dot exchange-only qubit dynamics in presence of magnetic and charge noises"
} |
Laboratorio de Óptica Cuántica, Universidad de los Andes, A.A. 4976, Bogotá, D.C., [email protected]@uniandes.edu.co We present the experimental implementation and theoretical model of a controllable dephasing quantum channel using photonic systems. The channel is implemented by coupling the polarization and the spatial distribution of light, that play, in the perspective of Open Quantum Systems, the role of quantum system and environment, respectively. The capability of controlling our channel allows us to visualize its effects in a quantum system. Differently from standard dephasing channels, our channel presents an exotic behavior in the sense that the evolution of a state, from a pure to a mixed state, shows an oscillatory behavior if tracked in the Bloch sphere. Additionally, we report the evolution of the purity and perform a quantum process tomography to obtain the χ matrix associated to our channel. (200.0200) Optics in computing; (230.5440) Polarization-selective devices; (260.5430) Polarization; (270.2500) Fluctuations, relaxations, and noise; (270.5565) Quantum communications . osajnl§ INTRODUCTION Many practical implementations of quantum-based applications must deal with the interaction of a quantum system and its environment, i.e. open quantum systems, Fig. <ref>(a). This interaction can be described, in general, by a non-unitary operation associated with a quantum channel and it is responsible for the appearance of decoherence in the system. In the field of quantum information, for example, decoherence changes or destroys the information encoded in a quantum system, producing an incorrect transmission, processing or storage of it <cit.>.Quantum channels can be classified depending on their effect in different input states. In the literature <cit.>, it is possible to find some specific examples: A depolarizing channel is one in which all pure states evolve towards the maximally mixed state. In an amplitude channel, all the pure states evolve towards a particular pure state that remains unaffected. A dephasing channel leaves two states, which are the opposite poles of the Bloch sphere, as decoherence-free states, while the rest of the pure states suffer decoherence.Controllable channels make possible to visualize the effects of decoherence in a quantum system and to study the prevention of the appearance of such decoherence <cit.>. Different types of channels that can be controlled have been performed using photonic systems. For example, depolarizing channels have been built using an arrangement of birefringent crystals <cit.> and via interferometric effects <cit.>. Amplitude channels have been performed in Ref. <cit.>, and dephasing channels have been implemented using birefringent materials <cit.>, a spatial light modulator (SLM) <cit.> or interferometric effects <cit.>.In this paper, we report the experimental implementation and theoretical model of a controllable dephasing channel. Unlike a standard quantum dephasing channel, our device induces decoherence in which a pure state does not evolve linearly towards a mixed state, when seen in the Bloch sphere, but follows a spiral behavior. In our experiment, the polarization degree of freedom of light acts as the quantum system, and the continuous variable, corresponding to the spatial degree of freedom of light, represents the environment. The characterization of our channel is done by using the Bloch sphere representation and by measuring the purity of the output state. Additionally, using quantum process tomography <cit.> (QPT) with a maximum-likelihood-estimation (MLE) algorithm <cit.>, we report the χ matrix associated to our quantum channel.§ THEORETICAL BACKGROUND In order to understand the working principle of our quantum channel, we start by presenting the theoretical model that describes its behavior. For this, consider a light beam that enters into the channel. Its polarization state can be represented by:|φ⟩=α|H⟩ + β|V⟩,where α and β are probability amplitudes, satisfying |α|^2 + |β|^2=1, and |H⟩ and |V⟩ are the horizontal and vertical polarization states, respectively. The channel couples such polarization state to the transverse momentum of light, whose state is given by|ξ⟩= ∫ dq⃗ f(q⃗) |q⃗⟩,where q⃗={q_x,q_y} denotes the transverse momentum of light and the function f(q⃗) indicates the transverse momentum distribution.As it has been mentioned, in our implementation the polarization and the transverse momentum of light represent the quantum system and the environment, respectively. The initial state, |Ψ⟩_in , that describes the beam entering into the quantum channel can be written as|Ψ⟩_in=|φ⟩⊗|ξ⟩ = α∫ dq⃗ f(q⃗) |H,q⃗⟩ + β∫ dq⃗ f(q⃗) |V, q⃗⟩. The channel is implemented by employing a polarizing tunable beam displacer (P-TBD) <cit.>, a device that splits an incoming beam into two parallel propagating beams with orthogonal polarizations as depicted in Fig. <ref>(b). The separation between the beams is restricted to they-direction and can be quantified by a tunable parameter, 2d_c, that represents the distance between the centroids of the two output beams. Therefore, the channel can be represented by a unitary operation, Û(d_c), that accomplishes the following transformations:Û(d_c)|H,q_y⟩ =e^i d_c q_y|H,q_y⟩ Û(d_c)|V,q_y⟩ =e^-i( d_c q_y+φ)|V,q_y⟩,with φ a generic phase difference that appears between the two output beams.The density matrix of the output polarization state, ρ̂_out^pol, can be written as ρ̂_out^pol=Tr_env{Û(d_c)|Ψ⟩_in⟨Ψ|_inÛ^†(d_c) },where Tr_env{∙} denotes the partial trace taken over the environment. Since our channel acts only in the y-direction, and q_x and q_y are independent, it is possible to write f(q⃗) ∝ f(q_y). In particular, for an input gaussian beam,f(q_y) = Ne^- w_y^2 (q_y-q_0y)^2/4,where w_y is the beam's waist,q_0y is the center of its transverse momentum distribution, and N is a normalization factor, such that ∫ |f(q_y)|^2 dq_y =1.Plugging Eq. (<ref>) into Eq. (<ref>) and usingEq. (<ref>), it is possible to writeρ̂_out^pol =[ |α|^2αβ^* e^-2d_c^2/w_y^2e^i (2d_cq_0y+φ); α^*β e^-2d_c^2/w_y^2e^-i (2d_cq_0y+φ) |β|^2; ].From this density matrix, it is clearly seen that the implemented channel changes the initial polarization state of the beam depending on the value of the parameter d_c. The off-diagonal terms contain the expression e^-2d_c^2/w_y^2e^i (2d_cq_0y+φ) or its conjugate. The term e^-2d_c^2/w_y^2 is responsible for the decay of a pure state into a mixed state, revealing the decoherence of the system induced by a standard dephasing channel <cit.>. However, the presence of e^i (2d_cq_0y+φ) indicates that our channel does not behave as a standard dephasing.The characterization of the channel is performed byobtaining the Stokes parameters, {S_0, S_1, S_2, S_3}, associated with a polarization state that passes through it. These parameters are represented in the Bloch sphere, and the corresponding purity can also be obtained. The Stokes parameters can be extracted from the polarization density matrix <cit.> given in Eq. <ref>,S_0= |α|^2 + |β|^2, S_1=|α|^2 - |β|^2, S_2(d_c)=2|α| | β^* |e^-2 d_c^2/ w_y^2cos(2q_0y d_c +φ+Φ), S_3(d_c)= 2|α| | β^* |e^-2 d_c^2/ w_y^2sin(2q_0y d_c +φ+Φ),where Φ=Φ_α - Φ_β with α=|α|e^iΦ_α and β=|β|e^iΦ_β.Figure <ref> shows the effect of our channel on various input pure polarization states when φ = π and they are coupled to an environmentcharacterized by a Gaussian distribution with w_y=0.88 mm and q_0y=10.6 mm^-1. The values of φ, q_0y and w_y are chosen to match the experimental parameters as will be described in sections (<ref>)-(<ref>). In Fig. <ref>(a), the dark regions represent the output polarization states in the Bloch sphere. Each polarization state is defined by a set of coordinates given by the Stokes parameters {S_1,S_2(d_c),S_3(d_c)}. The depicted spheres correspond to different settings of the parameter d_c, specifically, d_c=0, d_c=w_y/3, d_c=2w_y/3 and d_c=w_y. For the case d_c=0, only a dark sphere is observed, since the channel only induces a rotation maintaining the output state pure and therefore on the surface of the Bloch sphere. In contrast, when d_c increases, the dark region indicates that the poles of the sphere remain unchanged while the rest of the sphere shrinks towards the vertical axis, S_1. This behavior indicates that horizontal and vertical polarizations are decoherence free-states while any other input pure state becomes mixed. This particular way in which the Bloch sphere is shrinking is a signature of the fact that the proposed coupling corresponds to a dephasing channel.Another way to identify the type of channel is by means of the evolution of the purity of the output polarization state, P_out. Using the Stokes parameters, P_out can be written as P_out=Tr{(ρ̂_out^pol)^2}=(S_0^2+S_1^2+S_2^2+S_3^2)/2. Using Eq. (<ref>), Eq. (<ref>), Eq. (<ref>)andEq. (<ref>) the purity becomesP_out(d_c)=1/2(1+(|α|^2 - |β|^2)^2 + 4 |αβ^* |^2 e^-4 d_c^2/w_y^2). The behaviour of Eq. (<ref>) is depicted in Fig. <ref>(b) for five different initial polarization states defined in Table <ref>. The wide solid line corresponds to horizontal and vertical polarizations, the dashed line refers to the polarization states defined by {α=0.85-0.14i, β= -0.35 (1- i)} and {α=0.35(1+i), β=0.14-0.85i}, and the thin solid line corresponds to the purity of left circular polarization {α=1/√(2), β=-i/√(2)}. From these graphs, it is possible to see that if the input state has either horizontal or vertical polarization, the purity is one. This means that the channel does not affect the system and therefore there is no decoherence. On the other hand, for the other pure input states, the purity starts at one and monotonically decreases when d_c increases reaching a value P_out(d_c) ≈1/2(S_1^2+1) ford_c≫ w_y. According to the value of S_1 reported in Table <ref>, it is possible to see that for a state evolving in the equator the purity tends to 0.5, while for a state in the Tropic of Cancer or Tropic of Capricorn, the purity tends to 0.625.Figure <ref>(c) shows the evolution of an input left circular polarization state (located in the equator plane) with the parameter d_c. It is seen that, as d_c increases, the state tracks a spiral on the equatorial plane of the Bloch sphere. This spiral starts in the surface, for d_c=0, and ends up in the center. This type of evolution is not typical for a dephasing channel and gives an exotic characteristic to the implementation we report. The spiral evolution is due to the presence of oscillatory terms in the off-diagonal elements of ρ̂_out^pol (Eq. <ref>). These terms are caused by the fact that our environment corresponds to a Gaussiandistributioncentered at q_0y. Physically, q_0y is related to a small deviation of the incoming beam with respect to the z-direction <cit.>, given by an angle θ=q_0yλ/2 π with λ the wavelength of the light beam. A standard dephasing channel can be recovered by setting q_0y=0.There are two interesting ideas to notice from Fig. <ref>(c): first, at d_c=0 the output state has right circular polarization, even though the input state was initially prepared as left circular. This occurs because of the presence of the phase φ=π, that comes from the experimental implementation of the channel. Second, although Fig. <ref>(c) is for an input left circular polarization state, any other input pure state will follow the spiral behavior in a plane parallel to the equator of the Bloch sphere.§ EXPERIMENTAL IMPLEMENTATION Figure <ref> shows the setup for the implementation and characterization of our channel. Four steps can be clearly recognized. In the first step, an 808 nm-CWlaser (Thorlabs, CPS808) is coupled into a single mode fiber to obtain a Gaussian beam with a waist around w_y=0.88 mm and that can be considered collimated during the whole path of the experiment. In the second step, a polarizer is set to fix a vertical polarization; a quarter wave plate (QWP) and a half wave plate (HWP) are placed to obtain different polarizations that defined the input state by setting the values α and β in Eq. (<ref>). The third step in our setup is the channel implemented by employing a polarizing tunable beam displacer (P-TBD) <cit.>. It consists of a polarizing beam splitter (PBS) and two mirrors, M2 and M3, placed on an L-shaped platform that is mounted on a rotational stage. By rotating this platform, the separation 2d_c between the two emerging beams can be tuned as illustrated in Fig. <ref>(b). It is relevant to notice that due to the working principle of the P-TBD, the two beams that come out from it suffer a different amount of reflections in the PBS, M2 and M3. This fact justifies the introduction of φ in the theoretical model. The different amount of reflections results precisely in φ=π, explaining our choice of φ for the graphs of the theoretical model in Figure <ref>. In the fourth step, a polarization tomography analysis is implemented for different values of the separation d_c. This process is performed by sending the light through a HWP, a QWP and a PBS. The light transmitted by the PBS is focused, with a lens (f=25.4 mm), into a photodiode (Thorlabs FDS100) while the light coming from the reflecting output is neglected. The Stokes parameters are then recovered by performing intensity measurements in the photodiode for different settings of the HWP and QWP.§ RESULTS AND DISCUSSION The characterization of our channel is done by measuring how the Bloch sphere shrinks for different values of the parameter d_c when various input states are considered. The measurement of the Stokes parameters is done for the five different input polarization states defined in Table <ref>. Figure <ref> shows as dots the experimental data corresponding to {S_1,S_2(d_c), S_3(d_c)} for the values d_c=0, d_c=w_y/3, d_c=2w_y/3 and d_c=w_y. The dark region is the same as the one reported in the theoretical section, Fig.<ref>(a), and it is clearly seen that the experimental data are contained on it. From the way in which the shrinking of the Bloch sphere occurs according to Fig. <ref>, it is possible to conclude that indeed the channel presented in this work induces dephasing.The experimental results for the evolution of the purity, P_out(d_c), are shown inFig. <ref> for the five states of Table <ref>. The dots are experimental data corresponding to {S_1,S_2(d_c), S_3(d_c)} when d_c is scanned in the range [0, 1.42] mm in steps of 7.2 μm. The solid lines correspond to the theoretical modelaccording to Eq. (<ref>). As expected, the purity for the horizontal and vertical input states remain unchanged; for an input state in the Tropic of Cancer or the Tropic of Capricorn, P_out(d_c) tends to 0.625 while for left circular polarization,P_out(d_c) tends to 0.5. The deviation between the experimental data and the theoretical model is due totechnical features associated with the optical elements and the uncertainty on the rotation angle of the L-shaped platform of the P-TBD. The exotic behavior of our channel, in which a state suffers decoherence by following a spiral in the Bloch sphere, is corroborated in our experiment by tracking the evolution of a left circular input polarization state when the parameter d_c is varied. Figure <ref>(a) depicts this evolution in the Bloch sphere and its projection on the equatorial plane. The values of d_c are the same used in the measurement of the purity. The dots are experimental data, corresponding to the measurement of {S_1,S_2(d_c), S_3(d_c)},and the solid line is the theoretical model according to Eq. (<ref>), Eq. (<ref>) and Eq. (<ref>) using φ=π and q_0yas a fitting parameter with a value of q_0y^fit=10.6 mm^-1. The star indicates the output state that is measured at d_c=0 and the big arrow indicates the theoretical output state at d_c=0. The discrepancy between these two points is due to the same experimental uncertainty that was mentioned when discussing the experimental data for the purity in Fig. <ref>.As complementary measurements, the evolution of the other four states ofTable <ref> was also tracked. The data and theoretical model for input states in the Tropic of Cancer and Tropic of Capricorn are shown inFigs. <ref>(b)-<ref>(c) as dots and solid lines, respectively. From these graphs, it is clearly seen that the pure input states enter inside the Bloch sphere following a spiral parallel to the equator with a latitude defined by S_1. The lower plots are the projection of the corresponding planes. Figs. <ref>(d)-<ref>(e) reveal that, as expected for a dephasing channel, the horizontal and vertical input states remain in the poles. In order to complete the characterization of the described channel, we implemented a quantum process tomography method <cit.> to obtain its associated χ matrix. The complete positive map related to our channel is given by ρ̂_out^pol=∑_i,jχ_ijσ̂_i ρ̂_in^polσ̂^†_j, where ρ̂_in^pol= |φ⟩⟨φ|, with |φ⟩ defined in Eq. (<ref>), andσ̂_i are the matrices that span the space of ρ̂_in^pol that correspond, in this case, to the identity (σ̂_0=Î_2×2) and the Pauli matrices (σ̂_X, σ̂_Y and σ̂_Z). Figure <ref> shows thereal part of the experimental and theoretical χ matrices, χ_Exp and χ_The, for d_c=0 and d_c = 1.44 mm, corresponding to the initial and final values of d_c used in our experiment. To obtain χ_Exp, Eq. (<ref>) was solved, usingMLE with the experimental data. For χ_The, MLE was used with ρ̂_out^pol given by Eq. (<ref>) with α and β shown in Table <ref>. From Fig. <ref>(a), it is clearly observed that our channel does not induce decoherence when d_c=0 but generates only a rotation. This rotation is precisely the one that mentioned before as coming fromthe working principle of the channel (P-TBD). For d_c = 1.44 mm, Fig. <ref>(b) shows that the diagonal of the χ matrix has contributions from the identity and σ̂_Z. This fact can be related with the shrinking of the Bloch sphere depicted in Fig. <ref>(a), indicating that our channel indeed corresponds to a dephasing one.§ CONCLUSIONS In this paper, we have presentedthe theoretical model and the experimental implementation of a controllable dephasing channel. The channel is implemented using photonics systems and considering polarization as a quantum system and the environment simulated by the transverse momentum distribution of light. The implemented channel has been performed with the help of a Polarizing Tunable Beam Displacer, which permits the control of the decoherence that the polarization system suffers as a function of the tunable parameter, d_c. In order to identify the type of quantum channel that our setup describes, we have observed its effects in the polarization by means of the Bloch sphere representation, we have measured the evolution of the purity and we have performed a QPT protocol to identify the χ matrix. When the channel's parameter, d_c, is tuned, the Bloch sphere and the purity evolve in a manner that corresponds to a standard dephasing channel. Interestingly, the implemented channel exhibits also an exotic characteristic in which the evolution toward mixed states is done by following a spiral path. This deviation from the standard is due to the characteristics of the environment that are considered for this implementation. The results reported in this paper can be useful for decoherence suppression implemented via quantum error correction protocols in quantum information applications. § ACKNOWLEDGMENTS The author's acknowledge support from Facultad de Ciencias, Universidad de Los Andes. AV and MNP acknowledge support from FAPA project from Universidad de los Andes, Bogot, Colombia. | http://arxiv.org/abs/1710.09848v1 | {
"authors": [
"Daniel F. Urrego",
"Juan-Rafael Álvarez",
"Omar Calderón-Losada",
"Jiří Svozilík",
"Mayerlin Nuñez",
"Alejandra Valencia"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20171026180145",
"title": "Implementation and characterization of a controllable dephasing channel based on coupling polarization and spatial degrees of freedom of light"
} |
Current address: Department of Physics, Duke University,Durham, NC 27708 Department of Physics, University of Notre Dame, Notre Dame, IN 46556Department of Physics, University of Notre Dame, Notre Dame, IN 46556Department of Physics, University of Notre Dame, Notre Dame, IN 46556Department of Physics, University of Notre Dame, Notre Dame, IN 46556Current address: Department of Physics and Astronomy, Texas A & M University,College Station, TX 77843 Department of Physics and Astronomy, Northwestern University,Evanston, IL 60208Current address: Department of Energy and Hydrocarbon Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan Department of Physics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, JapanCurrent address: Department of Physics, Fudan University, Shanghai, China 200433 Department of Physics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, JapanDepartment of Physics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, JapanInstitut Laue-Langevin, 6 Rue Jules Horowitz, F-38042 Grenoble, FranceLaboratory for Neutron Scattering, Paul Scherrer Institute, CH-5232 Villigen, SwitzerlandCorresponding author: [email protected] Department of Physics, University of Notre Dame, Notre Dame, IN 46556Despite numerous studies the exact nature of the order parameter in superconductingremains unresolved. We have extended previous small-angle neutron scattering studies of the vortex lattice in this material to a wider field range, higher temperatures, and with the field applied close to both the ⟨ 100 ⟩ and ⟨ 110 ⟩ basal plane directions. Measurements at high field were made possible by the use of both spin polarization and analysis to improve the signal-to-noise ratio. Rotating the field towards the basal plane causes a distortion of the square vortex lattice observed forH∥⟨ 001 ⟩, and also a symmetry change to a distorted triangular symmetry for fields close to ⟨ 100 ⟩. The vortex lattice distortion allowsus to determine the intrinsic superconducting anisotropy between the c axis and the Ru-O basal plane, yielding a value of ∼ 60 at low temperature and low to intermediate fields. This greatly exceeds the upper critical field anisotropy of ∼ 20 at low temperature, reminiscent of Pauli limiting. Indirect evidence for Pauli paramagnetic effects on the unpaired quasiparticles in the vortex cores are observed, but a direct detection lies below the measurement sensitivity. The superconducting anisotropy is found to be independent of temperature but increases for fields ≳ 1 T, indicating multiband superconductvity in . Finally, the temperature dependence of the scattered intensity provides further support for gap nodes or deep minima in the superconducting gap. 74.70.Pq, 74.20.Rp, 74.25.Uv, 61.05.fgAnisotropy and multiband superconductivity inM. R. Eskildsen December 30, 2023 ==============================================§ INTRODUCTION The superconducting state emerges due to the formation and condensation of Cooper pairs, although the exact microscopic mechanism responsible for the pairing in different materials varies and in many cases remains elusive. In thecase of strontium ruthenate, multiple experimental and theoretical studies provide compellingevidence for triplet pairing of carriers (electrons and/or holes) and an odd-parity, p-wave order parameter,<cit.> which would classifyas an intrinsic topological superconductor.<cit.>This is supported by μSR,<cit.> Josephson junction,<cit.> and polar Kerr angle<cit.> measurements which show spontaneous broken time reversal symmetry below the critical temperature (). Additionally, Knight shift<cit.> and SQUID junction<cit.> measurements of the susceptibility indicate triplet pairing. At the same time, seemingly contradictory or inconclusive experimental results have left important open questions concerning the detailed structure and coupling of the orbital and spin parts of the order parameter.<cit.> Although studies ofunder strain do show a substantial increase of the critical temperature, the expected cusp at zero strain is not observed.<cit.> Low energy excitations indicate the existence of vertical line nodes in the superconducting gap, inconsistent with a p-wave order parameter.<cit.> Also, the first order nature of the upper critical field () at low temperature is suggestive of Pauli limiting,<cit.> and has been interpreted as evidence against equal spin pairing required for p-wave superconductivity.<cit.> Alternatively, it was suggested that the simple classification of either spin-singlet or spin-triplet pairing is not appropriate, due to strong spin-orbit coupling in .<cit.> Furthermore, recent work has attributed the suppression ofto so-called interorbital effects rather than Pauli limiting.<cit.>Superconducting vortices, introduced by an applied magnetic field, may serve as a sensitive probe of the superconducting state in the host material. Small-angle neutron scattering (SANS) studies of the vortex lattice (VL) have proved to be a valuable technique, often providing unique information about the superconducting order parameter including gap nodes and their dispersion,<cit.> multiband superconductivity,<cit.> Pauli paramagnetic effects,<cit.> and a direct measure of the intrinsic superconducting anisotropy ().<cit.> The latter quantity may be directly measured by the field-angle-dependent distortion of the VL structure from a regular triangular symmetry. In London theory, represents the anisotropy of the penetration depth.<cit.> In Ginzburg-Landau theory it also represents the anisotropy of the coherence length, which can arise from both superconducting-gap and Fermi-velocity anisotropy. A determination of is particularly relevantin materials where the upper critical field is Pauli limited along one or more crystalline directions, since theanisotropy may differ from the intrinsic superconducting anisotropy.Here we report SANS studies of the VL inwith magnetic fields close to the basal plane in order to investigate the superconducting anisotropy as well as possible effects of Pauli paramagnetism. In earlier work we found ≈ 60 at intermediate fields and low temperature (50 mK). This significantly exceeds the low-temperature upper critical field anisotropy = ^⊥ c/^∥ c≈ 20.<cit.> The Fermi surface inconsists of three largely two-dimensional sheets with Fermi velocity anisotropies ranging from 57 to 174,<cit.> andone would expect an upper critical field () anisotropy within this range.<cit.> The present work substantially extends the field range of our previous report, and also includes temperature dependent measurements. The temperature dependent intensity is consistent with gap nodes or deep minima in the order parameter. No temperature dependence ofis observed, but a field driven increase above 1 T indicates multiband superconductivity. While the discrepancy betweenandindicates Pauli limiting, no direct evidence for Pauli paramagnetic effects on the unpaired quasiparticles in the vortex cores was observed.This paper is organized as follows: in Section II we describe the SANS experimental details. Results are presented in Section III, focusing on the VL configuration and anisotropy, rocking curves, and a determination of the VL form factor. The implications of our results are discussed in Section IV, with an emphasis on the superconducting gap structure, Pauli limiting and Pauli paramagnetic effects, and evidence for multiband superconductivity. A conclusion is presented in Section V.§ EXPERIMENTAL DETAILSThe superconducting anisotropy was determined by small-angle neutron scattering (SANS) studies of the vortex lattice (VL).These studies simultaneously measure the VL form factor, h(Q). Results from five separate experiments are included in this report, performed at Institut Laue-Langevin (ILL) instruments D22 and D33,<cit.> and Paul Scherrer Institut (PSI) instrument SANS-I.The samesingle crystal with = 1.47 K was used for all the SANS experiments, and was also used in previous work.<cit.> The sample was mounted in a dilution refrigerator insert and placed in a horizontal-field cryomagnet. A motorized Ω stage rotated the dilution refrigerator around the vertical axis within the magnet, allowing measurements as the magnetic field was rotated away from the basal plane of the tetragonal crystal structure. The experimental configuration is shown schematically in Fig. <ref>(a). Measurements were performed with applied magnetic fields between 0.15 T and 1.3 T, applied at angles relative to the basal plane in the rangeΩ = 0.5^∘ to 6.9^∘, and with temperatures between 50 mK and 1.2 K. Figure <ref>(b) provides a summary of the measurements. For the SANS experiments performed at ILL the crystalline ⟨ 100 ⟩ axes were horizontal and vertical, while at PSI SANS-I the ⟨ 110 ⟩ axes were horizontal/vertical. The two configurations are denoted by respectively H⊥⟨ 100 ⟩ and H⊥⟨ 110 ⟩. Due to the smallness of Ω the applied field is also near-parallel to ⟨ 100 ⟩ or ⟨ 110 ⟩. The VL was prepared at low temperature by first ramping to the desired field (H) and rotating to the chosen field orientation (Ω), followed by a damped small-amplitude field modulation with initial amplitude 50 mT. This method is known to produce a well-ordered VL in , and eliminates the need for a time consuming field-cooling procedure before each SANS measurement.<cit.>The measurements used neutron wavelengths λ_n between 0.8 nm and 1.7 nm and a bandwidth Δλ_n /λ_n = 10%. A position sensitive detector, placed 11-18 m from the sample, was used to collect the diffracted neutrons. In order to satisfy the Bragg condition for the VL, the sample and magnet were tilted about the horizontal axis perpendicular to the field direction [angle φ in Fig. <ref>(a)].Some measurements on the ILL D33 instrument were performed with a polarized/analyzed neutron beam,<cit.> as indicated in Fig. <ref>(b) and denoted by “pol” in figure legends. This eliminates the need for background subtraction when measuring the VL spin flip scattering. For the unpolarized measurements, backgrounds obtained in zero field were subtracted from the data to clearly resolve the weak signal from the VL at high fields.§ RESULTSIn conventional VL SANS experiments the scattering is due solely to the modulation of the longitudinal component of B(r) in the plane perpendicular to the applied field direction, denoted h_z in Fig. <ref>(a). However, in highly anisotropic superconductors such asthere is a strong preference for the vortex screening currents to flow within the basal ab plane. In this case the associated transverse field modulation (h_x) becomes dominant for small, but non-zero, angles between the applied field and the basal plane.<cit.> It is the relatively large h_x that makes the present VL SANS measurements possible, since the signal due to h_z for in-plane fields is vanishingly small in .<cit.>§.§ Vortex Lattice ConfigurationIdeally,is determined from measurements with the applied field parallel to the crystalline basal plane.In this configuration, the primary VL Bragg peaks lie on an ellipse in reciprocal space with a major-to-minor axis ratio given by .However, as h_x, and thus the VL scattering intensity, vanishes when the field is exactlyparallel to the ab plane such measurements are not possible. Instead, we determine the VL anisotropy () with the field applied at an angle Ω with respect to the basal plane. Performing measurements at several angles it is possible to obtain = (Ω = 0) by extrapolation.The VL distortion due to the uniaxial anisotropy is illustrated in Fig. <ref>. For fields applied parallel to the c axis a square VL is observed at all fields.<cit.> Here the VL is oriented with the primary, first-order reflections along the ⟨ 110 ⟩ axis, Fig. <ref>(a). As the field is rotated towards the basal plane the square VL is distorted and may also undergo a symmetry change. The schematics in Fig. <ref>(b,c) show the position of VL reflections for H∥c corresponding to the two orientations of thecrystal used in the SANS measurements. In the first case (b), the sample is rotated around a vertical ⟨ 100 ⟩ axis, and the horizontal field will therefore always be perpendicular to this direction. We denote this by H⊥⟨ 100 ⟩. Correspondingly, the second case (c) is denoted by H⊥⟨ 110 ⟩.Fig. <ref>(d-f) show possible VL diffraction patterns that may be obtained as the field is rotated towards the basal plane. Each vortex carries a single quantum of magnetic flux = h/2e = 2068 Tnm^2, and as a result the reciprocal space unit cell area is conserved in all cases. Considering first H⊥⟨ 100 ⟩, the square VL may simply be distorted by an elongation along the Q_x direction and a compression along the Q_y direction (d), or the distortion may be accompanied by a transition to a distorted triangular symmetry (e). In the first case, the four first order and the four second order peaks will lie on two separate ellipses, indicated by the solid lines. In the second case there are six first order peaks lying on the same ellipse. The same two possibilities exist for the H⊥⟨ 110 ⟩ case, as shown in panels (e) and (f). In all casesis determined by the major-to-minor axis ratio of the relevant ellipse.To distinguish between the different VL configurations in Fig. <ref>(d-f) one should in principle measure the position of all the first order Bragg reflections. However, VL Bragg peaks that are not on the vertical axis (open circles) have scattering vectors almost parallel to h_x, and are effectively unmeasurable as only components of the magnetization perpendicular to the VL scattering vector will give rise to scattering.<cit.> This introduces an ambiguity as it is not possible to discriminate between (d) and (e) (or between (e) and (f)) based solely on the position of the Bragg peak along the short axis of the ellipse (solid circles). Experimentally, Bragg peaks are always observed on the vertical axis regardless of the crystal orientation. This makes the distorted square VL (d) unlikely as the observed peak would correspond to a second order reflection. For the H⊥⟨ 100 ⟩ case we therefore conclude that the VL undergoes a transition to a distorted triangular VL (e). For H⊥⟨ 110 ⟩ the magnitude of the scattering vectormakes the distorted square configuration (f) the most plausible, as will be discussed in more detail later.§.§ Vortex Lattice AnisotropyThe diffraction patterns in Fig. <ref> show the VL Bragg peaks used to determine the superconducting anisotropy in . The VL anisotropy is related to the magnitude of the minor axis scattering vector = Q_y (f). This is evident from Fig. <ref>(a-e), where /Q_0 increases ( decreases) as a constant applied field of 1.0 T is rotated away from the basal plane. Here, Q_0 = 2π (2 μ_0 H/√(3))^1/2 is the scattering vector for an isotropic triangular VL, where we have assumed that the magnetic induction B (vortex density) is equal to the applied magnetic field μ_0 H. The value of /Q_0 provides a direct measure of theas long as the VL configuration is known. In the case of a distorted triangular VL (Fig. <ref>(e), H⊥⟨ 100 ⟩) one finds = (Q_0/)^2. This relation is slightly modified for the distorted square VL with first order reflections on the vertical axis (Fig. <ref>(f), H⊥⟨ 110 ⟩): =√(3)/2 (Q_0/)^2. The minor difference between the two anisotropies is evident in Fig. <ref>(f) where the ellipse corresponding to the distorted triangular VL is shown by the dashed ellipse. Finally, if the VL for H⊥⟨ 100 ⟩ was a distorted square and the observed peaks were second order, the anisotropy would be given by =√(3)(Q_0/)^2. In such as case, using the expression for a distorted triangular VL would severely underestimateas shown by the dashed ellipse in Fig. <ref>(d). However, this would also yield a dramatically different VL anisotropy between the two field orientations, reinforcing the conclusion that the VL for H⊥⟨ 100 ⟩ does indeed have a distorted triangular symmetry. Finally we note that if one assumes a quantization of /2, as reported for mesoscopic rings of ,<cit.> the deduced values forwould double. We consider this an unrealistic scenario in the present case, with a macroscopic, homogenous sample. It would also causeto exceed the limit corresponding to a diverging anisotropy, as discussed in more detail in sect. <ref>.In addition to the Ω-dependence discussed above, a field dependence of the VL anisotropy was also found, shown in Fig. <ref>(f-j). In this case it is necessary to separate the effect of a changing superconducting anisotropy from the increasing vortex density due to the change in the applied field. To achieve this, the axes in Fig. <ref> have all been normalized by Q_0. Plotted in this fashion it is apparent thatincreases with increasing field (peaks moving closer to Q = 0). In contrast, no temperature dependence ofwas observed, as evident from Fig. <ref>(k-n), where /Q_0 remains constant within the precision of our measurements.§.§ Rocking CurvesAs an alternative to determining the VL anisotropy from the position of the VL Bragg peaks as discussed above, it is also possible to obtainfrom the so-called rocking curve. Figure <ref>(a) shows the evolution of the scattered intensity as a function of the rocking angle φ for a single VL Bragg peak (upper half of the detector) at two different fields of 0.75 T and 1.2 T.In a conventional VL SANS experiment the scattering is due to the longitudinal form factor h_z. As a result, the neutron undergoes non-spin flip (NSF) scattering, which gives rise to a single maximum in the rocking curve at a tilt angle φ_0 = /2k_0 given by Bragg's law in the small-angle limit, where k_0 = 2π/λ_n is the nominal neutron wave vector. However, no NSF scattering is observed for either rocking curve.Each neutron's spin (σ) will rotate adiabatically to be either parallel or antiparallel to the magnetic field at the sample position inside the magnet. The two different directions correspond to different nuclear Zeeman energies and lead to opposite shifts of the neutron wave vectork_↑(↓) = k_0√(1 ±Δε/ε_0),where the subscript in parentheses corresponds to the minus sign in the ±Δε term. Here, ε_0 = ħ^2 k_0^2/2m_n and Δε = γμ_N B,where m_n is the neutron mass, γ = 1.913 is the neutron gyromagnetic ratio and μ_N = eħ/2m_n = 31.5 neV/T is the nuclear magneton. For the fields and neutron wavelengths used in this work (k_↑^2 - k_↓^2) ≤ 2 × 10^-4k_0^2, and the difference between k_↑ and k_↓ is too small to be observed as a difference in φ_0 for NSF scattering. In contrast, spin flip (SF) scattering arising from h_x ⊥σ leads to two different scattering processes, shown schematically in Fig. <ref>(b). Since ≪ k_↑/↓, the small difference in the neutron wave vectors nonetheless leads to significantly different tilt angles.The Zeeman splitting of the rocking curve is clearly seen in Fig. <ref>(a) for 0.75 T. The 1.2 T data show a qualitatively similar behavior, except that only one of the maxima was rocked through the Bragg condition and the intensity is significantly reduced. For SF scattering Bragg's law is replaced by∓ (k_↓^2 - k_↑^2) - ^2 = 2k_↑(↓) sinφ_1(2),where the subscripts in parentheses correspond to the plus sign. From the splitting Δφ = |φ_1 - φ_2| one thus obtains ≈ (2 k_0/Δφ) (Δε/ε_0). The results of both methods of determining , and thereby , are indicated in Fig. <ref> by the lines (rocking curve) and open circles (detector image). Within experimental error the two methods agree, and henceforth the average is used.§.§ Superconducting Anisotropy Figure <ref>(a,b) shows the VL anisotropy as a function of Ω for applied fields of 1.0-1.3 T and 0.15-0.25 T, respectively. These measurements significantly expand previously reported results for 0.5 T and 0.7 T.<cit.> The low and high field cases are considered separately due to their qualitatively different field dependence. The high field data are fitted to the equation:= /√(cos^2 Ω + (sinΩ )^2)obtained for a 3-dimensional superconductor with uniaxial anisotropy.<cit.> Whileis a layered material the coherence length along the c axis, ξ_c = 3.3 nm is still several times greater than the Ru-O interlayer spacing, and Eq. (<ref>) was previously found to provide a reasonably good description of the data.<cit.> Numerical calculations based on the Eilenberger modelsuggest that this expression slightly underestimates .<cit.> However as numerical results are only available for a few values of Ω we shall rely on Eq. (<ref>). As shown in Fig. <ref>(a), this yields fitted values of the superconducting anisotropy = 72.4 ± 7.5 for the combined 1.0-1.1 T data, and 93 ± 23 for the 1.2 and 1.3 T data. Despite the large uncertainty on the fitted values ofthe 1.2-1.3 T data () exceeds the 1.0-1.1 T data over the entire measured Ω range, indicating that the superconducting anisotropy increases when approaching . This is also consistent with the fitted = 58.5± 2.3 obtained previously for fields of 0.5-0.7 T. <cit.>In contrast to the high field data discussed above, the measurements ofin the range of 0.15-0.25 T, shown in Fig. <ref>(b), deviate significantly fromEqn. (<ref>). Rather the measured VL anisotropy exceeds the expectation for a diverging .As will be discussed quantitatively later, this Base Line Excess (BLE) discrepancy may be due to multiband superconductivity or a difference between the nominal and actual value of Ω at low fields.In the low field case we instead obtain a lower limit onby averaging the measuredfor Ω≤ 1.2^∘, indicated by the shaded area in Fig. <ref>(b). The field dependence of the superconducting anisotropy for all magnetic fields is summarized in Fig. <ref>(c). On close inspection a deviation is also observed at intermediate Ω at intermediate<cit.> and high fields, although much less pronounced. Note that the BLE is not due an error in the sample alignment, as measurements of the scattered intensity (Fig. <ref>) at positive and negative Ω allow a precise orientation of the crystalline basal plane relative to the field direction.<cit.>The temperature dependence ofat 0.4 T is shown in Fig. <ref>(d) for both H⊥⟨ 100 ⟩ and H⊥⟨ 110 ⟩. The different magnitudes ofare due to differences in Ω.For H⊥⟨ 100 ⟩, the data is an average offor Ω values of 0.5^∘, 0.7^∘, and 0.9^∘, while H⊥⟨ 110 ⟩ was measured at a single Ω = 2^∘.For the 0.5^∘ - 0.9^∘ data, the sensitivity to changes inis ∼ 10. At larger Ω, thecurves for different merge as seen in Fig. <ref>(a,b). At Ω = 2^∘ the sensitivity is therefore reduced, and the uncertainty onis ∼ 20. Within these limits, the data in Fig. <ref>(d) indicates thatremains constant as T is increased from base temperature toward T_c2(0.4T) = 1.35 K.In contrast, theanisotropy as a function of temperature, (T), has been found to increase with increasing temperature from 20 at low temperature to 60 at .<cit.>The expected VL anisotropy if = (T) was calculated using Eq. (<ref>) and shown in Fig. <ref>(d).In both cases this lies noticeably below the measured . We note that the VL anisotropies for H⊥⟨ 110 ⟩ were obtained using the expression for a distorted square VL. From the average of these values we obtain = 61_-12^+20, in good agreement with the result from the Ω-dependence shown in Fig. <ref>(c). In contrast, values ofassociated with a distorted triangular VL exceed the limit for a diverging . As the previously discussed BLE practically vanishes above 0.25 T such a result would be unphysical, supporting the conclusion that for H⊥⟨ 110 ⟩ the VLhas a distorted square symmetry.§.§ Vortex Lattice Form FactorWe now return to the measurements of the transverse VL form factor, h_x. The integrated intensity of the Zeeman split Bragg peaks is obtained from rocking curves, such as the one shown in Fig. <ref>. Dividing the integrated intensity by the incident neutron flux yields the VL reflectivity, which is related to the form factor byR = 2πγ^2 λ_n^2 t/16 ^2|h_x|^2,where t is the sample thickness.<cit.> Here, the integrated intensity for the two maxima in the rocking curve are added, as each corresponds to half the incident flux (one direction of the neutron spin). For some measurements, a polarized/analyzed SANS configuration was used. Here, an incident beam polarized parallel or antiparallel to the applied field (σ in Fig. <ref>(b)) is scattered by the sample, and the spin of the outgoing neutrons is selectedusing a ^3He filter before detection.<cit.>By choosing opposite spin orientations before and after the sample, only neutrons undergoing SF scattering will be measured.<cit.> This is an effective method to measure the intensity due to h_x, as it suppresses the NSF scattering background.<cit.> The form factors obtained in this fashion are shown in Fig. <ref>. The same curve shape is used as a guide to the eye for all fields in panels (a-c). This illustrates how VL SANS measurements are possible within a narrow angular range, with H close to, but not perfectly aligned with, the basal plane. Both the width of the measurement “window” and the magnitude of the form factor decreases rapidly with increasing field, as clearly seen for the high field data in Fig. <ref>(a). At low fields, where the BLE is relevant, the curves are found to overlap at low Ω, Fig. <ref>(b).Finally, Fig. <ref>(c) shows the Ω dependence of the form factor at T = 750= 12. Here the magnitude of the form factor is reduced by a factor of 2.5 relative to the value at base temperature, but otherwise follows the same curve.The temperature dependence of h_x for two field orientations is shown in Fig. <ref>(d). While the form factors are in principle determined on an absolute scale the exact normalization varies slightly from one experiment to another due to differences in sample illumination, giving rise to minor systematic differences. In the present case the values forH⊥⟨ 100 ⟩ were multiplied by 1.1, to make the form factors overlap at base temperature. From this, one finds that the transverse form factors for the two different field directions follow the same temperature dependence within the precision of the measurements.Several theoretical models for the form factor exist, with the simplest analytical expressions obtained from the London model. In this case, the transverse form factor for the observed VL Bragg peaks is given by:<cit.> h_x=B λ^2 m_xz ^2/d d= (1 + λ^2 m_xx ^2) (1 + λ^2 m_zz ^2) - λ^4 m_xz^2^4m_xx =^-2/3sin^2 Ω + ^4/3cos^2 Ω m_zz =^-2/3cos^2 Ω + ^4/3sin^2 Ω m_xz = (^-2/3 - ^4/3) cosΩ sinΩ. Here λ = (λ_ab^2λ_c)^1/3 is the geometric mean of the penetration depths in the ab-plane and along the c axis. Using the zero temperature literature value λ_ab = 152 nm, <cit.> and λ_c = λ_ab with = 58.5, we find λ≈ 600 nm. This yields (λ )^2 ≈ 100 ≫ 1, and with all m_αβ in Eqs. (<ref>)-(<ref>) of at least order unity the form factor expression simplifies toh_x ≈B/(λ )^2m_xz/m_xxm_zz - m_xz^2.When using the London model, a correction due to the finite vortex core size is typically included by a Gaussian term exp [-c(ξ)^2], where ξ is the coherence length and c is a constant of order unity.<cit.> The zero-temperature value for the in-plane coherence length, estimated from the 75 mT upper critical field parallel to the c axis, is ξ_0 = (/2π H_c2^∥ c)^1/2 = 66 nm. For μ_0 H = B = 0.4T and the measured = 0.017 nm^-1, one gets a perfect agreement between the measured and calculated h_x at base temperature with a core cut-off constant c = 0.81. This shows the London model expanded with a core cut-off provides at least a qualitatively accurate estimates of the transverse form factor. That said, we have previously shown that it does not accurately describe the detailed Ω-dependence of h_x.<cit.>§ DISCUSSION §.§ Superconducting Gap StructureThe temperature dependence of the VL form factor reflects the structure of the superconducting gap in . As already discussed, the VL anisotropy remains constant for the measurements with H⊥⟨ 110 ⟩ at 0.4 T and Ω = 2.0^∘, shown in Fig. <ref>(d). The only temperature dependence will therefore be through the penetration depth and coherence length. From Eq. (<ref>) one finds that h_x ∝λ^-2, and the form factor is therefore proportional to the superfluid densityρ_s(t) = 1 - 1/4π t∫_0^2π∫_0^∞cosh^-2( √(ε^2 + Δ^2(t,ϕ))/2t) dϕd ε,where t = T/ is the reduced temperature and the dimensionless superconducting gap Δ(t,ϕ) is given in units of .<cit.> The superfluid density decreases with increasing temperature due to thermal excitation of quasiparticles, causing λ to increase. Obtaining information about a nodal gap structure requires measurements at temperatures T ≲/3 where the quasiparticle thermal excitation energies are much less than .<cit.> This is clearly satisfied in the present case, with a base temperature ∼/30.The gap function is separated into temperature-and momentum-dependent parts Δ(t,ϕ) = Δ_0(t) Δ_k(ϕ). For the temperature dependence we use the approximate weak coupling expression,<cit.>Δ_0(t) = Δ_0(0) tanh( 1.78 √(1/t - 1))where Δ_0(0) is the zero temperature amplitude of the gap. Replacing 1.78 by the more accurate π/Δ_0(0) will not affect the conclusion of the following analyis. In Fig. <ref> we show the results of fits to h_x for different angular dependences of Δ_k(ϕ).Here we focus on the difference between the data and the fits at low temperatures. In the absence of gap nodes, Δ_0(t) in the shaded region is nearly constant, andρ_s will therefore vary little as few quasi particles are excited across the superconducting gap. In contrast, ρ_s will decrease linearly with temperature near t = 0 if the gap has line nodes. We ignore the effect of a temperature dependent coherence length although this would be straightforward to include, multiplying ρ_s(t) by the previously discussed core correction and noticing that within the BCS theory ξ∝Δ_0(t)^-1. Including the core correction would not affect the calculated temperature dependence of h_x in any significant manner at low T.The transverse form factor saturates at low temperatures < 150 mK, suggesting a non-vanishing gap on all parts of the Fermi surfaces. However, a fit to a simple isotropic gap with Δ_k(ϕ) = 1 (s-wave) provides a poor description of the transverse form factor as shown in Fig. <ref>(a), regardless of whether the critical temperature is used as a fitting parameter or kept fixed. Furthermore the fitted Δ_0(0) is below the lower BCS weak coupling value of 1.76. A better agreement is obtained for a gap with line nodes, Fig. <ref>(b). Here we have used Δ_k(ϕ) = | sin (2ϕ)| for simplicity. The differences between this and a p-wave order parameter with accidental nodes are expected to be minor. While the nodal gap in the simplest form varies with temperature all the way to t = 0 (dashed line), the London approximation of a vanishing core size relative to the penetration depth will break down in the vicinity of the nodes and nonlocal corrections should be taken into account.<cit.> This leads to a cross-over to a slower temperature dependence below a characteristic temperature T^* = (Δ_0(0)/) (/κ), and yields a transverse form factorh_x ∝ 1 - (1 - ρ_s(t)) (+ T^*/) (T/T + T^*).As shown by the solid line in Fig. <ref>(b), this provides a good fit to the data throughout the entire low field region, although the fitted value of the cross-over temperature (0.33 K) is much smaller than the theoretical estimate T^* ≈ 1.3 K when one uses the literature value of κ = 2.3. Finally, a comparable but slightly better fit is obtained by an angular dependence of the gap with deepminima instead of nodes: Δ(t,ϕ) = Δ_1(t) + Δ_2(t) | sin (2ϕ)|, Fig <ref>(c). The fitted amplitudes for the nodal (2.39-2.54) or deep minima (2.52) gaps suggest strong coupling, and are in good agreement with results of scanning tunneling spectroscopy which found Δ_0(0) = 350 μ = 2.8.<cit.> The structure of the superconducting gap, and whether this varies between the three Fermi surface sheets, has been a topic of extensive discussions.<cit.> For a chiral p-wave order parameter gap nodes are not required by symmetry, and in the simplest case the gap is expected to be isotropic. However, numerous experiments have found evidence for either accidental nodes or deep minima in the superconducting gap from specific heat,<cit.> penetration depth measurements,<cit.> or ultrasound attenuation.<cit.> Our SANS results are fully consistent with this scenario, although we are not able to determine the location and orientation of the nodes/minima. More recently, an analysis of specific heat and thermal conductivity measurements put an upper limit on the gap minima ≃ 1% of the gap amplitude.<cit.> While the fits in Fig. <ref>(b) and (c) do not allow us to distinguish between actual nodes or deep minima(Δχ^2 = 7%), the latter yields a minima-to-amplitude ratio Δ_1/(Δ_1 + Δ_2) = 0.13. This exceeds by an order of magnitude the above mentioned upper limit obtained from measurements of the thermal conductivity.§.§ Possible Pauli Limiting and Pauli Paramagnetic EffectsThe striking difference betweenandindicates a strong suppression of the upper critical field inat low temperatures for H⊥c. This suggests Pauli limiting due to the Zeeman splitting of spin-up and spin-down carrier states by the applied magnetic field.<cit.> Further support for this comes from the temperature dependence ofwhich increases towardsas T →, indicating that the Pauli limiting of the in-plane upper critical field becomes progressively stronger at lower temperatures.In contrast, the lack of a temperature dependence of[Fig. <ref>(d)] is consistent withbeing a measure of the intrinsic superconducting anisotropy arising from the Fermi surfaces. In spin-triplet superconductors the order parameter is most conveniently described in terms of the d vector, directed along the zero spin projection axis where the configuration of the Cooper pairs is given by 1/√ 2(↑↓ + ↓↑).<cit.> Pauli limiting in the triplet case is therefore only possible if H∥d, which is inconsistent with the chiral superconducting state with d∥c proposed for .<cit.> We note, however, that Pauli limiting itself appears to be in disagreement with nuclear magnetic resonance and nuclear quadrupole resonance Knight-shift measurements (summarized in Ref. Maeno:2012ew), which suggest that the d vector rotates in the presence of a magnetic field such that d⊥H.Previously, we have used SANS measurements to obtain direct evidence for Pauli paramagnetic effects (PPEs) in superconducting TmNi_2B_2C and CeCoIn_5.<cit.> In these compounds, a strong coupling to the magnetic field leads to a polarization of the unpaired quasiparticle spins in the vortex cores, and thus a spatially varying paramagnetic moment commensurate with the VL.<cit.> This adds to the orbital field variation in the mixed state, giving rise to an increase in the total field modulation and hence the longitudinal form factor (h_z) with increasing field.<cit.> Recently, such effects were also found in(≃ 10) employing a measurement scheme with fields near parallel to the basal plane, analogous with the present work.<cit.> Inboth NSF and SF scattering were observed, and PPEs were inferred from the intensity which deviated significantly from the London model expectation.<cit.>In the present case of , no NSF scattering associated with h_z is observed. Rather, the transverse form factor decreases monotonically as the applied field is increased, as seen in Fig. <ref>(a,b) and in our previously published results. <cit.> That said, the London model (including a core correction) does not provide a quantitative description of the data and in particular the narrow range in Ω where a non-vanishing h_x is observed.<cit.> However, numerical solutions to the Elienberger equations that include PPEs do provide a qualitatively accurate description of the measured Ω-dependence of thetransverse VL form factor, and thereby further support for Pauli limiting in .<cit.> The numerical work also provides an estimate of the longitudinal form factor, which despite the PPE enhancement remains approximately two orders of magnitude smaller than h_x.<cit.> From the rocking curves shown in Fig. <ref>, we can provide an upper limit on h_z. Using the 1.2 T data as a reference, and estimating the minimum measurable NSF peak size at φ = φ_0 for any practical count time, we find the longitudinal form factor must exceed ∼ 0.06 mT to be observed. The failure to measure the NSF signal from the VL is thus consistent with the numerical calculations. An estimate of the longitudinal form factor can also be obtained from the experimentally determined magnetization jump at the first-order , Δ M = 0.074 ± 0.015 mT.<cit.> One expect h_z ∼Δ M/6,<cit.> indicating that the form factor is just below the value required to be measurable by our SANS measurements.In addition to the increase ofmentioned in Section <ref>, recent measurements of the upper critical field also found thatcan be decreased by more than an order of magnitude by strain.<cit.> This reduction is driven by a dramatic 20-fold increase offor fields along the c axis, accompanied by a more modest but still noticeable three-fold increase for in-plane fields. These changes are attributed to a reconfiguration of the Fermi surface and possibly a change in the order parameter. Complementary SANS studies would be of great interest, in order to explore the strain dependence ofand in relation to multiband superconductivity<cit.> as discussed below. §.§ Multiband Superconductivity The superconducting anisotropy determined from our SANS measurements differs dramatically from the upper critical field anisotropy at low temperature, ≃ 20.<cit.> Following our initial report,<cit.> this difference was confirmed by magnetic torque measurements performed in fields near parallel to the basal plane, which found a coherence length anisotropy ξ_a/ξ_c∼ 60.<cit.> We note that while there also is a subtle (∼ 3%) in-plane variation ofat low temperature,<cit.> we are not able to determine whether this is reflected in . This is due to the relatively poor resolution of our SANS measurements where the anisotropies for H ⊥⟨ 100 ⟩ and H ⊥⟨ 110 ⟩ are identical within the experimental error.A field dependent , such as the one seen in Fig. <ref>(c), is characteristic of a multiband superconductor. The superconducting anisotropy arises from the intrinsic anisotropy of the Fermi surface, ⟨ v_ab⟩ / ⟨ v_c ⟩. However, for multiband superconductorswill be a weighted average of the anisotropy for each band, according to their contribution to the superconducting state. As the applied field may suppress the superconductivity differently for each band, it may also change . A superconducting anisotropy that changes with field is thus a sign of multiband superconductivity, and has previously been observed in MgB_2<cit.> and .<cit.> We note that whilehas so far only been found to increase with increasing field, a decrease in the anisotropy would indicate multiband superconductivity by the same argument.In the case of , the Fermi surface has three bands denoted α, β and γ with anisotropies Γ_α = 116, Γ_β = 56.8, and Γ_γ =174.<cit.>Our results thus suggest that the least anisotropic β band is responsible for determining the superconducting anisotropy at low and intermediate fields, but also that the superconductivity on this band is suppressed above 1 T. This agrees with recent inelastic neutron scattering studies which found that the quasi two-dimentional γ band, and not the quasi one-dimensional α and β bands, to be primarily responsible for the superconductivity in strontium ruthenate.<cit.> Theoretically, the role of the individual bands and their interplay has been studied extensively.<cit.> While there is broad consensus that all contribute to the superconductivity, different models vary regarding which bands are predicted to be dominant. However, in most cases the effects of an applied magnetic field have not been considered in detail. Recently, Nakai and Machida proposed a model forbased on a dominant β band.<cit.> While this seems to be in disagreement with a > Γ_β at high fields, the model describes the sharp h_x(Ω) cut-off observed at low fields which is not possible using a single band.<cit.> A definitive understanding of how the superconductivity incorrelates with the individual bands is thus still lacking.Finally, we return to the anomalousΩ-dependence of the VL anisotropy at low fields (BLE), shown in Fig. <ref>(b), whereclearly exceeds the value expected for a divergingin the range 1.5^∘≤Ω≤ 5^∘. One possible explanation is provided by the above mentioned model based on a dominant β band.<cit.> Alternatively, this may be due to a rotation of the vortex direction away from the applied field and towards the basal plane. To quantify such an effect we define ΔΩ for each data point in Fig. <ref>(b), as the rotation required to shift the measured value ofonto the curve corresponding to = 58.5.Thus ΔΩ would be the “misalignment” angle between the nominal and actual VL direction. This is shown in Fig. <ref>.If a field rotation is responsible for the anomalous values of (Ω), ΔΩ will be related to the transverse magnetization, M_x ∝ h_x.<cit.> Since M_x ≪ H, we expect ΔΩ∝ h_x/H. The curves in Fig. <ref> correspond to those from Fig. <ref>(b), with each divided by its proper applied field. After dividing by H all three curves are scaled by the same factor. This is in reasonable agreement with ΔΩ(Ω), and we therefore consider a field rotation as the most likely explanation for the BLE. Since the ratio h_x/H decreases rapidly with increasing field, the“misalignment” effect is strongly suppressed at all but the lowest H. Nonetheless, to fully account for the VL behavior in strongly anisotropic superconductors, a fully three-dimenional treatment is desirable. Ideally, such a treatment should include realistic material parameters to explain the different VL configurations observed for H⊥⟨ 100 ⟩ and H⊥⟨ 110 ⟩. As an aside, we note that demagnetization effects will also provide a negligible change in the vortex lattice direction. However, from our measurements onwe estimate a variation ≤ 0.3^∘ over the relevant Ω range.<cit.> Additionally, the present experiments were performed using an elliptically cylindricalcrystal for which demagnetization effects will be much smaller than for the platelikesamples. § CONCLUSIONWe have studied the vortex lattice infor fields applied close to the basal plane, nearly parallel to the crystalline ⟨ 100 ⟩ and ⟨ 110 ⟩ directions. This significantly extends previous SANS measurements which were restricted to low temperature, intermediate fields and a single field rotation axis. Furthermore, SANS measures the bulk superconducting properties ofand allows us to simultaneously address a number of its features. The use of both spin polarization and analysis in neutron scattering studies of the VL provided an improved signal-to-noise ratio for studies of weak spin flip scattering.Rotating the field towards the basal plane causes a distortion of the square VL observed forH∥⟨ 001 ⟩, and in the case of H⊥⟨ 100 ⟩ also a symmetry change to a distorted triangular symmetry. This results in a VL configuration with first-order VL Bragg peaks along the rotation axis for both field orientations.The vortex lattice anisotropy greatly exceeds the upper critical field anisotropy of ∼ 20 at low temperature, suggesting Pauli limiting. An increasing anisotropy with increasing field indicates multiband superconductivity, with a value ofbetween 60 and 100 that suggests a suppression of superconductivity on the β band. In comparison, no temperature dependence of the anisotropy is observed, in striking contrast to . We also find that the angular dependence of the VL anisotropy deviates from a simple expression for a uniaxial superconductor, especially at low fields. A truly three-dimensional model, which includes the salient features relevant to strontium ruthenate, will be required to explain our data over the entire range of fields, field angles and temperatures.Finally, the temperature dependence of the form factor is consistent with either nodes or deep minima in the superconducting gap, in agreement with recent thermal conductivity measurements. We conclude by noting that a successful model for the superconducting state inmust provide an explanation for all the observations summarized above. § ACKNOWLEDGEMENTSWe acknowledge experimental assistance by D. Honecker and J. Saroni as well as useful discussions with M. Ichioka, V. G. Kogan, K. Krycka, and K. Machida. Funding was provided by the U.S. Department of Energy, Office of Basic Energy Sciences, under Award No. DE-FG02-10ER46783 (neutron scattering), and by the Japan Society for the Promotion of Science KAKENHI Nos. JP15H05852 and JP15K21717 (crystal growth and characterization). Part of this work is based on experiments performed at the Swiss spallation neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland. | http://arxiv.org/abs/1710.10310v1 | {
"authors": [
"S. J. Kuhn",
"W. Morgenlander",
"E. R. Louden",
"C. Rastovski",
"W. J. Gannon",
"H. Takatsu",
"D. C. Peets",
"Y. Maeno",
"C. D. Dewhurst",
"J. Gavilano",
"M. R. Eskildsen"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20171027193629",
"title": "Anisotropy and multiband superconductivity in Sr2RuO4"
} |
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